Structural Change and Economic Growth: Analysis within the “Partially Balanced Growth-Framework” INAUGURALDISSERTATION zur Erlangung des Grades eines Doktors der Wirtschaftswissenschaft (Dr. rer. pol.) des Fachbereichs Wirtschaftswissenschaft der FernUniversität in Hagen. vorgelegt von Dipl.-Volkswirt Denis Stijepic am 11. Oktober 2010
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Structural Change and Economic Growth:
Analysis within the
“Partially Balanced Growth-Framework”
INAUGURALDISSERTATION
zur Erlangung des Grades eines Doktors der Wirtschaftswissenschaft (Dr. rer. pol.)
des Fachbereichs Wirtschaftswissenschaft der FernUniversität in Hagen.
vorgelegt von Dipl.-Volkswirt Denis Stijepic
am 11. Oktober 2010
Erstgutachter: Univ.-Prof. Dr. Helmut Wagner
Zweitgutachter: Univ.-Prof. Dr. Alfred Endres
Drittgutachter: Univ.-Prof. Dr. Thomas Eichner
Tag der Disputation: 9. März 2011
II
ABSTRACT
The term “structural change” refers to changes in the sector-structure of an
economy, where “sectors” are some theoretical “groups” of goods and services
(e.g. agricultural sector, manufacturing sector, services sector). In fact, structural
change is one of the most striking empirical facts of the development process;
most prominent examples of structural change are “industrialization” and
“transition to a services economy”. Even more importantly, it is well known that
structural change has some key impacts on economy and society, especially on
(aggregate) economic growth.
Although structural change has been known for a long time, structural change
theory has not been a mainstream research topic, especially not in the growth
theory. Some new research introduced a new approach to studying structural
change, which is more in line with the mainstream growth theory. I name this
approach “partially balanced growth school” (“PBGP-school”). Broadly speaking,
this new school of structural change can be characterized upon two attributes (a
mathematical one and a theoretical one): (1) The concept of “partially balanced
growth” is used to study the differential-equation-systems of the theoretical
models. (2) The modelling framework may be regarded as “neoclassical” in many
ways.
I elaborate mathematical and theoretical foundations of the PBGP-school;
especially, I discuss the usage of partially balanced growth paths in structural
change modelling and the integration of structural change into the mainstream
neoclassical growth model (Ramsey-Cass-Koopmans-model). I systematize the
literature on structural-change-modelling and integrate/classify the new PBGP-
school into this scheme. Finally, I use the concepts of the PBGP-school for
analysing some actual economic topics related to structural change and (long-run)
economic growth. Especially, by using the PBGP-methods I analyse the Kuznets-
Kaldor-puzzle, the impacts of Offshoring on real GDP-growth and the effects of
demand-shifts associated with population ageing.
In fact, my work implies that the methods of the PBGP-school seem to be
valuable tools for analysing structural change. Furthermore, as I hope, my work
provides some new and interesting insights into structural change and economic
growth.
III
In Chapter I, I provide an introduction to my research. Subsequently in Chapters
II and III, I explain and discuss the mathematical and modelling foundations of
my research. Chapter IV includes a systematization of structural-change-
modelling-literature and the classification of the PBGP-school and of my
research. In Chapter V, I present my efforts on modelling the Kuznets-Kaldor-
Puzzle, Offshoring and Ageing by using the PBGP-methods. Finally, in Chapter
VI there is a summary of my work.
IV
TABLE OF CONTENTS CHAPTER I: Introduction
– Definitions, Overview, Motivation and Focus of the Analysis.............................1
1. Definition of sectors and of structural change (3) 2. Stylized structural-change-patterns: Kuznets-facts, Furastié-facts and
Baumol-facts (7) 3. Why does structural change take place? – The four main structural
change determinants ( 9) 3.1 Non-homothetic preferences across sectors (10) 3.2 Differences in TFP-growth across sectors (11) 3.3 Differences in output-elasticites of inputs across sectors (12) 3.4 Shifts in intermediates production across sectors (13)
4. Why does technology differ across sectors? (14) 5. Impact of structural change on the economy: focus on aggregate
growth (19) 6. The two schools of structural change modelling: Focus on the
neoclassical PBGP-school (22) 7. Further theoretical/analytical restrictions of the analysis (27) 8. General value of the models or: Which statements can be drawn
from the models and which not? (31) 9. Justification for the choice of topics that are analysed by using PBGP-
models (33)
CHAPTER II: Key Analytical Tool: Differential Equation-System Analysis
– Definitions, Analytical Concepts and Methods..................................................37
1. Basic terms (40) 2. Methods and key-concepts of differential-equation-system-analysis (44)
2.1 Methods of differential equation system analysis (44) 2.2. The qualitative approach to differential equation system analysis (46)
2.2.1 Dynamic equilibrium (46) 2.2.2 Stability of dynamic equilibrium (53)
2.2.2.1 Basic concepts and definitions of stability (53) 2.2.2.2 Methods for proving global and local stability of
equilibrium (58) 2.2.2.2.1 Local stability (58) 2.2.2.2.2 Global stability (60) 2.2.2.3 Transitional dynamics (71)
3. Optimal control (75) 3.1 The necessary conditions for a maximum (79) 3.2 Proof of sufficiency (80)
4. Summary: Step-by-step procedure in continuous-time growth-modeling (82) LIST OF SYMBOLS of CHAPTER II (84)
V
CHAPTER III: Key Modeling Approach: Integration of Structural Change
into a Neoclassical Growth Model
– Key Assumptions, Application of Analytical Tools, Challenges in Structural
Change Modeling, Application and Premises of PBGP-method........................87
1. An unbalanced growth model (90)
1.1 Assumptions (90) 1.2 Optimality conditions (93)
2. Analytical challenges to structural-change-analysis (95) 3. A balanced-growth multi-sector-model (98) 4. A truly “neoclassical” multi-sector growth-model (104) 5. Partially balanced growth (106)
5.1 An example of a partially balanced growth model (106) 5.2 Examples from the literature on how to create partially balanced
growth paths (usage of a priori and a posteriori knife-edge conditions) (110) 6. Validity of neoclassical models in the light of structural change
and the downside of knife-edge-condition use (115) APPENDIX (119) LIST OF SYMBOLS of CHAPTER III (121)
CHAPTER IV: Classification/Systematization of Structural Change
Literature and Classification/Subordination of own Research.....................123
1. Changes in capital structure (129) 2. Changes in intermediates structure (132) 3. Changes in consumption structure (137)
3.1 “Unbalanced” school of structural change (137) 3.2 “New” (PBGP) school of structural change (142)
4. Classification of own research (146) 5. Further aspects of classification (149)
5.1 Structural change induced by trade opening (structural change theory vs. trade theory) (149)
5.2 Factor reallocation between capital industries and consumption industries (Uzawa’s structural change) (151)
5.3 Factor-reallocation between the private sector and the public sector (152) 5.4 Factor-reallocation between the research sector and the consumption
sector (153) 5.5 Outsourcing of home production (factor reallocation between home-
sector and market-sector) (154)
VI
CHAPTER V: Application of the PBGP-Concept in Analysis of Structural
Change................................................................................................................155
PART I: A PBGP-Framework for the Analysis of the Kuznets-Kaldor-
Puzzle...................................................................................................................159
1. Introduction (163) 2. Stylized facts of sectoral structures (167)
2.1 Stylized facts regarding cross-sector-heterogeneity in production- technology (167)
2.2 Structural change determinants (168) 3. Model of neutral cross-capital-intensity structural change (169)
3.1 Model assumptions (169) 3.1.1 Production (169) 3.1.2 Utility function (172) 3.1.3 Aggregates and sectors (175)
3.2 Model equilibrium (178) 3.2.1 Optimality conditions (178) 3.2.2 Development of aggregates in equilibrium (178) 3.2.3 Development of sectors in equilibrium (182) 3.2.4 Consistency with stylized facts (184) 3.2.5 The relationship between structural change and aggregate-
dynamics (190) 4. A measure of neutrality of cross-capital-intensity structural change (195) 5. On correlation between preferences and technologies (203) 6. Concluding remarks (209) APPENDIX A (214)
APPROACH (1) (214) Necessary (first order) conditions for an optimum (214) Proof that sufficient (second order) conditions are satisfied (215)
APPROACH (2) (218) Producers (219) Households (220) Relationship between individual variables and economy-wide aggregates (224)
APPENDIX B (226) APPENDIX C (230) APPENDIX D (243) APPENDIX E (245) APPENDIX F (248) LIST OF SYMBOLS of PART I of CHAPTER V (252)
VII
PART II: A PBGP-Framework for Analyzing the Impacts of Offshoring on
Structural Change and real GDP-growth in the Dynamic
Context.................................................................................................................263
1. Introduction (265) 2. Model assumptions (270) 3. Optimum and equilibrium (277) 4. Effects of offshoring on growth of aggregates (280)
4.1 The overall impact on aggregate growth (281) 4.2 Impact channels and their relative importance (284)
5. The effects of offshoring on structural change (290) 6. Discussion and implications (297) 7. Concluding remarks (301) APPENDIX A (306) APPENDIX B (307) APPENDIX C (309) APPENDIX D (313) APPENDIX E (315) APPENDIX F (316) LIST OF SYMBOLS of PART II of CHAPTER V (317)
PART III: A PBGP-Framework for Analyzing the Impacts of Ageing on
Structural Change and real GDP-growth...........................................................321
1. Introduction (323) 2. Model assumptions (330)
2.1 Utility (330) 2.2 Production (333) 2.3 Numéraire (335) 2.4 Aggregates and sectors (337)
3. Model equilibrium (338) 3.1 Optimality conditions (338) 3.2 Aggregates (339) 3.3 Sectors (342)
4. Effects of ageing (343) 4.1 Partially Balanced Growth Path (PBGP) without ageing (344) 4.2 Ageing and cross-sector differences in TFP-growth (347) 4.3 Ageing and cross-sector differences in input-elasticities (352)
4.3.1 Productivity effect: Impacts and channels (353) 4.3.2 Additional impacts on GDP: The price-effect (359)
4.3.2.1 Transitional effects of ageing on GDP (359) 4.3.2.2 PBGP-effects of ageing (366)
4.3.3 Dynamic aspects (367) 5. Concluding remarks (368) APPENDIX A (373) APPENDIX B (379)
VIII
APPENDIX C (380) APPENDIX D (390) LIST OF SYMBOLS of PART III of CHAPTER V (391)
CHAPTER VI: Summary.................................................................................397
REFERENCES..................................................................................................403 Curriculum Vitae...............................................................................................419 “Erklärung laut §6(8) der Promotionsordnung” ...........................................421
IX
X
CHAPTER I
Introduction
- Definitions, Overview, Motivation and Focus of the Analysis -
The following chapter is aimed to provide an introduction to my research. Especially
in Section 1, some key definitions are provided. Sections 2-5 comprise some general
explanations about structural change and its impacts on the economy. In Sections 6-9,
I explain the theoretical and analytical focus and the justification of my research.
1
TABLE OF CONTENTS for CHAPTER I
1. Definition of sectors and of structural change ...........................................................3
2. Stylized structural-change-patterns: Kuznets-facts, Furastié-facts and Baumol-facts
........................................................................................................................................7
3. Why does structural change take place? – The four main structural change
determinants...................................................................................................................9
3.1 Non-homothetic preferences across sectors.......................................................10
3.2 Differences in TFP-growth across sectors .........................................................11
3.3 Differences in output-elasticites of inputs across sectors ..................................12
3.4 Shifts in intermediates production across sectors ..............................................13
4. Why does technology differ across sectors?............................................................14
5. Impact of structural change on the economy: focus on aggregate growth ..............19
6. The two schools of structural change modelling: Focus on the neoclassical PBGP-
school ...........................................................................................................................22
7. Further theoretical/analytical restrictions of the analysis ........................................27
8. General value of the models or: Which statements can be drawn from the models
and which not? .............................................................................................................31
9. Justification for the choice of topics that are analysed by using PBGP-models......33
2
1. Definition of sectors and of structural change In every economy several goods and services are produced and consumed. In general,
some of these goods and services share some common attributes. Hence, it can make
sense to group the goods and services that share common attributes (since, for
example, in this way it is easier to cope with the empirical data). In structural-change-
theory such groups of “similar” goods and services are named sectors. Hence, the
whole range of goods and services is subdivided into sectors, where each sector
contains goods and services that are similar regarding some attributes.
For example, the best known and the most basic sector-division of the economy is
(1) agriculture (primary sector)
(2) manufacturing (secondary sector)
(3) services (tertiary sector)
Note that the range of these sectors depends on the criterion that is used to formalize
this sector division. As discussed by Krüger (2008), p.335, and Wolfe (1955) there are
several examples:
• Fisher (1939, 1952) defines these sectors upon demand behaviour (where the
degree of necessity decreases and the income elasticity of demand increases
when approaching from sector (1) to sector (3))
• Wolfe (1955) categorizes these sectors upon the dominant factor in production
• Clark (1957) defines the range upon the nature of output of the sectors (e.g.
good vs. service).
Furthermore, the three-sector-division seems to be useful to explain the historical
development of the industrialized economies that will be discussed later (i.e. the
transition from agriculture to manufacturing and from manufacturing to services).
3
Another example of a (more disaggregated) sector-division, which I use in the
empirical study of Chapter V (Part I), is the sector-definition from the “Standard
Industrial Classification System”, which is used by the U.S. Department of Commerce
(Bureau of Economic Analysis). This definition includes the following sectors:
(1) Agriculture, forestry, and fishing
(2) Mining
(3) Construction
(4) Manufacturing
(5) Transportation and public utilities
(6) Wholesale trade
(7) Retail trade
(8) Finance, insurance, and real estate
(9) Services
This whole discussion implies that there are many several ways to subdivide an
economy into sectors, depending upon the attributes, which are used to group the
goods and services, and depending upon the questions, which are analysed. In my
research I do not choose any specific sector-division, in general. In fact, I study
models where production technology differs across sectors and where the degree of
disaggregation is not too high. (Furthermore, some models will assume that some
utility-function-parameters, e.g. the income elasticity of demand, differ across sectors,
as well.) My research does not require any other restriction on the definition of the
sectors. Hence, in fact every sector-definition, where the technology differs across
some sectors and the economy is not too disaggregated, is applicable to my research.
In the essay on the Kuznets-Kaldor-Puzzle (in Chapter V) I provide empirical
evidence, which shows that production technology differs strongly across sectors, in
4
general. (Intuitive explanations of cross-sector technology-disparities can be found in
Section 4 of the actual Chapter.)
To sum up this discussion I suggest the following definition:
Definition 1: Sectors are theoretical groups of goods- and services-varieties. Sectors
are defined such that
• technology differs across sectors; i.e. the “average” technology of sector i is
not the same as the average technology of sector j at least for some i and j; in
other words it is assumed that the production function, which is representative
of sector i, is not the same as the production function, which is representative
of sector j, at least for some i and j.
• the sector-division is exclusive, i.e. a good- or a service-type can be assigned
to only one sector; in other words, a sector does not include goods and
services that are assigned to another sector.
Now, we turn to the definition of the term structural change. The term structural
change, as it is used in my research, refers to a change in the sector structure of the
economy. For example, in the early stages of economic development agriculture
accounts for the largest part of real GDP, where in later stages of economic
development services constitute the biggest part of real GDP. (These empirical facts
are reviewed in Section 2 in detail.) That is, during the development process some
sectors become more important in comparison to other sectors; this is structural
change. Now, we have to find a measure of relative importance of sectors. In general,
5
we could use sector-shares of real GDP1 to measure the relative importance of
sectors. However, in the literature there is another measure of structural change,
which is more appropriate for the aims of my research: sectoral employment shares2.
Hence, if employment-shares of sectors change, structural change takes place
according to my definition. Note that, in general, the dynamics of sector-shares of
real GDP and the dynamics of sectoral employment shares are quite similar (in
empirical findings and in my models); hence, they are quite interchangeable indicators
of structural change. However, in the analysis of the relationship between structural
change and real GDP-growth sectoral employment shares are a more meaningful
structural change indicator: As we will see in Section 5, the cross-sector labour-
allocation determines the average (economy-wide) labour-productivity; hence, the
allocation of labour across sectors is a determinant of real GDP(-growth). Overall, our
definition of structural change allows us to study the impacts of structural change on
real GDP-growth in the most direct way. To sum up this discussion I provide the
following definition:
Definition 2: Structural change stands for the change in the cross-sector labor-
allocation, where
• sectors are defined upon Definition 1 and
• cross-sector labor-allocation is indicated by the sectoral employment shares.3
1 „Sector-i-share of real GDP“ means the real output of sector i divided by real GDP. 2 „Employment share of sector i“ means the number (or: working hours) of persons, which are employed in sector i, divided by the number (or: working hours) of persons, which are employed in the whole economy. 3 See the previous footnote.
6
2. Stylized structural-change-patterns: Kuznets-facts, Furastié-facts and Baumol-facts Structural change is a well-known empirical fact, which has been studied even since
the 1930ies. Empirical evidence on structural change can be found in the
papers/books by e.g. Fourastié (1969), Kuznets (1976), Maddison (1980), Elfring
(1989), Broadberry (1997), Kongsamut et al. (1997, 2001), Raiser et al. (2003), Ngai
and Pissarides (2004), Broadberry and Irwin (2006) and Schettkat and Yocarini
(2006). Useful sources for data on structural change are:
• for EU-countries data: EU KLEMS Project (http://www.euklems.net/)
• for US-data: U.S. Department of Commerce (Bureau of Economic Analysis)
(http://www.bea.gov/)
• for OECD-countries-data: OECD STAN Industry (http://www.oecd-
ilibrary.org/content/datacollection/stan-data-en).
In fact, all this empirical evidence implies some stylized facts of structural change.
Since structural change across sectors agriculture, manufacturing and services is the
most discussed one, I review the stylized facts about this structural change in this
section for the sake of completeness. I will show later that these stylized facts can be
satisfied by my models.
Kongsamut et al. (1997, 2001) formulate the following stylized facts, which they
name “Kuznets facts”:
a) The employment share of agriculture declines during the development process.
b) The employment share of manufacturing is constant during the development
process.
c) The employment share of services increases during the development process.
7
It should be noted that these stylized facts refer to the structural-change-patterns over
the last century. If longer periods are considered, the development of the
manufacturing-sector-employment-share is rather “hump-shaped” (as noted by Ngai
and Pissarides (2007) and Maddison (1980)). Hence, alternative stylized-facts of
structural change (which take account of the last two centuries or so) can be
formulated, which I name Furastié-facts:
a) The employment share of agriculture declines during the development process.
b) The employment share of manufacturing increases in early stages of development
(“industrialization”) and decreases in later stages of development (“tertialisation”).
c) The employment share of services increases during the development process.
I name these stylized facts after Jean Fourastié who discusses them in his book (see
Furastié (1969), pp.118ff). These stylized facts are also discussed as being the “three-
sector hypothesis” and have already been noted by Fisher (1939), according to Krüger
(2008), p.332.
Since, according to Definition 2, structural change across technologically distinct
sectors is in focus of my research, I should provide evidence that this sort of structural
change actually takes place. Empirical evidence on the fact, that in reality factors are
reallocated across technologically distinct sectors, is provided by: Close and
Schulenburger (1971), Baumol et al. (1985), Maddison (1987), p.666ff, Bernard and
Jones (1996), Broadberry (1997), Curtis and Murthy (1998), Foster et al. (1998),
Disney et al. (2003), Penderer (2003), Broadberry and Irwin (2006), Schettkat and
Yocarini (2006), UN (2006), Acemogly and Guerrieri (2008), Nordhaus (2008),
Valentinyi and Herrendorf (2008) and Duarte and Restuccia (2010).
8
In fact, these essays show that even the Kuznets- and Furastié-stylized-facts imply
that factors are reallocated across technologically distinct sectors, since agriculture,
manufacturing and services use different technology. (Even within these sectors
different technologies are used.) In fact, Wolfe’s (1955)-sector-division is based on
technological differences between sectors.
Because Baumol’s (1967)-work popularized the focus on sector-differences in
technology, I name the fact, that factors are reallocated across technology in reality,
Baumol’s stylized fact.
3. Why does structural change take place? – The four main structural change determinants When looking at the evidence on structural change, one question arises: why does
structural change occur? The literature has already dealt with this question. In fact,
there are four main types of cross-sector disparities that cause structural change. I
name them “structural change determinants” and they will be discussed now.
As proposed by Schettkat and Yocarini (2006), there are three main determinants of
structural change: cross-sector shifts in final demand (non-homothetic preferences),
cross-sector shifts in intermediates production (outsourcing), and cross-sector
differences in productivity growth. Note that differences in productivity growth can
arise due to differences in TFP-growth and due to differences in output-elasticities of
inputs across sectors as will be explained below. (Some further structural change
determinants, e.g. international trade, which are not in focus of my research, are
discussed in Chapter IV, Section 5.)
Empirical evidence on the impact of these determinants on structural change is
reviewed, e.g., by Schettkat and Yocarini (2006). Further evidence is provided in Part
I of Chapter V. I discuss the evidence on the structural change determinants in Part I
9
of Chapter V, since the proof of their relevance is an integral part of that Part, but is
less relevant for the other Parts of Chapter V.
The impacts of structural-change-determinants on structural change are jointly studied
in the model of Part I of Chapter V. Nevertheless, in the following, I explain how
each of these determinants causes structural change by itself and I provide some
references. Note that the following explanations are based on the long-run view of the
economy, where perfect cross-sector-mobility of factors and, in general, perfect
markets are assumed.
3.1 Non-homothetic preferences across sectors Non-homothetic preferences mean simply that income elasticity differs across goods.
Hence, when income increases, the demand does not increase uniformly across goods;
i.e. the demand for some goods increases more strongly in comparison to the demand
for other goods; i.e. demand is “shifted” across sectors, which produce these goods.
In general, it is argued that some sectors (e.g. agriculture) produce rather goods,
which are necessities (e.g. food); therefore, income-elasticity of demand is relatively
low regarding these sectors. It is argued as well, that other sectors (e.g. services)
produce rather goods which are luxury goods; hence, income-elasticity of demand is
rather high. For discussion, references and empirical evidence see e.g. Krüger (2008),
p.335, Schettkat and Yocarini (2006), pp.139ff, Laitner (2000), p.546, Curtis and
Murthy (1998). A micro-foundation of non-homothetic preferences in multi-sector
frameworks is provided by Foellmi and Zweimüller (2008) by using Engel’s Law.
The demand-shifts, which are caused by non-homothetic preferences, cause changes
in relative profitability of factor-use across sectors and thus result in factor
reallocation across sectors, i.e. structural change.
10
Of course, an impulse is necessary, which ensures that income increases over time. In
my models, this impulse comes from technological progress.
Non-homothetic preferences as a determinant of structural change are modeled by,
e.g., Kongsamut et al. (2001). Further literature, which studies the impact of non-
homothetic preferences in structural change frameworks, can be found in Section 3 of
Chapter IV.
3.2 Differences in TFP-growth across sectors If the growth rate of productivity differs across sectors, sectors can expand their
production at different rates for a given factor allocation. That is, production-
possibilities grow at different rates across sectors. Or the other way around: sectors
that feature relatively high productivity-growth-rates (“progressive sectors”) can
lower their prices over time more strongly in comparison to sectors with relatively
low productivity growth rates (“stagnant sectors”), for a given cross-sector factor
allocation and for a given profit rate.4 That is, the relative prices are changing.
Consumers respond to changes in relative prices; thus, demand is shifted across
sectors. These demand shifts cause factor reallocations similar to those discussed in
Section 3.1; thus, structural change arises.
Note, however, that there are two different forces regarding this factor reallocation:
On the one hand, the (relative) production possibilities increase in the progressive
sectors (in comparison to stagnant sectors). This effect implies that less factors are
required in the progressive sectors to produce a given amount of goods. – Effect 1
4 Of course, since we consider long-run perfect markets with perfect mobility, the profit rate is equal to zero.
11
On the other hand, (relative) demand increases in the progressive sector due to
relative price reductions. This effect implies that more factors are required for
production in progressive sectors. – Effect 2
Hence, whether factors are reallocated to progressive sectors or withdrawn from them
depends on which of the two effects is stronger. We know that Effect 2 depends on
the price elasticity of demand. If price elasticity of demand is relatively high (low),
households react very strongly (weakly) to price-changes. Therefore, if price elasticity
of demand is relatively high (low), Effect 2 is relatively strong (weak) and factors are
reallocated to (withdrawn from) the progressive sectors.
This argumentation implies that there must be a certain price elasticity which ensures
that Effects 1 and 2 are equally strong hence no factors are reallocated across sectors.
In fact, it can be shown that this is the case when price elasticity is equal to one (see
e.g. Ngai and Pisssarides (2007)).
The impact of cross-sector-TFP-differences on structural change is modeled by, e.g.,
Baumol (1967) and Ngai and Pissarides (2007). For further literature, see Chapter IV,
Section 3.
Now, the question arises why does TFP-growth differ across sectors. Regarding this
question and empirical evidence see Section 4.
3.3 Differences in output-elasticites of inputs across sectors Assume that output-elasticity of labor differs across sectors; this may be the case
when capital is introduced into production functions. Especially, assume that there are
labor-intensive sectors (i.e. sectors that feature high output-elasticity of labor) and
capital-intensive sectors (i.e. sectors that feature low output-elasticity of labor).
Furthermore, assume that the real wage-rate increases, ceteris paribus. Hence, the
average production costs in labor-intensive sectors increase more strongly in
12
comparison to the average production costs of capital-intensive-sectors, ceteris
paribus.5 Therefore, when the wage rate increases, the labor-intensive producers must
increase their prices more strongly in comparison to capital-intensive producers, for
given profit rates, ceteris paribus. That is, relative prices change. These relative price
changes cause factor reallocations in similar manner as in Section 3.2, i.e. structural
change arises. Like in Section 3.2, price-elasticity plays an important role for the
magnitude/direction of the resulting structural change.
Again, an impulse is necessary to increase the wage rate. In my models this impulse
comes from technological progress. (It is known from neoclassical growth theory, e.g.
the Ramsey-Cass-Koopmans-model, that an income-increase associated with
technological progress, causes an increase of the wage rate. In this respect my model
is the same; see, e.g., equation (33) in Part I of Chapter V. (For a discussion of the
Ramsey-Cass-Koopmans-model, see, e.g., Barro and Sala-i-Martin (2004), pp.85ff, or
Chapter III.)
For example, Acemoglu and Guerrieri (2008) provide a model, which explains
exactly how cross-sector differences in output-elasticities of inputs cause structural
change. For further literature, see Chapter IV, Section 3.
In Section 4, I provide intuitive arguments for why output-elasticites differ across
sectors.
3.4 Shifts in intermediates production across sectors Assume that each sector produces not only final goods, but intermediate goods as
well, which are used in the own production and in the other-sectors production.
5 The reason for this fact is simple: labor-intensive sectors use relatively much labor (in optimum); hence, a relatively large part of their average production costs is due to wage payments.
13
Differences in productivity-growth across intermediates production cause changes in
intermediate prices. Hence, cost-minimizing producers change their intermediate-
input-structure. Hence, there are changes in intermediate demand across sectors; they
cause factor-reallocation across sectors.
Models, which explain this channel exactly, are provided by, e.g., Fixler and Siegel
(1999) and Ngai and Pissarides (2007). Further literature can be found in Chapter IV,
Section 2.
The fact that each sector uses intermediates from other sectors is obvious and can be
seen from Input-Output-Tables of every country. It is obvious that the most of the
today’s very complex products are produced by using many different intermediates
from many different sectors. Just think about which resources, parts and services are
necessary to produce and sell a car.
4. Why does technology differ across sectors? Empirical evidence on technology-differences across sectors (output-elasticity of
inputs as well as TFP) is discussed in Part I of Chapter V. Why these differences exist
is quite obvious: The “nature of the final-product” differs across sectors strongly.
Therefore, the (physical) production processes, the resources which are used in
production, the sectoral market-structures, the degree of technology-spillovers from
other sectors/industries as well as transfer-process of the final-product to the
consumer differ across sectors strongly. In general, these differences affect the scope
for technological innovation, rationalization (substitution of labor by capital) and
division of labor for a given level of technological development. In fact, these
differences are (in part) the key criterion of sector-division according to Wolfe
(1955).
14
These aspects can be best understood by comparing the production process of a potato
and the “production process” of a service like counseling by a psychologist:
• The key restricting resource in the production process of a potato is land. Hence,
to some extent, the productivity increases in potato production are restricted by the
availability/extent of land and by growth rates of natural products (e.g. today a
potato still requires some time to grow). On the other hand, in psychological
counseling natural resources are rather unimportant. Hence, in psychological
counseling there is no scope for productivity improvements by increasing the
usability of the natural resource. The other way around, the productivity
improvements in counseling are not restricted by improvability of land-use or by
natural growth rates. (Rather, other facts restrict the productivity of counseling.)
These arguments are related to Wolfe (1955), p.414ff. To see how inclusion of
land in the agricultural production-function leads to different sectoral productivity
parameters, see, e.g., the model by Laitner (2000).
• The production of a potato includes a lot of “mechanical processes”. That is, a
human (or a machine) must move some physical matters from one place to another
in order to produce a potato, e.g. the seed must be spread and the potatoes must be
harvested. Hence, a lot of labor can be substituted by capital (= rationalization),
when relatively simple mechanical machines are invented. In contrast, mechanical
processes are not important for psychological counseling (except for the fact that
the patient must come to the psychologist). Hence, rationalization of the
counseling process by simple machine-innovations is very restricted or even
impossible. Hence, psychological counseling is necessarily labor-intensive. (This
argument is related to Wolfe (1955), pp.416ff, and Klevorick et al. (1995),
pp.187f. Klevorick et al. (1995) provide a study where they try to asses the
technological possibilities of several industries.). Therefore, in psychological
15
counseling is no/few scope for productivity-increases by technological innovation
related to capital. Of course, we can think of intelligent robots that have the fine
emotional sensibility of a human. However, this seems very futuristic.
(Furthermore, the question is whether such robots are regarded as machines or as
“humans”; in the latter case the usage of such robots would not decrease the labor-
intensity of counseling). To sum up: The rationalization of some industries/sectors
may require very fine/sophisticated technology developments (e.g. micro-chip),
which must be based on some basic mechanical/chemical/physical innovations
(steel, plastic, electricity). These basic innovations may rationalize the production
process of some sectors (potatoes); therefore, even in early stages of technological
development some sectors may be rationalized, especially sectors which include a
lot of mechanical processes. However, the industries/sectors, which require fine
technology improvement for rationalization, (psychological counseling) may
remain very labor-intense over the most phases of technological/economic
development (or “for ever”).
• The amount of technology-spill-overs from other sectors is as well determined by
the “nature of the product”. The production process of a potato was very enhanced
by some technological innovations of the manufacturing sector, where the
manufacturing sector is very technologically progressive due to the effects
discussed above/below among others. In contrast, psychological counseling
profited hardly from manufacturing-sector-advances. (This argument is related to
Klevorick et al. (1995), p.190f.). The strength of technology-spill-overs from
manufacturing to potato production is among others due to the fact that both
include a lot of “mechanical processes”.
• While a potato can be transferred to a consumer within few seconds,
psychological counseling services require the permanent personal contact of the
16
service provider to the “consumer”. This difference manifests the high labor
intensity of counseling services. In fact, in counseling services “labor is itself the
end-product” (Baumol (1967), p.416). This aspect has also another side (see also
Blinder (2007) for an interesting study regarding the following fact): In fact, the
psychologist could analyze its patient via life-stream via internet. However, a
masseur cannot transfer its service via communication and information
technologies. Hence, many industries, that require direct personal contact in order
to transfer the “final product” to the consumer, feature fewer productivity
improvements from to technological progress in information and communication
technologies.
• The different stages of potato production can be conducted by different
persons/producers/employees. Hence, productivity of potato production can be
increased by labor-division, by better organization of the production process and
by specialization on a specific stage of the production process (e.g. outsourcing or
labor-division a la Adam-Smith). In contrast, psychological counseling requires
that the largest part of the process is conducted by one person.
• Some further arguments/literature are/is discussed by Klevorick et al. (1995),
pp.186f, and Pavitt (1985), pp.365ff. These arguments are related to the market
structure (that is caused by nature of the final product) and its impacts on the
R&D-efforts of the firms. For example, depending upon the nature of the product,
the existing market structure may consist of large firms (potatoes), and hence more
R&D may be initiated by them in comparison to other environments (counseling)
where firms are rather small (and thus cannot cope with large R&D-sunk-costs).
Furthermore, the size of demand, the fraction of R&D-returns that a firm is able to
retain (large vs. low spillovers/externalities) and the type of competition depend
17
on the nature of the final product as well and can cause cross-sector differences in
R&D and thus in technology(-progress).
In fact, this discussion implies that there is a strong time-component in cross-sector
technology differences. It seems that there is a path of technologic development: In
the beginning, there are innovations, which rationalize mechanical processes.
Rationalization of mechanical processes is relatively simple and therefore stands at
the beginning of the technological development path. Industries/sectors (especially
manufacturing), where mechanical processes are a key component of the production
process, profit from these innovations. Over time, the rationalization of mechanical
processes progresses and the technologies that are used become more and more
sophisticated and the aggregated income increases. This whole process is named
industrialization. During this process the basis for fine technologies is constituted.
This basis and the increasing income (and thus increasing demand for luxury services)
open the door for technological progress in sectors/industries, where mechanical
processes are not the core of the production process, especially in some services
industries. This phase is often named tertiarisation.
In fact, what we learn from this whole discussion is that sectoral technologies diverge
during the industrialization (when comparing manufacturing and services). However,
during the tertialisation and especially in future the technologies could converge
again. The question is, whether the technological progress in manufacturing will be
strong enough to counteract this process. In fact, as discussed by Pavitt (1985) and
Klevorick et al. (1995), there seems to be a path dependence of technological progress
to some extent. That is, much of technological progress in past constitutes much of
future technological progress. However, the new possibilities which are opened to
18
technological progress in services by the development of the micro-chip could be
stronger and improve the services technology to unbelievable levels in future.
Overall, it is hard to find a strong reason for the assumption that future technology-
development-patterns will remain the same as in past.
Actually, this discussion is the basis for the long-run-independency-discussion in Part
I of Chapter V. I argue there (by using purely intuitive argumentation) that (for given
preference parameters) we have no reason to assume that high-income-elasticity-
sectors (like some services) will remain technologically inferior.
This seems to be a very interesting and valuable topic for future research. Note,
however, that I have not researched very much in this field; my research-focus was
rather on the implications of actual cross-sector technology differences, instead on the
reasons for and future development of these differences.
5. Impact of structural change on the economy: focus on aggregate growth Structural change has several effects on the society and the economy. For example,
• the Kuznets facts imply that an increasing part of manufacturing-sector-labor
is reallocated to the services sector; hence, the nature of working tasks and the
working environment changes, which may have some impact on the society;
e.g., Pugno (2006) argues that the economy-wide human capital may increase
due to this change in tasks (which may increase economic growth);
• since different sectors require different skills, structural change requires
changes in the education system, as noted by Blinder (2007);6
6 Especially, very different skill sets are required when comparing the manufacturing and the services sector.
19
• the reallocation of labor across sectors may cause short-run to medium-run
unemployment, and the rate of natural unemployment may be affected by the
strength of structural change in the medium-run;7 see e.g. Aronson et al.
(2004);
• structural change affects the growth rate of aggregates (e.g. the growth rate of
real GDP, aggregate capital and aggregate consumption expenditures), as I
will show soon.
For all these reasons it seems to be important to study and predict the changes in
structural change patterns and the changes in structural change strength. In my
research I focus solely on the study of structural change itself and on the impact of
structural change on the growth rate of aggregates. The other impacts (e.g. impacts on
unemployment) are not in focus of my research.
Empirical evidence on the impact of structural change on aggregate growth is
provided by, e.g., Robinson (1971), Madisson (1987), pp.666ff, Dowrick and Gemmel
(1991), Bernard and Jones (1996), Broadberry (1997,1998), Foster et al. (1998),
Berthélmy and Söderling (1999), Poirson (2000), Caselli and Coleman (2001),
Temple (2001), Disney et al. (2003), Penderer (2003), Broadberry and Irwin (2006),
UN (2006), Nordhaus (2008), Restuccia et al. (2008) and Duarte and Restuccia
(2010).
Now, I present a very short model, which is based on the model by Baumol (1967), to
explain why structural change has an impact on the growth rate of aggregates.
Assume a long-run growth model where two sectors (A and B) exist and where labor
is the only input-factor. The sectors differ by productivity:
7 Note that in the long run structural change cannot cause unemployment, since “long-run” is defined upon full cross-sector-mobility of labor.
20
(1) LAlY AA =
(2) LBlY BB =
(3) 1=+ BA ll
where ( ) is the output of sector A (B), ( ) is the employment share of sector
A (B), A (B) is the exogenous productivity parameter of sector A (B) and L is the
aggregate amount of labor. Equation (3) implies that we abstract from unemployment.
AY BY Al Bl
The real GDP (Y) is some weighted average of the sectoral outputs
(4) BA YaaYY )1( −+≡
where a is the weighting factor between the sectors. (Later, I will discuss it in detail.)
Inserting equations (1) and (2) into equation (4) yields
(5) [ ]BA BlaaAlLY )1( −+=
Remember that structural change means changes in and .Al Bl8 Equation (5) implies
that changes in and lead to changes in Y. Hence, structural change has an
impact on real GDP(-growth).
Al Bl
In fact, this is the quintessence of the relationship between structural change and
aggregate growth. Of course this is a very simple model; however, it shows why
structural change affects aggregate growth. In Chapters III and V, I analyse the
generalizations of this model. For example, I assumed here that the weighting factor a
is exogenous. However, sometimes, when real GDP is calculated with the “chain-
weights-method”, the weighting factor depends on the actual sectoral outputs and
hence on structural change. (For a simple explanation regarding this fact, see e.g.
Steindel 1995). However, I will show in Parts II and III of Chapter V that, despite this
fact, structural change still has an impact on real GDP-growth. Furthermore, capital
8 If A and B grow at different rates, it could be shown that structural change takes place in this model, like in the model by Baumol (1967). Furthermore, structural change could be generated by the assumption that preferences are non-homothetic across goods A and B.
21
and explicit assumptions about household-behaviour could be integrated into this
model. In this way it could be shown that structural change has an impact on the
growth rate of aggregate capital and consumption expenditures.
6. The two schools of structural change modelling: Focus on the neoclassical PBGP-school The largest part of mainstream growth theory is based on the concept of balanced
growth. That is, the most models feature assumptions which ensure the existence of a
balanced growth path (or: steady state). A balanced growth path is a trajectory where
all (relevant) variables grow at a constant rate. (For detailed discussion see Chapter
II.)
For example, the standard neoclassical growth models, like the Solow-model or the
Ramsey-Cass-Koopmans-model, generate an equilibrium that can be described by two
differential equations. These differential equations determine the growth paths of
consumption and capital. The assumptions of these models are such that this
differential equation-system features a stable (convergent) equilibrium growth path,
where capital and consumption grow at a constant rate. Hence, the equilibrium growth
path is a balanced growth path. (See also Chapters II and III for detailed explanations
and discussion.)
The convenient feature of the balanced growth approach is that the growth rates of the
variables are not state-dependent along the balanced growth path. That is, the growth
rates can be easily derived as functions of exogenous model parameters, and phase-
diagrams/simulations are only necessary to study the transitional dynamics. (For
details see Chapters II and III.) Furthermore, the balanced growth path is consistent
22
with the empirical evidence known as “Kaldor’s stylized facts of economic
development”.9 (See also Kongsamut et al. (2001)).
Therefore, it is not surprising that nearly the whole neoclassical growth theory (see
e.g. the book by Barro and Sala-i-Martin (2004)) is based on the balanced growth
concept.
Unfortunately, the theoretical literature implies that it is not easy to integrate
structural change into the balanced growth concept; see Chapter III. In the following I
provide examples of this literature. More literature-examples are provided in Chapter
IV.
• Baumol’s (1967)-model implies that the growth rate of aggregate output is not
constant, if there are at least two sectors (which differ by productivity-
growth). In contrast, the Solow-Model or the Ramsey-Kass-Koopmans-model
imply that the growth rate of aggregate output is constant along a balanced
growth path.
• Kongsamut et al. (2001) and Meckl (2002) show that neoclassical balanced
growth theory can be consistent with structural change only if some parameter
restrictions are assumed. However, they cannot provide any theoretical
rationale for these parameter restrictions.
• The paper by Acemoglu and Guerrieri (2008) implies that the neoclassical
balanced growth path does not exist as long as structural change takes place
and as long as sectors differ by output-elasticity of inputs.
Hence, structural change models imply that balanced growth theory is not necessarily
applicable as long as structural change takes place. This seems to be a serious critique
9 Kaldor facts are discussed in Part I of Chapter V.
23
point to balanced growth theory (neoclassical growth theory), since structural change
is one of the best known empirical facts.
Overall, it seems that there exist two schools of growth theory which contradict each
other: “balanced growth school” and “structural change school”. The balanced
growth school stands for the mainstream neoclassical growth theory. The models of
the balanced growth school are micro-founded (e.g. they use utility functions). The
structural change school stands for the structural change models where no balanced
growth paths exist (as long as structural change takes place), e.g. the models by
Baumol (1967), Echevarria (1997), Laitner (2000) and Acemoglu and Guerrieri
(2008). Note that the structural change school is not necessarily consistent with
Kaldor’s stylized facts. (Kaldor’s stylized facts require that capital, consumption and
output grow at a constant rate, i.e. the growth path must be balanced, while the
structural change school features unbalanced growth paths.) Furthermore, since the
structural change school does not rely on balanced growth paths, the analysis is
relatively complicated (which will be demonstrated in Chapter III). Therefore, the
models from the structural change school make either very simple assumptions (e.g.
capital is omitted in the model by Baumol (1967)), or simulations are necessary to
obtain the model results (e.g. in the models by Echevarria (1997) and Acemoglu and
Guerrieri (2008)); see also the literature discussion in Section 3.1 of Chapter IV.
Simulations are a very useful tool in economic modeling; however, they feature
several disadvantages:
• A (numerical) simulation seems to be like a black box in comparison to
analytically solvable models: That is, in relatively complex numerically solved
models it is difficult to understand why certain growth dynamics arise. Thus, it
requires a broad knowledge about analytically solvable models to guess which
factors led to certain dynamics. Therefore, developing analytically solvable
24
structural change models seems to be important: they can help to understand
the several channels along which structural change determinants affect
structural change and thus real GDP-growth; hence they can help to
understand numerically solvable models.
• The result of a numerical simulation is only applicable to the data which is
used. Hence, for example a structural change simulation for the USA, does not
say anything about structural change in Germany. Hence, for each
specification of parameters a new simulation is necessary. Therefore, it is
difficult to derive general theoretical results from numerical models. (See also
Barro and Sala-i-Martin (2004), p.113).
• Sometimes it is possible to derive the parameter-range for which the baseline
results of a numerical solution hold (however, these tests do not provide
100%-certainty). However, I have not seen such an approach in structural
change theory. Furthermore, the problem with such parameter range is that the
parameters which are displayed by the computer are solely numbers, but not
some parameter relations (equations). Hence, it is difficult to asses whether
these “numbers” will hold in future and it is difficult to derive some “micro-
foundation” for these “numbers”. In general, this problem does not appear in
analytically solvable models: if analytically solvable models feature some
parameter restrictions, these restrictions can be derived as parameter-equations
or -relations (like in the models by Kongsamut et al. (1997,2001)). Hence, it is
easier to derive a micro-foundation for these parameter restrictions. In fact, in
the essay about the Kuznets-Kaldor-Puzzle (Chapter V) I try to derive a micro-
foundation for the parameter-restrictions of the Kongsamut et al. (1997,2001)-
model.
25
So we can conclude that developing analytically solvable structural change models is
important for understanding structural change and for discussion about future
structural change.
Motivated by these challenges (in part), a new school emerged in the growth theory
(the “PBGP-school”) which attempts to merge the balanced growth school and the
structural change school by introducing the concept of “partially balanced growth”. A
“partially balanced growth path” (PBGP) features at the same time balanced growth
of aggregates and unbalanced growth of disaggregated variables. That is, along a
PBGP, aggregate output and aggregate capital grow at a constant rate and at the
same time structural change takes place (e.g. sectoral output shares change).
Hence, the convenient features of PBGP-analysis are:
• Like the neoclassical-growth-models, PBGP-models are analytically solvable.
Hence, we can derive in a convenient way relatively transparent explanations
and “general” theories about structural change (and aggregate growth). Many
aspects of structural change become easily analysable. This makes structural
change theory more transparent and more amenable to a larger group of
scientist (especially those who are familiar with neoclassical analysis.)
• The models are consistent with the Kaldor-facts (see also e.g. Kongsamut et
al. 2001).
• Since the PBGP is consistent with the equilibrium growth paths of the
“balanced-growth-school”, developing PBGP-models can help to reduce the
critique on the mainstream balanced-growth school (“Balanced growth school
is consistent with structural change.”)
However, it should be noted that the existence of a PBGP requires some restrictions in
the generality of the assumptions. Therefore, often a PBGP-model cannot depict all
26
the structural change channels. Nevertheless, relevant theoretical results can be
derived by using the PBGP-concept. This topic will be discussed in Section 8 in
detail.
To my knowledge, Kongsamut et al. (1997, 2001) are the first who introduced the
concept of PBGP-analysis into structural change theory. Several authors followed:
Meckl (2002), Ngai and Pissarides (2007) and Foellmi and Zweimüller (2008). (These
essays are discussed in Chapters III and IV). I focus on the PBGP-analysis of
structural change as well.
In this sense, my research aims to contribute to the development of the PBGP-school
of structural change by elaborating its foundations, classifying it and applying it to
several topics associated with structural change. Especially, I focus on three topics
which are associated with structural change and which are dealing with some key
stylized facts of economic development and/or some general macro-economic trends:
the Kuznets-Kaldor-puzzle, the impact of offshoring on structural change and
aggregate growth and the impact of ageing on real GDP-growth via structural change.
These topics are explained in Section 9. Note that I focused on these three topics,
since they seem to me “most” important (see Section 9). However, the PBGP-method
seems to be applicable to many more topics and it seems to be interesting for further
research to find such topics.
7. Further theoretical/analytical restrictions of the analysis I study structural change in the consumption goods sector. That is, I assume that there
are sectors that produce heterogenous consumption goods. On the other hand, it can
be assumed, e.g., that there are several types of capital and that these types of capital
are produced by different sectors. Hence, structural change in the capital-goods-sector
27
could be analysed as well. However, heterogeneous capital goods (and hence
structural change in the capital-producing sector) have been studied extensively in the
neoclassical endogenous growth literature (e.g. in models where physical capital and
human capital exists). Structural change in capital production and other types of
structural change are discussed in Chapter IV.
It should be noted here that all the models, which are developed in Chapters III and V,
are long run growth models. That is, I assume that
• there is perfect factor mobility across sectors
• prices are flexible and
• capital can be accumulated.
The inclusion of capital into analysis is a key to my research for two reasons:
(1) Some theoretical/empirical questions simply require the consideration of
capital in structural change analysis. For example, the analysis of the Kuznets-
Kaldor puzzle (see also the corresponding essay in Chapter V) requires a
model of structural change that takes capital accumulation into account, since
Kaldor-facts summarize mainly some empirical facts that are related to
capital. A model, where no capital exists, is comparable to only one Kaldor
fact: “the constancy of the output-growth-rate”. All the other Kaldor-facts (i.e.
increasing capital-intensity, a constant capital-to-output-ratio, a constant real
rate of return on capital and a constant income-distribution between capital
and labor) require the inclusion of capital into analysis.
(2) Capital-accumulation is still regarded as one of the key growth drivers. It
has been extensively studied in the neoclassical growth literature. Hence, in
general, it seems important to study the relationship between structural change
and capital accumulation and vice versa. Indeed, there seems to be a
28
relationship between capital-accumulation and structural change, as will be
discussed in Part II of Chapter V: First, since structural change affects the
productivity of factors, capital accumulation is affected by structural change,10
and we know from the neoclassical growth literature that capital accumulation
is important for aggregate growth. Second, the labor, which is available in an
economy, can be used for consumption-goods-production and for capital-
goods-consumption. Hence, if for some reasons labor is reallocated to capital-
production, it has to be withdrawn from consumption-goods-production. If a
smaller share of labor is used in the consumption-sector, all productivity
effects, which arise from labor-reallocation within the consumption-goods-
sector, become less important from the viewpoint of the economy as a whole.
Hence, the reallocation of labor between capital-goods-production and
consumption-goods-production has an important effect on the relationship
between consumption-goods-structural change and aggregate growth. This
argument will be of special importance in the Offshoring-essay and is also
discussed in Section 5 of Chapter IV.
(3) In reality, capital is used in production. Production functions with capital,
in general, generate (additional) structural change dynamics. For example, the
essay by Acemoglu and Guerrieri (2008) implies that (consumption-industries-
)structural change arises from the fact that capital and labor can be substituted
in production.
Beside of the fact that there are several consumption-goods sectors, all assumptions in
my research are very neoclassical: I assume neoclassical (representative) utility
10 Remember, that we know from neoclassical growth theory that the rate of capital accumulation depends on productivity growth; structural change affects productivity growth.
29
functions and production functions. In fact the aggregate structure of my models
coincides with the neoclassical growth models (see also Chapter III).
There are two reasons for using neoclassical assumptions:
(1) As explained above, it is interesting to know to what extent the (mainstream)
neoclassical growth theory is compatible with structural change. To reassess
neoclassical growth theory neoclassical assumptions are necessary.
(2) The assumptions of the neoclassical theory have been developed and studied
over a very long period of time. Therefore it is not surprising that they are very
convenient in analysis. They ensure that
• the optimization problems can be solved in an uncomplicated manner (e.g. by
using a Hamiltonian function; i.e. the sufficient conditions for the optimality
of the solution are ensured by neoclassical assumptions in my models; see also
Chapter II)
• the resulting equilibrium growth paths are stable (as is discussed in Chapter II)
• the stability analysis of the equilibrium growth paths is relatively simple (see
also Chapter II).
Furthermore, I have tried to use as simple functional forms as possible (without
reducing the generality of the key-model results): Especially, I use Cobb-Douglas
production functions and Cobb-Douglas-based utility functions. Cobb-Douglas
production functions are sufficient to include all key structural change determinants
which have been discussed above. (This is demonstrated in the models of Chapter III.)
Last not least, it should be noted that I assume that sectors use different technology
(see Section 1); however, these cross-sector differences in technology are not
explained endogenously in my models, but are assumed to be exogenously given.
30
Hence, I can only justify the cross-sector technology differences in my models by
pointing to the empirical evidence, which implies that these differences existed in
past. (For this empirical evidence see Section 2 of the actual chapter and Part I of
Chapter V). This is acceptable, since the largest part of my research is either about
explaining past developments (Kuznets-Kaldor-Puzzle) or about the proof of
existence of some effects in frameworks with technologically heterogeneous sectors
(Ageing and Offshoring models). However, the exogenity of technology parameters
restricts my results regarding future predictions to some extent. This topic is discussed
in the next section.
8. General value of the models or: Which statements can be drawn from the models and which not? My models are not aimed to predict/explain the overall dynamics of structural change.
Rather, they are theoretical constructs, which are used to isolate some theoretical
relationships (or: channels) between structural-change-determinants and cross-sector
factor-reallocation and between structural change and aggregate growth. The
structural change in reality cannot be explained only by my model-explanations, but
requires further explanations (which in part have not been found yet). The reason for
this fact is that PBGPs never depict all structural change channels at the same time, as
will be shown in the models of Chapter III. Hence, the overall dynamics of structural
change can only be explained by more complicated models that probably require
numerical solutions. (Furthermore, further research is required to isolate further
channels, in order to be able to construct numerical models more exactly.)
Overall, my models do not depict the reality, i.e. they are not descriptive models. For
example, I do not state that the economies are on a PBGP in reality. Also I do not
state that an average household behaves in the same manner as the representative
31
household in my models. The PBGP is only a theoretical construct that helps to take a
simple look at structural change.
For example, let us take the model about the Kuznets-Kaldor-Puzzle (Chapter V).
This model states that the Kuznets-Kaldor-Puzzle is solved along a PBGP. However, I
do not state that real economies are on the PBGP. I used the PBGP-concept, since it
allows isolating a certain type of structural change pattern (“neutral structural
change”), which is consistent with the Kaldor-Kuznets-Puzzle. That is, the model
helped me to recognise the distinctive feature of “neutral structural change”. By using
this distinctive feature I was able to test empirically to what extent is “neutral
structural change” an explanation of the Kuznets-Kaldor-Puzzle in reality. Hence, the
PBGP itself is not an explanation of the Kuznets-Kaldor-puzzle, but “neutral
structural change”, and relatively neutral structural change does not necessarily
require a PBGP.
Similar arguments apply to the other models: In Chapter V, I analyse offshoring and
ageing along PBGPs, since in this way they are easy to analyse. In this way I am able
to isolate some channels along which offshoring and ageing influence aggregate
growth. These channels exist even if the economy is not on the PBGP. Hence, again
the PBGP is not the explanation for these channels, but is only a technical help to
isolate these channels. For those reasons, the results of these papers hold, irrespective
of whether the real economies are on a PBGP or not.
Hence, the general relationship between (my) PBGP-models and numerical structural
change models is: PBGP-models help to recognize impact channels (and to test their
relative importance). Numerical models help to reproduce (and predict) overall
structural change patterns. The overall structural change patterns can be explained by
the results of several PBGP-models; furthermore, the theories from the PBGP-models
32
can help to asses whether the future-predictions of the numerical structural change
models are reasonable.
Last not least, remember that cross-sector technology-heterogeneity is not explained
endogenously in my models, as explained in Section 7. Hence, all the predictions of
my models regarding the future should be considered with caution. As discussed in
Section 4, it is possible that sector-technologies converge in (far) future, and hence
structural change becomes rather irrelevant regarding real GDP-growth. Therefore,
some of the effects studied in my models may become irrelevant in (far) future.
9. Justification for the choice of topics that are analysed by using PBGP-models As already mentioned, by using the PBGP-method I study three topics that are
associated with structural change: “Kuznets-Kaldor-Puzzle”, “Offshoring and
Structural Change” and “Ageing and Structural Change”. The criteria, which I have
used to choose these topics, are
• scientific interest (indicated by the number of topic-related publications in top-
ten economic journals)
• relevance of the phenomenon (e.g. whether it is a key macro-economic trend)
• applicability of the PBGP-concept
• whether additional results are obtainable.
Nearly all articles that study the Kuznets-Kaldor-Puzzle are published in top-five
economic journals: Kongsamut et al. (2001), Ngai and Pissarides (2007), Foellmi and
Zweimüller (2008) and Acemoglu and Guerrieri (2008)). Hence, this topic seems to
be of interest from the scientific point of view. Furthermore, the concept of PBGP has
been introduced by Kongsamut et al. (2001) to study the Kuznets-Kaldor-Puzzle;
33
hence, the PBGP-concept is applicable to this topic. Overall, my model on the
Kuznets-Kaldor-Puzzle contains all the key structural change determinants and it can
reproduce the empirically observable structural change patterns.
While the essay on the Kuznets-Kaldor-puzzle studies the traditional (or: “key”)
structural change determinants, my remaining essays study the impacts of the two key
(future) macro-economic trends that will probably have a relatively strong impact on
structural change (and thus growth) in future: offshoring and ageing.
Offshoring has been a very prominent topic in the political debate across Europe and
the United States in the 2000s, which induced a relatively extensive scientific debate,
indicated by a relative large number of publications in top journals. Therefore,
offshoring seems to be an important topic. The previous literature on offshoring is
primarily based on static models and/or models that do not include capital (in order to
keep the analysis traceable). The essay on offshoring adds to this discussion some
interesting dynamic effects of offshoring associated with structural change and
capital-accumulation. Nevertheless, the analysis in the offshoring essay remains
relatively traceable due to the application of the PBGP-concept.
Ageing is one of the key macro-economic trends in industrialized economies.
Nevertheless, the relationship between ageing and real-GDP growth via structural
change (i.e. via ageing induced cross-sector-demand-shifts) has been barely studied.
To my knowledge the only paper, that tries to model this relationship, is the one by
Groezen et al. (2005). Groezen et al. (2005) use very restrictive assumptions
regarding the consumption behaviour of the old people. By using the PBGP-concept I
was able, to generalize some of their assumptions and to show the existence of further
channels along which ageing affects structural change and thus real GDP-growth.
Of course, there seem to be many topics associated with structural change that need to
be analysed; they are left for further research.
34
In the following chapter, I provide some explanations, which are necessary to
understand the mathematical aspects of my research.
35
36
CHAPTER II
Key Analytical Tool:
Differential Equation System Analysis
- Definitions, Analytical Concepts and Methods -
The most of dynamic economic analysis can be described as three-step procedure (in
general, I follow this procedure as well):
First, assumptions are made about the institutional/physical structures within which
the agents act, i.e. assumptions on the framework, e.g. assumptions on production
functions, resource endowments and distribution, type of market, etc.
Second, assumptions are made on the behaviour of agents, e.g. rational behaviour,
perfect foresight, profit/utility maximization, etc.).
These two sorts of assumptions create a set of dynamic equations, which describe the
dynamic development of the model. For example, in neoclassical growth theories and
especially in all my research, the two sorts of assumptions are used to formulate
equilibrium conditions/postulates and dynamic maximization problems. The latter are
(often) solved by using optimal control techniques, especially the Hamiltonian
function, yielding some dynamic and static optimality conditions. These optimality
conditions as well as the equilibrium postulates establish a dynamic equation system.
In a third step the evolution of this dynamic equation system (and hence of the model)
is analysed. This analysis is conducted by using several methods and concepts of
dynamic-equation-system-analysis.
It should be noted that dynamic equation systems can be divide into differential
equation systems (where time is continuous) and difference equation systems (where
37
time is discrete). The methods of analysis (e.g. optimization, stability analysis, etc.)
differ between these two types of equation systems. In my research I prefer
continuous-time-models and therefore I discuss only the methods that are necessary
for differential equation system analysis. The technical aspects of difference equation
analysis can be found in the book by, e.g., Gandolfo (1996).
In this chapter I present some mathematical prerequisites, definitions and methods for
the analysis of differential equation systems. It should be noted that the use of the
terms, which are presented here, is not uniform across sciences, books and authors.
Therefore, the definitions, which I provide here, should be regarded as working-
definitions which I prefer to use. In my opinion a very nice introduction to dynamic
equation systems can be found in Gandolfo (1996). This book includes a very detailed
introduction and further aspects and literature on this topic. A less comprehensive and
maybe more intuitive basic discussion can be found in Chiang (1984), p.478ff. An
advanced introduction and more advanced topics can be found in Hahn (1967). I
restrict my discussion of differential equation systems only to the cases which are
directly relevant for my research. Exactly speaking, all my research deals with three-
dimensional, inhomogeneous, autonomous, ordinary, first-order differential equation
systems. In fact, first I explain what this term means (Section 1) and subsequently I
explain some aspects of the analysis of such a differential equation system (Section
2). In the third section of this chapter, I explain in short the optimal control problems
that arise in my models and their solution by the Hamiltonian function. In Section 4, I
summarize the whole discussion of this chapter by suggesting a step-by-step-
procedure in dynamic economic modelling. For further reading on all these topics I
recommend the book by Gandolfo (1996).
38
TABLE OF CONTENTS for CHAPTER II
1. Basic terms...............................................................................................................40
2. Methods and key-concepts of differential-equation-system-analysis......................44
2.1 Methods of differential equation system analysis..............................................44
2.2. The qualitative approach to differential equation system analysis ...................46
2.2.1 Dynamic equilibrium ..................................................................................46
2.2.2 Stability of dynamic equilibrium ................................................................53
2.2.2.1 Basic concepts and definitions of stability...........................................53
2.2.2.2 Methods for proving global and local stability of equilibrium............58
2.2.2.2.1 Local stability................................................................................58
2.2.2.2.2 Global stability..............................................................................60
2.2.2.3 Transitional dynamics..........................................................................71
3. Optimal control........................................................................................................75
3.1 The necessary conditions for a maximum .........................................................79
3.2 Proof of sufficiency............................................................................................80
4. Summary: Step-by-step procedure in continuous-time growth-modeling...............82
LIST OF SYMBOLS of CHAPTER II........................................................................84
39
1. Basic terms Assume that we have a variable y which is a function of time, i.e. . That is, y
is the dependent variable and t is the independent variable.
)(tyy =
Definition 1: A differential equation is an equation that contains a derivative with
respect to time.1
So if we go back to our example with y, we can postulate, that a differential equation
is an equation that contains a derivative of y with respect to t. For example,
(1) btayty += )()(&
is a differential equation, where the dot denotes the derivation of y with respect to
time t (a and b are some (exogenous) parameters that can be constant or time-
dependent).
Differential equations are widespread in sciences, because they can be used to
describe the development of variables over time. Hence, nearly every question that
deals with dynamics (e.g. the development of an economy or the route of a spaceship
in orbit) can be formulated and analysed by using differential equations. Many model
assumptions in economics (e.g. the assumption of intertemporal household-utility
maximization) result in differential equations. Thus, the analysis of models with such
assumptions requires the analysis of differential equations.
1 In fact the dependent variable need not being “time”, but can be everything else. If we had y=y(x), then a differential equation would be an equation that contains a derivative of y with respect to x. However, since in my research we only have differential equations with respect to time, Definition 1 seems to be a useful working definition.
40
To understand that differential equations describe the dynamic behaviour of their
variables consider the following fact: the differential equation (1) can be reformulated
(exactly speaking: “solved”) by using integral calculus such that we obtain
(1a) [ ] abeabyty at //)0()( −+=
(for details see e.g. Chiang (1984), p.143f). We can see now that this formulation
allows us to depict in a y-t space, which would allow us to see directly how y
evolves over time.
)(ty
A very important aspect of equation (1a) is that the development of depends
upon the initial value of , namely . That is, depending upon the initial value,
the differential equation (1) describes different growth paths of y. Later, this fact will
be of importance for Definitions (8) and (9).
)(ty
)(ty )0(y
Now, assume that we have two other variables that are dependent upon time, e.g.
and . Furthermore, assume that for each of these variables we a have
a differential equation and that the development of some variables determines the
development of other variables, e.g.
)(txx = )(tzz =
(2) )()()( tytxtx +=&
(3) )()()()( tdytaztcxtz ++=&
where a, c and d are some (exogenous) parameters.
Definition 2: A differential equation system is an equation system that consists of two
or more differential equations, describing the development of the dependent variables.
41
The number of dependent variables/equations denotes the dimension of the
differential equation system.
For example, equations (1)-(3) are a three-dimensional differential equation system.
Analysis of these three equations can reveal the development of the variables x, y and
z over time. Now, we could try to solve these equations (in a similar but much more
complicated way as we did to obtain (1a) from (1)) to study the dynamic behaviour of
y, x and z. However, in most cases this is not done, because the differential equation
systems are too complex or because the solutions are too complex and therefore have
too little intuitive meaning. Therefore, often instead of solving differential equation
systems, they are analysed by qualitative methods, which are discussed in the next
section. Remember that we have explained in the discussion of equation (1a) that the
growth path, which is described by a differential equation, depends upon the
(exogenously given) initial condition of the differential equation system ( ). The
same is true for the differential equation system. If we solved the differential equation
system we would see that the development of the variables y, x and z over time
depends upon the initial conditions , and .
)0(y
)0(y )0(x )0(z
Let us summarise all this discussion as follows:
Corollary 1: A differential equation system consists of
• dependent variables (e.g. , and ), )(ty )(tx )(tz
• independent variables (in all our discussion there is only one independent
variable, namely the time index, t) and
• exogenous parameters (e.g. a, b, c and d).
A differential equation system describes the development of the dependent variables
over time, i.e. a differential equation system determines a growth path of the
42
dependent variables. The shape of this growth path depends on the values of the
exogenous variables and on the initial state of the differential equation system (e.g.
, and ). )0(y )0(x )0(z
Note that the usual assumptions of economic analysis often produce two-dimensional
differential equation systems. For example, the assumptions of the Ramsey-Cass-
Koompmans-model create a two-dimensional differential-equation system, describing
the development of the two variables capital and consumption. The biggest part of the
analysis of the Ramsey-Cass-Koopmans model is then conducted by analysis of this
differential equation system. For the details, see e.g. Barro and Sala-i-Martin (2004),
p.85ff and also Wagner (1997), p.73ff.
Before approaching to the next section I may present some further (less important)
definitions:
Definition 3a: A differential equation system is inhomogenous, if at least one of the
involved differential equations is inhomogenous.
Definition 3b: A differential equation is inhomogenous, if it features an additive
exogenous parameter (e.g. btayty += )()(& is inhomogenous, while is
homogenous).
)()( tayty =&
In general the analysis of homogenous differential equation systems is easier.
Definition 4: In general, a differential equation system is autonomous, if its
exogenous parameters are not time-dependent (e.g. btayty += )()(& is autonomous,
while )()()( tbtayty +=& is non-autonomous). Exactly speaking, if the system is non-
43
autonomous, its dynamic equilibrium is dependent of time, i.e. depending upon time
the system converges to different dynamic equilibriums. See also Gandolfo (1996),
p.333.
Non-autonomous differential equation systems are relatively difficult to study, since
the points of convergence change over time within the phase space. This fact will be
of importance in the next section and in Chapter III.
Definition 5: A differential equation system is ordinary, if its dependent variables are
dependent upon only one independent variable, i.e. all dependent variables (y, x, z)
must be functions of time only (e.g. btayty += )()(& is an ordinary differential
equation, while bstaysty += ),(),(& is not ordinary, where s is an independent
variable); see. e.g. Gandolfo (1996), S.147.
Definition 6: A first-order differential-equation-system features only the first
derivatives with respect to time (e.g. btayty += )()(& is a first-order differential
equation, while btaytyty +=+ )()()( &&& is a second-order differential equation).
2. Methods and key-concepts of differential-equation-system-analysis
2.1 Methods of differential equation system analysis The analysis of differential equation systems can be conducted in three ways:
(1) solution of the differential equation system
(2) simulation of the differential equation system by using a computer
(3) qualitative analysis.
44
As mentioned in the previous section, the solution of a differential equation system
may be quite difficult and probably therefore approach (1) it is not very widespread in
growth theory. For a discussion of solution approaches and some examples of
economic application, see. e.g. Rommelfanger (2006) or Gandolfo (1996). In my
research I have not found any usage of this method by now. However, in general it
may be useful in structural change models. For example, since, as we will see soon,
the analysis of three-dimensional differential equation systems is quite difficult, in
some models it may be useful solving at least one of the three differential equations
and inserting it into the others. In this way, in some cases, it may be possible to
simplify a three-dimensional problem to a two-dimensional problem, which is easier
to analyse. However, by now I have not found a way to use this approach in my
analysis.
The simulation approach (2) is widespread in economic analysis; it became quite
popular due to progress in computer technology. In fact many (simple) differential
equation systems could be simulated on a standard PC. In general, a draw-back of this
approach is that its validity is restricted only to the parameter values that have been
used in simulation (see e.g. Barro and Sala-i-Martin (2004), p.113). Furthermore, it
may be quite difficult to derive some intuition about the functioning of a differential
equation system (e.g. which impact channels exist) from such a (“black box”)
simulation. Therefore, as mentioned many times, I do not use this method in my
research; my focus (namely the derivation of intuitive theoretical arguments in
structural change theory) requires rather analytical solutions.
I focus on the qualitative approach (3). Therefore, I will discuss it here in detail. This
approach is very widespread in (neoclassical) growth theory. The book by Barro and
Sala-i-Martin (2004) provides many examples of use of this method in neoclassical
growth theory. In fact this approach is based on finding dynamic equilibriums in
45
differential equation systems and proving their stability. Then, further analysis can be
conducted by analysing these dynamic equilibriums and by analysing the transition
period where the economy approaches to these dynamic equilibriums. Now, I provide
some definitions and methods which are used in this analysis.
2.2. The qualitative approach to differential equation system analysis The qualitative analysis starts with searching for a dynamic equilibrium of the
differential equation system. Hence, the question arises what a dynamic equilibrium is
and how to find it. These questions will be discussed in section 2.2.1. The convenient
feature of a dynamic equilibrium is that it is easy to understand; hence many intuitive
explanations can be drawn from it. However, as we will see in the next section, the
analysis of a dynamic equilibrium becomes more or less obsolete if the dynamic
equilibrium is unstable; therefore, the proof of dynamic equilibrium stability is
essential. Furthermore, since in most economic dynamic equation systems some
period of time is required until the dynamic equilibrium is reached (“transition
period”), it is important to analyse how the dynamic equation system behaves during
the transition period. The concepts and methods for analysing stability and the
transition period are introduced in section 2.2.2.
2.2.1 Dynamic equilibrium In general, the dependent variables of a differential equation system grow at different
and non-constant growth rates, and it is mostly impossible to recognise at first sight
how these growth rates develop. However, in some instances, for some parameter
values and/or at some points of time the growth rates of some of the variables of a
differential equation system may become constant and/or identical. Such singularities
46
of the differential equation system are important form the theoretical point of view,
since the differential equation system becomes intuitively understandable when some
of its variables grow at constant/identical growth rates. That is, when looking at such
a singularity we can see at first sight how the differential equation system develops
over time and we may be able do derive some economically intuitive explanations
from this singularity. (In Chapter III, I demonstrate/present several types of such
singularities, and, as I hope, it becomes there obvious what I mean when I say that a
differential equation system becomes easier to understand when some variables grow
at constant rate.)
On the other hand, if such a singularity lasts only for a moment, it may become
obsolete analysing it, since a lot of the analysis may be irrelevant for all the other
points of time.
Singularities, that feature constant rates of some variables and that last more than for
an instant of time, are often named dynamic equilibrium.
We may summarize this discussion in the following definition, which is in my opinion
sufficient to understand all the discussion in my research.
Defintion 7: A dynamic equilibrium is a dynamic state of a differential equation
system that satisfies the following requirements:
(1) some of the variables of the differential equation system grow at a constant
(identical) rate
(2) if the system is in the dynamic equilibrium, it remains in the dynamic equilibrium
provided that there are no shocks/parameter changes, that shift the system out of its
equilibrium.
47
Requirement (1) allows for the intuitive understandably discussed above.
Requirement (2) allows for relevance over time (timely relevant singularity),
discussed above.
Definition 7 allows for several types of dynamic equilibriums that are discussed in the
literature and that will be of relevance in my research; these are
(1) balanced growth path (“steady state”)
(2) asymptotically balanced growth path (“asymptotic steady state”)
(3) partially balanced growth path (“partial steady state”)
In the following, I provide only definitions of these sorts of dynamic equilibrium and
I discuss them briefly. In Chapter III, I apply these concepts to the reference model. I
hope that this will help to understand these concepts even more.
Definition 8: A balanced growth path (“steady state”) is a trajectory where all
dependent variables of a differential equation system grow at a constant and identical
rate. Provided that a balanced growth path exists in a differential equation system,
there is a set of finite initial conditions and parameters that puts the differential
equation system directly onto the balanced growth path.
For example, in our differential equation system (1)-(3) a balanced growth path is a
trajectory along which x, y and z grow at the same constant rate. (By now, we cannot
say whether such a trajectory really exists in the system (1)-(3); however, we will
discuss this topic soon.)
The second sentence from Definition 8 (namely the statement that only a certain
initial condition brings the differential equation system directly onto a balanced
growth path), is an important distinctive feature between balanced growth and
asymptotically balanced growth which will be discussed soon. Remember that we
48
have explained in the previous section (see also Corollary 1) that for each set of initial
conditions the differential equation system follows a different growth path. Hence, it
is not surprising that only one certain set of initial conditions brings the differential
equation system directly onto a balanced growth path.
Note that the distinction between a “balanced growth path” and a “steady state” is
rather linguistic/grammatical. A steady state is a balanced growth path where the
growth rate of variables is equal to zero. However, since a differential equation
system that is on a balanced growth path can be easily transformed into a differential
equation system that is in a steady state,2 from a mathematical point of view it is
irrelevant whether we use the term balanced growth path or steady state (although it is
little bit odd speaking about a steady state when a system is growing).
Note that the analysis in the neoclassical growth theory is based nearly exclusively on
the existence of balanced growth paths, which makes the balanced growth concept an
important one; see also Kongsamut et al. (1997, 2001); for examples see e.g. Barro
and Sala-i-Martin (2004).
Definition 9: An asymptotically balanced growth path (“asymptotic steady state”) is
a trajectory along which the differential equation system converges to a “final
dynamic state”. In the final dynamic state all dependent variables grow at a constant
and identical rate. There is no set of finite initial conditions and parameters that puts
the differential equation system directly into the final dynamic state. That is, for all
2 This can be easily done by dividing all variables of the differential equation system by an (auxiliary) exogenous variable that grows at the same rate as the variables of the differential equation system along the balanced growth path. This procedure is often done in neoclassical growth models by expressing the variables in “efficiency units”. In this case the original variables grow at a constant rate along the balanced growth path, while the variables in efficiency units are constant (i.e. are in a steady state) along the balanced growth path. For an example with the Ramsey-Cass-Koopmans-model, see Barro and Sala-i-Martin (2004), p.95ff or see APPENDIX C of Part I of Chapter V.
49
finite parameter and initial condition settings there must be an (infinitely lasting)
transition period to the final dynamic state.
The concept of asymptotically balanced growth is less widespread in the literature;
nevertheless, most structural change models feature some sort of asymptotically
balanced growth path, e.g. Kongsamut et al. (1997, 2001), Echevarria (1997) and
Acemoglu and Guerrieri (2008). In the reference model of Chapter III I that an
asymptotically balanced growth path exists.
Overall, the difference between a balanced growth path (steady state) and an
asymptotically balanced growth path (asymptotic steady state) can be explained as
follows: If a steady state exists in a differential equation system, the differential
equation system jumps right into this steady state, provided that the initial values (in
our example ) are set to some specific finite values
( ). (In general, these values can be derived from model parameters.)
That is, for there is no transition period and
the economy starts in the steady state and remains in the steady state (provided that
model parameters are not altered for t>0). The things are quite different for an
asymptotic steady state: there is no finite set that puts the economy
directly into the final dynamic state where y, x and z grow at a constant rate. That is,
for any finite setting of the economy must first go through a transition
period (which lasts infinitely). The final dynamic state where all variables grow at a
constant rate is reached after an infinite period of time.
)0(),0(),0( zxy
)0(),0(),0( *** zxy
)0()0(),0()0(),0()0( *** zzxxyy ===
)0(),0(),0( *** zxy
)0(),0(),0( zxy
50
Definition 10: A partially balanced growth path (“partial steady state”) is a
trajectory where at least one of the dependent variables of a differential equation
system and/or some transformations of these variables grow at a constant (and
identical) rate. Provided that a partially balanced growth path exists in a differential
equation system, there is a set of finite initial conditions and parameters that puts the
differential equation system directly onto the partially balanced growth path.
The key difference between a balanced and partially balanced growth path is that a
partially balanced growth path does not require that all variables grow at a constant
rate, i.e. some variables can grow at different and non-constant rates. Furthermore,
depending upon the actual purpose of the model and analytical requirements, the
partially balanced growth path can be defined upon the constancy of the growth rate
of only one or even more variables. In our example, a partially balanced growth path
may be defined in several ways, e.g.,
• it may be defined as a a growth path where y grows at constant rate but not x and
z
• or it may be defined as a growth path where y and z grow at an identical
constant rate but not x
• or we may define a transformation of variables x and y, e.g. n := f(x,y) where f is
a function of x and y, and we may define a partially balanced growth path as a
growth path where n and z grow at a constant rate.
The concept of partially balanced growth in structural change has been introduced by
Kongsamut et al. (1997, 2001) and used by e.g. Meckl (2002), Ngai and Pissarides
(2007), Foellmi and Zweimüller (2008). This concept is in focus of my research. I use
it in all the essays of Chapter V. As we will see in the next chapter, the advantage of
this concept is that it allows for structural change (since not all variables need to grow
51
at constant rate), while keeping the analysis traceable (due to the constancy of the
growth rates of some variables).
The fact, that we can define a dynamic equilibrium, does not imply that such an
equilibrium exists in a differential equation system. There may be some models where
none of the above dynamic equilibriums exists. Hence, in some sense the art of
modelling is to find intuitively and empirically reasonable assumptions which produce
differential equation systems where the sort of dynamic equilibrium, that is useful for
analysing a certain phenomenon, exists. I demonstrate in Chapter III by altering the
assumptions of my reference model that different sorts of dynamic equilibrium can
exist depending upon the model assumptions.
The last question that I discuss in this section is how I can find a dynamic equilibrium
in a differential equation system. This question is less relevant for my research, since I
try to formulate the assumptions a priori such that a dynamic equilibrium exists (or I
alter models where I know that a dynamic equilibrium exists.) However, for the sake
of completeness I discuss this topic in short. To my knowledge there are two ways to
find such an equilibrium:
(1) Simply look at the differential equation system. Often, differential
equations systems are simple enough to recognize that some variables grow at
constant rates under certain circumstances. For example, such an approach is
often chosen in the discussion of the Ramsey-Cass-Koopmans-model, see e.g.
Barro and Sala-i-Martin (2004), p.99. I will also demonstrate the usage of this
method in Chapter III several times.
(2) Sometimes, by simulating the model in a computer a stationary point
(“steady state”) to which the model converges can be identified graphically.
This can be used as a starting point for using approach (1).
52
If none of these approaches works (and if you are not a really bad scientist), probably
there does not exist the type of dynamic equilibrium that you have searched for.
However, probably most models approach asymptotically to some analytically
understandable “final state”. This state may be regarded as a dynamic equilibrium.
2.2.2 Stability of dynamic equilibrium As already mentioned a dynamic equilibrium is a quite nice thing. It allows us to
intuitively understand some of the model dynamics. However, outside of a dynamic
equilibrium other impact-channels may exist which dominate the impact-channels of
the dynamic equilibrium. That is, dynamic equilibrium dynamics may not represent
very well the transitional dynamics. Hence, it is important to study the transitional
dynamics of a differential equation system. Furthermore, if a dynamic equilibrium is
unstable, it appears unlikely that the dynamics of an economy during an arbitrary
period of time are well represented by dynamic equilibrium dynamics, since the
probability that the economy is outer dynamic equilibrium may be relatively high in
this case. Hence, to justify the focus on dynamic equilibrium analysis we must show
that a dynamic equilibrium is stable. In the following subsections I explain the
concepts of stability and methods that are used to prove stability and study transitional
dynamics (in my research).
2.2.2.1 Basic concepts and definitions of stability In literature there are several concepts of stability; sometimes the usage of terms is not
unified across books. Therefore, the definitions from this section may be regarded as
working definitions, which I use in my research. I do not try to formulate the
definitions as generally as possible, but I try to formulate them such that the
explanations in the essays of Chapter V are easy to understand. For extensive
53
discussion I recommend Hahn (1967), p.1ff and Gandolfo (1996), p.331ff. For the
purposes of my research the definition of saddle-path stability is essential, since the
differential equation systems that result from economic dynamic optimization
problems (which are in focus of my research) typically result in saddle-paths (see also
Gandolfo (1996), p.374f on this topic). Furthermore, to evaluate the generality of my
results the distinction between local (saddle-path-)stability and global (saddle-path-
)stability is essential.
Definition 11: A dynamic equilibrium is stable if the differential equation system
converges to the dynamic equilibrium when it is not in dynamic equilibrium. That is,
even if the initial conditions of the differential equation system are such that the
differential equation system does not start in dynamic equilibrium, the differential
equation system converges to the dynamic equilibrium.
This definition is important: We have no reason to assume that the initial conditions,
that are necessary to start in a dynamic equilibrium, are satisfied in reality. Hence, it is
important to show that the differential equation system converges to the dynamic
equilibrium (and hence hopefully in the long run the dynamic-equilibrium-dynamics
are dominant) if these initial conditions are not satisfied. If a dynamic equilibrium is
unstable, it may make no sense to study the dynamic-equilibrium-dynamics, since in
most cases we cannot expect that the economy will be close to the dynamic
equilibrium. Hence, the equilibrium dynamics become less interesting. Nevertheless,
sometimes it may make sense to study even unstable dynamic equilibriums: In fact
the key feature of a dynamic equilibrium is that some impact channels do not apply in
equilibrium and/or that several impact channels offset each other; hence, dynamic
equilibrium analysis is quite simple. It can be used to study those channels which
54
apply in the equilibrium and if these channels apply even outer-equilibrium, analysis
of unstable dynamic equilibriums can help to understand some important impact
channels of the model in a simple way.
Definition 12: The dynamic equilibrium of a 2-dimensional differential equation
system is saddle-path stable if the set of initial conditions, which ensures convergence
to the dynamic equilibrium, is given by a one-dimensional manifold (i.e. curve). This
curve is named saddle-path.
This definition is based on more general definitions by Gandolfo (1996), pp.373ff,
Acemoglu and Guerrieri (2008), pp.484ff and Acemoglu (2009), pp.269-273. I use the
simpler definition, since in all my research I reduce the differential-equation-stability
problems to two dimensions; hence, this definition is sufficient for understanding the
essays of Chapter V. All the essays of Chapter V work with this type of saddle-path
stability.
We can see from Definition 12 that saddle path stability is a relatively weak concept:
Not every set of initial conditions yield convergence to the dynamic equilibrium, but
only the sets that are given by the saddle-path-curve lead to convergence. Hence,
there are many sets of initial conditions that do not yield convergence to the dynamic
equilibrium. Therefore, further reasoning is necessary to ensure convergence to the
dynamic equilibrium. In neoclassical growth models, this reasoning is often based on
the assumption of a rational household and neoclassical production and utility
functions: Under these conditions, the representative household chooses to be on the
converging manifold (saddle-path), since other feasible growth paths are suboptimal.
For example, in the Ramsey-Cass-Koopmans-model it can be shown that the feasible
non-converging initial-condition sets yield growth paths that yield less consumption at
55
any point of time in comparison to the saddle-path; for details see Gandolfo (1996),
pp.384-386.
Nevertheless, the concept of saddle-path-stability is very widespread in (neoclassical)
growth theory. The reason for this fact is that neoclassical growth theory tries to
provide a microfoundation of their models by assuming some (representative) agents
(households/firms). This assumption normally results in optimal control problems (see
also Section 3) and the solutions of these problems in general yield differential
equation systems with the saddle-path feature; see also Gandolfo (1996), p.374f on
this topic. (For examples of such neoclassical growth models, see e.g. Barro and Sala-
i-Martin (2004).)
Definition 13: A dynamic equilibrium is globally stable, if every arbitrary set of
(finite) initial conditions induces convergence of the differential equation system to
the dynamic equilibrium.
Definition 14: A dynamic equilibrium is locally stable, if the initial conditions, which
are close to the dynamic equilibrium, yield convergence of the differential equation
system to its dynamic equilibrium.
We can see that local stability is a relatively weak concept. Local stability means that
differential equation systems which are close to their dynamic equilibrium converge
to their dynamic equilibrium. However, we do not know what happens if the
differential equation system is not close to its dynamic equilibrium. It may be globally
stable, but also it may be not.
In general, local stability is easier to prove in comparison to global stability. In the
next subsection the methods will be discussed. There are many papers that do not
56
manage it to prove global stability. For example, Acemogly and Guerrieri (2008) give
only a proof of local stability of their dynamic equilibrium. By now, in endogenous
growth theory there does not exist a general proof of global stability for models with
multiple capital goods; merely local stability has been proven by now.
Note that, in general, the fact, that only local stability can be proven, does not imply
that the model is globally instable. It is simply relatively complicated to prove global
stability; locally stable models can be globally stable or globally unstable. Hence,
such a model is not useless. If it can be shown by empirical evidence (e.g. by a
simulation) that the model dynamic-equilibrium dynamics describe the reality well,
the model may be regarded as “acceptable”. Furthermore, in some cases the proof of
global stability may be irrelevant since it is not necessary for the key-argumentation.
For example, the proof of stability is not very important in my model from the
Kuznets-Kaldor-Puzzle-essay (Chapter V), since the model is not aimed to be
descriptive, but seeks only to show the existence of certain structural change patterns.
The relevance of these structural change patterns can be shown either theoretically
(by showing global stability of these patterns) or empirically (by showing that these
patterns are a relatively big part of actual structural change). In fact, in the essay on
the Kuznets-Kaldor-puzzle both ways are chosen.
Note that models, which are merely simulated on a computer, are in most cases not
feasible of showing that the growth path that they have simulated features some sort
of global validity/stability. There may be some techniques (e.g. sensitivity analysis)
that can help to get a notion of what happens when parameters change. However,
these techniques are rather not feasible of really proving some sort of global
validity/stability of the model results.
In the essays of Chapter V I prove global saddle-path-stability. That is, I show that
there exists a saddle path to the dynamic equilibrium (for every value of the initial
57
state variable) and that the representative household decides to be on this saddle path.
Hence, for every set of (finite) initial conditions the system converges to the dynamic
equilibrium.
2.2.2.2 Methods for proving global and local stability of equilibrium As mentioned there are two stability concepts: local and global stability. Local
stability is easier to prove; however, a proof of global stability is desirable, although
not always feasible. Now, I introduce the methods which I use in my research;
however, I also explain briefly some alternative methods, especially methods of
proving global stability.
2.2.2.2.1 Local stability To prove local stability I use a linear approximation approach. That is, the differential
equation system is linearly approximated around its dynamic equilibrium and then
stability of this linear approximation is proven. This approach works only if the
differential equation system is linearly approximable around its dynamic equilibrium.
As we will see, fortunately the models that I use are linearly approximable.
Furthermore, since the linear approximation is only a good approximation in the
neighbourhood of the dynamic equilibrium (point of approximation), this proof of
stability has only local validity. In the following I present a recipe for proving local
stability by using linear approximation (for further explanations regarding this recipe
see e.g. Acemoglu (2009), pp.269-273). This recipe is used by e.g. Acemoglu and
Guerrieri (2008), pp.484f, to show the local saddle-path-stability in their structural
change model. In Chapter V, I demonstrate how this recipe works in the models about
Ageing and the Kuznets-Kaldor-puzzle. The following steps need to be done to prove
local stability:
58
1. First, the differential equation system must be reformulated such that the
dynamic equilibrium is a steady state. (That is, by using an auxiliary variable
the dependent variables of the system must be expressed in “efficiency units”;
see also the discussion of the Definition 8 and footnote 2 in Section 2.2.1.)
2. Subsequently, it has to be shown that the determinant of the Jacobian
evaluated at the steady state is different from zero. If this can be shown, we
know that the behaviour of the differential equation system can be linearly
approximated around the steady state (i.e. the steady state is “hyperbolic”,
because of the Grobman-Hartman Theorem); see also Acemoglu (2009),
p.926.
3. The number of negative eigenvalues of the Jacobian evaluated at the steady
state must be equal to the number of state-variables of the system. (State
variables are the variables that have an exogenous initial condition in optimal
control problems; see also Section 3 on this fact). In this case there is a saddle-
path in the neighbourhood of the steady state. All economies starting at this
saddle-path will converge to the steady state.
4. The question is whether for given initial value(s) of the state variable(s) the
economy will be on the saddle path (i.e. whether control variables in the initial
point of time will correspond to the control variables that lie on the saddle-
path for given initial state variables)3. Remember that I have discussed in the
previous section (during the discussion of Definition 12) that the economy
needs not necessarily starting on the saddle-path. However, I have mentioned
there as well that by using neoclassical assumptions this problem can be
solved. A mathematically exact formulation and proof of necessary and
sufficient conditions for starting on the saddle-path is given by Acemoglu
3 For distinction between control and state variables; see Section 3.
59
(2009), p.257 (Theorem 7.14); see Acemoglu (2009), p.272. In fact these
conditions are satisfied if we assume neoclassical production/investment
structures and neoclassical utility functions4 and if we assume the satisfaction
of a transversality condition (which is in general presumed in most
neoclassical models);5 see also Acemoglu (2009), p.269. Simply speaking, I
can ensure that my models are on the saddle-path if I take some typically
neoclassical assumptions. This among others is one reason for using
neoclassical assumptions in structural change modelling.
Corollary 2: If the determinant of the Jacobian of the differential equation system
evaluated at the steady state is non-singular and has as much negative eigenvalues as
the differential equation system has state variables, then the steady state is locally
saddle-path stable. In this case we know that the system can converges to the steady
state if the initial conditions do not deviate much from the steady state.
2.2.2.2.2 Global stability To my knowledge there are two ways of proofing global stability: by using phase
diagrams and by using Ljapunov’s Second method.
4 Exactly speaking the utility function must be concave and the dynamic constraint must be concave. The latter is concave if the sectoral production functions are concave and if the dynamic investment constraint is assumed to have the form like in the Ramsey-Cass-Koopmans model. These assumptions ensure that the Hamiltonian function is concave, which is a requirement for starting on the saddle path; see also Acemoglu (2009), p.269. My models feature all of these assumptions. Therefore, I am able to show that the Hamiltonian is indeed concave in my models; see e.g. APPENDIX A of the Kuznets-Kaldor-Puzzle-essay. 5 I also presume that the necessary transversality condition holds in my models. In the context of the Ramsey-Cass-Koopmans-model, such transversality conditions are intuitively explained by ruling out Ponzi-games. That is, the representative household is not allowed to cover its interest payments on debt by borrowing more and more perpetually. That is, in the limit the value of household assets must be non-negative; see also Barro and Sala-i-Martin (2004), p.89. Since the aggregate structure of my models is regarding these facts the same as the structure of the Ramsey-Cass-Koopmans model, this argumentation can be applied to my models as well; see also Section 3.
60
The latter method may be regarded as the ultimate way, since it is applicable to
(nearly) any differential equation system. However, the drawback of this method is
that in most cases it is quite difficult to apply it (especially it seems to require a
certain amount of “ingenuity” to find “Ljapunov’s distance function”, which is
necessary to prove global stability; see Gandolfo (1996), p.411). Furthermore, this
method has originally been created to prove global stability and not global saddle-
path-stability. Remember, that we have mentioned in section 2.2.2.1 that the optimal
control problems in my research produce saddle-paths. Hence, to apply Ljapunov’s
Second Method I would have to alter it such that it proves global saddle-path
stability. For these reasons and due to the fact that I have found a way to analyse the
three-dimensional stability problems of my research in phase diagrams, I do not use
Ljapunov’s Second Method. Therefore, I do not discuss it here in more detail. For a
discussion of Ljapunov’s Second Method, see e.g. Gandolfo (1996), pp.407ff, or
Hahn (1967) pp.93ff. Overall, Ljapunov’s Second Method seems to be valuable and
there seems to be a lot of potential to this method in economics, especially since it has
not been used widely by now in economics and since it is applicable to nearly any
differential equation system. Further research on the applicability of this method in
economics would (hopefully) allow us to analyse higher-dimension differential
equation systems, which would allow us to make more general assumptions in
economic modelling. The drawback of this method (namely the difficulty of finding
Ljapunov’s distance functions) could be reduced, if this method was used more
frequently (in this way experience regarding useful/applicable Ljapunov’s distance
functions in economic models could be acquired).
Since I focus on phase diagram analysis, I explain it here in more detail. I try to give
here some general notion on how phase diagrams are used in stability analysis.
However, it is very difficult to explain the usage of phase diagrams without a concrete
61
example i.e. a model. Therefore, after reading the following lines I suggest you to take
a look at the models on the Kuznets-Kaldor-Puzzle and Ageing in Section V where
phase diagram analysis is applied.
The following steps must/should be done to prove the stability of a two-dimensional
differential equation system in phase space:
1.) The dynamic equilibrium of each of the differential equations of the
differential equation system should be found.
2.) The differential equation system must be reformulated such that the dynamic
equilibrium is a steady state. (That is, by using an auxiliary variable the dependent
variables of the system must be expressed in “efficiency units”; see also the
discussion of the Definition 8 and footnote 2 in section 2.2.1.)
3.) For each of the variables, the curve (locus) along which the variable is in
steady state must be depicted in the phase space. For example, if we have a two-
dimensional differential equation system (y, x), the loci in phase space can look as
follows (actually they look like this in the Ramsey-Cass-Koompmans model):
62
Figure 1: Exemplary phase space for a two-dimensional differential equation system (y, x)
x
y
combinations of x and y for which y is in steady
combinations of y and x for which x is in steady state
II
III
I
IV
4.) We can see that the steady state loci, divide the phase space into several
sections (in our example these are the sections I-IV). By studying the differential
equations of the corresponding differential equation system it can be entangled
which dynamics forces rule in each of the sectors.6 That is, for each of the
sections it can be said, whether x increases or decreases and whether y increases
or decreases over time. In general this is illustrated by small arrows, as shown in
the following Figure (where the arrows are only examples and are drawn like in
the Ramses-Cass-Koopmans-model):
6 For detailed explanations of how this is done, see e.g. the stability analysis of the essays from Section V, especially the essay on Kuznets-Kaldor-puzzle or the essay on Ageing; for textbook-type explanations related to the set-up of the phase-diagram of the Ramsey-Cass-Koopmans-model, see e.g. Barro and Sala-i-Martin (2004), pp.99ff.
63
Figure 2: Exemplary phase space for a two-dimensional differential equation system (y, x)
x
y
combinations of x and y for which y is in steady
combinations of y and x for which x is in steady state
II
III
I
IV S
.
These small arrows, which stand for the forces within each section, indicate in
which direction the dependent variables of the differential equation system are
moving over time, provided that these dependent variables are in the range of the
space of the section. For example, as long as the system is in section IV it will
move upwards and in direction of the y-axis over time, i.e. x will decrease and y
will increase over time. It should be noted that the closer the system is to one of
the steady state loci, the weaker/slower is the motion that is ruled by the
corresponding locus. The steady-state locus of y rules the motion of y and the
steady state locus of x rules the motion of x. The closer the system is to the
steady-state-locus of y (x), the slower the motion of y (x). For example, if the
system is in section III, the downward motion (i.e. the decrease in y) is ruled by
the steady-state-locus of y. Hence, the closer the system comes to the steady-state-
locus of y the weaker the change in y over time.
Furthermore, the intersections of the two steady state-loci are the steady states
(dynamic equilibriums) of the system: only at the intersections of the steady state
64
loci both variables of the differential equation system are in steady state. Hence, S
is a steady state. It should be noted that beside of S there may be some further
steady states depending upon the model; e.g. if the Figure 2 depicted the Ramsey-
Cass-Koopmans-model the origin is a steady state among others. This point will
be discussed in step 5.
5.) Now, after the forces within the sections have been analysed and the dynamic
equilibriums (steady states) have been spotted, the analysis of the phase space is
completed and the stability analysis can start. Stability analysis means that we
look at each of the steady states in the phase diagram, and try to judge whether the
system moves towards them over time (which would imply stability) or away
from them (which would imply instability). For example, let us look at S as a
steady state. We can see that, if the system is in sections II and IV, it will not
converge to S, since the arrows imply movement away from S. On the other hand,
since the arrows in sections I and III point rather to the steady state, they imply
that there must be a path (a curve) which leads the differential equation system to
the steady state S. According to our definitions from Section 2.2.2.1, this implies
that the steady state S is saddle path stable. For those, who do not find this
argument convincing, I can recommend the method of local stability proof: In the
previous section we have provided a mathematical way to show that indeed there
is only a one-dimensional manifold that leads to the steady state. In general, in
addition to the graphical proof of saddle-path stability, the local stability proof is
provided by many authors, as a sort of approbation/confirmation of the graphical
results. We depict this discussion in the following figure:
65
Figure 3: Exemplary saddle-path of a two-dimensional differential equation system (y, x)
x
y
combinations of x and y for which y is in steady
combinations of y and x for which x is in steady state
II
III
I
IV S
. saddle-path
x(0)
y(0)
y’(0)
In Figure 3 the saddle-path goes through the origin and approaches infinity. Thus,
this figure implies that for every x at any point of time (e.g. for x=x(0)) there is a
corresponding y (in our example y(0)) that puts the differential equation system
onto the saddle-path. Once the system is on the saddle-path, it “travels” along this
saddle-path to the steady state (provided that there are no model-parameter
changes). This is consistent with our definition of saddle-path stability from
Definition 12 (Section 2.2.2.1).
6.) However, as can be seen from Figure 3, saddle-path-stability implies that for
every x there is an infinite number of y’s, which do not bring the economy onto
the saddle-path, and hence into the steady state. We can see from Figure 3 that,
e.g., the point (x(0),y’(0)) does not lead the system to the steady state (the arrows
imply that from this point the system moves left downward). That is, there is an
infinite number of initial points/conditions which do not inudce convergence to
the steady state. Hence, to justify the focus on steady state analysis we have to
show that these initial conditions are not relevant and that the system always starts
66
at the saddle-path. This is where normally economically intuitive arguments are
used in (neoclassical) growth theory. Hence, we cannot approach without giving
our variables an intuitive meaning. Instead of discussing here an explicit example,
which would be quite lengthy, I refer to the Parts I and III of Chapter V as
examples for how it can be ensured that the system starts on the saddle-path.
Furthermore, see e.g. Barro and Sala-i-Martin (2004), p.103, or Gandolfo (1996),
p.386, for the corresponding discussion in the standard Ramsey-Cass-Koopmans-
model. In the following, I briefly illustrate their arguments in order to give you a
notion of the kind of arguments that are used in general.
For example, Figure 3 depicts the phase space of the Ramsey-Cass-Koopmans-
model, where y is interpreted as consumption and x as capital, and where a
representative household seeks to maximize its life-time-utility. In this model it is
argued/shown that all growth paths that start below the saddle-path (e.g. the path
which starts with initial condition (x(0),y’(0))) yield less consumption for any
given capital level at any point of time; see e.g. Gandolfo (1996), p.386. Hence, it
would be suboptimal for the household to choose such a growth path. On the other
hand, it can be shown that all growth-paths that start above the saddle-path hit the
y-axis in finite time, which yields a down-ward-jump in consumption due to the
Inada-conditions, which is suboptimal (intuitively spoken, jumps in consumption
are always suboptimal in neoclassical growth frameworks, which is due to
decreasing marginal utility of consumption); for details see e.g. Barro and Sala-i-
Martin (2004), p.103. Hence, the only optimal strategy for the household is to start
on the saddle-path, which ensures that the economy will approach to the steady
state.
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If these six steps are successfully completed the stability analysis of a differential
equation is completed and we hopefully know whether the (relevant) steady state is
globally stable.
It should be noted that the stability analysis by the use of phase diagrams has two
essential drawbacks:
First, the proof of stability by using a phase diagram is only applicable to two-
dimensional differential equation systems. To my knowledge, if you have a higher-
dimensional differential equation system, there is only one way to make it analyzable
within a phase diagram: You could try to solve some of the differential equations of
your differential equations system and to insert them into the other differential
equations. In this way a higher-dimension stability problem could be reduced to a
two-dimension stability problem (hence phase-diagram-analysis would become
applicable). However, this approach may not be applicable for three reasons:
(I) Solving differential equations is often rather an art than the application of
concrete receipts, since it requires integral calculus. Hence, sometimes we may
not be able to solve a differential equation (within reasonable period of time).
(II) Since the solution of a differential equation has normally some time
dependent exogenous terms (see e.g. in Section 1 equation (1a) as a solution of
equation (1)), it can happen that the resulting two-dimensional differential
equation system is non-autonomous. As mentioned in Section 1 (during the
discussion of Definition 4) and as we will see below, the analysis of non-
autonomous differential equation systems can be quite difficult (since it
includes time-varying dynamic-equilibrium-loci; see below).
(III) If we solve a differential equation and the solution is an implicit function,
it may happen that the solution of this differential equation cannot be inserted
into the other differential equations. Hence, we cannot reduce the three-
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dimensional differential equation system to a two-dimensional differential
equation system. I have had this problem in my essays on the Kuznetzs-
Kaldor-Puzzle and Ageing (see there on more explanations of this problem).
However, I have found a way to deal with this problem:
Try to find a two-dimensional transformation of your higher-dimensional
system and study the stability of this-two dimensional equation system. For
example, if you have a three-dimensional differential equation system (y, x, z)
where the differential equation for z can be solved and results in an implicit
function of z, you could define a new variable n = f(x, z). Then, by using this
definition and the solution for z, you could reformulate your original
differential equation system (y, x, z) into a two-dimensional differential
equation system (y, n), which can be analysed by using a phase diagram. This
approach features several requirements
a) It requires the solution of a differential equation (in our example z).
a) It has to be proven, that the steady state of the differential equation
system (y, x, z) has a unique coincidence with the steady state of the
differential equation system (y, n). In other words, it has to be shown
that the differential equation system (y, n) is in steady state, if and only
if the differential equation system (y, x, z) is in steady state and vice
versa. Only in this case the proof that the steady state of (y, n) is
globally stable implies that the steady state of (y, x, n) is globally
stable.
b) In general, a transformation of a three-dimensional differential
equation system into a two-dimensional differential equation system
leads to loss of information. Hence, the phase space (y, n) may include
some economically contra-intuitive spaces. Hence, the phase space (y,
69
n) may be restricted, which may be difficult to handle (what happens
when the restriction is reached?)
c) The “appropriate” transformation n = f(x, z) may be difficult to find.
It has to satisfy requirement IIa, while solving the problem that z is an
implicit function.
Second, a phase diagram provides only qualitative results. In stability analysis this is
not such a big problem. However, in the analysis of the transition period (see the next
section) this may be a drawback: We cannot derive the growth rates of the variables
during the transition period from a phase diagram. We can only obtain information
about the qualitative development patterns of the variables (whether they increase or
decrease during the transition period, monotonically, cyclically, etc.).
In previous sections I have often referred to the difficulty of analysing non-
autonomous differential equation systems (see also Definition 4). In the main part of
my research (Section V) all stability analysis is about autonomous differential
equation systems. Nevertheless, for the sake of completeness and since non-
autonomous systems arise in structural change models (e.g. in the reference model of
Chapter III and in the essay by Kongsamut et al. 2001), I explain these difficulties
briefly in the following: A key feature of non-autonomous differential equation
systems is that the steady-state-curves are moving over time in phase space. Hence,
e.g. in Figure 3, the range of the sections I-IV and the steady state S would move over
time. It is possible to show stability in such frameworks, provided that the steady-
state-curves move “monotonously” (see e.g. the analysis by Kongsamut et al. 2001);
however, it may not always be possible. For example, the reference-model from
Chapter III generates a non-autonomous differential equation system. However, this
system is that complicated that it seems impossible to disentangle the movement of
the steady state curves. A further interesting difference between an autonomous and
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non-autonomous phase diagram is: As shown in Figure 3, in autonomous saddle-path-
stable differential equation systems there is only one saddle-path and if the economy
starts on the saddle-path it remains on the saddle-path during the transition to the
steady-state, irrespective of at which place of the saddle-path the economy started.
That is, there is only one path to the steady state. In non-autonomous differential
equation systems there is a unique saddle-path for each initial condition set. That is,
depending on the initial state, the way/path to the dynamic equilibrium is different.
All in all phase diagrams are a very useful tool that plays the central role in
neoclassical growth theories.
2.2.2.3 Transitional dynamics By now all the discussion focused on dynamic equilibriums. However as mentioned
many times, the economy need not being in dynamic equilibrium. We have seen in the
previous section that there is only one combination of initial conditions that lets the
economy start directly in dynamic equilibrium (namely the combination given by the
point S in Figure 3). For all other initial points/condition there is a transition period
during which the economy converges to the dynamic equilibrium (along a saddle-
path). Obviously, it is very difficult to find economically intuitive arguments that the
economy starts in S. Therefore, I have provided methods in the previous section that
can show that the dynamic equilibrium is stable. When we know that the dynamic
equilibrium is stable, we can expect that after some period of time the dynamic
equilibrium dynamics will become dominant. In fact, if the dynamic equilibrium is
stable the dynamic equilibrium forces seem to be most important (since they are the
only trends that are not transitional but persistent “forever”).
71
Furthermore, empirical evidence can be used to answer the question whether the
dynamic-equilibrium forces are dominant in reality. For example, one reason for the
focus of the neoclassical theory on dynamic equilibrium analysis is that the
neoclassical dynamic equilibrium is often consistent with some stylized facts of
dynamics in industrialized countries (see e.g. the discussion of the Kaldor facts in the
beginning of the essay on the Kuznets-Kaldor-Puzzle in Chapter V).
Nevertheless, there might be (theoretical) cases where the transitional dynamic are
dominant for a very long period of time7 and/or where empirical evidence implies that
transitional dynamics rule in reality. Anyway it is interesting to know how the
economy develops during the transition period (e.g. whether the transitional dynamics
are very different in comparison to the equilibrium dynamics, which could help us,
e.g., to disentangle some empirical puzzles).
The methods from the previous section can help us to study transitional dynamics. In
fact, the phase diagram depicts the transitional dynamics. For example, in Figure 3 we
can see how the variables develop during the transition to the steady state. For
example, an economy, that starts in section I, moves upward to the steady state along
the saddle path. Hence, both variables (y and x) are increasing over time during the
transition period. Furthermore, it is self-evident that the growth-rates of the dependent
variables (y and x) during the transition period are not the same as in steady state: In
steady state the system remains in point S over time (i.e. y and x are constant), while
during transition the system moves (i.e. y and x change over time). Now, we could
study whether the growth rates of y and x are higher or lower during transition in 7 In fact there is a way to asses the speed of convergence to the dynamic equilibrium in theoretical models. For example, Barro and Sala-i-Martin (2004), pp.56ff, provide a method. In fact their measure is similar to the measure of radioactive decay in physics/chemistry: They calculate the “half-life of convergence”, i.e. “the time that it takes for half the initial gap [i.e. the gap between initial state and steady state] to be eliminated”. However, I do not use this method, since the speed of convergence is irrelevant for my research. For example, I can show by empirical evidence that dynamic-equilibrium-dynamics rule in my model on the Kuznets-Kaldor-puzzle. In detail, I focus on the study of industrialized countries in this model, and there the dynamic-equilibrium-dynamics of the Ramsey-Cass-Koopmans-model seem to rule; see also essay on the Kaldor-Kuznets-Puzzle in Chapter V.
72
comparison to the steady state and whether the growth rates are increasing or
decreasing during the steady state. To do so we need an example. Instead of creating
here a lengthy example, I refer to Section V, where especially in the model on Ageing
these questions are discussed. Furthermore, for a textbook-discussion of these
questions in the context of the Ramsey-Cass-Koopmans-model, see e.g. Barro and
Sala-i-Martin (2004), pp.105ff.
From Figure 3 we can see that the transitional dynamics are monotonous and
continuous, i.e. the saddle-path is continuous and monotonous. That is, e.g. when
starting from section I, the dependent variables are increasing monotonously and
continuously over time. Furthermore, note that it can be shown that the duration of the
transition period is infinite in Figure 3: The growth rates of y and x become smaller
and smaller as the system approaches the steady state S.8 Hence, in fact the steady
state is “never reached”. However, this does not matter: The closer the system is to
the steady state, the more similar is the system-behaviour to the steady-state. Hence, if
the system is relatively close to the steady state, it behaves quite similar as in steady
state.
In fact, these transitional dynamics (monotonous, continuous and infinitely lasting
transition dynamics) are representative of the transitional dynamics in all my models,
as we will see in Chapter V. There, I apply the methods of this section and study the
transitional dynamics in more detail by studying the models (as examples).
As mentioned in the previous section a drawback of studying transitional dynamics
only with phase diagrams is that phase diagrams deliver only qualitative results. For
example, from a phase diagram we do not learn by how much the growth rate during
the transition period is higher/lower in comparison to the steady state growth rate.
8 Remember that I have explained in the previous section (in point 4.)) that the closer the system comes to the steady-state-loci, the slower/weaker the movement/dynamics become.
73
Note that we have shown in the previous section that there are also some methods for
proving local stability. In fact these methods can be used to study the transition close
to the dynamic equilibrium as well. Remember that the proof of local stability
requires showing that the dynamics around the dynamic equilibrium is adequately
approximated by a linear differential equation system. Hence, the transitional
dynamics close to the dynamic equilibrium are qualitatively the same as those of a
linear differential equation system. Therefore, if we prove local stability, we know
that the transitional dynamic are monotonous, continuous and infinitely lasting, as
above. However, since the proof of local stability has only local validity, by using
local stability methods we cannot say what the transitional dynamic are when the
system is far away from the dynamic equilibrium.
Last but not least, there is another way to study some transitional dynamics:
simulations. By simulating the system on a computer we can see how the differential
equation system develops from the initial state to the steady state. A drawback of this
method is that it requires finding meaningful/adequate parameters for the model. This
is often difficult, since some model parameters may be very theoretical and difficult to
measure/estimate. Furthermore, an estimation does not provide any general result: the
results from a simulation with some specific parameter values need not being valid for
other parameter values. Hence, authors often try to run simulations for several
parameter constellations to show that at least the qualitative results of the simulation
are valid for many other reasonable parameter constellations. Hence, de facto, the
simulation approach to transitional dynamics can provide only general qualitative
results. Furthermore, these qualitative results are not necessarily of general validity,
since by using many different parameter constellations it can never be proven that the
qualitative results hold for all relevant parameter constellations.
74
3. Optimal control In all my models and in the most part of neoclassical growth theory it is assumed that
there exists a representative household that seeks to maximize its life-time utility by
consuming. The representative household has a time preference and faces a dynamic
restriction: at every point of time the household can consume and invest a part of its
wealth; the more the household consumes the less remains for investment. Since
investment is associated with capital accumulation and thus with expansion of future
production/consumption possibilities the household faces a dynamic optimization
problem: the decision on consumption quantities at one instant of time affects not
only actual utility, but also the future consumption possibilities and thus future utility.
Hence, the rational household tries to choose a plan/program of consumption
quantities that maximizes its life-time utility, while taking into account that more
consumption today is associated with less consumption tomorrow. In other words, we
have a maximization problem, where a target function is maximized subject to a
dynamic constraint. Such problems are often named optimal control problems.
The solution of such problems has already been derived by mathematicians. In fact
they have derived a “recipe” on how to solve such an optimization problem. While the
derivation of this recipe requires rather the skill level of an intermediate to advanced
mathematics student, the application of the recipe is taught to beginning to
intermediate economics students. Therefore, I will not derive the recipe (since it
would be redundant); instead I will simply explain in short how to use the recipe.
It should be noted that the method of solution of a dynamic maximization problem
(i.e. the recipe) depends on whether a model of continuous or discrete time is
assumed. In my research I use continuous time models and the corresponding solution
method (recipe) focuses on the “Hamiltonian”. (In discrete time models the Bellman-
equation is used rather; see e.g. de la Fuente (2000), pp.549ff). Introduction to
75
dynamic optimization, optimal control problems, application of the Hamiltonian and
heuristic derivation of the Hamiltonian can be found, e.g., in the books by Gandolfo
(1996), pp.374ff, de la Fuente (2000), pp.549ff and 566ff, and Barro and Sala-i-
Martin (2004), pp.604ff.
Now, I describe the procedure for solving a dynamic optimization problem in
continuous time by using a Hamiltonian. I restrict the discussion only to the case that
is relevant for my research: I assume that there is only one dynamic constraint. Let us
assume the following maximization problem:
(4) s.t. ( )∫∞
−
021 ),(),(),...(),(max dtettytxtxtxu t
mρ
(5a) ( )ttytxtxtxfty m ),(),(),...(),()( 21=&
(5b) 0)0( yy =
(.)u is the target function and (5a) is the dynamic constraint. Only the initial
condition for the state variable (5b) is exogenously given. The initial condition for the
control variable can be chosen by the household. This is in fact the reason for the fact
that later in stability analysis of the resulting equilibrium we need some intuitive
argumentation to show that the household will actually chose the initial condition for
the control variable such that the economy starts on the saddle-path (see section
2.2.2.2.2, step 6).
Furthermore, in general some non-negativity constraints and other restrictions (e.g.
Inada-conditions) are imposed on the variables. These further constraints have to be
considered in the phase-diagram analysis of the Hamiltonian optimum (see later).
76
Variables, that are determined by a dynamic constraint (i.e. ), are named state
variables. The other variables (which do not have a dynamic constraint a priori), i.e.
,… , are named control variables.
)(ty
)(1 tx )(txm
The maximization problem (4),(5) can be solved by maximizing the (“current value”)
Hamiltonian.
The (“current value”) Hamiltonian (H) for the problem (4),(5) is given by (compare
also Gandolfo (1996), p.375):
(6)( )( ) ( )ttytxtxtxftttytxtxtxu
tttytxtxtxH
mm
m
),(),(),...(),()(),(),(),...(),(),(),(),(),...(),(
2121
21
ψψ
+=
where )(tψ is an auxiliary variable named co-state variable. In the final set of
optimality conditions this variable does not appear. In fact, this variable has a similar
meaning like the Lagrange multiplier in constrained static optimization: it is the
shadow price of the restriction, i.e. it implies by how much the utility would increase
if the dynamic constraint was slacked/relaxed by one marginal unit. Exactly speaking,
)(tψ is the value of one additional marginal unit of y at time t expressed in utility
units, i.e. if at time t one additional unit of y was available, the utility in time t would
increase by )(tψ units; compare Barro and Sala-i-Martin (2004), p.607. Hence, )(tψ
is the “current value” of one marginal unit of y in time t.
Furthermore, we have to assume that a transversality condition is satisfied. This
among others is necessary to ensure that the optimality conditions, which are derived
from the Hamiltonian, are not only necessary but also sufficient (compare Acemoglu
(2009), pp.268f, de la Fuente (2000), p.572, and Gandolfo (1996), p.376). The
transversality condition is given by
77
(7) 0)()(lim =−
∞→
t
tetyt ρψ
In fact, this transversality condition has an intuitive meaning: Remember that we have
just explained that )(tψ can be interpreted as the shadow price of y. Hence, )()( tytψ
is the (current) value of y that exists at time t. If we “discount” this value with the
time preference rate (i.e. multiply )()( tytψ with ), we obtain the present value of
y (i.e. the value of y expressed in units of utility at time 0). Now, think of as a
“final state”. That is, the representative household does not live beyond . Hence,
is the present value of y that is left over at the end of households
“life”. That is, implies that the household does not leave over
anything after its “death” (compare also Barro and Sala-i-Martin (2004), p.611). This
seems to be reasonable: Why should the household leave over any resources for the
time after its existence: if the household consumes these resources in its life time it
can draw utility from them; otherwise these resources would be wasted, which would
be suboptimal. Exactly speaking, if I do not exist in infinity why should I leave over
some resources for infinity, when I benefit from consumption only in my life time.
For these reasons is never larger than zero, in models which I use, i.e.
in models with rational households and perfect foresight. Furthermore, note that
cannot be negative under usual conditions as well: y is a real resource
(in my models this real resource is capital); hence, it cannot be negative; furthermore,
since (at least in my models) the real resources bring utility and no disutility, the value
of these resources
te ρ−
∞→t
∞=t
t
tetyt ρψ −
∞→)()(lim
0)()(lim =−
∞→
t
tetyt ρψ
t
tetyt ρψ −
∞→)()(lim
t
tetyt ρψ −
∞→)()(lim
)(tψ cannot be negative.9
9 See also section 2.2.2.2.1 and there point 4.), footnote 5, for a similar explanation proposed by Barro and Sala-i-Martin (2004).
78
In fact, the solution of the maximization problem (4),(5) by using a Hamiltonian is a
two step procedure: First, the necessary conditions for an optimum of (4),(5) are
formulated; second, it is shown that these necessary conditions are also sufficient;
hence, a household that acts according to the necessary and sufficient optimality
condition maximizes its life time utility.
3.1 The necessary conditions for a maximum The necessary conditions, which are derived from the Hamiltonian (6), are given as
follows:
(8) 0)(
(.),...0)(
(.),0)(
(.) !!
2
!
1
=∂∂
=∂∂
=∂∂
txH
txH
txH
m
(9) )()()(
(.) !tt
tyH ψρψ &−=∂∂
As mentioned above, I do not prove the validity of these optimality conditions; this
has already been done by mathematicians. Heuristic proofs of these conditions can be
found in e.g. in Barro and Sala-i-Martin (2004), pp.606ff, and de la Fuente (2000),
pp.567ff.
Optimality conditions (8) are often referred to as intratemporal optimality conditions,
since they determine the optimal allocation of budget across goods for a given point
of time. Condition (9) (and condition (5)) is often referred to as intertemporal
optimality condition since it determines the optimal allocation of budget across time.
Conditions (5), (7), (8) and (9) describe the optimal path of variables, ,… ,
and
)(1 tx )(txm
)(ty )(tψ . If representative household acts according to these conditions, the
Hamiltonian (6) is maximized and the life time utility (4) is maximized. In general, in
79
economic analysis conditions (7), (8) and (9) are restructured further (where equation
(5) is used as well). In doing so, )(tψ is eliminated from the equations and a (m+1)-
dimensional differential equation system is obtained that describes the development of
the state variable ( ) and control variables ( ,… ). (For examples of how
this restructuring is done, see e.g. the essays on the Kuznets-Kaldor-Puzzle and
Ageing from Chapter V.) This differential equation system is then analyzed regarding
the existence and stability of a dynamic equilibrium by using the methods from the
previous sections.
)(ty )(1 tx )(txm
3.2 Proof of sufficiency In general, the optimality conditions (5), (7), (8) and (9) can only be regarded as
necessary conditions for a solution of the maximization problem (4),(5). That is, by
now we do not know whether the household really maximizes its life-time utility if it
acts according to conditions (5), (7), (8) and (9). However, there are several cases
where we can be sure that these necessary conditions are sufficient as well. In these
cases we can be sure that the household maximizes its life-time utility by acting
according to conditions (5), (7), (8) and (9). To my knowledge there are two such
cases:
(1) The target function and the dynamic constraint are both concave in
( ,… , ); see e.g. de la Fuente (2000), p.575, and also Acemoglu
(2009), p.269, and Barro and Sala-i-Martin (2004), p.610. In this case,
conditions (5), (7), (8) and (9) are also sufficient conditions for solving
problem (4),(5). Hence, by proving concavity of the target function and the
dynamic constraint, we can prove the sufficiency of our necessary conditions.
This approach is very useful for problems with only one state variable (y) and
)(1 tx )(txm )(ty
80
only one control variable (x), since then the target function and the dynamic
constraint are functions of only two variables (x and y) in general, and the
concavity can be proven by calculating some second partial derivatives of the
functions (for detailed description of the proof of concavity of functions with
two independent variables, see e.g. Kamien and Schwarz (2000), p.300).
Nevertheless, even in this simple case, the proof can be quite lengthy and
complicated. However, since in general in most neoclassical growth models it
is a priori assumed that the utility function and the production function are
concave, the proof of concavity of the target function and the dynamic
constraint is almost given a priori; see e.g. Acemoglu (2009), p.268f, for how
simple this proof becomes. However, showing concavity of the target function
and the dynamic restriction features some drawbacks as a proof of sufficiency:
(a) This approach becomes the more complicated the more state and/or control
variables are involved. Showing concavity of functions with three or more
independent variables requires showing that the corresponding Hessian matrix
is negative (semi)definite, which requires calculating determinants, which in
turn becomes the more complex the more independent variables are included.
(b) Concavity of the target function and the dynamic constraint are stronger
than necessary conditions (see e.g. de la Fuente (2000), p.575, and Barro and
Sala-i-Martin (2004), p.610). That is, even if not both, the target function and
the dynamic constraint, are concave, the optimality conditions (5), (7), (8) and
(9) can be sufficient. In this case the following approach may be useful.
(2) It can be shown that optimality conditions (5), (7), (8) and (9) are sufficient
for solving the problem (4),(5) by using the Arrow-Kurz-criterion; see e.g.
Barro and Sala-i-Martin (2004),p.610, Gandolfo (1996), p.376, Acemoglu
81
(2009), p.257, and de la Fuente (2000), pp.575/577. This sufficiency proof
includes three steps:
(a) Maximize the Hamiltonian (6) with respect to the control variables
for given state variable ( ), co-state-variable (y ψ ) and time (t). In fact
this results in the optimality conditions (8).
(b) Insert these optimality conditions (8) into the Hamiltonian (6). The
resulting Hamiltonian ( H~ ) is only a function of the state variable ( ),
co-state-variable (
y
ψ ) and time (t), i.e. . )),(),((~~ tttyHH ψ=
(c) Show that is concave in the state variable ( ) for
given co-state-variable (
)),(),((~ tttyH ψ )(ty
)(tψ ) and time (t). It is well known that
is concave in for given )),(),((~ tttyH ψ )(ty )(tψ and t, if
0))((
~2
2
<∂∂
tyH .
If 0))((
~2
2
<∂∂
tyH , the optimality conditions (5), (7), (8) and (9) are sufficient for
solving the problem (4),(5). A proof of the validity of this sufficiency criterion
can be found in de la Fuente (2000), p.575f.
I use the first procedure whenever I can. However, I have to use the second procedure
(Arrow-Kurz-criterion) in PART I of CHAPTER V.
4. Summary: Step-by-step procedure in continuous-time growth-modeling The methods and concepts from this chapter are applied to a reference model (and
thus explained further) in the following chapter. Before doing so, I provide a short
summary of Section II. In this way a more or less general procedure in (continuous-
82
time) growth modeling is presented. This procedure will be applied in the following
sections.
Analytical procedure in further modeling:
1. Assumptions are made about the environment in which the agents act and about the
behavioral patterns of the agents (aims).
2. These assumptions are used to formulate maximization problems.
3. The maximization problems are solved by using a Hamiltonian:
a) The necessary optimality conditions are derived.
b) Sufficiency of the optimality conditions is shown.
4. The necessary and sufficient conditions from the Hamiltonian maximization are
restructured such that (economically) intuitive differential equation systems result.
5. These differential equation systems are analyzed regarding the existence and
stability of a dynamic equilibrium.
6. Economically intuitive results are derived from the dynamic equilibrium analysis,
e.g. impact channels.
7. The transition period is analyzed and economically intuitive results are derived for
the transition period.
In the following chapter, I discuss the modeling foundations of my research.
83
LIST OF SYMBOLS of CHAPTER II * Indicates the steady-state or dynamic equilibrium value(s) of the
corresponding variable.
(.)H Hamiltonian.
(.)~H Maximum of the Hamiltonian with respect to the control variables.
S The steady-state-point in a phase diagram.
a Exogenous parameter of a differential equation (system).
b Exogenous parameter of a differential equation (system).
c Exogenous parameter of a differential equation (system).
d Exogenous parameter of a differential equation (system).
(.)f A function.
n A function (transformation) of variables x and y.
s Independent variable of a differential equation (system).
t Time index. (Independent variable of a differential equation (system).)
(.)u Target function of an optimal control problem.
)(tx Dependent variable of a differential equation (system); function of time.
)(),...(1 txtx m Control variables of an optimal control problem.
)(ty Dependent variable of a differential equation (system); function of time and/or
state variable of an optimal control problem.
)0('y An initial level of in a phase diagram that does not induce convergence
to the steady state.
)(ty
0y Initial condition for the state variable of an optimal control problem.
)(tz Dependent variable of a differential equation (system); function of time.
84
ρ Time-preference rate.
)(tψ Co-state variable.
85
86
CHAPTER III
Key Modeling Approach: Integration of Structural Change into a Neoclassical Growth Model
- Key Assumptions, Application of Analytical Tools, Challenges in Structural
Change Modeling, Application and Premises of PBGP-method -
In this chapter I present a “relatively general”1 model to explain several concepts and
questions that are of importance for understanding all the previous explanations
regarding the importance of my research. The assumptions of the model are nearly the
same as in the third model by Kongsamut et al. (1997). However, the approach in
model analysis is quite different. (In their third model, Kongsamut et al. (1997)
analyze an equilibrium growth path which features only a constant real interest rate
(growth rates are not constant along this growth path); I analyze other dynamic
equilibriums.) The following model includes the most key-assumptions, that are used
in the main part of my research (namely in the essays of Chapter V). Furthermore, all
the models, which exist in the neoclassical structural change school by now, can be
derived as special cases of the following model. Therefore, the model, which is
presented in the following, seems to be a good starting point and a good reference
model. By using this model I explain
1.) the application of the concepts of unbalanced growth, asymptotically balanced
growth, partially balanced growth and balanced growth,
2.) the standard mainstream neoclassical growth model and how structural change
affects its validity
1 “Relatively general” means here that all key structural change determinants are included; however, I use relatively simple functional forms to keep the discussion traceable.
87
3.) the general analytical challenges to structural change modeling, namely the
difficulties in understanding the dynamics of such models (transitional as well as
“equilibrium dynamics”) and
4.) what is the key to generating partially balanced growth (“a priori” and “a
posteriori” knife-edge conditions regarding model parameters).
During the discussion of the latter point I also demonstrate how the previous literature
generates partially balanced growth paths.
88
TABLE OF CONTENTS for CHAPTER III
1. An unbalanced growth model ..................................................................................90
1.1 Assumptions.......................................................................................................90
1.2 Optimality conditions.........................................................................................93
2. Analytical challenges to structural-change-analysis................................................95
3. A balanced-growth multi-sector-model ...................................................................98
4. A truly “neoclassical” multi-sector growth-model ................................................104
5. Partially balanced growth ......................................................................................106
5.1 An example of a partially balanced growth model ..........................................106
5.2 Examples from the literature on how to create partially balanced growth paths
(usage of a priori and a posteriori knife-edge conditions) .....................................110
6. Validity of neoclassical models in the light of structural change and the downside
of knife-edge-condition use .......................................................................................115
APPENDIX................................................................................................................119
LIST OF SYMBOLS of CHAPTER III.....................................................................121
89
1. An unbalanced growth model
1.1 Assumptions All model assumptions are quite the same as in the standard one-sector Ramsey-
(Cass-Koopmans-)model, i.e. they are very “neoclassical”.2 That is, we have a long-
run growth model with perfect markets3 and with a rational representative household
with perfect foresight. The only difference is that I assume the existence of multiple
(heterogeneous) sectors.
The representative household maximizes the following utility function by consuming
heterogeneous goods ),...1( mi =
(1) , ∫∞
−=0
21 ),...,( dteCCCuU tmttt
ρ 0 >ρ
where
(2) ( ) ∑∑ =>+== i
ii
m
i
it
iti
mtt SCCCu 1;0,)ln(,...
1
1 βββ
where t is the time index. denotes the “market consumption” of good i (i.e.
consumption of goods that are purchased on the market);
itC
ρ is the time preference
rate. We can see that the preference function (1) is quite the same as the functions
used in standard growth literature (i.e. it is time-separable). The only difference is that
there exist multiple goods. To meet the requirements regarding the structural change
determinants (mentioned in Chapter I) the instantaneous utility function u(.) must be
non-homothetic across goods i=1,…m and the price elasticity must be different from
one. A function that satisfies these requirements is given by (2). This function is very
similar to the one used by Kongsamut et al. (1997, 2001). I have decided for this
2 For detailed explanations of the standard Ramsey-(Cass-Koopmans)-model see e.g. Barro and Sala-i-Martin (2004), 85ff. However, in this chapter I provide some explanations on this model as well. 3 Perfect markets means that there are no information asymmetries or information delays (information is instantaneously available and processed and all agents react immediately to changing conditions), all markets are polypolistic and producers are marginalistic (price takers) there are no entry barriers.
90
function, since it is analytically very convenient (i.e. a lot of intersectoral dynamics
can be determined by the setting of the constant parameters itS ). i
tS are exogenous. If
a itS is negative, it can be interpreted as the basic need regarding good i (e.g. food,
basic education). If a itS is positive, it can be interpreted as an endowment regarding
good/service i, e.g. a household that can repair cars has some positive endowment
regarding the service “car repairing”. Furthermore, itS could stand for some free
services and grants that are provided (and guaranteed) by the government. In this
case, if some itS are assumed to be negative, they can be interpreted as goods/services
that have to be provided to the government (a sort of “tax”), e.g. military service;
Some itS could be assumed to be constant and/or equal to zero.
Since income-elasticity and price-elasticity of demand differ across goods i and are
different from unity (as long as not all 0=itS ), the preferences allow for structural
change caused by non-homothetic preferences and relative-price-changes.4
Each of the goods is produced by a polypolistic sector. Each sector produces its
output by a Cobb-Douglas production function )( itY
(3) miBgBnk
BKnBY i
tiitii
t
it
it
tit
it
it
i
,...1,,10, =∀=<<⎟⎟⎠
⎞⎜⎜⎝
⎛= &α
α
where I have normalized the aggregate amount of labor to unity. represents the
aggregate amount of capital; and represent respectively the fraction of capital
and labor devoted to sector i; is a sector-specific technology-parameter that grows
at the exogenous, sector-specific and constant rate . Equation (3) implies that the
TFP-growth rate differs across sectors. Furthermore, since
tK
itk i
tn
itB
ig
iα differ across sectors,
4 See Section 3 in Chapter I.
91
the output-elasticity of labor differs across sectors. Thus, again all requirements
regarding the structural change determinants are satisfied by these production
functions.
All capital and labor have to be used in production
(4) ∑∑ ==i
it
i
it nk 1;1
That is, there is no unemployment and I assume full labor mobility across sectors,
reflecting the long-run-character of the model.
Like Kongsamut et al. (1997, 2001) and Ngai and Pissarides (2007), I assume that
only sector m produces capital (and consumption goods). That is, the output of sectors
is used for consumption only and the output of sector m is used for
consumption and as capital:
mi ≠
(5) mttt
mt CKKY ++= δ&
(6) miCY it
it ≠∀= ,
where δ is the depreciation rate. This assumption seems to be reasonable at low
disaggregation levels, since in reality for example the manufacturing sector produces
consumption goods and capital goods.
Like Kongsamut et al. (1997, 2001) and Ngai and Pissarides (2007), I define
aggregate output and aggregate consumption-expenditures as follows )( tY )( tE
(7) ; ∑≡i
it
itt YpY ∑≡
i
it
itt CpE
where denotes the relative price of good i. Sector itp mi = is numéraire
(8) 1=mtp
92
1.2 Optimality conditions When there is free mobility of factors across sectors, the intratemporal and
intertemporal optimality conditions for this model are given by
(9) iCuCu
KkYKkY
nYnYp m
t
it
tit
it
tmt
mt
it
it
mt
mti
t ∀∂∂∂∂
=∂∂∂∂
=∂∂∂∂
= ,/(.)/(.)
)(/)(/
)(/)(/ and
ρδ −−∂∂
=−)( t
mt
mt
m
m
KkY
uu&
,
where . These optimality conditions can be obtained by maximizing
equations (1)-(2) subject to equations (3)-(8), by using the Hamiltonian. A proof is in
the APPENDIX.
mtm Cuu ∂∂≡ /(.)
These conditions imply the following equations, describing the development of
aggregates and sectors:
Equilibrium Aggregate-Behavior
(10) m
mmmt
mt
mmt
mt
mtmtt n
kkn
KBYα
αα αα ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−+= − 1)()( 1
(11) tttt EKYK −−= δ&
(12) ρδαρδ −−=−−∂∂
=++
mt
mt
t
tm
tmt
mt
tt
tt
kn
KY
KkY
VEVE ~
)(
&&
(13) tmm
ti
iimti
ii
m
tmt
mt
Y
EVW
kn
~)1(
11
αα
βααβα
α −
⎟⎠
⎞⎜⎝
⎛−−⎟
⎠
⎞⎜⎝
⎛+
+−=∑∑
Equilibrium Sector-Behavior (represented by employment shares)
(14) ( )
( ) ( )mi
KBnk
SY
VEnii
i
tit
i
i
m
mmt
mt
it
t
tti
m
iit ≠∀
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
+−−
=−
,
11
~)1()1(
1 ααα
αα
αα
βαα
93
(15) ( )t
mt
t
ttmttmt Y
SY
VEKKn ~~ −+++
=βδ&
where
(16) mmm mt
mtt
mtt nkKBY ααα )/()()(~ 1−≡
(17) ∑≡i
it
itt SpV
(18) ∑⎟⎟⎠
⎞⎜⎜⎝
⎛≡
i
it
it
itti
t
iti
t i
nBkKB
SW α
(19) i
nBkK
nBkK
pi
m
it
it
itt
i
mt
mt
mtt
mit ∀
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
= −
−
,1
1
α
α
α
α
(20) i
i
m
mmt
mt
it
it
nk
nk
αα
αα
−−
=1
1
The equation-system (10)-(20) describes the dynamics of our economy when all
markets are in equilibrium. Any further analysis of the model (e.g. the analysis of
structural change patterns) has to deal with these equations.
I omit here the intuitive discussion of equations (10)-(20), since this is only a
reference model which will be altered in the following sections and in doing so I will
explain more and more the intuition behind these equations. Nevertheless, here are
just two short intuitive explanations:
1.) Note that equations (10)-(12) are nearly the same as in the “normal” one-
sector Ramsey-model (thus the intuitive explanation of these equations is
nearly the same as well), beside of the fact that they contain the terms
and . The latter terms reflect the fact that we have here multiple sectors and
mt
mt nk /
tV
94
that reallocations between these sectors affect the behavior of aggregates (Y, K
and E). Hence, a lot of analysis of multi-consumption-goods models can be
done by focusing on the aggregate equations and the terms which reflect the
impact of sectoral reallocation on aggregates (i.e. and ). Only if
sectoral behavior is of interest (and mostly it is) sectoral equations (14) and
(15) need to be analyzed in more detail.
mt
mt nk / tV
2.) The dynamic equation system (10)-(20) implies that in general the
equilibrium development of the variables is unbalanced. That is, in general the
variables do not grow at constant rate and in general the variables grow at
different rates (which contradicts in some respect to the mainstream
neoclassical growth school as we will see). Hence, this is an unbalanced
growth model, i.e. structural change takes place (i.e. e.g. employment share are
changing). For some parameter values, however, (partially) balanced behavior
may arise. This fact will be discussed in the following sections.
Now, I discuss several points by using this model as an example for structural change
models.
2. Analytical challenges to structural-change-analysis Now, the question arises how the variables of interest (e.g. the employment shares)
develop over time. To answer this question, we could try to simulate the model by
using a computer or try to get some answers on this question by an analytical
approach. As already mentioned many times, the focus of my research is on the
analytical approach.
Only by looking at the equation-system (10)-(20) we cannot obtain much information
about the dynamics: the sectoral variables are dependent upon each other and upon
95
the aggregate variables and vice versa, making it very difficult to disentangle the
dynamics of the system at first sight. (These difficulties arise despite the fact that I
kept the model quite simple by, e.g., using as simple functional forms as possible and
by using many simplicity-promising assumptions like perfect markets.) As explained
in Chapter II, the typical approach to analytical analysis of such problems is studying
the existence and the stability of a hopefully existing dynamic equilibrium and by
studying the transitional dynamics (i.e. the way to this dynamic equilibrium) by using
e.g. phase diagrams. To do so, we have to study a three-dimensional differential
equation system consisting of variables E, K and V. The development of these
variables is primarily given by equations (11), (12) and (17), where all the remaining
equations of the system (10)-(20) have to be inserted into these equations to make the
equation-system (11)-(12)-(17) only functions of the variables E, K and V. We could
try this; however, it seems not recommendable for several reasons:
1.) The equation-system is very complicated and seems to be difficult to
interpret; thus even if we managed it to transform the dynamic-equation
system (10)-(20) into a differential equation system with only three equations,
we would probably get only little economic intuition from it.
2.) In general, we can say that the equilibrium growth path of the model is
unbalanced, i.e. the growth rates of Y, E and K are not constant and
employment shares change (since the equations from above do not give us any
reason to believe that at least some variables grow at the same rate for general
parameter settings). However, it seems to exist an asymptotic steady state5 in
this differential equation system. That is, as time goes on, the equation system
converges to a “state” where all variables grow at the same constant growth
5 see also Definition 9 in Chapter II.
96
rate6. In this steady state, the system becomes easier to understand (see also
the next section); however, no structural change takes place in this steady
state, since all variables and thus the employment shares grow at a constant
rate. (For a detailed analysis of such a steady state, see the next section.)
Hence, looking only at this asymptotic steady state does not help us very much
to understand the structural change dynamics.
3.) To understand the structural change we would have to analyze the
transitional dynamics (i.e. the way to the asymptotical steady state). However,
the differential equation system is non-autonomous due to non-homotheticity
of preferences (non-homothetic preferences are required as a structural change
determinant); hence, the analysis of transitional dynamics in a phase-diagram
is difficult or even not feasible. Furthermore, we even cannot use a phase
diagram for analysis of transitional dynamics, since phase diagrams cannot be
used for three-dimensional differential equation systems (and I see no way of
simplifying the differential equation system to only two dimensions); see also
Section 2.2.2.2.2 in Chapter II.
All these points are not very encouraging regarding my plans: even in such a simple
model we cannot study structural change analytically. The only way seems to simplify
the model. To do so there are two approaches:
1.) I can omit capital from analysis and restrict analysis to only two sectors.
This has been done in the traditional structural change school, e.g. by Baumol
(1967). In this case the model becomes quite simple and relatively intuitively
understandable even without computer simulations. I have presented this
6 This can be seen from equations (1) and (2): Since the constant parameters iS become relatively unimportant with increasing consumption quantities, the utility structure becomes more and more like a Cobb-Douglas-utility, provided that consumption increases (see also Kongsamut et al. (2001)). With Cobb-Douglas utility there is no structural change and a balanced growth path exists in this model (the proof of this fact is provided in the following section).
97
approach in Chapter I. However, as explained in Chapter I, including capital
into analysis is important (at least for the questions which I am dealing with,
e.g. for offshoring or for joining neoclassical growth theory with structural
change). Therefore, I focus on the second way, which is:
2.) Finding parameter restrictions or assumptions which ensure that the
structural change analysis keeps being traceable even with capital. This is what
I name “using a priori or a posteriori knife-edge conditions for generating
partially balanced growth paths” along which structural change can be
analyzed analytically.
Before discussing the latter approach, I present in the next section a special case: In
the model from above I set the a priori assumptions (or: a priori knife-edge
conditions) such that structural change is restricted completely. In this way we obtain
a quite neoclassical multi-sector model, which we can use to explain the mainstream-
concept of balanced growth in more detail and thus explain the neoclassical
mainstream approach to long-run growth analysis.
3. A balanced-growth multi-sector-model To explain the concept of balanced growth we have to change our assumptions such
that no structural change takes place, i.e. all sectors grow at a constant rate. As
explained in the previous section a balanced growth path (see Definition 8 from
Chapter II) does not exists in the model from Section 2 for general parameter setting.
However, the parameter setting could be changed such that a balanced growth path
exists. The following parameter knife-edge restriction ensures balanced growth:
(21) iS it ∀= ,0
98
This knife-edge condition reduces our utility function (2) to a logarithmic Cobb-
Douglas-function:
(2’) ( ) ∑∑ =>== i
ii
m
i
iti
mtt CCCu 1;0,ln,...
1
1 βββ
When keeping all other assumptions the same, the equilibrium dynamic equation
system (10)-(20) becomes:
Equilibrium Aggregate-Behavior
(10’) m
mmmt
mt
mmt
mt
mtmtt n
kkn
KBYα
αα αα ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−+= − 1)()( 1
(11’) tttt EKYK −−= δ&
(12’) ρδα −−= mt
mt
t
tm
t
t
kn
KY
EE ~&
(13’) tmm
ti
iim
mt
mt
Y
E
kn
~)1(1
αα
βαα
−
⎟⎠
⎞⎜⎝
⎛−
−=∑
Equilibrium Sector-Behavior (represented by employment shares)
(14’) miYEn
t
ti
m
iit ≠∀
−−
= ,~)1()1( β
αα
(15’) t
tmttmt Y
EKKn ~βδ ++
=&
where
(16’) mmm mt
mtt
mtt nkKBY ααα )/()()(~ 1−≡
(17’) 0=tV
(18’) 0=itW
99
(19’) i
nBkK
nBkK
pi
m
it
it
itt
i
mt
mt
mtt
mit ∀
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
= −
−
,1
1
α
α
α
α
(20’) i
i
m
mmt
mt
it
it
nk
nk
αα
αα
−−
=1
1
Lemma 1: The equation system (10’)-(20’) features a unique balanced growth path
along which and YEK ,, Y~ grow at the constant rate and where are constant
for .
mg in
i∀
Proof: In the following a variable with a “^” denotes the growth rate of the
corresponding variable. According to Definition 8 from Chapter II, a balanced growth
path requires that all variables grow at a constant rate. We start with searching for a
growth path where E and K grow at a constant rate:
(21) . ˆ constK =
(22) .ˆ constE =
(12) and (22) imply
(23) 0ˆˆˆ~=−−+ KknY mm
Equation (16) implies that )ˆˆˆ()1(~ KkngY mmmmm −−−−= αα ; hence, it follows due to
(23) that
(24) mmm gKknY =++−= ˆˆˆ~
(21) and (24) imply
(25) .ˆˆ constkn mm =+−
(13), (22) and (24) imply
(26) mgE =ˆ
100
(16) and (24) imply
(27) mgK =ˆ
(10), (25) and (27) imply
(28) mgY =ˆ
Equations (14) and (15) together with (24), (26), (27) and (28) imply that the
employment-shares are constant. Q.E.D.
Lemma 2: The balanced growth path from Lemma 1 is saddle-path stable.
Proof: The dynamic equation system (10’)-(20’) is simply a special case of the
models from Chapter V (especially of the models about the Kuznets-Kaldor-puzzle
and Ageing). Hence, the proof of global saddle-path stability from these models
applies here too. Therefore I omit it. See there for details. Q.E.D.
Lemma 1 implies that no structural change takes place along the balanced growth
path. However, we could take a look at the transition period. Equations (14) and (15)
imply that during the transition period (i.e. when YE ~/ is not constant) structural
change takes place primarily between the capital-producing sector m and the other
(consumption-goods-only-producing) sectors.7 In fact, this sort of structural change is
already well known. As will be explained in Chapter IV, this sort of structural change
has already been studied in neoclassical-like frameworks, especially by Uzawa
(1964), and is rather not in focus of my research. This sort of structural change arises
from the fact that during the transitions period (in contrast to the steady state) the
savings-rate is changing and hence the investment-to-consumption ratios are changing
7 Equation (14) implies that even during the transition period the employment shares of all sectors
grow at identical rate. Hence, relative employment between the industries ( ) remains stable.
mi ≠ jt
it nn /
101
as well (this can be seen from the standard growth models, e.g. Solow-model or
Ramsey-Cass-Koopmans-model)8. Therefore, correspondingly, factors are reallocated
across these industries (investment and consumption industries). Furthermore, it is
well known from the standard growth theory that whether the savings-rate is
increasing or decreasing during the transition period depends upon whether the initial
capital level (in efficiency units) is larger or lower in comparison to the steady-state-
capital-level (in efficiency units). In this way we could find out whether factors are
reallocated from the capital production to the consumption production. However, in
this section these considerations are rather uninteresting.
What we should learn from this section is that even if there are multiple
technologically distinct sectors a balanced growth path can exist and structural change
need not taking place necessarily. The reason for the existence of this balanced
growth path is simply the very restrictive assumption on the preferences. Cobb-
Douglas preferences are homothetic. Hence, no structural change arises from the
demand side, which allows for balanced growth. Furthermore, Cobb-Douglas-
preferences feature a unitary price-elasticity. Hence, changes in relative prices cause
one-to-one changes in relative consumption. This fact hinders structural change:
Remember that we have seen in Section 3.2 and 3.3 of Chapter I that cross-
technology-disparities cause changes in relative production possibilities of the sectors
and that the changes in relative production possibilities are reflected by relative
prices. If consumption reacts one-to-one to relative price changes, the change in
production possibilities is exactly covered by the change in demand (which arises due
to relative price changes). Hence, labor reallocation across sectors is not necessary to
meet demand changes or production possibility changes. (On impact of relative price
8 For a discussion of these models see e.g. Barro and Sala-i-Martin (2004).
102
changes on demand and change in production possibilities due to cross-sector
technology-disparity, see the Sections 3.2 and 3.3 in Chapter I.)
Note that the aggregate behavior of this model is not directly comparable to the one-
sector Ramsey-Cass-Koopmans-model for two reasons:
(1) In the corresponding Ramsey-Cass-Koopmans model equations (10’), (11’)
and (12’) would be the same, but would be equal to one. Along the
aggregate balanced growth path of our model this fact is rather irrelevant,
since is simply a constant factor along the balanced growth path. Thus,
the dynamics of our model along the balanced growth path are (quite) the
same as the dynamics of the standard Ramsey-Cass-Koopmans model along its
balanced growth path. However, during the transition period responds
to the changes in the savings rate (cf. equation (13)). Therefore, the
transitional dynamics are (quantitatively) different from the standard Ramsey-
Cass-Koopmans model. (Qualitatively, the transitional dynamics are quite the
same: monotonous and continuous.)
mt
mt nk /
mt
mt nk /
mt
mt nk /
(2) Equation (19) implies that relative prices change all the time in this model.
We have defined our aggregate output (Y) in manufacturing terms (cf.
equation (8)), i.e. manufacturing is numéraire. In reality and in the Ramsey-
Cass-Koopmans-model output is rather measured in some other compound
numéraire. (See also the detailed discussion in the Offshoring- and Ageing-
model in Chapter V). Hence, the output of our model is not comparable to the
output of the standard Ramsey-Cass-Koopmans model.
All in all the results of this chapter imply that the results of the standard Ramsey-
Cass-Koopmans model are not necessarily consistent/reconcilable with the existence
103
of multiple technologically distinct sectors, even if structural change does not take
place. The aggregate development of our model becomes the same as the standard
Ramsey-Cass-Koopmans model only if technologically homogenous sectors are
assumed. This is demonstrated in the next section.
4. A truly “neoclassical” multi-sector growth-model In addition to Cobb-Douglas preferences (cf. (2’)), we assume now that sector-
technologies are identical, i.e. and titi BB == ,αα ggi = , it,∀ . In this case the
production functions become
(3’) migBBnk
BKnBY tti
t
it
t
titt
it ,...1,,10, =∀=<<⎟⎟
⎠
⎞⎜⎜⎝
⎛= &α
α
When keeping all other assumptions the same, the equilibrium dynamic equation
system (10)-(20) becomes:
Equilibrium Aggregate-Behavior
(10’’) αα )()( 1ttt KBY −=
(11’’) tttt EKYK −−= δ&
(12’’) ρδα −−=t
t
t
t
KY
EE&
(13’’) 1=mt
mt
kn
Equilibrium Sector-Behavior (represented by employment shares)
(14’’) miYEn
t
ti
it ≠∀= ,β
104
(15’’) t
tmttmt Y
EKKn βδ ++=&
where
(16’’) tt YY =~
(17’’) 0=tV
(18’’) 0=itW
(19’’) ipit ∀= ,1
(20’’) mt
mt
it
it
nk
nk
=
We can see at first sight that the aggregate structure of this model (especially
equations (10’’), (11’’) and (12’’)) is the same as the structure of the standard
Ramsey-Cass-Koopmans model with Cobb-Douglas production function and
logarithmic utility. Hence, the proof of existence and saddle-path-stability of the
balanced growth path from the standard Ramsey-Cass-Koopmans-model applies here
as well. In fact, this model behaves the same as the Ramsey Cass-Koopmans model
during the transition period and along the balanced growth path. Note that for this
result the assumption of Cobb-Douglas utility is not necessary; any other homothetic
neoclassical utility function would yield the same result.
The results regarding structural change in this section are the same as in the previous
section: structural change takes place only during the transition period between
capital-production and consumption-goods-production.
All in all, what we learn from these sections is that in fact the results of the standard
neoclassical growth model are compatible with the existence of multiple sectors only
if all sectors have identical production functions. As discussed in Chapters I and V
105
several times, empirical evidence implies that sector technologies differ significantly.
So far, the results of the neoclassical growth theory seem not to be compatible with
multiple technologically-heterogeneous sectors. This seems to be a challenge to
neoclassical growth theory to some extent. However, we will discuss this topic later in
detail (in Section 6).
5. Partially balanced growth
5.1 An example of a partially balanced growth model The models from previous sections were not very useful for studying structural
change analytically: The quite general model from Section 1 and 2 featured no
structural change in its asymptotic steady state, while being such complicated that the
transitional dynamics (which allow for structural change) are not examinable
analytically. I simplified this model strongly by using knife-edge parameter-
restrictions in Sections 3 and 4. However, these restrictions hindered structural change
(in steady state and during transition) strongly. In this section we will see that there is
a way between these two extremes. By assuming some less restricting knife-edge
restrictions I create a partially balanced growth path. Along this growth path the
model features structural change while still being analytically comprehensible. In fact,
the trick is finding such a “less restricting” knife-edge parameter restriction. I have
found one: Assume that model parameters are such that
(29) tWV tt ∀== ,0
In this case the model equations (10)-(20) become:
Equilibrium Aggregate-Behavior
(10’’’) m
mmmt
mt
mmt
mt
mtmtt n
kkn
KBYα
αα αα ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−+= − 1)()( 1
106
(11’’’) tttt EKYK −−= δ&
(12’’’) ρδα −−= mt
mt
t
tm
t
t
kn
KY
EE ~&
(13’’’) tmm
ti
iim
mt
mt
Y
E
kn
~)1(1
αα
βαα
−
⎟⎠
⎞⎜⎝
⎛−
−=∑
Equilibrium Sector-Behavior (represented by employment shares)
(14’’’)
( ) ( )mi
KBnk
SYEn
ii
i
tit
i
i
m
mmt
mt
it
t
ti
m
iit ≠∀
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
−−
=−
,
11
~)1()1(
1 ααα
αα
αα
βαα
(15’’’) t
mt
t
tmttmt Y
SY
EKKn ~~ −++
=βδ&
where
(16’’’) mmm mt
mtt
mtt nkKBY ααα )/()()(~ 1−≡
(17’’’) 0=tV
(18’’’) 0=itW
(19’’’) i
nBkK
nBkK
pi
m
it
it
itt
i
mt
mt
mtt
mit ∀
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
= −
−
,1
1
α
α
α
α
(20’’’) i
i
m
mmt
mt
it
it
nk
nk
αα
αα
−−
=1
1
Lemma 3: A unique partially balanced growth path exists in the dynamic system
(10’’’)-(20’’’), where aggregates ttt YKE ~,,( and grow at the constant rate
and is constant. Furthermore, structural change takes place along this
)tY mg
mt
mt nk /
107
equilibrium growth path, i.e. the employment shares of sectors change. Note,
however, that here equations (14’’’) and (15’’’) imply that the employment shares
change in the dynamic equilibrium in contrast to Lemma 1.
)( itn
Proof: Note that the dynamic equation systems (10’’’)-(20’’’) and (10’)-(20’) are
identical. Hence, the proof of the existence and uniqueness of the dynamic
equilibrium from Lemma 1 applies here. Q.E.D.
Lemma 4: The partially balanced growth path from Lemma 3 is saddle-path stable.
Proof: See proof of Lemma 2. Q.E.D.
We can see that parameter restriction (29) creates an aggregate dynamic system
(equations (10’’’)-(13’’’) and (16’’’)) that is identical to the aggregate dynamic
system of Section 3 (equations (10’)-(13’) and (16’)), which was created by parameter
restriction (21). However, at the disaggregate level the two models differ: the model
from the current section features much more sectoral dynamic (cf. (14’’’) and (15’’’),
since iS need not being equal to zero to satisfy parameter restriction (29). Therefore,
in the model from the current section structural change takes place even in the
dynamic equilibrium. (Furthermore, in contrast to Section 3, the (transitional)
structural change dynamics are “richer”, i.e. factors are not only reallocated between
capital-production and consumption-goods-production, but also between the
consumption goods sectors).
We can see that the structural change dynamics and the factors which cause them are
quite easily identifiable in the dynamic equilibrium: Since in equations (14’’’) und
(15’’’) the first terms respectively and are constant and since mt
mt nk / tY~ and K grow
108
at constant rate (cf. Lemma 3), the following is true for the structural change
patterns in the dynamic equilibrium
mg
• Sector-m-employment share increases or decreases monotonously, depending
upon the sign of mS . Furthermore, the speed of change is determined by . mg
• In sectors , the direction of structural change depends upon the sign of mi ≠
iS as well, and the patterns are monotonous. Furthermore, the speed of this
decrease is determined primarily by the magnitude of the term
miii gg αα +− )1( .
Note that these structural change patterns are (primarily quantitatively) different
during the transition period: Since during the transition is not constant and mt
mt nk /
tY~ and K grow at non-constant rates different from , the structural change patterns
may be weaker or stronger, depending on from where the economy starts (below or
above the partial steady state) and depending upon how close the economy is to the
steady state. Furthermore, in some cases even the direction of structural change may
change during the transition period: We know that in dynamic equilibrium the
denominator of the second term on the right-hand side of equation (14’’’) is
increasing (cf. Lemma 3). However, during the transition period this term may be
decreasing, for example if the economy starts above the partial steady state and the
capital level must be reduced during the transition period. These points could be
analysed in more detail by studying the phase diagram
mg
9 and the development of
etc. However, these questions are not in focus of this chapter. mt
mt nk /
Note that during the transitional period an additional structural-change-channel arises:
As discussed in Section 3, the structural change patterns modelled by Uzawa (1964)
9 The phase diagram of this model can be derived in similar way as the phase diagrams of the models about the Kuznets-Kaldor-Puzzle and Ageing (see Chapter V). In fact the phase diagram of this model is qualitatively the same as the phase diagrams of the Kuznets-Kaldor-Puzzle and Ageing models.
109
arise. That is, additionally, factors are reallocated between the consumption-goods-
production and capita-goods-production. (This can be seen, e.g., from equation
(14’’’): since the savings rate “ tt YE ~/ ” is not constant during transition, employment
shares change due to change in the savings rate.). This point could be studied in more
detail, as well, by studying the phase diagram etc. However, I omit this discussion
here, since these questions are not in focus of my research.
What we should learn from this section is that very complicated and not intuitively
understandable models can be made understandable by “clever” usage of cross-
parameter restrictions. In this way we have obtained a quite understandable model
that can be used for many analytical topics. In other words, by clever usage of knife-
edge conditions a model can be created that inherits the best of two worlds: It has the
analytical transparency of a balanced growth model and rich sectoral dynamics of an
unbalanced growth model.
5.2 Examples from the literature on how to create partially balanced growth paths (usage of a priori and a posteriori knife-edge conditions) It should be noted that I have not been the first that uses knife-edge conditions in
structural change modelling to create intuitively understandable models. In fact, they
have been used since ever. I like thinking of knife-edge conditions as being divisible
into two classes:
(1) A priori knife-edge conditions. This means that right from beginning some
knife-edge restrictions are imposed by using restrictive assumptions regarding
utility and production. For example, in Section 3 I assume Cobb-Douglas
utility. The model-results (especially the existence of a balanced growth path)
110
exist only with the Cobb-Douglas utility. More general assumptions would not
allow for balanced growth. For example, I have shown in Sections 1 and 2 that
with Stone-Geary-preferences no balanced growth path exists. With CES-
preferences a balanced growth path would not exist in the model as well. (This
can be seen from the model by Acemoglu and Guerrieri (2008).) Therefore, we
can say that balanced growth is a knife-edge case, and I have obtained it in
Sections 3 and 4 by imposing a priori knife-edge restrictions (i.e. restrictive
assumptions). A priori knife-edge conditions are widespread. In fact, every
model uses some less general functional forms (e.g. time separable
preferences, identical households), which ensure the existence/transparency of
the model-results. A priori conditions may be justified by empirical evidence.
For example, if the empirical evidence implies that the elasticity of
substitution is equal to 1, the usage of a Cobb-Douglas function as a priori
knife-edge condition may be justified.
(2) A posteriori knife-edge conditions. These knife-edge conditions are not
imposed by assumption of specific functional forms, but are imposed on
specific functional forms. For example, restriction (29) cannot be expressed a
priori as a functional form. That is, we have to presume a functional form (like
(2)) and then impose this restriction on it. In fact, all a priori knife-edge
conditions can be expressed as a posteriori knife-edge conditions. For
example, assuming a CES function and imposing a posteriori that substitution
elasticity is equal to one, is the same as assuming a priori a Cobb-Douglas
function. In general, a posteriori knife-edge conditions could be justified by
showing empirically that they are satisfied.
I have introduced this distinction since it seems very important to understand the
following discussion. We will see that previously some authors have imposed a priori
111
knife-edge conditions and some authors have imposed a posteriori knife-edge
condition. The latter have been “criticized” by the former for using knife-edge
conditions. This criticism seems, however, pointless: in fact both of them are using
knife-edge conditions. This fact becomes especially clear, when we see that all the
previous literature is only special cases of the model of Section 1. All these special
cases are obtained by assuming some knife-edge conditions. In general, a knfe-edge
condition is a severe restriction whether it is imposed a priori or a posteriori. What
matters is that it does not contradict empirical evidence; however, in fact, all previous
authors use knife-edge conditions that are clearly not supported by empirical evidence
or that no evidence has been provided for.
Kongsamut et al. (2001) use a posteriori knife-edge conditions. Their model is a
special case of the model from Section 5.1. Simply assume tiggii ,∀=∧=αα in
model from section 5.1 and you obtain the model by Kongsamut et al. (2001). Hence,
their model features as well a partially balanced growth path which exists only if an a
posteriori condition is satisfied. However, their structural change dynamics are less
rich, since their sectoral production functions are nearly identical (their production
functions differ across sectors only by a constant multiplicative parameter).
Meckl (2002) integrates the demand side of the model from Section 1 into an
endogenous growth model. His results regarding the existence of a partially balanced
growth path are quite the same as those by Kongsamut et al. (2001), since the utility
modelling and the a posteriori knife-edge conditions are quite the same as in the
model by Kongsamut et al. (2001). Nevertheless, his model is valuable for explaining
the role of indeterminacy for creating partially balanced growth.
Ngai and Pissarides (2007) use a priori knife-edge conditions. Their model is nearly
identical to the model of Section 3. As mentioned in Section 3, the assumption of
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Cobb-Douglas preferences in this model restricts structural change very much (due to
unitary price elasticity). Therefore, Ngai and Pissarides (2007) use CES-preferences,
which feature price elasticity different from one. Furthermore, they restrict the model
from Section 3 further to allow for partially balanced growth: they assume
tii ,∀=αα . (It has been shown by Acemogu and Guerrieri (2008) that a partially
balanced growth path does not exist in this model, if not tii ,∀=αα .) For details of
the model by Ngai and Pissarides (2007), see the Offshoring model in Chapter V.
Foellmi and Zweimüller (2008) use a quite different framework in comparison to the
previously discussed authors. In fact, to ensure that partially balanced growth exists in
their model Foellmi and Zweimüller (2008) assume identical production functions
across sectors, which is again an a priori knife-edge condition. The model from
Section 5.1 can be used to see how this knife-edge condition works. Simply set
tiBBii ,∀=∧=αα in the model from Section 5.1. Nevertheless, the very valuable
contribution of the paper by Foellmi and Zweimüller (2008) is that their model
features the maximal degree of disaggregation; hence, it provides a lot of micro-
foundation for less disaggregated structural change models.
Note that knife-edge conditions are not only being used to create partially balanced
growth paths, but are used as well to make (simulatory) structural change models,
where no partially balanced growth paths exist, more intuitively understandable.
Examples are:
• The model by Acemoglu and Guerrieri (2008) is nearly identical to the model
of Section 3, which features a priori knife-edge restrictions. As mentioned in
Section 3, the assumption of Cobb-Douglas preferences in this model restricts
structural change very much (due to unitary price elasticity). Therefore,
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Acemoglu and Guerrieri (2008) use CES-preferences, which feature price
elasticity different from one. For this reason, no partially balanced growth
path exists in the model by Acemoglu and Guerrieri (2008), but only an
asymptotically balanced growth path. Nevertheless, their a priori knife-edge
restriction (usage of homothetic preferences instead of non-homothetic
preferences) makes the model easier to understand intuitively; nevertheless,
they require a simulation to show some results.
• The model by Baumol (1967) uses a priori restrictions as well. Baumol omits
capital from analysis and assumes only two sectors. In this way he was able to
prove his arguments with his simple model without any simulation, despite
the fact that there is no partially balanced growth path in his model.
• The second and third model by Kongsamut et al. (1997) could be mentioned
here as well. They are special cases of the model from Section 5.1. Their
second model results by simply assuming tii ,∀=αα in the model from
section 5.1. The assumptions of their third model are nearly identical to the
assumptions of Section 5.1. However, in both of their models they impose an
a posteriori knife-edge condition that is different from that of Section 5.1
(equation (29)). Therefore, a partially balanced growth path does not arise in
their model, but only a growth path with a constant real interest rate.
Therefore, among others, their model is still very difficult to study (e.g. they
do not provide a stability analysis or an interpretation of structural change).
• …
We learn from this discussion that, in fact, all essays from the partially balanced
growth school use severe knife-edge restrictions to create intuitively understandable
models. In fact, all of these restrictions either contradict empirical evidence (e.g.
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tii ,∀=αα , or tiBBi ,∀= , or tiggi ,∀= contradicts empirical evidence as discussed
in Chapter I), or are not empirically proven (e.g. the a posteriori knife-edge
restrictions).
6. Validity of neoclassical models in the light of structural change and the downside of knife-edge-condition use We have seen in the previous sections that a balanced growth path, as studied in the
neoclassical mainstream theory, is not easy to obtain by using neoclassical modeling
techniques when taking structural change into account. The model from Sections 1
and 2 does not reproduce a balanced growth path. Only strong knife-edge parameter
restrictions can ensure that despite structural change the neoclassical feature of
balanced growth is achieved. Hence, from this point of view the neoclassical growth
theory seems to be “inconsistent” with structural change, since its results can be
obtained only by imposing very restrictive knife edge conditions.
However, the usage of knife-edge conditions in economic modelling is not new, and
every model features some knife-edge conditions, whether it is a time-separable
utility, logarithmic utility, Cobb-Douglas-functions or cross-parameter restrictions in
the manner of restriction (29). Any model has to simplify in some way to become
understandable. Therefore, to say it in the words of Acemoglu (2009), pp.702f: when
comparing the advantages of such knife-edge conditions (i.e. comprehensibility), the
shortcomings of such conditions are “worth nothing”.
The key shortcoming of restrictions like (29) is that we have no reason to assume that
these conditions hold in reality. Hence, when some features of the model (especially
features associated with partially balanced growth) are used to explain some empirical
questions, the explanation of the model will always be that the knife-edge condition
115
(29) is the actual explanation of the empirical question. Hence, to make such models
more usable, we need a micro foundation for such restrictions.
As an example for this fact, the essay on the Kuznets-Kaldor-Puzzle (see Chapter V)
may be considered: To make the things short: the Kuznets-Kaldor-Puzzle implies that
the reality behaves as being close to a partially balanced growth path from Lemma 3.
Hence, we could use the model from this section and Lemma 3 as an explanation for
the Kuznets-Kaldor-Puzzle. However, the partially balanced growth path in this
model exists only due to the assumption of restriction (29): without this restriction a
partially balanced growth path does not exist in our model, and hence the model is not
consistent with/an explanation of the Kuznets-Kaldor-Puzzle. Therefore, the actual
explanation for the Kuznets-Kaldor-Puzzle is restriction (29): Only if we find an
intuitive explanation for this restriction, the partially balanced growth path of the
model (together with the intuitive explanation of restriction (29)) is the explanation of
the Kuznets-Kaldor-Puzzle. In fact, this is what I do in the essay on the Kuznets-
Kaldor-Puzzle: I argue that independency between preferences and technologies yield
the satisfaction of restriction (29) and we show that preferences and technologies are
independent in realty in part. (Of course, my empirical proof of independency is only
a first step; much more theoretical and empirical work has to be done to provide a
solid theoretical basis for the assumption of independent preferences and
technologies.)
Furthermore, it should be noted that, in general, any usage of knife-edge conditions
eliminates some impact channels (provided that the knife-edge-conditions make the
model simpler). For example, by imposing knife-edge conditions in the previous
models, we have always eliminated some structural change patterns: e.g. in Sections 3
and 4 structural change between the consumption-goods-sectors has been eliminated
by imposing the knife-edge conditions; e.g. in Section 5.1 structural change between
116
the capital-goods-production and consumption-goods-production has been eliminated
in dynamic equilibrium by using the knife-edge conditions. This fact should be kept in
mind when using knife-edge-conditions, and the model-results should always be
examined upon whether they are only due to the knife-edge conditions or whether
they have more general meaning. In the essays from Section V, I have always tried to
focus on the results that have rather general meaning (i.e. results which are not
restricted in validity by the validity of the knife-edge condition); otherwise, as in the
essay on the Kuznets-Kaldor-Puzzle, I have tried to provide some empirical evidence
for the validity of the knife-edge condition, to asses the validity of the model results.
However, it seems that often decades are required to discuss whether the results, that
are derived by a knife-edge condition, are general enough. The best example is the
neoclassical growth theory. This theory started in the 1950ies by assuming hyper-
rational identical households with time-separable preferences (which are knife-edge
conditions), etc, and today we still have a discussion about these assumptions, their
implications and the appropriateness of their usage.
All in all, from the viewpoint of the neoclassical growth theory it seems important to
find intuitive/theoretical explanations for the knife-edge conditions which are
necessary to make it consistent with technologically heterogeneous sectors. In the
essay on the Kuznets-Kaldor-Puzzle a first step in this direction is done by showing
that the required knife-edge conditions may be explained by the assumption of
consumers that do not care about the production process. However, of course there
seems to be a lot of further research necessary to completely find a micro-foundation
for these knife-edge conditions.
In this way we may also get a notion of the micro-economic presumptions that are
required by standard neoclassical growth literature. Searching deeper and deeper into
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the microeconomics for foundation of the knife-edge conditions can reveal the real
individual behavior that underlies neoclassical growth theory.
In general, all my models contribute to justifying neoclassical growth theory, since I
show that a lot of structural change dynamics is consistent with aggregate growth
paths that are similar to the neoclassical growth paths.
In the following chapter I systematize the literature on structural transformation and
classify my own research.
118
APPENDIX Efficient allocation of labor across sectors requires that the values of marginal
productivity of labor are equal across sectors (when there is free mobility of labor
across sectors), which implies:
(A.1) minYnY
pp
it
it
mt
mt
mt
it ≠∀
∂∂∂∂
= //
The representative household maximizes its lifetime-utility given by (1) and (2)
subject to equations (3)-(8). The corresponding Hamiltonian is given by:
(A.2) ( ) )(,..., 21 mtt
mtt
mttt CKYCCCuH −−+= δψ
where
121 ,..., −mttt CCC are given by (cf. (6)), miYC i
tit ≠∀= ,
mttt YYY ,..., 21 are given by (3),
mtk is given by (cf. (4)), ∑
≠
−=mi
it
mt kk 1
and is given by (cf. (4)). mtn ∑
≠
−=mi
it
mt nn 1
Control variables are: and . is state variable. ,,..., ,,..., 121121 −− mttt
mttt kkknnn m
tC tK
Thus, optimality conditions are given by:
(A.3) ⇔≠∀=∂∂
∂∂
+∂∂
∂∂
=∂∂ mi
kkK
KkYK
KkY
Cu
kH
it
mt
tt
mt
mt
ttt
it
it
it
it
0)()(
(.) !ψ
(A.4) miKk
YKk
YCu
tmt
mt
tt
it
it
it
≠∀∂∂
=∂∂
∂∂
)()((.) ψ
⇔≠∀=∂∂
∂∂
+∂∂
∂∂
=∂∂ mi
nn
nY
nY
Cu
nH
it
mt
mt
it
tit
it
it
it
0(.) !ψ
(A.5) minY
nY
Cu
mt
mt
tit
it
it
≠∀∂∂
=∂∂
∂∂ (.) ψ
119
⇔=−∂∂
=∂∂ 0(.) !
tmt
mt C
uCH ψ
(A.6) tmtC
u ψ=∂∂ (.)
(A.7) ( ) ( ) tttmt
tmt
mt
tmi
it
tit
it
itt
kKk
YkKk
YCu
KH ρψψδψψ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
∂∂
+∂∂
∂∂
−=∂∂
− ∑≠
&!(.)
From (A.4)-(A.6) follows that:
(A.8) minYnY
CuCu
it
it
mt
mt
mt
it ≠∀
∂∂∂∂
=∂∂∂∂
(.)(.)
(A.9) minYnY
KkYKkY
it
it
mt
mt
tit
it
tmt
mt ≠∀
∂∂∂∂
=∂∂∂∂
)()(
Inserting first (A.4) and then (4) into (A.7) results in:
(A.10) ( ) t
t
tmt
mt
KkY
ψψρδ&
−=+−∂∂ )(
(9) follows from (A.1), (A.6), (A.8), (A.9) and 8. Q.E.D.
120
LIST OF SYMBOLS of CHAPTER III ^ Denotes the growth rate of the corresponding variable.
itB Parameter indicating technology/productivity level of sector i at time t.
(exogenous)
itC Consumption of sector-i-output at time t; indicates how much of the output of
sector i is consumed at time t.
E Aggregate consumption expenditures; index of overall consumption-
expenditures of the representative household
(.)H Hamiltonian.
K Aggregate capital; i.e. the amount of capital that is used for production in the
whole economy.
itS Parameter of the utility function; closely related to the utility of . May be
interpreted as minimum consumption regarding good i (e.g. subsistence level),
if positive. May be interpreted as “natural” endowment of good i, if negative.
(exogenous)
itC
U Life-time utility of the (representative) household.
tV Auxiliary variable. (Function of other model-variables.)
tW Auxiliary variable. (Function of other model-variables.)
Y Aggregate output; index of economy-wide output-volume.
tY~ Auxiliary variable. (Function of other model-variables.)
itY Output of sector i at time t.
ig Growth rate of labor-augmenting technological progress in sector i.
i Index denoting a sector.
121
j Index denoting a sector.
itk Capital-share of sector i at time t; indicates which share of aggregate capital
(K) is used in sector i.
itl Employment-share of sector i at time t; indicates which share of aggregate
labor is used in sector i.
m Number of sectors.
t Index denoting time.
(.)u Instantaneous utility-function.
mu First derivative of u(.) with respect to . mtC
iα Parameter of the Cobb-Douglas production function of sector i; is equal to
output-elasticity of labor in sector i. (exogenous)
iβ Parameter of the utility function; closely related to the utility of . itC
δ Depreciation rate on capital (K). (exogenous)
ρ Time-preference rate. (exogenous)
)(tψ Co-state variable.
122
CHAPTER IV
Classification/Systematization of Structural Change Literature and Classification/Subordination of own
Research
In this Chapter, I suggest a classification/systematization of structural change
literature and show the position of my research within this system. Exactly speaking, I
review here only the formal/mathematical modelling efforts, since the focus of my
research is on mathematical structural change modelling. Essays, where structural
change is discussed rather in verbal/anecdotal/empirical manner (which is especially
true for the early structural change literature), have been reviewed in Chapter I.
In general, there are many ways to classify/systematize the structural change
literature. My systematization is designed such that it becomes clearly visible how my
research differs from other research, as I hope. Although it is usual to discuss the
categorization of the literature relatively at the beginning of an essay, I discuss it in
this chapter, since I think that the previous Chapters II and III are necessary to really
understand the following categorization.
Note that this chapter is aimed to be a systematization of literature, which is only
loosely related to the applications of my research in Chapter V. Additional literature,
which is directly relevant to Chapter V, is discussed in the essays of Chapter V, since
only in this way the argumentation in the essays can be made clear.
Furthermore, note that structural change is a word with many meanings in economics.
For example, institutional changes, changes in the behaviour of the agents, changes in
regulatory policy, etc, are often named “structural change”. However, such imprecise
and too general definitions of the word “structural change” are not practicable in
model-oriented research, since many of the facts, which are named “structural
123
change” in the literature, require completely different methods of analysis, models
and/or time horizons (and thus are not related to my research). Therefore, I categorize
here only the literature that deals with long-run (trend) changes of real production
structures; especially, there must be some reallocation across real production sectors.
This literature may be similar to my research at least regarding the analytical methods
and many model assumptions. Nevertheless, we will see that the field of literature that
analyses the questions, which I analyse, and that uses the methods, which I use, is
quite small.
I suggest the following classification, where the class of my approach is emphasised:
Figure 1: A systematization of structural change literature
Long-run Changes in Real Sectoral Production Structures
Changes in Capital Structure Changes in Intermediate Structure
Changes in Consumption Structure
Traditional (unbalanced) School New (partially balanced) School
In the following I discuss this figure. Before doing so let me point to three other
criterions by which the structural change literature can be systematized parallely to
the system of Figure 1:
(1) Structuralist Approach vs. Neoclassical Approach to Structural Change
Modelling. This distinction is inspired by Wagner’s (1997) systematization of
124
development theories (see there p.38ff). The structuralist approach presumes
some structural rigidities. These rigidities are, however, not micro-founded.
That is, some (rather: many) relationships are assumed to be exogenously
given; however, there are no other (mathematical) models that explain what
determines these relationships and why the relationships are such as they are in
reality (i.e. the argumentation for these parameter values is rather “verbal”). In
contrast, the neoclassical approach aims to provide micro-foundation of as
much relationships as possible. Especially, this micro-foundation is based on
the assumption of more or less rational agents and capital accumulation plays
often a crucial role. In fact, none of the structural change models is clearly
structuralist or neoclassical. However, there are clear tendencies regarding
whether the model features rather neoclassical assumptions (hence, a lot of
neoclassical micro-foundation applies) or rather some exogenously given
relationships. We will see that my approach is rather neoclassical.
(2) Analytical Approach vs. Simulation Approach to Structural Change
Modelling. As mentioned many times there is always a trade-off in economic
modelling: more general models are often less intuitively understandable and
therefore require simulations to disentangle their dynamics; to make a model
intuitively understandable simplifying assumptions are required often, which
reduce the generality of the model. Again, the distinction between analytical
and simulation models is not discrete. That is, in every model some results can
be derived analytically, irrespective of how complicated the model is. As
mentioned a lot of times, I focus on analytical analysis of models.
(3) Degree of Disaggregation. The degree of disaggregation is very important
regarding the question what the adequate assumptions for a model are. To my
knowledge, all authors that model low degree of aggregation (e.g. sector-
125
division: agriculture, manufacturing and services) assume perfect markets and
especially perfect factor mobility in the long run. However, the higher the
degree of disaggregation, the more important become such things as
monopoles, oligopoly, strategic behaviour of agents, etc. Furthermore, at very
low degree of disaggregation the stylized facts of structural change across
sectors (see Chapter I) are not relevant, but other stylized facts (regarding
structural change across product-varieties) are relevant, e.g. product-life-
cycles. Overall, at very high degree of disaggregation (e.g. when looking at
individual entrepreneurs/product-varieties) completely other assumptions and
model-results are required in comparison to the assumptions and results of
cross-sector-structural change models. Furthermore, the most authors, who
analyse models with high degree of disaggregation (e.g. Schumpeterian
models), do not draw any references to structural change across broad sectors
like manufacturing and agriculture. (However, there are exemptions, like
Foellmi and Zweimüller (2008).) In general, it requires further theoretical
reasoning/models to explain how the micro-reallocations at high degree of
disaggregation (“structural change across product-varieties”) are related to the
reallocations at less disaggregated level (“structural change across sectors”).
(For example, Foellmi and Zweimüller (2008), p.1322, develop a theory that
the services sector may include rather new product-varieties, since services
satisfy “less urgent needs”.) My research focuses on reproducing structural
change patterns across sectors (and not across product-varieties), i.e. I focus on
low degree of disaggregation.
In the following discussion, mostly, I classify the models as structuralist or
neoclassical and as analytical or simulation models. However, as mentioned above
126
these classifications imply solely a tendency; every model is structuralist in some
sense and every model is analytically understandable to some degree. Furthermore,
although I study structural change rather at low degree of disaggregation, in every
section I also mention some models that feature high degree of disaggregation.
127
TABLE OF CONTENTS for CHAPTER IV
1. Changes in capital structure ...................................................................................129
2. Changes in intermediates structure ........................................................................132
3. Changes in consumption structure .........................................................................137
3.1 “Unbalanced” school of structural change.......................................................137
3.2 “New” (PBGP) school of structural change.....................................................142
4. Classification of own research ...............................................................................146
5. Further aspects of classification.............................................................................149
5.1 Structural change induced by trade opening (structural change theory vs. trade
theory) ....................................................................................................................149
5.2 Factor reallocation between capital industries and consumption industries
(Uzawa’s structural change) ..................................................................................151
5.3 Factor-reallocation between the private sector and the public sector ..............152
5.4 Factor-reallocation between the research sector and the consumption sector .153
5.5 Outsourcing of home production (factor reallocation between home-sector and
market-sector) ........................................................................................................154
128
1. Changes in capital structure This point refers to the fact that there are many heterogeneous capital goods in the
real economy, which are produced by different sectors/technologies. For example, the
simplest case is the assumption that there is (physical) capital and human capital.
However, physical capital and human capital can be subdivided further, depending
upon which technology is used to produce them or depending upon the output-
elasticity of the respective capital-sort.
Changes in capital demand and/or differences in production technologies across
capital goods can yield structural change across capital producing sectors similar to
structural change modelled in Chapter III. However, in contrast to the models of
structural change of Chapter III, changes in the capital structure are closely related to
endogenous growth, as shown in the neoclassical endogenous growth literature. In
fact, there is a large body of neoclassical growth literature dealing with the existence
of heterogeneous capital goods. In some of this literature, the heterogeneous capital
goods are produced by technologically heterogeneous sectors; hence, these models
depict structural change across capital-goods sectors. An example of such models is
the Uzawa-Lucas-model with physical capital and human capital that are produced by
two technologically distinct sectors (for discussion see e.g. Barro and Sala-i-Martin
(2004), pp.247ff). More general models (i.e. models where many heterogeneous
capital goods are produced by technologically heterogeneous sectors) are presented
by, e.g., Benhabib and Nishimura (1979), Kaganovich (1998) and Takahashi (1992,
2008). (See there for further literature on this topic.) In general, these models are
treated rather in analytical manner, and of course they are rather neoclassical.
It should be noted that in all my models there is only one capital good and only one
capital-producing sector. (Usually, sector m is producing capital.) That is, structural
change across capital goods and all its impacts are exogenous in my model (i.e. all the
129
impacts of this sort of structural change are contained in the exogenous parameters,
like the technology parameters of sector m). The reasons for the fact that I do not
model structural change across capital sectors, but across consumption sectors, is
simple: First, I have not found any research topic that is feasible for me in this field.
Structural change across consumption goods sectors seemed to me relatively less
elaborated in comparison to structural change across capital goods sectors. The
second reason is rather technical: the mathematical challenges of studying such
models, as will be explained now.
Beside the fact that structural change across capital-goods can generate endogenous
growth in contrast to my models, there is a key analytical difference, which is related
to the dynamic systems that are created by multi-capital-goods-models. In general,
every capital good that is assumed to exist in a model creates a dynamic restriction
(i.e. a dynamic capital accumulation equation). The discussion in Section 3 of Chapter
II has shown that every dynamic constraint establishes a state variable. The dimension
of the dynamic system (that results from the Hamiltonian optimality conditions) is
equal to the sum of the number of control and state variables. Hence, the more capital
goods are included into a model the higher the dimension of the dynamic system that
has to be analysed in final analysis. I have explained in Sections 2 and 3.2 of Chapter
II that this analysis becomes the more complicated the higher the dimension of the
dynamic system is. (Especially, stability analysis, study of transitional dynamics and
proof of sufficiency of necessary Hamiltonian optimality-conditions become quasi
“not feasible”.)
The nice thing in my models (where only heterogeneous consumption goods exist) is
that the final dynamic system, which has to be analysed, features maximally 3
dimensions, irrespective of how much consumption goods are assumed to exist. (In
fact, this has been demonstrated in the models of Chapter III.) The reason is, that the
130
number of consumption goods, in general, constitutes the number of control variables;
and the dynamics of control variables can, in general, be described by one dynamic
equation (and several static equations) when using the Hamiltonian optimality
conditions. At the same time, the number of dynamic equations, which describe the
development of state variables, cannot be reduced. Hence, the dimension of the final
dynamic system is, in general, at least equal to the number of the state variables.
Therefore, increasing the number of capital goods always creates higher-dimensional
dynamic systems in comparison to increasing the number of consumptions goods,
and, as explained above, this makes the systems very difficult to analyse. Therefore
among others, global stability of relatively general heterogeneous-capital-sector-
models has not been proven by now, to my knowledge (see e.g. Takahashi (2008),
p.48). Furthermore, due to the analytical complexity of these models, structural
change across capital-sectors (during the transition) is very difficult to study (in
models with three or more capital goods).
Nevertheless, a lot of “interesting” research seems to be possible in this field at more
disaggregated level: At low degree of disaggregation, capital goods are produced
primarily by one sector, i.e. the manufacturing sector (see e.g. Kongsamut et al.
2001). Therefore, for explaining the stylized facts of structural change across broad
sectors (like agriculture, manufacturing and services) cross-capital structural change
may be less interesting. However, for explaining the structural change within the
manufacturing sector, cross-capital structural change seems very important. Stylized
facts about the labour-reallocation that is induced in the manufacturing sector by
cross-capital structural change and corresponding mathematical models may be
formulated. However, by now, I have not researched much in this field; hence I am
not sure to what extent the existing literature can be adapted to study these topics.
131
2. Changes in intermediates structure In reality, the output of each sector is not only produced by using capital and labour
but also by using intermediate products that are produced in other sectors (by using
capital and labour and other intermediates); furthermore, in general, each sector
produces intermediates for other sectors as well. Hence, the production structure of an
economy is a huge cob-web of relations between sectors, where each sector is supplier
and receiver of intermediate services and goods.
In the static context, this cob-web can be depicted by input-output tables, where the
relations between the sectors are depicted by constant empirically measured input-
coefficients. (On discussion of these input-output tables and their practical use, see
e.g. Pasinetti, 1988, p.57ff.) Input-output-tables may be analysed dynamically as well,
where it is assumed, that the input coefficients are constant over time. However, there
is little intuition behind dynamic input-output tables, since the input-coefficients are
not micro-founded in any sense (furthermore, we have no reason to believe that input
coefficients are constant in the long-run). Therefore, such dynamic input-output tables
are rather not useful as theoretical structural change models, but are rather of interest
for empirical “short-run” questions, like the Leontjef-systems; for discussion, see e.g.
Pasinetti (1988), pp.77ff. The latter may be known to most growth theoreticians by
the relatively restrictive production function, which is derived from them (“Leontjef
production function” or “limitational production function”), which is a special case of
the (in Neoclassics) well known CES-production function (see e.g. Maußner and
Klump, 1996, pp.56ff). An application of Lenontjef-tables in structural change
analysis is provided by, e.g., Greenhalg und Gregory (2001).
In models, which omit intermediates, it is implicitly assumed that for each good the
amounts of capital and labour (that are used in final and intermediate production) are
summed up and inserted into a production function that depicts the sectoral output as
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a function of capital and labour only; the eventual productivity effects of intermediate
restructuring are assumed to be implicitly contained in the (exogenous) productivity
parameters of the sectors.
Of course, it is interesting to disentangle how the intermediate structure affects the
sectoral production functions and in this way provide a more detailed “micro”-
foundation of the sectoral parameters. This is done, among others, by the “Sraffa-
Pasinetti-school” of structural change: The starting point of this school are input-
output-systems, which provide a detailed description of the cob-web of intermediate
suppliers and receivers. In fact, these quasi purely descriptive input-output-systems
are filled/micro-founded with economic intuition, which is based primarily on
classical economic theories (e.g. Ricardian assumptions, like subsistence wages, etc.);
see e.g. Pasinetti (1988), pp.53, 71ff, 95, and Harris (1982), p.28f. Then, the input-
output-systems are transformed to vertically integrated sectors. In this way, for each
good the whole amount of embodied labour (used in final and intermediate
production) is obtained. (For mathematical derivation of vertically integrated sectors
see Pasinetti (1988), p.97.) Overall, a relationship is constituted between “sectoral
production functions” (which are functions of labour only) and the cob-web of
intermediate relations from the input-output-tables; compare also Harris (1982), p.31.
The analysis is primarily analytical.
These (rather static) concepts are generalized/complemented/extended (among others
by alternative derivations of “sectoral production functions” and by including
dynamic aspects) by Passineti (1981, 1993), Andersen (2001), Gualerzi (2001),
Godwin and Punzo (1987), etc. An overview/discussion of these efforts can be found
in the essays by Harris (1982), Malinvaud (1995), Nayak et al. (2009), Krüger (2008),
Punzo (2006) and Landesmann and Stehrer (2006).
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Beside of these quite general approaches (“many sectors”, “many intermediate
linkages”), which are based on input-output-tables, there are also some two-sector
models, where only one of the sectors provides intermediates to sector-external firms,
for example Fixler and Siegel (1999), Oulton (2001), Sasaki (2007) and Restuccia et
al. (2008). Despite their simplicity, these models provide interesting/relevant results
regarding the question what impacts have (changes in) intermediate linkages on the
sectoral productivity parameters. These models are rather structuralist simulation
models. (However, many intuitive results can be derived from them even without
simulation, since they are relatively “simple”.)
Last (not) least, neoclassical endogenous growth theory (including “Schumpeterian
endogenous growth models”) features as well models with multiple intermediate
products. For description of such models see, e.g., Grossman and Helpman (1991),
pp.43ff, Aghion and Howitt (1998), pp.85ff, Barro and Sala-i-Martin (2004),
pp.285ff. These endogenous growth models are rather analytically solvable and
feature a high degree of disaggregation, i.e. they analyse structural change across
product-varieties. However, due to very restrictive preference and technology
assumptions, the models are, in general, not very useful for studying structural
change, since no structural change takes place (across technologically heterogeneous
sectors); see also Krüger (2008), pp.340ff and Montobbio (2002), p.410. It should be
noted that there are few exemptions: These are the models by Meckl (2002) and
Foellmi and Zweimüller (2008), which will be discussed in Section 3.2, and the model
by Aghion and Howitt (1998), pp.86ff. The latter features unbalanced expansion of
intermediate-variety-producers (see there p.87). However, when the model is
extended to include capital in production, all sectors expand at the same rate, i.e. there
is no structural change anymore (see there p.95 and see also Krüger (2008), p.341).
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The study of intermediate restructuring is not in focus of my research; respectively,
my research is not adequate for studying intermediate restructuring in detail. I include
intermediate sectors in all of my models (for the sake of generality and to show how
much intermediate restructuring can be integrated into structural change models by
my approach). However, I restrict the functional forms, which rule the dynamics of
intermediate restructuring, severely by assuming Cobb-Douglas intermediate indices.
In this way a big part of intermediate restructuring is eliminated (as can be seen, e.g.,
in the Offshoring-model and in the Kuznets-Kaldor-model from Chapter V, or in the
model by Ngai and Pissarides (2007)). In other words, the biggest part of intermediate
restructuring is exogenous in my models (i.e. the effects of intermediate restructuring
are rather represented by the changes in the exogenous model parameters, e.g.
technology parameters). Overall, studying intermediate restructuring requires
probably other models than mine.
I restrict (endogenous) intermediate restructuring in my models, since I have not
found a way to make it consistent with partially balanced growth, by now. (As
mentioned many times, I focus on the partially balanced method.) In fact, it seems
that only very restrictive assumptions on intermediate structures are feasible with
partially balanced growth (within the neoclassical framework), as shown by Ngai and
Pissarides (2007). As can be seen in the Offshoring- and Kuznets-Kaldor-model of
Chapter V, Cobb-Douglas-intermediate structures can be feasible with partially
balanced growth; however, they eliminate a lot of intermediate restructuring (as
elaborated especially in the Kuznets-Kaldor-model, Lemmma 7/Appendix E).
Overall, integrating more complex intermediate sector assumptions into my models
would make simulations necessary to derive the key model results.
It may appear contradictory that I study offshoring in Chapter V (where intermediate
restructuring is an integral part of offshoring), while stating in the actual section that
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my models are not adequate for studying intermediate restructuring. However, a
closer look at the Offshoring-model shows that there is actually no contradiction: The
key mechanism in the Offshoring-model is: offshoring (i.e. import of intermediates)
increases productivity-growth; this increase in productivity-growth accelerates capital
accumulation and therefore slows down final-goods structural change. That is, in
some sense the restructuring of intermediates is exogenous in my model: after
opening of international borders intermediates are imported from abroad (i.e. the
intermediate structure changes immediately). The proof, that this opening increases
productivity (and thus capital accumulation), does not actually require complicated
modelling of intermediate sector structure. (In fact the only necessary assumption is
that at least some sectors use intermediates.) In this sense, I do not study the long-run
impact of offshoring on intermediate restructuring, but I simply show the existence of
a long-run impact of offshoring on final-goods structural change. Therefore, too
simple intermediate sector assumptions can only affect the quantitative results of my
model (“How strong is the structural change slow-down?”); however, they do not
affect the qualitative result (namely the fact that consumption-goods-structural change
is slowed-down by offshoring). Since, anyway, my models are not designed to
produce quantitatively good results1, complicated modelling of the intermediate
sector is not necessary in the Offshoring essay.
Of course, it may be interesting to integrate offshoring into models, where
intermediate structures are more complicated than my intermediate structures. This
topic is already studied by some researchers (on literature, see the Offshoring-essay in
Chapter V). However, I guess that, nevertheless, there are still open questions
regarding this topic, especially, regarding the implications of offshoring for (longer-
run) dynamics of intermediate restructuring. 1 In fact, in most cases only simulation-models of structural change can produce quantitatively adequate results; see also Chapter I.
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Again, it should be noted that I have focused on consumption goods structural change
(instead of focusing on intermediate restructuring) only for practical reasons: I have
not found enough questions in intermediate restructuring that are feasible for me.
3. Changes in consumption structure More and more, we come closer to the actual topic of my research: change in
consumption structures. In general, in my research I refer only to this sort of
restructuring as “structural change”. In reality, there are several sectors that produce
consumption goods (e.g. agriculture, manufacturing and services). The reallocation of
labour between these sectors is named structural change (in my research). For
example, the term structural change comprises the fact that the labour share of
agriculture is decreasing and the labour share of services is increasing (in
industrialised countries). In general, the literature on (consumption) structural change
can be divided into an “unbalanced approach” and a relatively new “partially balanced
growth approach”. The difference between these two approaches is rather
methodological, as we will see. The methods of the new approach are rather familiar
with the methods of (mainstream) neoclassical growth theories. Hence, this approach
may be amenable to a larger part of researchers in comparison to the unbalanced
approach. Therefore, among others, the partially balanced growth approach has the
potential for bringing the structural change theory closer to the mainstream.
3.1 “Unbalanced” school of structural change Traditionally, structural change theory is “unbalanced”, since structural change
requires (per definition) unbalanced expansion of sectors (i.e. structural change means
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that sectors expand at different (non-constant) rates). Hence, balanced growth paths
do not exist in “unbalanced school”-models (for general parameter settings), i.e. the
“unbalanced school” focuses on analysis of unbalanced growth paths.
Remember that I have discussed in Chapter II that balanced growth paths are
advantageous, since they make a lot of analysis simple/intuitively understandable. In
fact, I have demonstrated in Chapter III that a balanced growth model is quite easy to
understand (see there Sections 3 and 4), while the corresponding unbalanced growth
model is very difficult to study (see there Sections 1 and 2).
Probably therefore among others, the “unbalanced school” has never been regarded as
mainstream theory of economic development. Rather, the neoclassical growth
theories, which base their analysis heavily on balanced growth paths, were
“mainstream”. Note that neoclassical growth theories feature as well models with
multiple sectors; however, there is no relevant structural change within these models,
due to restricting assumptions (as explained in the previous section and as will be seen
below). That is, the neoclassical growth models are rather “balanced”.
Note that, furthermore, “unbalanced school”-models feature (often) asymptotically
balanced growth paths.
In the following I provide some literature references. This literature features models
where the focus is on the study of unbalanced growth paths and where the degree of
disaggregation is relatively low. Since the models are unbalanced, they rather require
simulations to disentangle their dynamics. However, often many interesting results
can be derived from these models even without simulations.
On the one side, there are models where nearly all key results can be derived
analytically, due to simplicity of assumptions, e.g Baumol (1967) and Gundlach
(1994). The tools, which are used in this literature to increase the degree of simplicity
(and hence to increase the degree of analytical derivability of results), are:
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(1) restricting the analysis to only two sectors and/or
(2) omitting capital accumulation from analysis and/or
(3) omitting (mathematical/explicit) micro-foundation of demand
behaviour and/or
(4) omitting some structural change determinants (e.g. by assuming
homothetic preferences, identical output-elasticities of inputs across
sectors).
In fact, Baumol (1967) uses all of these tools.
On the other side, there are models which, in fact, require simulations to derive their
key results. These models use only few or none of the simplifying tools from above.
For example, Echevarria (1997, 2000) uses none of the tools from above, and the
resulting model features quite complex dynamics/simulations.
Between these two extremes there is a lot of literature. In the following I subdivide
this literature according to three criterions:
• Which structural change determinants are included (demand-side- vs.
supply-side-structural change determinants; see also Chapter I)?
• Are demand patterns micro-founded (in the sense of utility
maximization by a representative household)?
• Is capital included into analysis?
I use these three criterions, since in this way I can subdivide the literature into groups
that are more or less comparable to my research, as we will see below. These groups
are:
Baumol (1967) (in conjunction with Baumol et al. (1985)) and Beissinger (2000)
provide models where only supply-side-determinants are considered, capital is
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omitted and demand patterns are not micro-founded. Hence, these models are rather
structuralist and analytically interpretable.
Jensen and Larsen (2004), Zuleta and Young (2007) and Acemoglu and Guerrieri
(2008) focus on supply-side-determinants of structural change as well; however, they
include capital into analysis and a micro-foundation of demand patterns (utility
maximization). Hence, these models are rather micro-founded in neoclassical way.
Buera and Kaboski (2009b) provide a “micro-founded” model, where structural
change is caused by the demand-side. They omit capital accumulation and assume
quite restrictive Leontjef-production functions (in part), which makes the model more
“structuralist” again.
The remaining literature includes both, demand-side- and supply-side-determinants of
structural change. Demand-side structural change is caused by some sort of non-
homothetic preferences. These preferences are mostly based on Stone-Geary
preferences in this literature. Supply-side structural change is caused primarily by
cross-sector differences in technological progress. Some models include also other
supply-side-structural change determinants, like cross-sector differences in input-
elasticites of output.
This literature can be systematised as follows.
First, there is a group of rather structuralist models that do not include capital and
utility maximization into analysis, e.g. Gundlach (1994), Notarangelo (1999),
Mickiewicz and Zalewska (2001) and Raiser et al. (2003).
The second group, includes utility maximization; however, capital is omited from
analysis; e.g. Appelbaum and Schettkat (1999), Messina (2003), Pugno (2006)
(includes human capital), Duarte and Restuccia (2010), Rogerson (2008) and Boppart
(2010)..
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The third group may be rather described as “neoclassical” approach to structural
change modelling, since demand patterns are “micro-founded” by utility-
maximization and capital is included into analysis. The papers belonging to this group
are, e.g., Echevarria (1997, 2000), Laitner (2000) (includes land), Golin et al. (2002),
Greenwood and Uysal (2005), Bah (2007) (includes land), Golin et al. (2007) and
Buera and Kaboski (2009a).
In fact, the papers from group three are, if at all, comparable to my research, since, I
include capital and micro-foundation into analysis, like they. Some of these papers
use more restrictive assumptions on sectors and technologies or additional
assumptions (inclusion of land) in comparison to me. However, the key difference to
my research is simply that I use another method of analysis (PBGP), which makes
analytical study of structural change possible. In contrast, the papers of group three
may be regarded as simulation models. In general, if all the assumptions, which I use
in my models, were integrated into the models of group three, all the models of group
three would become such complicated, that nearly no intuitive results could be
derived from them (without simulations).
Note that the theories from Section 2, e.g. Fixler and Siegel (1999), Oulton (2001),
Sasaki (2007) and Restuccia et al. (2008), could be mentioned in this section as well,
since they also include heterogeneous consumption goods. In fact, they
complement/micro-found the results of the theories of the actual section.
Furthermore, in fact, many of the neoclassical endogenous growth theories, that
feature heterogeneous intermediates, could be interpreted as having heterogeneous
final consumption goods (see e.g. Grossman and Helpman (1991), p.46). Therefore,
they could be discussed in this section as well. However, as explained in the previous
section, these theories are not very useful for studying structural change, due to some
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very restricting assumptions (see there for details). (These theories feature very high
degree of disaggregation.)
Last not least, there is some “evolutionary structural change literature” (with very
high degree of disaggregation). Like evolutionary literature in general, evolutionary
structural change literature departs strongly from the assumptions which are used in
other models discussed in this chapter. For example: there are no standard utility and
production functions; quite a lot of assumptions are structuralist (sometimes proven
by empirical laws); bounded rationality and heterogeneous agents are assumed in part;
some “laws”, that are known from biological evolutionary theory (sorting and
selection), are shown to be true regarding industry-behaviour; etc. These
characteristics can be seen in the models by, e.g., Montobbio (2002), Saviotti and
Pyka (2004) and Metcalfe et al. (2006). An extensive discussion of the evolutionary
structural change literature and further references are provided by Krüger (2008),
p.344ff.
The evolutionary approach is a relatively new approach to structural change theory. It
may be very promising. However, in contrast to the approach discussed in the next
section, a lot of mathematical micro-foundation seems to be inexistent by now and
often simulations are necessary due to complexity of assumptions; see e.g. Krüger
(2008), p.345.
In general, Krüger (2008) and Pugno (2006) seem to be very useful references for
alternative structural change literature overviews.
3.2 “New” (PBGP) school of structural change Finally, we have arrived at the group of literature, to which my research belongs. The
key feature of the PBGP-school is that it tries to create analytically interpretable
models, without using the simplification tools (1)-(4) from the previous section.
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Hence, this group can analyse some questions analytically, which cannot be analysed
analytically by using the models from the previous section. Especially, many
questions, that require the consideration of capital in structural change analysis, can
be analysed by using the PBGP-approach. In general, all models from the previous
sections, that include capital into analysis, become such complicated that simulations
are necessary to disentangle their dynamics. Especially therefore, the PBGP-school
seems very useful. (I discuss in Chapter I, why it is important to include capital into
analysis of structural change.)
The PBGP-school uses, in general, neoclassical assumptions. Furthermore, these
assumptions are restricted to some extent (by usage of knife-edge-conditions) to
ensure the existence of partially balanced growth paths, see also Chapter III.
As discussed in Chapter III, the development of structural change is quite intuitively
understandable along a PBGP even without simulations. Hence, the PBGP-school
seems to be especially useful for deriving (qualitative) theories of structural change.
The quantitative aspects of structural change rather cannot be derived from the PBGP-
school, since, as discussed in Chapters I and III, the simplifying assumptions, which
are necessary to ensure the existence of a PBGP, restrict some structural change in
general. This opinion is supported by the empirical study by Buera and Kaboski
(2009a), which shows that the models by Kongsamut et al. (1997, 2001) cannot
generate sufficiently strong structural change patterns. Especially, they argue that the
usage of Stone-Geary preferences restricts the quantitative adequacy of the
Kongsamut et al.-models. (In fact, the quantitative restrictiveness of Stone-Geary-
preferences has already been mentioned by Samuelson (1948). I use Stone-Geary-
based preferences in my Kuznets-Kaldor-essay as well.) As can be seen from the
models of Chapter III, Stone-Geary preferences are very useful for generating PBGPs,
since the demand-dynamics, which are created by Stone-Geary-preferences, can be
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controlled by the preference parameters in a very detailed and direct fashion. (In
detail, the knife-edge conditions, that ensure the existence of PBGPs in the models
from Chapter III, are functions of the Stone-Geary-parameters. Other non-homothetic
utility functions, e.g. the one used by Echevarria (1997, 2000), do not allow for such
knife-edge conditions; hence, they do not allow for PBGPs; see also Meckl (2002),
footnote 2.)
The models of the PBGP-school have already been discussed in Chapter III regarding
the knife-edge-conditions that they use. In the following, I provide an overview that
discusses this literature regarding the structural change determinants that are used:
Kongsamut et al. (2001) study structural change by employing demand side structural
change determinants (Stone-Greary-preferences).
Meckl (2002) integrates this utility structure into an endogenous model with
intermediate production, similar to the models of Grossman and Helpman (1991),
which allows showing that neoclassical endogenous growth theory can be consistent
with structural change to some extent. However, Meckl (2002) does not include
capital-accumulation into analysis. Therefore, he omits a lot of (consumption
industries) structural change dynamics that have been shown to be important by
Acemoglu and Guerrieri (2009).
Ngai and Pissarides (2007) study supply-side structural change determinants.
Especially, they analyse the effects of cross-sector differences in TFP-growth.
Furthermore, they also analyse to what extent their model is feasible with
intermediate restructuring. (In fact, they show that the PBGP is not really consistent
with rich intermediate restructuring dynamics in their model.) Furthermore, since they
assume identical output-elasticites of inputs across sectors, their model omits the
structural change dynamics that have been shown to be important by Acemoglu and
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Guerrieri (2008). Ngai and Pissarides (2008) extend this model for demand-side
structural change (in detail, they include outsourcing of home production).
Last not least, Foellmi and Zweimüller (2008) study the PBGP of a demand-side
structural-change-model with very high degree of disaggregation. Their model
provides a lot of interesting micro-foundation for the other less-disaggregated PBGP-
models of structural change.
Note that all of these models do not consider the structural change patterns, which
have been studied by Acemogly and Guerrieri (2008). The latter show that structural
change arises if output-elasticity of inputs differs across sectors, provided that capital
is accumulated. In fact, their model implies that this sort of structural change is in
general not consistent with PBGPs. In the essay on the Kuznets-Kaldor-puzzle I show
that these structural change patterns are consistent with PBGPs provided that
preferences and technologies are independent.
It should be mentioned that there are two further models that actually do not fit into
the PBGP-school, but which approach of analysis is similar to the PBGP-school in
some sense: the second and third model by Kongsamut et al. (1997). In these models
the authors do not search for a growth path where some aggregates behaviour is
balanced (PBGP) but simply search for a growth path where the real interest rate is
constant. They derive the necessary knife-edge conditions for the existence of this
growth path. However, the difference of a growth path with a constant real interest
rate in comparison to the (other) PBGPs is that it is still very difficult to disentangle
the structural change dynamics along such a growth path. That is, a constant-real-
interest-rate-growth-path is a more general approach to structural change analysis in
comparison to the (other usual) PBGPs and therefore it is more difficult to study
analytically. As can be seen from the discussion by Kongsamut et al. (1997), in their
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second and third model neither the stability of the equilibrium growth path can be
shown, nor clear intuitive description of structural change can be derived.
4. Classification of own research My research belongs clearly to the PBGP-school (Section 3.2). In fact, as explained in
Chapter I, my research is devoted to the elaboration of the foundations and the
exploration of the applicability of the PBGP-school to questions associated with
structural change. Many of my assumptions are very similar to the models of the
PBGP-school, especially they are very neoclassical. In general the aggregate structure
of my model is very similar to the Ramsey-Cass-Koopmans model (see also Chapter
III on several aspects of this model). In the following, I compare my models (from
Chapter V) to the models of the previous section:
The model on the Kuznets-Kaldor puzzle is close to the third model by Kongsamut et
al. (1997). In contrast to all other PBGP-frameworks (Kongsamut et al., Ngai and
Pissarides, Meckl and Foellmi and Zweimüller), it features all key structural change
determinants: non-homothetic preferences, cross-sector differences in TFP-growth,
cross-sector differences in output-elasticites of inputs (combined with capital
accumulation) and outsourcing. (Outsourcing is not modelled in the models by
Kongsamut et al. (1997).) I use the very simple/restrictive intermediate structure
suggested by Ngai and Pisssarides (2007) for PBGP-frameworks. The key differences
in comparison to the third model by Kongsamut et al. (1997) are:
(1) I introduce another economy structure: there are multiple subsectors and each
subsector decides for one production technology. For simplicity, there are only two
production technologies available in the model. When the subsectors are aggregated,
each sector can result in using more than only one technology.
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(2) I use a slightly another preference structure, which allows to determine whether
preferences are independent from technology or not.
(3) I use another dynamic equilibrium concept: as discussed in the previous section,
Kongsamut et al. (1997) analyse a constant-real-interest-rate growth path in their third
model, which makes their analysis quite difficult; I use a PBGP along which
aggregates (capital, output and consumption) grow at a constant rate.
I use this framework, since it allows me to describe the concept of independency and
its implications for structural change. Furthermore, this framework is general enough
for the question that I analyse, i.e. it features all the key structural change
determinants.
In fact, one interesting aspect of the Kuznets-Kaldor model is that it shows for the
first time in the literature that the structural change patterns studied by Acemoglu and
Guerrieri (2008) can be consistent with PBGPs. Furthermore, to some extent it
provides an intuitive explanation for the many knife-edge conditions that are used in
the PBGP-school: It shows that independency of preferences and technologies can be
a “micro-foundation” of these knife-edge-conditions. It should be noted that
previously it has been mentioned by Foellmi and Zweimüller (2008) that some sort of
independency may be useful for generating PBGPs. However, they have not
studied/proven this argument in detail.
In fact, the model on Offshoring is an extension of the work by Ngai and Pissarides
(2007). I choose this simple framework, since it was good enough to show the results,
that I was seeking for. More complex assumptions (e.g. the inclusion of all structural
change determinants) are not necessary for the arguments of this essay. The only
necessary assumptions to prove the key-results of the offshoring model are: capital
accumulation, cross-sector differences in TFP-growth and usage of intermediates; and
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these are exactly the assumptions that the model by Ngai and Pissarides (2007)
features. Of course, to study offshoring I introduce intermediate trade into this model.
(Furthermore, I simplify the model’s intermediate structure further, to adapt it for my
needs.) Moreover, I introduce and justify a real GDP-measure in this model. The
GDP-measure is based on the GDP-measures used in reality. The impacts of structural
change on GDP are, in general, not adequately described by the original model by
Ngai and Pissarides (2007); for discussion see the essay on Offshoring.
The essay on Ageing contains two models: a complex and a simplified version. The
complex version is similar to my model on the Kuznerts-Kaldor-puzzle except for
three facts:
(1) The preference structure in the Ageing models features additional demand shifts
which depend upon the growth rate of the old population.
(2) Each sector uses only one technology. The assumption that a sector uses several
technologies is not necessary to show the key-model-results, but would complicate the
analysis enormously.
(3) Similar to the Offshoring-model, I introduce a measure of real GDP into the
Ageing-model.
The simpler version of the Ageing model assumes that output-elasticities are equal
across sectors. Hence, this model rules out the structural change patterns analysed by
Acemoglu and Guerrieri (2008). This simplifying assumption makes the model much
easier to understand and, at the same time, it still allows for some interesting results
regarding the impacts of Ageing via structural change.
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5. Further aspects of classification The above systematization of structural change literature (Figure 1 and Sections 1, 2
and 3) represents only one way to systematize the literature. Depending upon the
definition of structural change and the actual focus of research, alternative
systematizations may be useful. During my research on my topic I have found several
papers/topics that seem to me somehow related to my topic and/or that seem to be
analysable with similar methods/models as my topic. In the following I discuss these
aspects as possible alternative systematizations of the “structural change literature”.
This discussion should help to further classify my research and to explain which
facts/channels are studied explicitly in my models and which not.
5.1 Structural change induced by trade opening (structural change theory vs. trade theory) Traditionally, the theory (of factor reallocation) has always been divided into trade
theory and structural change theory. Simply speaking, the former analyses the impacts
of trade-opening onto domestic factor allocation; the latter analyses the relationship
between the traditional/domestic structural change determinants and domestic factor
(re)allocation. It is obvious that one must specialise in one or another, since trade-
theory itself constitutes a large body of literature and studying this whole literature
constitutes an own topic. The focus of my research is on structural change theory and
not trade theory. An overview of trade-theoretical results related to sectoral
reallocation is provided by, e.g., Barry and Walsh (2008).
It may seem contradictory that I state here that I do not focus on trade theory, while I
present in Chapter V a model on Offshoring. However, actually it is not contradictory.
A closer look at the essay on Offshoring reveals that the effects of offshoring in my
model can be divided into two types: Transitional effects and PBGP-effects. After
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opening of borders, some reallocations are induced during a transition period and the
economy is converging to a PBGP. The reallocations during the transition period are
in fact those that are studied in the trade literature: some domestic production is
substituted by foreign production, some domestic production is increased by exports
and, in general, these reallocations cause an increase in productivity of the domestic
economy. This is nothing new and is well known from trade theory. I explain these
effects for the sake of completeness in my essay, and I have no ambitions regarding
the study of these effects, since they are extensively studied in the trade theory.
Instead, the key-results of my essay are related to the PBGP. By analysing the
resulting PBGP we can find the following impact channel of offshoring: The
productivity increase by offshoring (which has just been explained) accelerates capital
accumulation, and thus makes the consumption industries less relevant in comparison
to the capital industries (in terms of employment). Hence, the traditional/domestic
structural change in consumption industries becomes less relevant for real GDP-
growth. We can see that this key result is clearly related to traditional structural
change theory and is rather not studied in trade theory. Therefore, the Offshoring-
essay is a contribution to structural change theory. In fact it analyses how the
traditional relationship between structural change and productivity growth is affected
by one big macro-trend (i.e. globalisation). However, as I believe, it won’t hurt trade
theorists, if they take a look at my Offshoring-essay. As discussed in the Offshoring-
essay, it could point to impact-channels of trade that are omitted in standard
Offshoring theory by now (due to omit of capital accumulation).
In general, it can be questioned whether trade has an impact on sector structure at low
degree of disaggregation. Only if a country specialises in a specific sector, sector
structure of the economy is affected by trade; otherwise, only the speed of structural
change and/or the sectoral productivity growth parameters are affected. (Regarding
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the latter see, e.g., the study by Fagerberg (2000)). Some evidence implies indeed that
trade seems to be a relatively unimportant determinant of sector structure at low
degree of disaggregation in the past in the industrialized countries (see e.g. Rowthorn
and Ramaswamy (1999)).
Nevertheless, there are authors who believe that, in general, structural change should
be analysed in open economy settings and who provide corresponding models (e.g.
Matsuyama (2009), Hsieh und Klenow (2007)). In fact, although I focus on the
traditional approach to structural change analysis, my Offshoring-essay is a
contribution to such an open economy theory of structural change, since it draws a
relationship between standard structural change theory (impact of domestic structural
change determinants) and globalization.
5.2 Factor reallocation between capital industries and consumption industries (Uzawa’s structural change) In general, the whole production of the economy can be divided into capital-goods
production and consumption goods production. These two sectors may feature
different production functions and factors may be reallocated between them. Models,
which study the reallocation of factors between the capital-sector and the
consumption-sector, where the two sectors differ by production technology, include,
e.g., Uzawa (1964) and Boldrin (1988) and (in open economy setting) Hsieh und
Klenow (2007).
In my models there are always some sectors that use a different production
technology in comparison to the capital producing sector. Uzawa’s structural change
occurs only during the transition period of my models; along the PBGP there is no
reallocation between consumption and capital industries (since the savings rate is
constant along the PBGP); see also Sections 3 and 5 in Chapter III. Therefore and
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since I focus primarily on PBGP-analysis in my essays, Uzawa’s structural change is
rather not in focus of my research issues. However, in the Offshoring-essay from
Chapter V, Uzawa’s structural change is the connecting link between transitional
effects (studied in the trade theory) and PBGP-effects (studied in the structural change
theory) of Offshoring. As explained in Section 5.1, the transitional effects cause a
productivity increase, which accelerates capital accumulation. Hence, Uzawa’s
structural change is induced (i.e. factors are reallocated from consumption-goods
production to capital-goods-production). In fact, this effect causes the slow-down of
structural change (across the consumption-goods industries) along the PBGP. (For
details, see also the Offshoring-essay in Chapter V.)
5.3 Factor-reallocation between the private sector and the public sector In general, the economy-wide labour is employed in the private and public sector. It is
an interesting question, what the (long-run) reallocation patterns of labour between
these two sectors are, especially if productivity growth differs between the two
sectors. In fact, this is a “classic” topic in economics that has been studied even since
1883 more or less directly under the title “Wagner’s law”. (“Wagner’s law” refers to
an increase in the share of the public sector in national income; see, e.g., Oxley
(1994), p.286.) Discussion/application of Wagner’s law and in general factor
reallocation between the public and private sector can be found in the essays by Vogt
(1973), Oxley (1994), Kongsamut et al. (2001) and Simpson (2009).
In my research I have not explicitly modelled the public sector. However, it may be
interesting for further research to analyse to what extent my models could be
interpreted as including a public sector and whether interesting/new results are
generated by them regarding the public-private-sector reallocation.
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5.4 Factor-reallocation between the research sector and the consumption sector Especially in some neoclassical endogenous growth theories, it is assumed that there
is a research sector which produces in some sense the technological progress. Hence,
the society has to decide on the division of factor-use between the research sector and
the consumption sector.
For example, in the model by Romer (1990), p.S83, the economy has to decide on the
allocation of human capital between the research sector and the consumption sector;
in the model by Meckl (2002), p.250, the economy has to decide on the allocation of
(exogenous) factors between the consumption and research sector. In the dynamic
equilibriums of these models there is no factor reallocation between the research
sector and consumption sector (see Romer (1990), p.S90, and Meckl (2002), p.253).
Unfortunately, both essays provide no results regarding the factor reallocation during
the transition period: Romer (1990) does not attempt to do such an analysis (see there
p.S90); Meckl’s (2002)-model features no transition period (see there p.261, footnote
14).
In contrast, the model by Barro and Sala-i-Martin (2004), p.303, features a transition
period; however, they do not explicitly discuss the allocation of factors between
research and consumption during the transition period. As far as I can see, the rate of
innovation is changing during the transition period of their model (see the phase
diagram on p.304 of their book); therefore, the costs of research and development are
changing (cf. equation 6.36 on p.303 of their book); therefore, the allocation of
resources between research and development is changing.
However, I have not studied this topic in detail. Especially, it should be noted that in
my research neither endogenous growth nor a research sector are modelled explicitly.
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5.5 Outsourcing of home production (factor reallocation between home-sector and market-sector) In fact, some services can be “produced” at home or bought on the market, e.g.
repairing a car, cutting hair, cooking a meal, etc. If home services are outsourced to
the market (e.g. going to a restaurant instead of cooking and eating at home), the
household gets more leisure time and the market demand increases. Eventually, some
of the leisure time is not only used for fun, but additional labour is supplied on the
market. Overall, outsourcing of domestic services is associated with demand changes
(increase in demand for services in comparison to demand for manufactured goods)
and eventually with labor supply changes. Hence, factors are reallocated between
domestic and market production (and across sectors). Essays that study this process
are provided by, e.g., Ngai and Pissarides (2008) and Buera and Kaboski (2009b).
In my research, this outsourcing process is not explicitly modelled, but may be
regarded as being implicitly depicted in the development of exogenous model
parameters.
In the following Chapter V, I present the application of my research, where the
PBGP-method is applied to problem-analysis associated with the Kuznets-Kaldor-
Puzzle, Dynamic Effects of Offshoring and Structural Change Effects of Ageing.
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CHAPTER V
Application of the PBGP-Concept in Analysis of Structural Change
In the following, I analyse several questions that are associated with structural change
by using the PBGP-method. Especially, I analyse the Kuznets-Kaldor-Puzzle, the
dynamic effects of Offshoring and the structural change impacts of population
Ageing. The justification of these topics is given in Chapter I.
In fact, this chapter provides four models, which can be used to analyse many
questions associated with structural change. (The essay on ageing contains 2 models;
therefore Chapter V contains four models.). For example, the exogenous demand
shifts in the ageing models need not necessarily being interpreted as resulting form
ageing, but may be interpreted in another way (e.g. as resulting from outsourcing of
home production). That is, the models become even more valuable, if we regard them
merely as mathematical models, which can be filled with intuition as necessary.
Remember that I have discussed in Chapters II and III that it is very difficult to find
assumptions that ensure the existence of a PBGP and that it is even more difficult to
find assumptions that additionally make the proof of sufficiency of Hamiltonian
conditions and global stability of the PBGP feasible (when all structural change
determinants are included into analysis). Hence, I provide in the following nice
mathematical a-priori-solutions, which can be adapted to analyse some new economic
questions associated with structural change. Depending upon the question, the
adequate model may be chosen, where the complex model on ageing features the
richest structural change dynamics and thus the strongest impacts on real GDP-
growth.
155
Before approaching to the models, I provide here just a short explanation of the
meanings of the different postulate-types that I use. As I know, these are not
uniformly used across sciences and scientist; therefore, here follows an explanation of
how I use them:
A Lemma is a rather unimportant postulate, which is aimed to help proving more
important postulates.
A Theorem is a very important postulate. In general, theorems are the key-results of
the paper.
A Proposition is a rather unimportant postulate, which is not used in proving other
postulates.
A Corollary is a summary of the results from other postulates or an interpretation of
another postulate(s).
Hence, if you have no time or fun in reading mathematical proofs, just focus on the
Theorems, Propositions and Corollaries.
156
TABLE OF CONTENTS for CHAPTER V (Detailed tables of contents are provided in each PART.)
PART I: A PBGP-Framework for the Analysis of the Kuznets-Kaldor-
Puzzle.........................................................................................................................159
PART II: A PBGP-Framework for Analyzing the Impacts of Offshoring on
Structural Change and real GDP-growth in the Dynamic Context............................263
PART III: A PBGP-Framework for Analyzing the Impacts of Ageing on Structural
Change and real GDP-growth....................................................................................321
157
158
PART I of CHAPTER V
A PBGP-Framework for the Analysis of the Kuznets-Kaldor-Puzzle
The Kuznets-Kaldor stylized facts are one of the most striking empirical
observations about the development process in the industrialized countries: While
massive factor reallocation across technologically distinct sectors takes place, the
aggregate ratios of the economy behave in a quite stable manner. This implies that
cross-technology factor reallocation has a relatively weak impact on the
aggregates, which is a puzzle from a theoretical point of view.
I apply the PBGP-method to this puzzle, since the PBGP can be defined such that
the Kaldor-Kuznets-facts are satisfied. Hence, the study of this PBGP and the study
of the conditions, which are required for the existence of this PBGP, seem to be
predestined for a discussion of the Kuznets-Kaldor-puzzle. I provide a model that
can explain the Kuznets-Kaldor-puzzle by independent preferences and
technologies along a PBGP. Furthermore, I show by empirical evidence that this
model is in line with 55% of structural change.
The model which I present here is a modification of the reference-model from
Chapter III (Section I). That is, it is still a multi-sector Ramsey-Cass-Koopmans
model. However, as I will explain in detail later, I modify the reference model as
follows: I restrict the number of technologies and sectors for simplicity (only two
technologies and three sectors). Furthermore, I add intermediate production
structures (intermediate products) to discuss some issues related to them.
Moreover, to discuss the “independency between preferences and technologies” I
159
add subsectors to each sector and I introduce a slightly different utility structure in
comparison to the reference model.
160
TABLE OF CONTENTS for PART I OF CHAPTER V
1. Introduction .........................................................................................................163
2. Stylized facts of sectoral structures .....................................................................167
2.1 Stylized facts regarding cross-sector-heterogeneity in production-
technology .................................................................................................167
2.2 Structural change determinants ...........................................................168
3. Model of neutral cross-capital-intensity structural change..................................169
3.1 Model assumptions ..............................................................................169
3.1.1 Production...............................................................................169
3.1.2 Utility function .......................................................................172
3.1.3 Aggregates and sectors ...........................................................175
3.2 Model equilibrium ...............................................................................178
3.2.1 Optimality conditions .............................................................178
3.2.2 Development of aggregates in equilibrium ............................178
3.2.3 Development of sectors in equilibrium ..................................182
3.2.4 Consistency with stylized facts...............................................184
3.2.5 The relationship between structural change and aggregate-
dynamics..........................................................................................190
4. A measure of neutrality of cross-capital-intensity structural change ..................195
5. On correlation between preferences and technologies ........................................203
6. Concluding remarks.............................................................................................209
APPENDIX A..........................................................................................................214
APPROACH (1): .......................................................................................214
Necessary (first order) conditions for an optimum..........................214
Proof that sufficient (second order) conditions are satisfied ...........215
APPROACH (2) ........................................................................................218
Producers .........................................................................................219
Households ......................................................................................220
Relationship between individual variables and economy-wide
aggregates ........................................................................................224
APPENDIX B..........................................................................................................226
APPENDIX C..........................................................................................................230
APPENDIX D..........................................................................................................243
APPENDIX E..........................................................................................................245
161
APPENDIX F ..........................................................................................................248
LIST OF SYMBOLS of PART I of CHAPTER V .................................................252
162
1. Introduction As shown by Kongsamut et al. (1997, 2001), the development process of the
industrialized countries during the last century satisfies two types of stylized facts:
“Kuznets facts” and “Kaldor facts”. Generally speaking, Kuznets facts state that
massive structural change takes place during the development process.1 Especially,
in the early stages of economic development factors are primarily reallocated from
the agricultural sector to the industrial sector and in later stages of development
factors are primarily reallocated from the manufacturing sector to the services
sector. (It has also been shown, that structural change takes place at more
disaggregated level.) On the other hand, the Kaldor facts state that some key
aggregate measures of the economy are quite stable during the development
process; especially, the aggregate capital-to-output ratio and the aggregate income
shares of capital and labor are quite stable whereas the aggregate capital-to-labor
ratio increases (at a fairly constant rate) in the industrialized countries.2 That is, the
growth process seems to be “balanced” at the aggregate level. As discussed by
Kongsamut et al. (2001) and Acemoglu and Guerrieri (2008), the coexistence of
Kuznets and Kaldor facts seems to be a puzzle, since strong factor-reallocations
across sectors in general imply that Kaldor-facts are not satisfied (“unbalanced”
growth of aggregates). In fact it has been shown in Chapter III that in general the
1 Papers that provide empirical evidence for the massive labor reallocation across sectors during the growth process are e.g. Kuznets (1976), Maddison (1980), Kongsamut et al. (1997, 2001) and Ngai and Pissarides (2004). Kongsamut et al. (1997, 2001) formulate the following stylized facts of structural change for the last hundred years: 1.) the employment share of agriculture decreases during the growth process; 2.) the employment share of services increases during the growth process; 3.) the employment share of manufacturing is constant. Ngai and Pissarides (2007) note that the development of the manufacturing employment-share can be regarded as “hump-shaped” in the longer run. See also Chapter I for detailed discussion of the stylized facts. 2 In detail, Kaldor’s stylized facts state that the growth rate of output per capita, the real rate of return on capital, the capital-to-output ratio and the income distribution (between labor and capital) are nearly constant in the long run; capital-to-labor ratio increases in the long run. It is widely accepted that these facts are an accurate shorthand description of the long run growth process (at the aggregate level) in industrialized countries. A discussion of these facts can be found in the paper by Kongsamut et al. (1997, 2001) and in the books by Maußner and Klump (1996) and Barro and Sala-i-Martin (2004).
163
behavior of aggregates is unbalanced, as long as structural change takes place.
Therefore, I name the empirically observable coexistence of Kuznets and Kaldor
facts “Kuznets-Kaldor-puzzle”.
The literature, which deals with the Kuznets-Kaldor-Puzzle more or less explicitly,
includes Kongsamut et al. (1997, 2001), Meckl (2002), Foellmi and Zweimueller
(2008), Ngai and Pissarides (2007), Acemoglu and Guerrieri (2008) and Boppart
(2010).
We learn from this literature, in general, that the solution of the Kuznets-Kaldor-
Puzzle in neoclassical growth frameworks requires the use of some knife-edge
conditions. In fact, all papers used very severe restrictions to solve the Kuznets-
Kaldor-Puzzle: all of them omitted some structural change determinants (which is
the same as imposing some implicit knife-edge conditions) and/or imposed some
explicit knife-edge parameter restrictions (like Kongsamut et al. (1997, 2001) and
Meckl (2002)). Such (implicit and explicit) knife-edge conditions are severe
restrictions, if their validity is not proven by empirical and/or theoretical reasoning,
as discussed in Section 6 of Chapter III. (For a discussion of structural change
determinants see section 2.2.)
I include all key structural change determinants into analysis and then try to
analyze whether the knife-edge conditions, which are required for the solution of
the Kuznets-Kaldor-Puzzle, are empirically reasonable. Furthermore, I point to a
possible theoretical micro-foundation of these knife-edge conditions.
The starting point of my analysis is the following fact: The key challenge to
solving the Kuznets-Kaldor-Puzzle is already known since Baumol (1967): If
production technology differs across sectors, the reallocation of factors across
sectors causes unbalanced growth, i.e. Kaldor-facts are not satisfied.
Then, I approach as follows:
164
First, I show that Kaldor facts can be satisfied despite the fact that factors are
reallocated across technologically distinct sectors. In this sense my results postulate
that structural change across technology can be irrelevant regarding the
development of aggregate ratios. I name this type of factor reallocation “neutral
(cross-capital-intensity) structural change”. Of course, the existence of neutral
structural change requires some knife-edge conditions (which will be analyzed
below). Previously, Ngai and Pissarides (2007) have shown that neutral structural
change can arise when all sectors have the same capital-intensity. However,
Acemoglu and Guerrieri (2008) have shown that their results do not hold if capital-
intensities differ across sectors, i.e. they show that in this case growth is in general
unbalanced. In some sense, my result contradicts Acemoglu and Guerrieri (2008),
since neutral structural change arises despite the fact that capital-intensities differ
across sectors in my model. I am able to obtain my results, since, in contrast to
Acemoglu and Guerrieri (2008), I assume a utility function that has non-unitary
price elasticity of demand (i.e. each good has its own specific price elasticity) and
since I assume that at least one of the three sectors uses two technologies. (As I
will discuss in my essay, the latter assumption is consistent with empirical
evidence, which postulates that e.g. the services sector is quite technologically
heterogeneous.) Furthermore, in contrast to Acemoglu and Guerrieri (2008), I
model sectors that feature non-constant output-elasticities of inputs.
Second, I study the empirically observable patterns of structural change and
analyze whether they were neutral or non-neutral. In this sense, I analyze implicitly
whether the knife-edge conditions, which ensure the satisfaction of the Kuznets-
Kaldor-facts in my model, are given in reality. I develop an index of neutrality of
structural change and show with the data for the US between 1948 and 1987 that
about 55% of structural change was neutral structural change. Hence, neutrality of
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structural change seems to be a relatively large explanatory variable regarding the
Kuznets-Kaldor-puzzle. I argue as well that this result applies to the most of the
previous literature, implying that the previous literature can explain (maximally)
55% of structural change.
Third, I argue that low (no) correlation between preference parameters and
technology parameters can explain the prevalence (existence) of neutral structural
change in reality (my model).3 I also argue that the assumption of uncorrelated
preferences and technologies may be theoretically reasonable in long run growth
models. In this sense, the independency between preferences and technologies can
be a theoretical foundation of the knife-edge conditions that are necessary for the
solution of the Kuznets-Kaldor-Puzzle.
In the next section (section 2) I provide some evidence on sectoral structures that
are observed in reality, in order to provide an empirical basis for my discussion and
model assumptions. Then, in section 3, I provide a PBGP-model of structural
change in order to show the existence of neutral structural change. (There I also
generalize some of the model results in Proposition 4.) Section 4 is dedicated to the
empirical analysis, where among others I develop an index of neutrality of
structural change and analyze the cross-capital-intensity structural change patterns
in detail. In section 5 I discuss the assumption of low correlation between
technology and preferences. Finally, in section 6 I provide some concluding
remarks and hints for further research.
3 It should be noted here that previously it has been mentioned by Foellmi and Zweimueller (2008) that some type of independency between technology and preferences may be useful for generating aggregate balanced growth. However, this topic has not been studied further by them.
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2. Stylized facts of sectoral structures 2.1 Stylized facts regarding cross-sector-heterogeneity in production-technology Empirical evidence implies the following stylized facts of sectoral production
functions:
1. TFP-growth differs across sectors. Empirical evidence implies that TFP-growth
rates differ strongly across sectors. For example, Bernard and Jones (1996) (pp.
1221f.), who analyze sectoral TFP-growth in 14 OECD countries between 1970
and 1987, report that e.g. the average TFP-growth rate in agriculture (3%) was
more than three times as high as in services (0.8%). Similar results are obtained by
Baumol et al. (1985), who report the TFP-growth-rates of US-sectors between
1947 and 1976.
2. Capital intensity differs across sectors. Empirical evidence implies that factor
income shares differ strongly across sectors (hence, capital intensities differ
strongly across sectors as well4). For example, Kongsamut, Rebelo and Xie (1997)
provide evidence for the USA for the period 1959-1994. Their data implies that, for
example, the labor income share was relatively high in manufacturing and
construction (around 70%) in this period. At the same time, e.g. the labor income
share in agriculture, finance, insurance and real estate was relatively low (around
20%). Similar results for the USA are obtained by Close and Shulenburger (1971)
for the period 1948-1965 and by Acemoglu and Guerrieri (2008) for the period
1987-2004. Some new evidence for the USA (presented by Valentinyi and
Herrendorf (2008)) supports these results as well. Gollin (2002) (p. 464) analyzes
4 If labor income shares (or: output elasticities of labor) differ across sectors, it follows that capital intensities differ across sectors as well, since in optimum capital intensity is determined by factor prices and by output elasticities of capital and labor. We will see later that this is true within my model.
167
the data from 41 countries reported in the U.N. National Statistics. He confirms
that factor income shares vary widely across sectors.
A model that analyzes structural change across sectors should be consistent with
these “stylized” facts of sectoral production functions. This is especially important,
since these stylized facts have an impact on structural change (and hence on
aggregate balanced growth), as we will see now.
2.2 Structural change determinants As discussed in Chapter I, there a four main determinants of structural change. I
recapitulate them here, since they are important for the following discussion:
1. Non-homothetic preferences across sectors – relevance for structural change
analyzed empirically and theoretically, e.g., by Kongsamut et al. (1997, 2001).
2. Differences in TFP-growth across sectors – empirical relevance for structural
change shown, e.g., by Baumol (1985); theoretical relevance for structural change
shown, e.g., by Ngai and Pissarides (2007).
3. Differences in capital intensities across sectors – relevance for structural change
analyzed empirically and theoretically, e.g., by Acemoglu and Guerrieri (2008).
4. Shifts in intermediates production across sectors – relevance for structural
change analyzed empirically and theoretically, e.g., by Fixler and Siegel (1998).
So I can conclude that all these determinants influence the structural change
patterns. Since the aggregate economy is the weighted average of its sectors, the
aggregate behavior depends on the structural change patterns. That is, all four
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structural change determinants influence the behavior of the aggregate economy.
Hence, only if I include all four structural change determinants into a model, I can
adequately analyze whether balanced growth with respect to aggregates can coexist
with structural change.
3. Model of neutral cross-capital-intensity structural change 3.1 Model assumptions
3.1.1 Production I assume an economy where two technologies exist (the model could be modified
such that it includes more technologies; the key results would be the same). The
technologies differ by capital intensity (i.e. output elasticities of inputs differ across
technologies) and by total factor productivity (TFP) growth. TFP-growth rates are
constant and exogenously given. Goods ni ,...1= are produced in the economy.
Goods are produced by using technology 1 and goods are
produced by using technology 2 ( . I assume that three inputs are used for
production: capital (K), labor (L) and intermediates (Z). All capital, labor and
intermediates are used in the production of goods
mi ,...1= nmi ,...1+=
)mn >
ni ,...1= . The amount of
available labor grows at constant rate ( ). Since I want to model TFP-growth, I
assume Hicks-neutral technological progress. It is well known that the existence of
a balanced growth path in standard balanced growth frameworks requires the
assumption of Cobb-Douglas production function(s) when technological progress
is Hicks-neutral. (Later, we will see that the aggregate production function
“inherits” the attributes of sectoral production functions along the PBGP, i.e. the
Lg
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aggregate production function is of type Cobb-Douglas.) These assumptions imply
the following production functions:
(1) miZzKkLlAY iiii ,...1,)()()( == γβα
where .;0,,;1 constgAA
A ==>=++&
γβαγβα
(2) nmiZzKkLlBY iv
iii ,...1,)()()( +== μχ
where .;0,,;1 constgBB
B ==>=++&
μνχμνχ
(3) ∑∑∑===
===n
ii
n
ii
n
ii zkl
1111;1;1
(4) .constgLL
L =≡&
where denotes the output of good i; and denote respectively the fraction
of labor, capital and intermediates devoted to production of good i;
iY ii kl , iz
K is the
aggregate capital; aggregate labor; L Z aggregate intermediate index. Note that I
omit here the time index. Furthermore, note that the index i denotes not sectors but
a good or a group of similar goods. I will define sectors later.
Of course, it is not “realistic” that there are only two technologies and that some
goods are produced by identical production functions. However, every model
simplifies to some extent and it is only important that the simplification does not
affect the meaningfulness of the results. My assumption is only a “technical
assumption”, which is necessary to make my argumentation as simple as possible.
My key arguments (namely the existence of neutral structural change) could also
170
be derived in a framework where each good is produced by a unique production
function. (I show this fact in Proposition 4.) However, it would be much more
difficult to formulate the independency assumptions (which are formulated in the
next subsection). Instead of the simple restrictions, which I use in the next
subsection, I would have to derive complex restrictions which would not be such
transparent. Anyway, later my focus will be on the analysis of only three sectors
(which are aggregates of the products i=1,…n); thus, two technologies are
sufficient to generate technological heterogeneity between these three sectors. In
this sense, I have introduced technological diversity into my framework in the
simplest manner (by assuming that there are only two technologies).
It may be easier to accommodate with my assumption of only two technologies by
imagining that an economist divides the whole set of products of an economy into
two groups (a technologically progressive and a technologically backward) and
estimates the average production function for the two groups. Such approaches are
prominent in the literature: e.g. Baumol et al. (1985) and Acemoglu and Guerrieri
(2008) approach in similar way in the empirical parts of their argumentation.
Furthermore, note that much of the new literature on the Kuznets-Kaldor-puzzle
assume very similar sectoral production functions (e.g. Kongsamut et al. (2001)
and Ngai and Pissarides (2007)) or assume even identical sectoral production
functions (e.g. Foellmi and Zweimueller (2008)). Hence, my assumption of only
two (completely distinct) technologies is an improvement in comparison to some
previous literature. Note that the empirical study of my paper (section 4) uses the
more general assumption, i.e. each good is produced by a unique production
function.
I assume that all goods can be consumed and used as intermediates. Furthermore, I
assume that only the good m can be used as capital. (Note, that the model could be
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modified such that more than one good is used as capital e.g. in the manner of Ngai
and Pissarides (2007).) This assumption implies:
(5) mihCY iii ≠∀+= ,
(6) KKhCY mmm δ+++= &
where denotes consumption of good i; iC δ denotes the constant depreciation rate
of capital; is the amount of good i that is used as intermediate input. ih
I assume that the intermediate-inputs-index Z is a Cobb-Douglas function of ’s
which is necessary for the existence of a PBGP (see Ngai and Pissarides 2007):
ih
(7) ∏=
=n
ii
ihZ1
ε
where ∑=
=∀>n
iii i
11;,0 εε
3.1.2 Utility function I assume the following utility function, which is quite similar to the utility function
used by Kongsamut et al. (1997, 2001):
(8) , ∫∞
−=0
1 ),...( dteCCuU tn
ρ 0>ρ
where
(9) ⎥⎦
⎤⎢⎣
⎡−= ∏
=
n
iiin
iCCCu1
1 )(ln),...( ωθ
172
(10) ∑=
=m
ii
1
0θ
(11) ∑+=
=n
mii
1
0θ
where U denotes the life-time utility of the representative household and iω , iθ
and ρ are constant parameters. In contrast to the model by Ngai and Pissarides
(2007), the assumption of logarithmic utility function (equation (9)) is not
necessary for my results, i.e. I could have assumed a constant intertemporal
elasticity of substitution function of the consumption composite in equation (9).
We can see that this utility function is based on the Stone-Geary preferences.
Without loss of generality I assume that iθ s are not equal to zero and that they
differ across goods i. The key reason why I use this utility function is that it
features non-unitary income-elasticity of demand and non-unitary price-elasticity
of demand. That is, each good has its own income elasticity of demand and its own
price elasticity of demand (as long as iθ differs across goods). For example, the
good i=4 has another price elasticity of demand than good i=7 (provided that
74 θθ ≠ ). Due to this feature, I can determine the own price elasticity and the own
income elasticity for groups of goods. For example, by setting the iθ in a specific
pattern I can determine that the (average) price elasticity of demand for the goods
i=7,…14 is larger than for goods i= 56,…79.
This is the key to my argumentation about preference and technology correlation
later: By setting parameter restrictions (10) and (11) I determine that
173
1.) on average, the income elasticity of demand for technology-1-goods is not
larger or smaller in comparison to the income elasticity of demand for technology-
2-goods.
2.) on average, the “relative price elasticity of demand” (i.e. elasticity of
substitution) between technology-1-goods and technology-2-goods is equal to one.
Hence, preferences and technologies are not correlated on average. This means for
example, that demand for some of the goods that are produced by technology 1 can
be price-inelastic and for some of the technology-1-goods price-elastic, while at the
same time the demand for some goods that are produced by technology 2 can be
price-elastic and for some of the technology-2-goods price inelastic. However, on
average, the elasticity of stubstitution between technology-1-goods and
technology-2-goods is equal to unity.
This restriction (equations (10) and (11)) reduces the generality of my model.
Nevertheless, for my further argumentation it does not matter. It is simply a
technical assumption in order to show in the simplest manner the existence of
neutral-cross-capital-intensity structural change. That is, due to this assumption I
can pursue my analysis along a PBGP, which is technically simple. Without this
assumption, I would have to numerically solve the model and the distinction
between neutral and non-neutral cross-capital-intensity structural change would be
quite difficult. Nevertheless, I will discuss theoretical reasonability of this
restriction later and I will show empirically that the largest part of structural change
is in line with this restriction.
Overall, my utility function allows for structural change caused by all structural
change determinants: In general the goods have a price elasticity of demand that is
different from one (as discussed above). Hence, changing relative prices can cause
structural change in this model (see also Ngai and Pissarides 2007 on price
174
elasticity and structural change). Intertemporal elasticity of substitution differs
across goods i and is not equal to unity, despite of the fact that equation (9) is
logarithmic. Equations (8)-(11) imply that the utility function is non-homothetic
across goods i, i.e. income elasticity of demand differs across goods i (depending
on the parameterization of the iθ ’s).
3.1.3 Aggregates and sectors I define aggregate output (Y), aggregate consumption expenditures (E) and
aggregate intermediate inputs (H) as follows:
(12) ; ; ∑=
≡n
iiiYpY
1∑=
≡n
iiiCpE
1∑=
≡n
iiihpH
1
where denotes the price of good i. I chose the good m as numéraire, hence: ip
(13) 1=mp
Note that in reality the manufacturing sector is not the numérarire in the real GDP
calculations. Hence, my definition of aggregate output Y is not the same as real
GDP. However, the choice of numérarie is irrelevant when discussing ratios or
shares (see e.g. Ngai and Pissarides (2004, 2007)), since the numérarire of the
numerator and the denominator of a ratio offset each other. Therefore, I focus my
discussion on the shares and ratios in my paper (e.g. aggregate capital-intensity,
capital-to-output ratio, income-share of capital and labor), where the numérarire
choice is irrelevant. My results regarding the other Kaldor-facts, which are dealing
175
with the development of the real-GDP-growth rate and the real interest rate, should
be considered with caution. However, as discussed by Barro and Sala-i-Martin
(2004), the constancy of the real interest rate (as a Kaldor fact) may anyway be
questionable. Furthermore, as shown by Ngai and Pissarides (2004, 2007) the real
GDP as measured in reality and the real GDP in manufacturing terms seem to
behave quite similar. Therefore, possibly my results regarding the real GDP
development may be to some extent related to the real GDP as measured in reality.
Last but not least I have to define the sectors of our economy. Without loss of
generality I assume here that there are three sectors which I name for reasons of
convenience (according to the tree sector hypothesis): agriculture, manufacturing
and services. Furthermore, I assume that without loss of generality
• agricultural sector maai <<= 1;,...1
• manufacturing sector includes goods nsmsai <<+= ;,...1
• services sector includes goods nsi ,...1+= .
Hence, the agricultural sector uses only technology 1, the manufacturing sector
uses technology 1 and 2 and the services sector uses only technology 2. Note, that
this whole division is not necessary for my argumentation, neither the naming of
the sectors. I could also assume that the capital-producing manufacturing sector
uses only one technology (and the services sector both technologies). I could even
assume that there are more sectors (and more technologies). In all these cases my
key results would be the same. Furthermore, note that the assumption that a sector
uses both technologies is plausible. For example, the service sector includes
services that feature high TFP-growth and/or high capital intensity, e.g. ICT-based
services, as well as services that feature low TFP-growth and/or low capital
intensity, e.g. some personal services like counseling and consulting (for discussion
and empirical evidence see e.g. Baumol et al. 1985 and Blinder 2007). Similar
176
examples can be found in the manufacturing sector (e.g. a traditional clock maker
vs. a car producer). Furthermore, my sector-division implies that only sector M (the
manufacturing sector) produces capital. This is consistent with the empirical
evidence, which implies that most capital goods are produced by the manufacturing
sector (see e.g. Kongsamut et al. 1997).
According to my classification, I can define the outputs of the agricultural, services
and manufacturing sector ( , and ) and the consumption expenditures
on agriculture, manufacturing and services ( , and ) as follows:
.agrY .manY .serY
.agrE .manE .serE
(14) ∑∑∑+=+==
≡≡≡n
siiiser
s
aiiiman
a
iiiagr YpYYpYYpY
1.
1.
1. ;;
(15) ∑∑∑+=+==
≡≡≡n
siiiser
s
aiiiman
a
iiiagr CpECpECpE
1.
1.
1. ;;
Furthermore, note that employment shares ( , and ), capital shares
( , and ) and intermediate shares ( , and ) of sectors
agriculture, manufacturing and services are given by:
.agrl .manl .serl
.agrk .mank .serk .agrz .manz .serz
(16)
∑∑∑
∑∑∑
∑∑∑
+=+==
+=+==
+=+==
≡≡≡
≡≡≡
≡≡≡
n
siiser
s
aiiman
a
iiagr
n
siiser
s
aiiman
a
iiagr
n
siiser
s
aiiman
a
iiagr
zzzzzz
kkkkkk
llllll
1.
1.
1.
1.
1.
1.
1.
1.
1.
;;
;;;
;;;
177
3.2 Model equilibrium
3.2.1 Optimality conditions I have now specified the model completely. The intertemporal and intratemporal
optimality conditions can be obtained by maximizing the utility function (equations
(8)-(11)) subject to the equations (1)-(7) and (12)-(16) by using e.g. the
Hamiltonian. When there is free mobility of factors across goods and sectors these
(first order) optimality conditions are given by:
(17) ihZ
ZzY
ZzYZzY
KkYKkY
LlYLlYp
im
m
ii
mm
ii
mm
ii
mmi ∀
∂∂
∂∂
=∂∂∂∂
=∂∂∂∂
=∂∂∂∂
= ,)()(/
)(/)(/)(/
)(/)(/
(18) iCuCup
m
ii ∀
∂∂∂∂
= ,/(.)/(.)
(19) ρδ −−=− ruu
m
m&
where and mm Cuu ∂∂≡ /(.) )(/ KkYr mm ∂∂≡ is the real interest rate (see
APPENDIX A for proofs). I show in APPENDIX A that these are the sufficient
conditions for an optimum (together with the transversality condition).
3.2.2 Development of aggregates in equilibrium To be able to derive some theoretical arguments from the model, we have to insert
equations (1) to (16) into optimality conditions (17) to (19) in order to transform
the optimality conditions into some explicit functions of model-variables and
model-parameters. To get an impression of how this is done, see the corresponding
derivations in my Ageing-model (especially APPENDIX A of PART III of
178
CHAPTER V). In fact the derivations there are very similar to the derivations
which are necessary to obtain the following equations. Therefore, I present the
following equations, which describe the optimal aggregate structure of the
economy, without explicit proof:
(20) HEKKY +++= δ&
(21) qqq
m
m KGLlkY −
⎟⎟⎠
⎞⎜⎜⎝
⎛= 1~
(22) ρδβ −−⎟⎟⎠
⎞⎜⎜⎝
⎛= −−
−
111
qqq
m
m KGLkl
EE&
(23) ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
m
m
klccYH 21
~γ
(24) YHc
YEc
kl
m
m ~~1 43 −−=
where
(25)
m
m
klcc
YY65
~
+≡
(27) 0)1(1
)1(>
−−−+−
≡μεεγγνεμεβq
(28) εμεγ
γ
ε
εμνεε ε
χγαμ
χβαν
αχγ
−−−
=
−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ∏
)1(1
1
1n
ii
iBAAG
(29) ∑+=
≡n
mii
1
εε
179
and
21 1 cc −≡ ,
χβανχγαμ
−
−≡
1
12c ,
∑
∑
=
+⎟⎟⎠
⎞⎜⎜⎝
⎛−≡ n
ii
n
mi
c
1
13 1
ω
ω
αχ
χβαν , ∑
+=⎟⎟⎠
⎞⎜⎜⎝
⎛−≡
n
miic
14 1 ε
αχ
χβαν ,
and 65 1 cc −≡
χβανχα
−
−≡
1
16c .
Note that G grows at positive constant rate, q is positive and 1<ε .5
Equations (20)-(28) look actually more complicated than they are. As we will see
soon they are quite the same as in the standard one-sector Ramsey-Cass-
Koopmans-model6 or Solow-model. The key difference is that my equations
feature the term , which reflects the impact of cross-capital intensity
structural change on the development of aggregates. However, before discussing
these facts I start with my definition of an equilibrium growth path which is quite
similar as the definition used by Ngai and Pissarides (2007).
mm kl /
Definition 1: A partially balanced growth path (PBGP) is an equilibrium growth
path where aggregates (Y, Y~ , K, E and H) grow at a constant rate.
Note that this definition does require balanced growth for aggregate variables.
However, it does not require balanced growth for sectoral variables (e.g. for
sectoral outputs). Hence, it allows for structural change.
5 The term within the {}-brackets in equation (28) grows at constant positive rate since ε is positive and smaller than one (see equation (29)). Furthermore, the exponent of the {}-brackets is positive as well, since 1)1( <+− εμεγ (a weighted average of numbers that are smaller than one (γ and μ ) is always smaller than one). As well, q>0, since 1)1( <+− εμεγ . 6 For a discussion of the Ramsey-model see e.g. Barro, Sala-i-Martin (2004) pp. 85ff.
180
Lemma 1: Equations (20) to (28) imply that there exists a unique PBGP, where
aggregates (Y, Y~ , K, E and H) grow at constant rate and where is
constant. The PBGP-growth rate is given by
*g mm kl /
LBA gggg +
+−+−
=χεγαεμεγεμ
)1()1(* .
Proof: See APPENDIX B.
Proposition 1a: A saddle-path, along which the economy converges to the PBGP,
exists in the neighborhood of the PBGP.
Proposition 1b: If intermediates are omitted (i.e. if 0== μγ ), the PBGP is locally
stable.
Proof: See APPENDIX C.
Proposition 1 ensures that the economy will approach to the PBGP even if the
initial capital level is not such that the economy starts on the PBGP.
Proposition 2: Along the PBGP the aggregate dynamics of the economy are
represented by the following equations: ; and EKKY ++= δ&ˆ qq KLGY −= 1~ˆ
ρδλ −−=KY
EE ˆ&
, where G is a parameter growing at constant rate (“Hicks-
neutral technological progress”),
~
Y denotes aggregate output without
intermediates production (i.e. Y-H) and λ is a constant (see APPENDIX B for
details of these parameters).
Proof: See APPENDIX B.
181
In fact Proposition 2 implies that the aggregate structure of our economy is quite
the same as the structure of the standard Ramsey-Cass-Koopmans- or Solow-model
(with Cobb-Douglas production function and logarithmic utility).
Now, the question arises, whether structural change takes place along the PBGP. I
discuss this question in the following.
3.2.3 Development of sectors in equilibrium By inserting equations (1) to (16) into optimality conditions (17) to (19), the
following equations that describe the optimal sector structure of the economy
(represented by the employment shares) can be obtained:
(30a) ∑=
+Λ=a
iiagragr Y
l1
.. ~1 θ
(30b) ∑∑+=+=
Γ++Λ=s
mii
m
aiimanman Y
l11
.. ~1 θθ
(30c) ∑+=
Γ+Λ=n
siiserserl
1.. θ
where
(31a) ∑∑
∑=
=
= +≡Λa
iin
ii
a
ii
agr YH
YE
1
1
1. ~~ ε
ω
ω
(31b) Y
KKYH
YE s
mii
m
aiin
ii
s
mii
m
aii
man ~~~11
1
11.
δεαχε
ω
ωαχω
++⎟
⎠
⎞⎜⎝
⎛++
+≡Λ ∑∑
∑
∑∑+=+=
=
+=+=&
182
(31c) ∑∑
∑+=
=
+= +≡Λn
siin
ii
n
sii
ser YH
YE
1
1
1. ~~ ε
αχ
ω
ω
αχ
(31d) εμεεεμβ
γμμ
χγαμ
χβαν −−(−
+−+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≡Γ
)11)1(
/
1vv
m
mv
LlKkL
AGB
Again, to get an impression of how these equations can be derived, see the
derivations in my Ageing-model (especially APPENDIX A of PART III of
CHAPTER V).
Note that ... ,, sermanagr ΛΛΛ and Γ can be easily derived as functions of exogenous
parameters along the PBGP.7 However, I omit here the explicit proof, since it is
trivial and irrelevant for further discussion (for a sketch of the proof see footnote
7).
Lemma 2: Structural change takes place along the PBGP. That is, the employment
shares of sectors agriculture ( ), manufacturing ( ) and services ( ) are
changing along the PBGP.
.agrl .manl .serl
7 In APPENDIX B (equation (B.17)) I have derived as function of exogenous model
parameters. This function can be used to derive
mm kl /
Y~ and Y as functions of exogenous model parameters by using equations (21) and (25). Then, when I have Y~ and as functions of
exogenous model parameters, I can derive mm kl /
H as a function of exogenous model parameters by using equation (23). Finally, I can use Y and H to derive E as function of exogenous model parameters (via equation (20); note that the initial capital endowment is exogenously given;
hence 0K
K can be calculated by using and the equilibrium growth rate of capital , where
is a function of exogenous model parameters as shown in Lemma 1). When I have ,
0K *g *g
mm kl / Y~ , K
and E as functions of exogenous model parameters, I can derive ... ,, sermanagr ΛΛΛ and as functions of exogenous model parameters.
Γ
183
Proof: This Lemma is implied by equations (30) and (31). Note that ,
and are constant along the PBGP (due to Lemma 1);
.. , managr ΛΛ
.serΛ Y~ grows at rate
along the PBGP (see Lemma 1).
*g
Γ decreases at constant rate along the PBGP. The
latter fact comes from Lemma 1 and equation (28). Note that G/A grows at positive
constant rate; see equation (28) and footnote 5. Furthermore, note that the exponent
εμγεεεμβ
−−(−+−
+ )11)1( v
v is positive, since 1)1( − + εμ <εγ as explained in footnote 5. Q.E.D.
Now, the remaining exercise is to show that along the PBGP my model is indeed
consistent with all the stylized facts mentioned in the introduction and section 2 of
my paper.
3.2.4 Consistency with stylized facts
Lemma 3: The PBGP of my model satisfies the Kaldor facts regarding the
development of the great ratios. That is, the aggregate capital intensity (K/L) is
increasing; the aggregate capital-income-share ( or ), the
aggregate labor-income-share ( or
YrK / )/( HYrK −
YwL / )/( HYwL − ) and the aggregegate
capital-to-output ratio (K/Y or K/(Y-H)) are constant (where r is the real rate of
return on capital and w is the real wage rate).
Proof: The constancy of K/Y and K/(Y-H) as well as the increasing capital-intensity
(K/L) are directly implied by Lemma 1. Since I assume perfect polypolisitic
markets, the marginal productivity of capital (of labor) in a sub-sector i is equal to
the real rate of return on capital (real wage rate) for all i. This implies for example
for : mi =
184
(32) KY
kl
KkYr
m
m
m
m~
)(β=
∂∂
=
(33) LY
LlYwm
m~
)(α=
∂∂
=
Hence, Lemma 1 and equations (32) and (33) imply that YrK ,
HYrK−
, YwL and
HYwL−
are constant. Q.E.D.
Note that there are two further Kaldor-facts: namely Kaldor stated that the
aggregate volume of production grows at a non-decreasing rate and that the real
rate of return on capital is constant. As discussed in section 3.1, due to numéraire
choice I cannot say whether these two Kaldor-facts are satisfied approximately in
my model. However, as mentioned before, the constancy of the real interest rate
seems to be rather not a fact in reality. Furthermore, the results by Ngai and
Pissarides (2004, 2007) imply that the aggregate output expressed in manufacturing
terms (as in my model) behaves quite similar as the aggregate output that is
measured in reality (by using some compound numéraire). Hence, my model could
be consistent with a constant real rate of aggregate output.
Lemma 4: Along the PBGP the development of sectoral employment shares over
time (equations (30)-(31)) can be monotonous (monotonously increasing,
monotonously decreasing or constant) or non-monotonous (“hump-shaped” or
“U-shaped”), depending on the parameterization of the model.
Proof: This Lemma is implied by equations (30)-(31). In the proof of Lemma 2 I
have shown that , and .. , managr ΛΛ .serΛ are constant along the PBGP, Y~ grows at
185
rate along the PBGP (see Lemma 1) and *g Γ decreases at constant rate along the
PBGP. Hence, since and Y~/1 Γ grow at different rates, equation (30b) implies that
the development of the manufacturing-employment-share over time ( ) can be
non-monotonous, provided that has not the same sign as . That is, it
can be hump-shaped or U-shaped depending on the parameterization. Hence, the
model can reproduce a “hump-shaped” development of the manufacturing-
employment share over time, which has been emphasized by Ngai and Pissarides
(2007) and Maddison (1980). Note that only sectors, which use at least two
technologies, can feature non-monotonous development of their employment share
over time. However, as discussed in section 3.1 the manufacturing sector (i.e. the
capital producing sector) need not using two technologies, i.e. the model could be
set up such that the agricultural sector or the services sector uses two technologies.
Hence, in fact any of the sectors could feature non-monotonous dynamics of its
employment-share over time. The proof that
.manl
∑+=
m
aii
1
θ ∑+=
s
mii
1
θ
• can be monotonously increasing, monotonously decreasing or constant, .agrl
• can be monotonously increasing or monotonously decreasing, and .manl
• can be monotonously increasing, monotonously decreasing or constant .serl
is obvious when taking into account that , and can be
negative, positive or equal to zero respectively. Q.E.D.
∑=
a
ii
1
θ ∑∑+=+=
s
mii
m
aii
11
, θθ ∑+=
n
sii
1
θ
Lemma 5: Agriculture, manufacturing and services have different production
functions in my model. Especially, the optimal capital intensity differs across these
sectors.
186
Proof: Since I assumed that agriculture (services) uses only technology 1 (2) its
production function is represented by technology 1 (2). Hence, we know that the
technology (especially the TFP-growth-rate and the capital-intensity) differ across
agriculture and services. Furthermore, manufacturing uses both technologies.
Hence, the average manufacturing technology is a mix of technology 1 and 2.
Hence, the representative production function of the manufacturing sector is
different in comparison to the services sector or the agricultural sector which each
use only one technology. Nevertheless, since I have an emphasis on the cross-
capital-intensity structural change, let us have a close look on the capital-intensity
(LlKk
LlKk
man
man
agr
agr
.
.
.
. , and LlKk
ser
ser
.
. ), the output-elasticity of labor ( .agrλ , .manλ and .serλ )
and the output-elasticity of capital ( .agrκ , .manκ and .serκ ) in each sector:
(34) χβαν
χβαν
LlKk
LlKk
LlKk
LlKk
LlKk
LlKk
m
m
ser
ser
m
m
man
man
m
m
agr
agr =≠⎟⎟⎠
⎞⎜⎜⎝
⎛+=≠=
.
.
.
.
.
. 1
(35) χλ
χα
αλαλ ==≠+
==≠==
∑ ∑+= +=
.
..
1 1
.
.
..
.
..
ser
serserm
ai
s
miii
man
man
manman
agr
agragr Y
Lwl
ll
lY
LwlY
Lwl
(36) νκ
χαχβαν
βκβκ ==≠+
+==≠==
∑ ∑
∑ ∑
+= +=
+= +=
.
..
1 1
1 1
.
..
.
..
ser
serserm
ai
s
miii
m
ai
s
miii
man
manman
agr
agragr Y
Krk
ll
ll
YKrk
YKrk
(Note output elasticity of factors is equal to the factor-income shares due to the
assumption of perfect markets and perfect factor mobility.) Overall, capital
intensities and output-elasticities of inputs differ across sectors agriculture,
manufacturing and services. Q.E.D.
187
Lemma 6: Along the PBGP the factor reallocation across the agricultural,
manufacturing and services sector is determined by cross-sector-TFP-growth
disparity, by cross sector capital-intensity-disparity and by non-homothetic
preferences.
Proof: As discussed above, the TFP-growth rates and the capital-intensities differ
across the sectors agriculture, manufacturing and services; see also Lemma 5.
Equations (30)-(31) (and equations (21) and (28)) imply that cross-sector-
differences in TFP-growth-rates and cross-sector-differences in output-elasticities
of inputs (which determine the capital-intensities) determine the strength of the
factor reallocation between the sectors agriculture, manufacturing and services.
Especially, they affect the sectoral employment shares ( , and ) via the
terms
.agrl .manl .serl
Y~ and Γ , which are among others functions of the parameters that
determine the sectoral TFP-growth rates and sectoral capital intensities (see
equations (21), (31d) and (28) and Lemma 5).
Furthermore, equations (8) to (11) imply that preferences are non-homothetic
across sectors agriculture, manufacturing and services. A detailed proof is in
APPENDIX D, where I show among others that the terms , and
determine the pattern of non-homotheticity across sectors agriculture,
manufacturing and services. Equations (30)-(31) imply that this non-homotheticity
determines the strength and direction of structural change (via terms ,
and ). Q.E.D.
∑=
a
ii
1θ ∑∑
+=+=
s
mii
m
aii
11, θθ
∑+=
n
sii
1θ
∑=
a
ii
1
θ
∑∑+=+=
s
mii
m
aii
11, θθ ∑
+=
n
sii
1θ
188
Lemma 7: Intersectoral outsourcing (i.e. shifts in intermediates production across
sectors) takes place along the PBGP. That is, along the PBGP manufacturing-
sector-producers shift more and more intermediates production to services-sector-
producers (i.e. changes), provided that services-sector-production becomes
cheaper and cheaper (or less and less expensive) in comparison to manufacturing-
sector-production (i.e. provided that relative prices change), and vice versa. Any
direction of relative price changes (and hence any direction of intermediate-
production shifts between the manufacturing and the services sector) can be
generated along the PBGP, depending on the parameterization.
ji hh /
Proof: See APPENDIX E.
Theorem 1: The PBGP satisfies simultaneously the following stylized facts:
• Kaldor-facts regarding the development of the great ratios,
• Kuznets facts regarding structural change patterns,
• “stylized facts regarding cross-sector-heterogeneity in production-
technology” (see section 2 as well), and
• empirical evidence on structural change determinants in industrialized
countries (see section 2).
Proof: The consistency of the PBGP with the Kaldor facts is implied by Lemma 3.
Note that empirical evidence on structural change between agriculture,
manufacturing and services in industrial countries implies the following stylized
facts for the development of the employment shares over the last century:
• the agricultural sector featured a monotonously decreasing employment
share,
• the services sector featured a monotonously increasing employment share,
and
189
• the manufacturing sector featured a constant or “hump-shaped” employment
share (depending on the length of the period considered).
These stylized facts have been formulated by Kongsamut et al. (1997, 2001); on
the “humped shape” of the manufacturing-employment share see e.g. Ngai and
Pissarides (2004, 2007) and Maddison (1980). In the proof of Lemma 4 I have
shown that my model can reproduce these stylized facts regarding the development
of the agricultural, manufacturing and services employment shares. Hence, the
PBGP is consistent with the Kuznets-facts.
The consistency of the PBGP with the “stylized facts regarding cross-sector-
heterogeneity in production-technology” is shown in Lemma 5, where I show that
production technology differs across agriculture, manufacturing and services in my
model.
Finally the consistency of the PBGP with the empirical evidence on structural-
change-determinants in industrialized countries is shown in Lemmas 6 and 7.
Q.E.D.
3.2.5 The relationship between structural change and aggregate-dynamics Now I turn to the question about the relationship between structural change and
aggregate growth, i.e. I ask how structural change affects aggregate growth, which
is important for understanding the Kuznets-Kaldor-puzzle. In the following I will
show that there are two types of cross-capital-intensity structural change, which are
distinguished according to their impact on the aggregate structure of the economy.
Definition 2: The term “cross-capital-intensity structural change” stands for
factor reallocation across sectors that differ by capital intensity.
190
It can be shown that
(37) ⎟⎟⎠
⎞⎜⎜⎝
⎛++=≡ .
.
..
.
..
.
.ser
ser
serman
man
managr
agr
agr
m
m
m
m lllkll
λκ
λκ
λκ
αβ
where .agrλ ( .agrκ ), .manλ ( .manκ ) and .serλ ( .serκ ) are respectively the income-share
of labor (capital) in sectors agriculture, manufacturing and services. Equation (37)
follows from the assumption of factor mobility across sectors and from the
assumption of perfect markets.
Equation (37) and Lemma 1 imply that there are two sorts of cross-capital-
structural change:
(1) Cross-capital-intensity structural change where l is not constant. Lemma 1
implies that the economy is on a PBGP, only if is constant; furthermore,
equation (37) implies that the constancy of
mm kl /
l is required for the constancy of
. Hence, as long as mm kl / l is not constant, the economy is not on a PBGP and the
Kaldor-facts are not satisfied (exactly). That is, this type of structural change is not
compatible with the Kaldor facts (unless structural change is very weak such that
its impact via l is weak which would imply that Kaldor facts are approximately
satisfied).
(2) Cross-capital-intensity structural change that is compatible with a constant l .
Hence, an economy can be on a PBGP, even when cross-capital-intensity factor
reallocation takes place, provided that this factor reallocation is such that l = const.
(see also Lemma 1).
So I can give the following definition and theorem:
191
Definition 3: “Neutral structural change” stands for cross-capital-intensity
structural change that satisfies the following condition:
(38) ...
..
.
..
.
. constllll serser
serman
man
managr
agr
agr =⎟⎟⎠
⎞⎜⎜⎝
⎛++≡λκ
λκ
λκ
Theorem 2: Along the PBGP, the cross-capital-intensity structural change
(between agriculture, manufacturing and services) is “neutral” in the sense of
Definition 3.
Proof: Note that I have shown in Lemma 5 that sectors agriculture, manufacturing
and services differ by technology, and especially by capital intensity and by output-
elasticities of inputs/income-shares of inputs. Lemma 2 implies that structural
change takes place across these sectors. Equation (37), Definition 3 and Lemma 1
(necessity of a constant for a PBGP) imply the rest of the theorem. Q.E.D. mm kl /
Theorem 3: Neutral structural change is an explanation for the Kuznets-Kaldor-
Puzzle in my model.
Proof: Remember that the Kuznets-Kaldor-puzzle was about the empirical
question why cross-capital-intensity structural change is compatible with the
stability of the great ratios (Kaldor facts). Theorem 2 implies that neutral-cross-
capital-intensity structural change takes place along the PBGP, while Theorem 1
shows that the PBGP is consistent with the Kaldor facts. Thus, Kaldor-facts are
satisfied, since cross-technology structural change needs not necessarily to
contradict the Kaldor facts, which is satisfied in my model only neutral cross-
capital-intensity structural change patterns. Furthermore, Theorem 1 shows the
192
generality of my proof: neutral cross-technology structural change is not only
consistent with the Kaldor facts about the great ratios but also with the other
stylized facts which are relevant for the analysis of the relationship between
structural change and aggregates. Hence, Theorem 1 shows that I solved the
Kuznets-Kaldor-puzzle under consideration of the most important structural change
determinants and under assumption of sectoral cross-technology disparities
observed in reality. Q.E.D.
The convenient feature regarding latter two theorems is that I can use them to test
my theory empirically: I can calculate l , and then decompose which share of
structural change does not change the value of l and which share of structural
change changes the value of l . In this way I can evaluate the quantitative
significance of my model-explanation for the Kuznets-Kaldor-Puzzle, since my
explanation focuses only on structural change that does not change l (due to
Theorem 2).
However, before doing so I show two further interesting results
Proposition 3: The output-elasticity of inputs ( .manλ , .manκ ) is not constant in the
manufacturing sector along the PBGP, but changes according to the amount of
inputs used in this sector.
Proof: This is implied by equations (35) and (36). Note, that any sector that uses
two technologies has a non-constant output-elasticity of inputs in my model setting.
Q.E.D.
193
This result is interesting: in fact it implies that observed technology changes in
sectors need not necessarily resulting form technological progress at sector level,
but can also result from structural change. Of course this requires that sectors use
several technologies, which seems to be a reasonable assumption. This fact could
be of importance for further research, especially when analyzing endogenous
technological progress at sector level. That is, Proposition 3 implies that such
research will require considering technology change at sector level with caution,
since some technology change may not result from technological progress at sector
level.
As argued in section 3.1 I assume that there are only two technologies in my
model, but that there is an arbitrary number of subsectors. Hence, some subsectors
have to use identical technologies. As explained there, I use this assumption to
explain the concept of “uncorrelated preferences and technologies” in a traceable
way, which will be of interest later in this paper. However, the assumption of partly
identical production functions is not necessary for the key results of the actual
section: the following proposition shows that the key result of this section (namely
for the existence of neutral cross-capital-intenstity structural change) can be
derived even all (sub-)sectors have completely different production functions.
Proposition 4: Generalization of my results: In a framework where
• all sub-sectors (i) have sub-sector-specific production functions,
• sub-sector production functions are general neoclassical production
functions
• and intermediate production is omitted
a necessary condition for neutrality of cross-capital-intensity structural change
and for the satisfaction of Kaldor-facts is
194
(39) .~ constlli i
i =≡ ∑ λ
where iλ is the output-elasticity of labor in subsector i which is equal to the labor-
income share in sector i.
Proof: See APPENDIX F.
4. A measure of neutrality of cross-capital-intensity structural change In the previous section, I have presented a model that explains the Kuznets-Kaldor-
puzzle with a certain structural change pattern which I name “neutral structural
change”. In Theorem 2 and Proposition 4 I have shown that this structural change
pattern must satisfy condition (38). Due to lack of data I cannot consider
intermediates production explicitly. Therefore, I assume that capital and labor are
the only inputs in the production function in this section. In this case condition (38)
transforms into condition (39).
In proposition 4 I have generalized the validity of condition (39) to a more general
framework than that that of section 3. Hence, the development of this condition is
not only of interest for my model, but for all models that analyze PBGP’s.
I can use condition (39) to asses to what extent neutral structural change takes
place in reality.
For the calculations in this section I use the data for the U.S.A., which is available
at the web-site of the U.S. Department of Commerce (Bureau of Economic
Analysis). I use the U.S.-Gross-Domestic-Product-(GDP)-by-Industry-Data, which
195
is based on the sector-definition from the “Standard Industrial Classification
System”, which defines the following sectors:
(1) Agriculture, forestry, and fishing
(2) Mining
(3) Construction
(4) Manufacturing
(5) Transportation and public utilities
(6) Wholesale trade
(7) Retail trade
(8) Finance, insurance, and real estate
(9) Services
My calculations are based on the data for the period 1948-1987. Uniform data for
longer time-periods is not available, since the “Standard Industrial Classification
System” has been modified over time (hence, the sector definition after 1987 is not
the same as the sector definition before 1987).
To calculate the sectoral labor income shares ( iλ ) I divided “(Nominal)
Compensation of Employees” by “(Nominal) Value Added by Industry” in each
sector. The sectoral employment shares ( ) are calculated by using the sectoral
data on “Full-time Equivalent Employees”. (This approach is similar to that used
by Acemogu and Guerrieri (2008)).
il
Figure 1 depicts the development of , calculated by these data: l~
196
Figure 1: Development of over time l~
1,65
1,7
1,75
1,8
1,85
1,9
1,95
2
2,05
2,1
2,15
1948
1950
1952
1954
1956
1958
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
We can see that is decreasing and not constant. The question is, how small the
decline of is. The decline in could have been much stronger or much weaker.
If the decline is relatively small, I could postulate that is “approximately”
constant from a theoretical point of view, hence the model of neutral structural
change would be relatively good in explaining the Kuznets-Kaldor puzzle. Hence, I
have to develop an index which indicates how strong the decline is. In the
following I develop such an index. This index is based on calculating the strongest
possible decline in and then relating the actual decline to it.
l~
l~ l~
l~
l~
Any actual l~ can be expressed as a unique combination of neutral and “maximally
non-neutral structural change”. “Maximally non-neutral structural change” is the
pattern of factor reallocation that causes the maximal decline in l~ for a given
amount of reallocated labor over a period. Hence, maximally non-neutral structural
change is a diametric concept of neutral structural change: while neutral structural
change is defined upon no change in l~ , maximally non-neutral structural change is
197
defined upon maximal change in l~ . This allows me to create an index that shows
us whether a given amount of reallocated labor has been reallocated rather in the
neutral way or rather in the maximally non-neutral way. According to this
discussion the following relation must be true:
(40) max)~()~)(1()~( lIlIl Nneutral
Nactual Δ+Δ−=Δ
where is a weighting factor between neutral and maximally non-neutral
structural change, i.e. it indicates whether structural change was rather neutral or
non-neutral; if =1, structural change is maximally non-neutral over the
observation period; if =0 structural change is neutral over the observation
period.
NI
NI
NI
actuall )~(Δ measures the change in l~ that really took place between 1948
and 1987; measures the maximal change in lmax)~( lΔ ~ , that would be
(hypothetically) possible with the given amount of cross-sector factor reallocation
between 1948 and 1987, i.e. max)~( lΔ stands for “completely non-neutral structural
change”. neutrall )~(Δ measures the change in l~ that is caused by neutral structural
change. Since per definition is equal to zero, I can rearrange the
condition from above as follows:
neutrall )~(Δ
(41) max)~(
)~(llI
actual
N ΔΔ
≡
where max)~( lΔ and actuall )~(Δ are defined as follows:
198
(42) ∑∑ −=−≡Δi i
i
i i
iactual lllll 1948
1948
1987
1987
19481987~~)~(
λλ
(43) ∑∑ −=−≡Δi i
i
i i
i lllll 1948
1948
1987
max1987
1948max
1987max ~~)~(
λλ
where , , and denote respectively the employment share of
sector i in 1948, the employment share of sector i in 1987, the labor-share of
income in sector i in 1948 and the labor-share of income in sector i in1987.
stands for the hypothetical employment share of sector i, which would result, if the
labor, which has been reallocated between 1948 and1987, were reallocated in such
a manner that the maximal decrease in
1948il
1987il
1948iλ
1987iλ
max1987il
l~ was accomplished between 1948 and
1987. That is, the ’s stand for the hypothetical factor allocation in 1987,
which yields the maximally non-neutral structural change between 1948 and 1987.
max1987il
Last but not least, since my definition of requires knowing how much labor
has been reallocated between 1948 and 1987, I propose the following index of
observable factor reallocation between 1948 and 1987:
max1987il
∑ −≡Δi
ii lll 19481987
21
This measure indicates how much labor has been reallocated between 1948 and
1987. This measure is set up as follows: First, the change in the employment share
in each sector is measured. The absolute values (modulus) of these changes are
summed up (otherwise, without taking absolute values, that sum of the sectoral
changes would always be equal to zero, since ∑ =i
il 1 per definition). Since the
199
change in the employment share in one sector has always a corresponding change
in the employment shares of the other sectors (labor is reallocated across sectors),
the sum of the absolute values of the changes must be divided by two to avoid
double-counting.
It is possible that between 1948 and 1987 in some sectors the employment share
increased first and decreased then. Hence, the pure difference would
indicate less reallocation than actually took place. My index of factor reallocation
( ) neglects such non-monotonousity in sectoral employment shares. Hence, it
underestimates the real amount of labor reallocated between 1948 and 1987.
Therefore, my index underestimates the neutrality of structural change: if more
labor were reallocated during the period, the hypothetical maximal change in
19481987ii ll −
lΔ
NI
l~
( ) would be larger; hence, would be smaller, which would imply more
neutrality. Overall, for these reasons, my index indicates less neutrality than
actually is.
max)~( lΔ NI
NI
Note that it is important that my measure of maximally non-neutral structural
change ( max)~( lΔ ) is based on the actual amount of reallocated labor ( ). In this
way I distinguish between strength and direction of structural change. Strength of
structural change implies how much labor has been reallocated (e.g. as measured
by ). The direction of structural change implies how the labor has been
reallocated across technology. Neutrality of structural change is not related to
strength but only to direction, since condition (39) can be satisfied by more or less
strong structural change patterns. What counts for satisfying condition (39) is the
direction of structural change. If there is no significant direction of structural
change (39) is satisfied. Therefore, when calculating the neutrality index it is
important to be cautious about not defining
lΔ
lΔ
max)~( lΔ such that it features stronger
200
structural change than actual structural change is. Therefore, I calculate by
using the actual amount of reallocated labor (
max)~( lΔ
lΔ ).
The data that I need for my calculations is given in the following table:
Table 1
Sector 1948/1 iλ 1987/1 iλ 1948il 1987
il
(8) 5.248981966 3.997781119 0.039609477 0.077711379
(1) 6.874359747 3.921756596 0.05019623 0.019310549
(2) 2.62541713 3.240100098 0.024056398 0.008630482
(5) 1.632072868 2.20691581 0.099835263 0.063265508
(6) 1.937362752 1.72651328 0.062648384 0.070192118
(7) 1.988458748 1.649066345 0.141770435 0.191092947
(3) 1.495168451 1.505702087 0.056228499 0.059919498
(4) 1.505805486 1.447391372 0.376011435 0.229516495
(9) 1.681140684 1.444831355 0.149643878 0.280361023
Now, by using these data, I have to do the following steps to calculate : NI
1.) Calculate the amount of reallocated labor between 1948 and (1987), which
results in 0.23. ≈Δl
2.) Calculate max1987~l . According to my definition of max
1987~l , I have to do the following
steps:
a.) Find the sector that has the smallest . This is actually sector (9). 1987/1 iλ
b) Make a ranking of the remaining sectors according to their . This ranking
is given by (8)-(1)-(2)-(5)-(6)-(7)-(3)-(4), where sector (8) has the largest
and sector (4) has the smallest in this ranking.
1987/1 iλ
1987/1 iλ
1987/1 iλ
201
c) By using the ranking from b) reallocate the labor from the sectors with the
largest to sector (9). I first use the whole amount of labor, that has been
employed in sector (8) in 1948, then the whole amount of labor, that has been
employed in sector (1) in 1948, and so on, stepping up in the ranking until I have
hypothetically reallocated the whole
1987/1 iλ
≈Δl 0.23. Hence, I obtain the following
maximally non-neutral factor allocation for the year 1987
Table 2
Sector max1987il
(8) = 0
(1) = 0
(2) = 0
(5) = 0
(6) = = 0.046969461 )( 1948)8(
1948)5(
1948)2(
1948)1(
1948)6( llllll −−−−Δ−
(7) = = 0.141770435 1948)7(l
(3) = = 0.056228499 1948)3(l
(4) = = 0.376011435 1948)4(l
(9) = = 0.37902017 ll Δ+1948)9(
3.) The rest of the calculations is quite simple: by inserting the data from Tables 1
and 2 into equations (41)-(43), we can obtain . NI
My calculations imply an index = 0.45. This implies that actual structural
change was slightly closer to its neutral extreme than to its non-neutral extreme. In
NI
202
other words, the actual structural change between 1948 and 1987 was by 55%
neutral and by 45% maximally non-neutral.
In this sense, my model can explain 55% of the structural change between 1948
and 1987.
Note that my measure underestimates the neutrality of structural change. That is, in
reality more than 55% of structural change can be regarded as neutral. There are
two reasons: as discussed above, my measure assumes monotonousity of factor
reallocation; furthermore, as will be discussed close to the end of next section, the
period, which I used for analysis, is quite short and structural change is more
neutral over very long periods of time.
5. On correlation between preferences and technologies In section 3.1 I have assumed that preferences and technologies are uncorrelated in
my model. In detail, I have assumed that
• on average the income elasticity of demand is equal when comparing
technology-1-goods and technology-2-goods
• on average the elasticity of substitution is equal to unity when comparing
technology-1-goods and technology-2-goods.
In the following I will discuss the rationale for these assumptions. I focus here on
the elasticity of substitution, but the corresponding arguments apply for the income
elasticity of demand.
Assuming that the relative price-elasticity of demand between two goods is
different from unity implies that the household has a certain preference for the one
good over the other: Imagine that there are only two goods (good A and good B). If
the relative price of the good A increases by one percent and the relative demand
203
for this good decreases by less than one percent, good A is regarded as more
important than good B by the household in the dynamic context. That is, the price
change causes a weaker reaction than it would be if the two goods were regarded as
equivalents. Only if two goods are regarded as equivalents, a one-percent-change
in the relative price between these goods would yield a (minus) one-percent-change
in the demand-relation between these goods (hence, elasticity of substitution
between these goods being equal to one).
Now, the same argument could be applied to two groups of goods (group A and
group B): if the household regards the two groups as equivalents, the average
elasticity of substitution between the two groups is equal to unity. Otherwise, we
would have to postulate that on average group A includes goods that are preferred
over group B (or the other way around).
Now, imagine that the whole range of products in an economy is divided into two
groups according to their production technology. Group A includes goods that are
regarded as technologically progressive and group B includes goods that are
produced by a backward technology. Furthermore, let us make the following
assumptions:
(a) The household doesn’t know anything about the production process, i.e. the
household’s preference depends only on the “objective taste” of the goods (but not
on the knowledge that the good is produced at e.g. high-capital-intensity).
“Objective taste” means the taste which depends only on the physical/chemical
properties of the good or on the basic properties (i.e. actual quality) of the service,
but not on the knowledge about the production process of the good or service. For
example, if two goods are produced by different capital intensities, but if the two
goods are basically the same (i.e. have the same physical and chemical properties),
the objective taste of the two goods is the same. A further example is the following
204
experiment: imagine that a live concert is recorded and then later replayed as a
playback to a similar audience (while the original musicians pretend performing
music). The labor-intensity of the original concert is higher in comparison to the
playback concert, since pretending is easier (i.e. less labor-intense) in comparison
to performing live music. The objective taste of the two concerts would be the
same. (However, the “subjective taste” of the two concerts would differ, if the
audience knew that the second concert is only a playback.)
(b) The “objective taste” of a good is on average not dependent on the technology
that is used to produce it. That is, some very tasty goods are produced by
progressive technology and some very tasty goods are produced by backward
technology; as well, some less tasty goods are produced by progressive technology
and some less tasty goods are produced by backward technology.
With these assumptions we would conclude that on average group-A-goods are not
preferred over group-B-goods and group-B-goods are not preferred over group-A-
goods. That is, the groups are regarded as equivalents; hence, on average the
elasticity of substitution between these two groups will be close to one (according
to the discussion above).
Now let us make a further assumption:
(c) I look only on the averages over very long periods of time and I assume that
there are many technologies and goods.
Hence, from this perspective due to the law of large numbers the elasticity of
substitution between the two groups is equal to unity.
In other words, if preferences and technologies are uncorrelated (i.e. if the taste
does not depend on production technology), the household behavior will not
display any preference for the technology-level (group A or group B), provided
205
that very long periods of time are considered and provided that there are many
goods.
This is what I assumed in section 3.1: I assumed that there are two technologies
and that there are many goods that are produced with these technologies and that
the preference structure does not display any preference for a certain technology.
This is what I did by assumptions (10) and (11). These assumptions ensure that on
average the relative price-elasticity between technology-1-goods and technology-2-
goods is equal to unity.
Now the question is whether the assumptions (a), (b) and (c) are suitable in long
run growth models.
Assumption (c) seems not to be problematic, since the long-run growth theory is
anyway based on analyzing long-run-averages (e.g. the time preference rate is
assumed to be constant in standard neoclassical growth models). Furthermore,
since I look at very long run, any accidental correlations between technology and
preferences, which may arise from a relatively low number of products, may as
well offset each other over the period’s average.
Assumption (b) is less problematic in comparison to assumption (a). In fact, the
technological progress during the last century has implicitly shown that the basic
physical/chemical properties of a good are not necessarily dependent on its capital-
intensity. In industrialized countries nearly all goods featured some technological
progress that substituted labor by capital, while the basic physical properties of the
goods remained the same basically. The most obvious example is agriculture. Food
has for the most part the same basic physical properties today as earlier in the
century, while the capital-intensity of agriculture increased significantly. Such
developments are also apparent in manufacturing (e.g. regarding the increasing
capital-intensity of car-production) and services (e.g. cash-teller-machines).
206
Furthermore, today we can imagine for nearly every good or service a relatively
realistic technology that could substitute the labor by capital, without changing the
basic physical properties of the good. It is not plausible to assume that in the very
long run technological progress is restricted to certain types of goods. In the last
two decades many service-jobs, which were regarded as labor-intensive, were
replaced by computer-machines and the substitutability of human by machines in
services is increasing. Hence, when developing a long run theory of structural
change, the dependency between technology and certain types of goods (and hence
certain preferences) seems to be difficult to defend. Therefore, overall, the
assumption that the “objective taste” of a good is independent of the capital-
intensity of the production process seems to be acceptable to some degree,
especially when assuming (c).
It is more difficult to evaluate assumption (a) a priori. Assumption (a) requires that
the representative household behaves like he doesn’t know about the actual capital
intensity of a good, i.e. it is required that the household’s demand reaction to a
price and/or income change is based only on physical/chemical properties of a
good. What we know from basic microeconomics (e.g. form the discussion about
“Giffen-goods”) is that the price elasticity (and income elasticity) depends on the
basic physical/chemical properties of the good, i.e. whether the physical/chemical
properties of a good are such that it is feasible to satisfy the basic needs of a
household. (The price elasticity for such goods is low.) On the other hand, there is
also a discussion about a “snob” effect, where some very labor-intensity services
(like a full time servant) are used to signal the wealth of the household. Such
services have a relatively high income-elasticity and price-elasticity. However, as
well, there are many high-capital-intensity-goods that have high price-elasticity of
demand and high income-elasticity of demand, like very expensive cars. Hence,
207
there is both: capital-intensive and labor-intensive goods that feature a relatively
high price-elasticity and a relative high income-elasticity. My model requires that
on average (i.e. when looking at the average of all consumption goods) the income
(price) elasticity of demand does not depend on the capital-intensity of a good.
Last not least, the increasing complexity of the products and of the production
process, international outsourcing and increasing variety of products make it
increasingly unlikely that the household has clear information about the capital-
intensity of a large part of its consumption bundle.
All in all, the empirical evidence from the previous section implies that the
assumption of no/low correlation between technology and preferences can explain
a part of the Kuznets-Kaldor-puzzle. The fact that there is some correlation
between technology and preferences results probably from the fact that assumption
(a) has not been satisfied over the time-period of my sample. That is, probably high
labor-intensity of a service has been regarded as an aspect of quality and/or luxury.
Hence, high-labor-intensity services have probably had high income-elasticity of
demand on average, which caused the correlation between technology and
preferences in the past.
The fact that there has been some correlation between preferences and technologies
in my sample does not necessarily imply that we can presume such correlation in
future:
I analyzed only a 40 year period. This is a very short period to satisfy assumption
(c) and to study growth theory empirically in general. Remember that Kaldor-facts
(which I seek to explain in my paper) do not necessarily apply to such a short
period. The probability is very high that over such a short period “accidental”
correlation between technology and preferences arises, which does not persist over
the long run. It seems that this was the case: The technological innovation between
208
1940 and 1980 allowed to a big part an increase in capital-intensity in non-service-
sectors (such as manufacturing and agriculture). That is, the technological break-
throughs were such that they were easy to implement in non-services sectors but
they were hardly implementable in the services sector8. Hence, if services have
high income-elasticity of demand, some correlation between technology and
preferences may have been arisen due to such biased technological progress.
However, new sorts of technological break-through occurred after this period,
especially in the information and communication technology. Such break-throughs
have increased the capital-intensity in the services sector and have a high potential
for increasing the capital-intensity of the services sector drastically (e.g. by
progress in computers and robotics, which is implementable in services).
Hence, my empirical results probably over-estimate the long-run degree of
correlation between preferences and technologies; the long-run correlation between
preferences and technologies is probably very low or even inexistent. In this sense,
my model of independent preferences and technologies predicts quite well the
future structural change impacts on aggregates.
6. Concluding remarks In this essay I have searched for a solution of the Kuznets-Kaldor-puzzle. In fact,
the Kuznets-Kaldor-puzzle states that aggregate ratios behaved in a quite stable
manner in industrialized countries, while at the same time massive factor
reallocation took place across sectors, which differ by technology (and especially
by capital-intensity).
8 Of course, the term “services” means here rather personal services (i.e. services which require face-to-face contact, e.g. counselling) and rather not such services as transportation. The latter featured strong increases in capital-intensity. See for example Baumol et al. (1985) on discussion and empirical evidence about progressive and stagnant services.
209
For the first time in the literature, I have shown that a PBGP can exist even when
factors are reallocated across sectors that differ by capital intensity. I name the
cross-capital-intensity structural change that is compatible with a PBGP “neutral
structural change”.
To test the actual neutrality of structural change I developed an index of neutrality.
In fact, my measure of neutrality indicates the weighting between two measures
( )neutrall~Δ and ( )max~lΔ . ( )neutral
l~Δ measures the hypothetical change in l~ that would
result, if the empirically observed amount of reallocated labor ( ) were
reallocated in the neutral way.
lΔ
( )max~lΔ measures the hypothetical change in l~ that
would result, if lΔ were reallocated in the maximally non-neutral way. Hence, the
weighting between these two measures implies how much labor has been
reallocated in the neutral way and how much labor has been reallocated in the non-
neutral way between 1948 and 1987. This index implies that 55% of structural
change can be regarded as neutral. I provided also some theoretical/verbal
arguments which imply that over the (very) long run significantly more than 55%
of the structural change is neutral (see section 5).
I also made a first step towards a micro-foundation of neutrality of structural
change by showing that neutral structural change can arise if preferences and
technologies are uncorrelated. Therefore, my neutrality index could also be
interpreted as an index of correlation between technology and preferences. In this
sense, my empirical findings imply that the correlation between preferences and
technologies is rather low. (Exactly speaking, the actual correlation was closer to
the extreme of “no correlation” than to the extreme of “maximal correlation”).
Note that I could try to assess the degree of correlation between preferences and
technologies in an alternative way: First I would have to estimate the price
210
elasticity of demand, the income elasticity of demand and the production functions
for all sectors and then I would have to try to somehow figure out the degree of
correlation between the estimated preference and technology parameters. This
approach would be problematic for two reasons:
(1) Estimation of preference parameters (and especially of income elasticity of
demand) is very difficult, since there are problems in measuring the changes in
quality of goods and services. Hence, it is difficult to isolate whether demand for a
good increased due to relatively high income-elasticity of demand or due to an
increase in quality of the service; see e.g. Ngai and Pissarides (2007).
(2) Even if I could measure the preference and technology parameters exactly there
would be a problem in defining a measure of correlation between preferences and
technologies, since we have actually two sorts of preference parameters (income
elasticity of demand and price elasticity of demand). Hence, if we have two
economies (A and B), which are identical except for their correlation between
income elasticity and technology and between price elasticity and technology, it
would be difficult to say in which economy the correlation between preferences
and technologies is lower: For example, if the correlation between income elasticity
and technology is slightly lower in country A in comparison to country B and if the
correlation between price elasticity and technology is slightly lower in country B in
comparison to country A, we could not say whether preferences and technologies
are more or less correlated in country A in comparison to country B. My approach
omits this problem by focusing on the factor reallocation across technology, which,
as modeled in my paper, reflects the degree of correlation between preferences and
technologies.
Furthermore, note that my empirical findings are valid for all the literature that
analyses structural change along PBGP’s (and where capital is included into
211
analysis): I have shown in Proposition 4 that every PBGP, that satisfies the Kaldor-
facts (exactly), must feature neutral structural change. Hence, we can say that the
papers by Kongsamut et al. (2001), Ngai and Pissarides (2007) and Foellmi and
Zweimueller (2008) are compatible with 55% of structural change observed.
Overall, my explanation for the Kuznets-Kaldor-puzzle is the following: There is a
certain degree of independency between technologies and preferences. As
discussed in the previous section, over the very long run such independency comes
from the assumption that the household’s consumption decisions are based on the
physical and chemical properties of the goods, but not on the capital-intensity (i.e.
households are not interested in the production process of the consumption goods
but only on the “taste” of the goods). If preferences and technologies are
uncorrelated (or independent), structural change patterns can arise that satisfy all
the empirical observations associated with the Kuznets-Kaldor-puzzle (especially
factors are reallocated across sectors that differ by capital intensity). I show that
this explanation is compatible with 55% of the structural change.
The remaining task is to answer the question why the remaining 45% of the
structural change are compatible with the Kuznets-Kaldor-Puzzle. One answer may
be that these 45% are quantitatively small hence their aggregate impact is relatively
low (in comparison to the other aggregate-growth determinants, e.g. technological
progress) at least at the level of stylized facts. In fact, this is implied by the paper
by Acemogly and Guerrieri (2008). However, there may be other explanations as
well. For example, the aggregate effect of these 45% of structural change may be
offset by the aggregate effects of other growth determinants, e.g. some sort of
“economy-wide technological progress” may have accelerated between 1948 and
1987 which would have offset the (negative) impacts of non-neutral structural
change. Further research could analyze this question in more detail. Furthermore, it
212
seems interesting to search for other micro-foundations of neutral structural
change: I explained the parameter restrictions, which are necessary for the
existence of neutral structural change, by uncorrelated preferences and
technologies; however, there are certainly other micro-foundations that can explain
these parameter restrictions.
213
APPENDIX A There are two approaches to solve my model, which are known from the literature
on the Ramsey-Cass-Koopmans model: (1) I can assume that there is a social
planer who maximizes the welfare of the representative household (“benevolent
dictator”); or (2) I can assume that there are many marginalistic households and
entrepreneurs who maximize their life-time utility and profits in perfect markets.
Both ways of solution lead to the same first order optimality conditions. I explain
approach (1) in short and focus on the approach (2).
APPROACH (1):
Necessary (first order) conditions for an optimum The benevolent dictator maximizes the utility function of the representative
household (equations (8)-(11)) subject to the equations (1)-(7) and (12)-(16).
The Hamiltonian for this control problem is given by:
( ) )(,..., 21 mmmHn hCKYCCCuHAM −−−+= δψ
where Hψ is the co-state variable.
The variables of this Hamiltonian are determined as follows:
iC are given by mihYC iii ≠∀−= , (cf. (5)),
iY are given by (1) and (2),
Z is given by (7)
mk is given by (cf. (3)), ∑≠
−=mi
im kk 1
ml is given by (cf. (3)), ∑≠
−=mi
im ll 1
mz is given by (cf. (3)) ∑≠
−=mi
im zz 1
214
Control variables are:
m
n
nmm
nmm
nmm
Chh
zzzzzkkkkk
lllll
,,...,,...,,...,,,...,,...,
,,...,,...,
1
1121
1121
1121
+−
+−
+−
K is state variable.
As explained in Section 3 of Chapter II, the first order optimality condition can be
derived by
• setting the first derivatives of the Hamiltonian with respect to the control
variables equal to zero
• setting the first derivative of the Hamiltonian with respect to the state
variable equal to HH ψρψ &− .
Then after some algebra, the first order optimality conditions (17)-(19) can be
obtained. Q.E.D. I omit this derivation, since it is trivial. See, e.g., the
APPENDIX of Chapter III for an example on how this can be done.
Proof that sufficient (second order) conditions are satisfied Note that the proof that the first order conditions are sufficient for an optimum is
quite difficult in this APPROACH (1). Especially the proof of concavity in Step 1
becomes quite “impossible” as we will see. Therefore, the following proof of
sufficiency of the optimality conditions may be regarded as incomplete. As we
will see, in APPROACH (2) this problem does not arise.
To prove the sufficiency of these necessary conditions I use the Arrow-Kurz-
criterion. For a description of this criterion see Section 3.2 in Chapter II. In the
following I apply the steps described there. Note that in the following I omitted
215
intermediates production, i.e. 0== μγ , for simplicity. (Analogous results can be
obtained with intermediate production.)
Step 1: Maximize the Hamiltonian with respect to the control variables for
given state variable, co-state variable and time.
In fact, this implies 0!=
∂∂
mCHAM and (17). The latter together with (1), (2), (3) and
(12) implies (21), (24) and (25).
From (9) and (17) I obtain
(A.1) ip
CC ii
mm
m
ii ∀+
−= θθωω
Inserting this equation into (12) yields
(A.2) m
mmCEω
θ−=
Inserting (A.1) and (A.2) into (9) yields after some algebra:
(A.3) ωω +−= nn pEu lnln(.)
where ∑+=
≡n
miin
1
ωω and ∑≡i
ii )ln(ωωω
and where I have obtained from (1), (2) and (17)
(A.4) νβ
αχνχ
νβ
χα
−
−⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛≡
m
mn Ll
KkABp /1
Equations (9) and (A.2) and condition 0!=
∂∂
mCHAM imply
(A.5) EH1
=ψ
Now note that we have just derived the first order conditions for a maximum.
These conditions are sufficient only if the Hamiltonian is jointly concave in the
216
control variables for given state variable, co-state variable and time. This requires
determining the signs of the first minors of the Hessian determinant of the
Hamiltonian (with respect to the control variables for given state variable, co-state
variable and time); see e.g. Chiang 1984, p.336. Since we have an arbitrary (and
large) number of state variables this becomes impossible (at least for me), due to
the difficulties in calculating determinants. (Note that sometimes these difficulties
do not arise if the Hessian is a diagonal matrix. However, in my model it is not.)
Therefore, Step 1 may be regarded as incomplete. I have not researched for a
solution of this problem, since, as mentioned above, the model can be solved by
using APPROACH (2). The proofs of sufficiency in APPROACH (2) are feasible
for me.
Step 2: Insert the optimality conditions from Step 1 into the Hamiltonian, in
order to obtain ),,(~~ timeKMAHMAH Hψ= !
Inserting (A.3) and (A.5) into the Hamiltonian yields:
(A.6) ⎟⎟⎠
⎞⎜⎜⎝
⎛−−++−⎟⎟
⎠
⎞⎜⎜⎝
⎛= KYptKMAH
HHnn
HH δ
ψψωω
ψψ 1ln1ln),,(~
where can be derived as (implicit) function of np Hψ by using equations (A.4),
(24), (21) and (A.5) (remember that in equation (24) H=0, due to (23) and
0== μγ ) and
where Y can be derived as (implicit) function of Hψ by using equations (25), (21),
(24) and (A.5) (again, remember that in equation (24) H=0, due to (23) and
0== μγ ).
217
Step 3: Show that is concave in K for given ),,(~~ tKMAHMAH Hψ= Hψ and
time, by showing that 0)(
),,(~2
2
<∂
∂K
tKMAH Hψ .
This step is quite lengthy and includes calculating implicit derivatives, but straight
forward. After some algebra it can be shown that
(A.7) 0)1(
1
)()(
),,(~2
22
2
<+
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+
−−=
∂∂ β
αβ
αχωψ
m
m
m
m
nH
lk
lkK
KtKMAH
This relation is true, since equation (24) implies that 0)(1 <>−m
m
lk
if 0)(<>−αχ
(remember that in equation (24) H=0, due to (23) and 0== μγ ).
Since we have accomplished all three steps, the Arrow-Kurz-criterion implies
that conditions (17)-(19) are sufficient for an optimum (together with the
transversality condition). Q.E.D. (Remember, however, that there are some
difficulties in Step 1, as explained there.)
APPROACH (2) As mentioned above in this section I assume that there exist many marginalistic
and identical households and producers. (Of course the producers are identical
within a sector, while they differ across sectors.) The assumption of marginalistic
agents implies that all agents consider the prices and factor prices as exogenous;
i.e. all agents are “price-takers”. The prices, factor prices and quantities are
determined by laws of (aggregate) demand and (aggregate) supply on the
corresponding markets (where market clearing is assumed).
218
This interpretation of the Ramsey-Cass-Koopmans model is suggested by Cass
(1965) and it is well known in the literature (see any book on growth economics,
e.g. Barro and Sala-i-Martin (2004), pp.86ff).
Remember, however, that although APPROACH (1) and APPROACH (2) interpret
my model in different ways, both approaches yield the same first order optimality
conditions (and results in general).
For simplicity I omit intermediates production in this section, i.e. I set 0== μγ .
(Analogous results can be obtained with intermediate production.)
Producers Since I have assumed that each sector is polypolistic and since there is perfect
mobility of factors across sectors, we know that the value of marginal factor-
productivity in each sector must be equal to the (economy-wide) factor-price, i.e.
(A.8) iwLl
Ypi
ii ∀=∂∂ ,
)(
(A.9) irKk
Ypi
ii ∀=∂∂ ,
)(
where is the real wage rate and w r is the real rate of return on capital; see also,
e.g., Kongsamut et al. (2001). These conditions can be obtained by maximizing the
sector-profit function { }KrkLwlYp iiii −− with respect to factor inputs and
, while sector demand, sector-price and factor-prices are exogenous. (That is,
the sector behaves like a price-taker; the reason for this fact is that all entrepreneurs
of the sector are price-takers. This fact could be proved by modeling explicitly each
sector as consisting of identical marginalistic profit-maximizing producers; then
conditions (A.8) and (A.9) could be obtained by calculating the first-order
Lli
Kki
219
conditions for profit-maximization of each individual producer and by aggregating
over all producers of a sector.)
We know that the wage rate and the rental rate of capital are equal across sectors
due to the following fact: differences in factor-prices across sectors are eliminated
instantly by cross-sector factor-migration due to the assumption of perfect cross-
sector factor-mobility.
Equations (A.8) and (A.9) imply jiLl
Yp
LlYp
j
jj
i
ii ,,
)()(∀
∂∂
=∂∂ and
jiKk
Yp
KkYp
j
jj
i
ii ,,
)()(∀
∂∂
=∂∂ . This in turn implies for mj = due to (13):
(A.10) iKkYKkY
LlYLlYp
ii
mm
ii
mmi ∀
∂∂∂∂
=∂∂∂∂
= ,)(/)(/
)(/)(/
which is part of optimality condition (17) (Q.E.D.).
Inserting (1) and (2) into (A.10) yields
(A.11) mipi ,...1,1 ==
(A.12) nmipLlKkABp n
m
mi ,...1,/1 +=≡⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
νβαχ
νχ
νβ
χα
Households In this section the index ι denotes the corresponding variable of the individual
household. For example, while E stands for consumption expenditures of the
whole economy, ιE stands for consumption expenditures of the household ι . I
assume that there is an arbitrary and large number of households ( x,...1=ι ),
sufficiently large to constitute marginalistic behavior of households. Hence, it
follows from equations (8)-(11) that each household has the following utility
220
(A.13) , ιριιι ∀= ∫∞
− ,),...(0
1 dteCCuU tn 0>ρ
where
(A.14) ι θ ωιιιι ∀⎥⎦
⎤⎢⎣
⎡−= ∏
=
,)(ln),...(1
1
n
iiin
iCCCu
(A.15) ∑ =
=m
ii
1
0ιθ
(A.16) ∑+=
=n
mii
1
0ιθ
Furthermore, each household has the following dynamic constraint:
(A.17) ιδ ιιι ∀−−+= ,)( EWrLwW&
where ιW is the wealth/assets of household ι , ιE are consumption expenditures of
household ι and L is the (exogenous) labor-supply of household ι . The latter
implies that each household supplies the same amount of labor at the market.
The dynamic constraint (A.17) is standard (compare for example Barro and Sala-i-
Martin (2004), p.88). It implies that the wealth of the household increases by labor-
income and by (net-) interest-rate-payments and decreases by consumption
expenditures.
Note that I assume that the labor supply of each household is exogenously
determined.
In line with (12), consumption expenditures of a household are given by
(A.18) ι ιι ∀= ∑ ,i
iiCpE
Each household maximizes its life-time-utility (A.13)-(A.16) subject to its dynamic
constraint (A.17). Since this optimization problem is time-separable (due to the
221
assumption of separable time-preference and marginalistic household), it can be
divided into two steps; see also, e.g., Foellmi and Zweimüller (2008), p.1320f:
1.) Intratemporal (static) optimization: For a given level of consumption-budget
( ιE ), the household optimizes the allocation of consumption-budget across goods.
2.) Intertemporal (dynamic) optimization: The household determines the optimal
allocation of consumption-budget across time.
Intratemporal optimization:
The household maximizes its instantaneous utility (A.14)-(A.16) subject to the
constraint (A.18), where it regards the consumption-budget ( ιE ) and prices ( ) as
exogenous. (Remember that the household is price-taker.) The corresponding
Lagrange-function is given by
ip
ιψθ ιιωιι ∀⎥⎦
⎤⎢⎣
⎡−−⎥
⎦
⎤⎢⎣
⎡−= ∑∏
=
,)(ln1 i
iiL
n
iii CpECLG i
where Lψ is the LaGrange-multiplier (shadow-price).
The first order necessary optimality conditions are given by
(A.19) ιψθ
ωιι ,,0 ip
C iLii
i ∀=−−
These conditions are also sufficient for an optimum (maximum), since the target
function is concave and the restriction linear. (The non-negativity constraints are
studied in the phase diagram in APPENDIX C.)
From (A.19) and (13), we have
(A.20) ιθθωω ι
ιιι ,, i
pCC i
i
mm
m
ii ∀+
−=
Inserting (A.20) into (A.18) yields
222
(A.21) ιω
θιι ∀
−= ,
m
mmCE
Intertemporal optimization
Inserting (A.11), (A.12), (A.20) and (A.21) into (A.14)-(A.16) yields after some
algebra:
(A.22) ιωωι ∀+−= ,lnln(.) nn pEu
where ∑+=
≡n
miin
1
ωω and ∑≡i
ii )ln(ωωω
Now, we have determined the instantaneous utility as function of consumption-
budget (and prices). (Remember that the household is price-taker, i.e. prices are
exogenous from the household’s point of view.) Inserting (A.22) into (A.13) yields
(A.23) ( ) ιωω ριι ∀+−= ∫∞
− ,lnln0
dtepEU tnn
Thus, the intertemporal optimization problem is to optimize (A.23) subject to the
dynamic constraint (A.17). This is a typical optimal control problem. The
Hamiltonian for this problem is as follows:
(A.24) [ ] ιδψωω ιιι ∀−−+++−= ,)(lnln EWrLwpEHAM Hnn
where Hψ is the co-state variable. ιE is control-variable and ιW is state variable.
The prices ( ) and factor prices ( w and np δ−r ) are regarded by the household as
exogenous (since the household is marginalistic and thus price-taker.) Remember
that L is exogenous. It may be confusing that is time varying (while being
regarded as exogenous in the optimal control problem of the household). However,
this fact does not prevent us from using the Hamiltonian, since the Hamiltonian
function allows in general that time enters the target function explicitly (i.e. via
np
223
exogenous “parameters”); see e.g. Gandolfo (1996), p.375, on a general
formulation of the control-problems that are solvable by using the Hamiltonian.
The first order optimality conditions are given by 0!=
∂∂
ιEHAM and
HHWHAM ρψψι +=∂∂
− &!
. These conditions imply (after some algebra) that
(A.25) ιρδι
ι
∀−−= ,rEE&
Note that this first order condition is also a sufficient condition for an optimum.
This can be immediately concluded from the Hamiltonian. Equation (A.25) has the
same concavity features as the Hamiltonian of the standard one-sector Ramsey-
Cass-Koopmans model. Especially, the target function is concave in the control-
variable and the restriction (i.e. the term within the squared brackets) is linear in
the control and the state variable. Therefore, we know that the Hamiltonian is
concave; therefore, the optimality conditions are sufficient. Q.E.D.
Relationship between individual variables and economy-wide aggregates Aggregate consumption expenditures are given by
(A.26) ∑∑ ==i
iiCpEE ι
ι
ι
where the following relation holds:
(A.27) iCC ii ∀=∑ ,ι
ι
There is no unemployment, i.e.
(A.28) ∑=ι
LL
Last not least, since the wealth/assets can only be invested in production-capital
(K), the following relation must be true
224
(A.29) ∑=ι
ιWK
(see also, e.g. Barro and Sala-i-Martin (2004), p.97). That is, all assets are invested
in capital (capital-market-clearing).
Furthermore, the “subsistence needs” of the whole economy are simply equal to the
sum of the subsistence needs of its individuals, i.e.
(A.30) iii ∀=∑ ,ι
ιθθ
Equation (A.20), (A.27) and (A.30) imply
(A.31) ip
CC ii
mm
m
ii ∀+
−= ,θθωω
This equation corresponds to equation (18). Q.E.D. Exactly speaking, inserting
(9) into (18) yields (A.31).
(A.25) and (A.26) imply
(A.32) ρδ −−= rEE&
This equation corresponds to equation (19). Q.E.D. Exactly speaking, (19) can
be transformed into (A.32) by using (1), (2), (8)-(12), (17) and (18).
225
APPENDIX B Equations (20) to (29) are relevant for aggregate analysis. Now let us search, like in
the “normal” Ramsey model, for a growth path where E and K grow at constant
rate, i.e.
(B.1) EgEE !=
&
(B.2) KgKK !=
&
Equations (B.1) and (22) imply that
(B.3) .constLL
ll
KK
kk
m
m
m
m =−−+&&&&
Requirement (B.3) and equation (21) imply that
(B.4) .~~
constYY=
&
(B.2) and (B.3) imply
(B.5) ( ) .// constlklkmm
mm =&
(B.1), (B.2) and (B.4) imply
(B.6) ( ) .~/
~/ constYEYE
=&
and ( ) .~/
~/ constYKYK
=&
Equations (B.2), (20) and (25) imply
(B.7) YKg
YH
YE
klcc K
m
m ~)(~~65 δ+++=+
226
Solving equation (24) for YH~ and inserting it into equation (B.7) yields after some
algebra:
(B.8) YKg
kl
cc
YE
cc
cc K
m
m ~)(1~11
46
4
3
45 δ++⎟⎟
⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=−
Remember that and Kgcccc ,,,, 6543 δ are constants. Furthermore, note that
(B.5) and (B.6) imply that m
m
kl
YE ,~ and
YK~ grow at constant rate. Hence, equation
(B.8) can be satisfied at any point of time only if m
m
kl
YE ,~ and
YK~ are constant (i.e.
they grow at the constant rate zero), i.e.
(B.9) ..,~ constklconst
YE
m
m == , .~ constYK=
Equations (B.9), (23), (25) imply
(B.10) .~~
constHH
YY
KK
YY
EE
=====&&&&&
Q.E.D.
Let denote the constant growth rate from equation (B.10). Hence, (B.9), (B.10),
(21) and (26) imply
*g
(B.11) Lgq
GG
g +−
=1
*
&
Inserting equations (27) and (28) into equation (B.11) yields after some algebra:
(B.12) LBA gggg +
+−+−
=χεγαεμεγεμ
)1()1(*
Q.E.D.
227
Note that in all the calculations from above I searched for an equilibrium growth
path where E and K grow at constant rate. As a result I obtained that H grows at
constant rate along this growth path. Hence, I can treat H like exogenous
technological progress along this growth path. Let HYY −≡ˆ . In this case equation
(20) can be written as follows:
(B.13) EKKY ++= δ&ˆ
Q.E.D.
Inserting equations (21), (23) and (25) into HYY −≡ˆ yields:
(B.14) qq KLGY −= 1~ˆ
where ⎟⎟⎠
⎞⎜⎜⎝
⎛−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛≡
m
m
q
m
m
klcccc
lkGG )(~
2615 γγ grows at constant positive rate due to
(B.9) (G~ grows at constant positive rate). Q.E.D.
Inserting equation (B.14) into equation (22) yields:
(B.15) ρδλ −−=KY
EE ˆ&
where ⎟⎟⎠
⎞⎜⎜⎝
⎛−+−≡
m
m
m
m
klcccc
kl )( 2615 γγβλ is constant due to (B.9). Q.E.D.
Equations (B.14) and (B.15) include the term . This term is constant along
the equilibrium growth path and can be derived as function of model parameters by
setting equation (22) equal to and solving afterwards for
mm kl /
*g mm kl / :
(B.16) KLGg
lk q
q
m
m −−
⎟⎟⎠
⎞⎜⎜⎝
⎛ ++= 1
11
1*
βρδ
228
Note that the term LG q−11
is a function of exogenous parameters and grows at rate
(see equation (B.11) for ). K grows at rate along the equilibrium growth
path as well (see Lemma 1). Hence, the term
*g *g *g
KLG q−1
1
is constant along the
equilibrium growth path, so that I can rewrite equation (B.16) in terms of initial
values of exogenous parameters (the index zero denotes the initial value of the
corresponding variable):
(B.17) 0
011
0
11
*
)(KLGg
lk q
q
m
m −−
⎟⎟⎠
⎞⎜⎜⎝
⎛ ++=
βρδ
where q, and are given by equations (27), (28) and (B.12). Q.E.D. 0G *g
I have shown now that along an equilibrium growth path where E and K grow at
constant rate H grows at constant rate as well and is constant. When this
fact is taken into account, the economy in aggregates is represented by equations
(B.13)-(B.15). These equations are similar to the Ramsey-model regarding all
relevant features; hence, they imply that this equilibrium growth path exists and is
unique. Q.E.D.
mm lk /
229
APPENDIX C First, I show by using linear approximation that the saddle-path-feature of the
PBGP is given (Proposition 1a). Then I prove local stability by using a phase
diagram (Proposition 1b).
Existence of a saddle-path (Proposition 1a) First I rearrange the aggregate equation system (20)-(29) as follows:
(C.1) Kq
ggEK
kl
kl
K GL
q
m
m
q
m
m ˆ)1
(ˆˆ)(ˆ−
++−−+⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
δβα&
(C.2) q
ggK
kl
EE G
Lq
q
m
m
−−−−−⎟⎟
⎠
⎞⎜⎜⎝
⎛= −
−
1ˆ
ˆˆ
11
ρδβ&
(C.3)
αχεγεμ
ωαβ
ανχββνεγεμ
+−
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−+−
=
−
1
ˆ
ˆ1 q
m
mq
n
m
m klK
E
kl
where aggregate variables are expressed in “labor-efficiency units”, i.e. they are
divided by qLG −11
; hence qLG
KK−
≡1
1ˆ and
qLG
EE−
≡1
1ˆ . Furthermore, is the
growth rate of G given by (28) and
Gg
∑+=
≡n
miin
1ωω .
These equations imply that and have the following values along the
PBGP
EK ˆ,ˆmm kl /
(C.4) *
11
*ˆ⎟⎟⎠
⎞⎜⎜⎝
⎛= −
m
mq
kl
K σ
(C.5) *
11
1*ˆ⎟⎟⎠
⎞⎜⎜⎝
⎛+= −−
m
mqqq
kl
E ρσασ
230
(C.6) σ
βρωανχβεμαχγα
ωανχβεμβνγββα
n
n
m
m
kl
)()(
)()(*
−+−+
−−−+=⎟⎟
⎠
⎞⎜⎜⎝
⎛
where
qg
g GL −+++
≡
1ρδ
βσ
where an asterisk denotes the PBGP-value of the corresponding variable.
The proof of local saddle-path-stability of the PBGP is analogous to the proof by
Acemoglu and Guerrieri (2008) (see there for details and see also Acemoglu
(2009), pp. 269-273, 926).
First, I have to show that the determinant of the Jacobian of the differential
equation system (C.1)-(C.2) (where is given by equation (C.3)) is different
from zero when evaluated at the PBGP (i.e. for
mm kl /
*** ,ˆ,ˆ
⎟⎟⎠
⎞⎜⎜⎝
⎛
m
m
kl
EK from equations
(C.4)-(C.6)). This implies that this differential equation system is hyperbolic and
can be linearly approximated around *
** ,ˆ,ˆ⎟⎟⎠
⎞⎜⎜⎝
⎛
m
m
kl
EK (Grobman-Hartman-Theorem;
see as well Acemoglu (2009), p. 926, and Acemoglu and Guerrieri (2008)). The
determinant of the Jacobian is given by:
(C.7) EK
KE
EE
KK
EE
KE
EK
KK
J ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
∂∂
∂∂
−∂∂
∂∂
=
∂∂
∂∂
∂∂
∂∂
=&&&&
&&
&&
The derivatives of equations (C.1)-(C.2) are given by:
231
(C.8)
Ekl
kl
qKE
qg
gKkl
EE
Kkl
kl
Kqkl
KqEKE
Ekl
kl
qkl
qKEK
qg
gKkl
kl
qkl
qK
kl
kl
KqKK
m
mq
m
mq
GL
m
m
m
mq
m
mqq
m
mq
m
mq
m
m
q
m
mq
GL
m
mq
m
m
q
m
mq
q
m
m
q
m
mq
ˆ)1(ˆˆ
1ˆ
ˆˆ
ˆˆ)1(ˆ)1(ˆ
ˆˆ
1ˆ)1(ˆˆˆ
1ˆ)1(ˆ
ˆˆˆ
1
11
11
2
1
1
11
∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−−−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
∂∂
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
∂∂
−∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−
++−∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=
∂∂
−
−
−
−
−
−
−
−
−−−
−−−
−−
−
β
ρδβ
β
βα
δβα
βα
&
&
&
&
where the derivatives of equation (C.3) are given by
(C.9)
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎠⎞
⎜⎝⎛ −++⎟
⎠⎞
⎜⎝⎛ −+
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
=∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎠⎞
⎜⎝⎛ −++⎟
⎠⎞
⎜⎝⎛ −+
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−=∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
−
−
−
+
−
−
q
m
mq
n
q
m
mq
n
m
m
q
m
mq
n
q
q
m
m
nm
m
klK
Eq
klK
Eq
Kkl
klK
Eq
Kkl
Ekl
1
1
1
1
1
ˆ
ˆ111
ˆ
ˆ
ˆ
ˆ
ˆ111
ˆˆ
ωαβ
ανχβααμχγε
ααμχγε
ωαβ
ανχβ
ωαβ
ανχβααμχγε
ααμχγε
ωαβ
ανχβ
232
Inserting the derivatives (C.8) and (C.9) into (C.7) and inserting the PBGP-values
from equations (C.4)-(C.6) yields after some algebra the following value of the
determinant of the Jacobian evaluated at the PBGP:
(C.10) ( )
q
m
mq
n
nn
klK
Eq
KEq
J
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
⎥⎦
⎤⎢⎣
⎡−
+−−−=
1**
*
*
*
*
)ˆ(
ˆ)(ˆ
ˆ)1(
ωβανχβα
σβ
ωανχβαρωανχβ
where ( ) 0>−+≡ αμχγεαα and q is given by equation (27).
This equation can be transformed further by using equations (27) and (C.4)-(C.6):
(C.11) ( )[ ]
αβ
βαα
ωανχββσα
2*
*
*
* ˆˆ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
−−−=
m
m
n
kl
KE
J
where ( ) 0>−+≡ βμνγεββ . Note that *
*
ˆˆ
KE and
*
⎟⎟⎠
⎞⎜⎜⎝
⎛
m
m
kl are positive and are given
by equations (C.4)-(C.6). Furthermore, note that following relations, which are
useful for deriving equation (C.11), are true: εμγγβα )(1 −+−=+ , βα
β+
=q
(from (27)) and σ
βρωανχβα
ωανχβββα
n
n
m
m
kl
)(
)(*
−+
−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ (from (C.6)).
We can see that the determinant evaluated at PBGP is different form zero. Hence,
the PBGP is hyperbolic. Furthermore, equations (C.10) and (C.11) imply that
0* <J . (Equation (C.10) implies that 0* <J , if 0>−ανχβ ; equation (C.11)
implies that 0* <J , if 0<−ανχβ as well.)
Our differential equation system consists of two differential equations ((C.1) and
(C.2)) and of two variables ( E and K ), where we have one state and one control-
233
variable. Hence, saddle-path-stability of the PBGP requires that there exist one
negative (and one positive) eigenvalue of the differential equation system when
evaluated at PBGP (see also Acemoglu and Guerrieri (2008) and Acemoglu (2009),
pp. 269-273). Since 0* <J we can be sure that this is the case. ( 0* <J can exist
only if one eigenvalue is positive and the other eigenvalue is negative. If both
eigenvalues were negative or if both eigenvalues were positive, the determinant
*J would be positive.) Therefore, the PBGP is locally saddle-path-stable, i.e.
Proposition 1a is proved. Q.E.D.
Local stability (Proposition 1b)
In the following, I omit intermediates for simplicity, i.e. I set 0== μγ .
Furthermore, I study here only the case where output-elasticity of capital in
investment goods industries (i=m) is relatively low in comparison to the output-
elasticity of capital in the consumption goods industries ( mi ≠∀ ), i.e. I assume
αχ < . This is consistent with the empirical evidence presented and discussed in
Valentinyi and Herrendorf (2008) (see there especially p.826). Note, however, that
the qualitative stability results for the other case (i.e. αχ > ) are the same.
To show the stability-features of the PBGP, the three-dimensional system (C.1)-
(C.3) has to be transformed into a two dimensional system, in order to allow me
using a phase-diagram. By defining the variable m
m
lkK
≡κ , the system (C.1)-(C.3)
can be reformulated as follows (after some algebra):
(C.12) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
+++−= −
βρδβκ β
1ˆˆ
1 GL
gg
EE&
234
(C.13)
β
ββ
κω
βχα
ρκωαβ
χακβ
δκ
κκ
E
Egg
n
nG
L
ˆ1
1ˆ
)1
( 11
−+
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−
−++−
=
−−
&
where ∑+=
≡n
miin
1ωω
I can focus attention on showing that the stationary point of this differential
equation system is stable: The discussion in APPENDIX B implies that κ and E
are jointly in steady state only if K , E and are jointly in steady state and
that
mm lk /
K , E and are jointly in steady state only if mm lk / κ and E are jointly in
steady state. Therefore, the proof of stability of the stationary point of system
(C.12)-(C.13) implies stability of the stationary point of system (C.1)-(C.3). Hence,
in the following I will prove stability of the stationary point of system (C.12)-
(C.13).
It follows from equations (C.12) and (C.13) that the steady-state-loci of the two
variables are given by
(C.12a)
β
βρδ
βκ
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−+++
==
11
*
1
:0ˆˆ
GL
ggEE&
(C.13a) κκω
αβχαρ
βδκ
κκ
β
β
κ−
−
= −−
−++−
==1
1
0
1
)1
(ˆ:0
n
GL
gg
E &
&
Now, I could depict the differential equation system (C.12)-(C.13) in the phase
space ( ). Before doing so, I show that not the whole phase space ( ) is
economically meaningful. The economically meaningful phase-space is restricted
by three curves ( ), as shown in the following figure and as derived below:
κ,E κ,E
321 ,, tt RRR
235
Figure C.1: Relevant space of the phase diagram
κ
E
1R
30=tR
20=tR
Only the space below the 1R -line is economically meaningful, since the
employment-share of at least one sub-sector i is negative in the space above the
1R -line. This can be seen from the following fact:
It follows from equations (1), (2), (3) and (17) after some algebra that
(C.14) ∑+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
n
mii
m
m lkl
11
χβανχβ
Note that αχανχβ −=− when 0== μγ .
Since, cannot be negative (hence , ) this equation implies that il 101
≤≤ ∑+=
n
miil
(C.15) χβαν
<m
m
kl
Inserting equation (24) into this relation yields
(C.16) βκωχ
α
n
ER 1ˆ:1 <
236
(remember that in equation (24) H=0, due to (23) and 0== μγ ).
Hence, the space above 1R is not feasible. When the economy reaches a point on
1R , no labor is used in sub-sectors i=1,…m. If I impose Inada-conditions on the
production functions, as usual, this means that the output of sub-sectors i=1,…m is
equal to zero, which means that the consumption of these sectors is equal to zero.
Our utility function implies that life-time utility is infinitely negative in this case.
Hence, the household prefers not to be at the 1R -curve. Note that actually the 1R -
curve is only an outer limit: Since we have existence-minima in our utility
function, the utility function becomes infinitely negative when the consumption of
one of these goods falls below its subsistence level. Hence, even when the
consumption of all goods is positive, it may be the case that the utility function is
infinitely negative due to violation of some existence minima. Therefore, the actual
constraint is somewhere below the 1R -curve. However, this fact does not change
the qualitative results of the stability analysis.
Now I turn to the and -curves. I have to take account of the non-negativity-
constraints on consumption (
2tR 3
tR
iCi ∀> 0 ), since our Stone-Geary-type utility
function can give rise to negative consumption. By using equations (A.1), (A.2)
(A.11) and (A.12) from APPENDIX A and equations (27) and (28) the non-
negativity-constraints ( iCi ∀> 0 ) can be transformed as follows (remember that I
assume here 0== μγ ):
(C.17) miLA
Ei
i ,...11ˆ1 =
−>
αωθ
(C.18) nmiLBA
E vi
i ,...111ˆ2 +=⎟⎟
⎠
⎞⎜⎜⎝
⎛−> −− β
αβν
χ
κνβ
χβαν
ωθ
237
This set of constraints implies that at any point of time only two constraints are
binding, namely those with respectively the largest i
i
ωθ− . Hence, the set (C.17),
(C.18) can be reduced to the following set:
(C.19) αω
θ1
2 1ˆ:LA
ERj
jt
−>
where mii
i
j
j ,...1=−>
−ωθ
ωθ
and mj ≤≤1 .
(C.20) βαβν
χ
κνβ
χβαν
ωθ
−−⎟⎟⎠
⎞⎜⎜⎝
⎛−> v
x
xt
LBAER 11ˆ: 2
3
where nmii
i
x
x ,...1+=−
>−
ωθ
ωθ
and nxm ≤≤+1
These constraints are time-dependent. It depends upon the parameter setting
whether or whether is binding at a point of time. In Figure C.1 I have
depicted examples for these constraints for the initial state of the system. Only the
space above the constraints is economically meaningful, since below the
constraints the consumption of at least one good is negative. Last not least, note
that equations (C.19)/(C.20) imply that the -curve and the -curve converge to
the axes of the phase-diagram as time approaches infinity.
2tR 3
tR
2tR 3
tR
Now, I depict the differential equation system (C.12)-(C.13) in the phase space
( ). κ,E
238
Figure C.2: The differential equation system (C.12)-(C.13) in the phase-space for
βδχαρω
αβ
−++
<−
1
1)( G
Ln
gg
κ
E
*κ
0=κ& 0ˆ =E&
S
saddle-path
κ T poleκ
0=κ&
0κ
R
Note that I have depicted here only the relevant (or: binding) parts of the
restriction-set of Figure C.1 as a bold line R.
As we can see, the 0=κ& -locus has a pole at β
χαρωαβκ
−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=1
1
)(n
pole .
The phase diagram implies that there must be a saddle-path along which the system
converges to the stationary point S (where S is actually the PBGP). The length of
the saddle-path is restricted by the restrictions of the meaningful space
(bold line). In other words, only if the initial
321 ,, tt RRR
κ ( 0κ ) is somewhere between 0κ and
239
κ 9, the economy can be on the saddle-path. Therefore, the system can be only
locally saddle-path stable. Now, I have to show that the system will be on the
saddle-path if κκκ << 00 . Furthermore, I have to discuss what happens if 0κ is
not within this range.
Every trajectory, which starts above the saddle-path or left from 0κ , reaches the
1R -curve in finite time. As discussed above, the life-time utility becomes infinitely
negative if the household reaches the 1R -curve. These arguments imply that the
representative household will never choose to start above the saddle path if
κκκ << 00 , since all the trajectories above the saddle-path lead to a state where
life-time-utility is infinitely negative.
Furthermore, all initial points that are situated below the saddle-path or right from
κ converge to the point T. If the system reaches one of the constraints ( )
during this convergence process, it moves along the binding constraint towards T.
However, the transversality condition is violated in T. Therefore, T is not an
equilibrium. To see that the transversality condition is violated in T consider the
following facts: The transversality condition is given by , where
32 , tt RR
0lim >−
∞→
t
tKe ρψ ψ
is the costate variable in the Hamiltonian function (shadow-price of capital; see
also APPENDIX A). By using the equations from APPENDIX A this transversality
condition can be reformulated such that we obtain: 01
lim 1 >−
−−−−
→∞ βδβκ β G
Lt
gg ,
which is equivalent to:
β
βδ
βκ
−
∞→
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−++
<
11
1
limG
Lt gg
. However, equation (C.13a)
9 Note that κ must be somwhat smaller than depicted in this diagram, since, as discussed above,
1R -curve is only an „outer limit“.
240
implies that in point T in Figure C.2
β
βδ
κ
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−++
=
11
1
1G
Lgg
. Hence, the
transversality condition is violated if the system converges to point T.
Overall, we know that, if κκκ << 00 , the household always decides to be on the
saddle-path. Hence, we know that for κκκ << 00 the economy converges to the
PBGP. In this sense, the PBGP is locally stable (within the range κκκ << 00 ).
If the initial capital is to small ( 00 κκ < ), the economy converges to a state where
some existence minima are not satisfied (curve 1R ) and thus utility becomes
infinitely negative. This may be interpreted as a development trap. For example,
Malthusian theories imply that in this case some part of the population dies, which
would yield an increase in per-capita-capital (and hence an increase in 0κ ).
On the other hand, if initial capital-level is too large ( κκ >0 ), all trajectories
violate the transversality condition. Therefore, in this case, the representative
household must waste a part of its initial capital to come into the feasible area
( κκκ << 00 ).
Furthermore, note that there are always some happenings that reduce the capital
stock, e.g. wars (like the Second World War) or natural catastrophes. These
happenings could shift the economy into feasible space ( κκκ << 00 ). These
thoughts could be analyzed further in order to develop a theory that the Second
World War is the reason for the fact that many economies satisfy the Kaldor-facts
today.
241
The alternative is to assume that the transversality condition needs not to hold
necessarily. In this case the point T would be an equilibrium. All economies, that
start at κκ >0 , would converge to this equilibrium. However, I have no idea of
how I could omit the transversality condition. We know that the transversality
condition implies that the value of capital is not allowed to be negative at the
household’s death (at infinity). In the actual model, there seems to be no adequate
theory of allowing for the violation of the transversality condition.
Note that Figure C.2 depicts the phase diagram for parameter constellations, which
satisfy the condition
βδχαρω
αβ
−++
<−
1
1)( G
Ln
gg. For parameter constellations,
which satisfy the condition
βδχαρω
αβ
−++
>−
1
1)( G
Ln
gg, the discussion and the
qualitative results are nearly the same. The only difference is that the 0=κ& -locus
is humpshaped (concave) for poleκκ < . However, all the qualitative results remain
the same (local stability of PBGP for some range κκκ << 00 and “infeasibility”
for 00 κκ < and κκ >0 ). Q.E.D.
242
APPENDIX D It follows from the optimality condition (18) that
(D.1) ipEC i
in
ii
ii ∀+=
∑=
θω
ω
1
For the sake of simplicity I consider only the non-homotheticity between the
services sector and the conglomerate of the agriculture and manufacturing sector.
Inserting equation (D.1) into equations (15) yields (remember equation (10)):
(D.2) 21.. dEdE managr +=+
(D.3) 43. dEdEser +=
where ... seragrmanagr EEE +=+
∑
∑
=
=≡ n
ii
s
ii
d
1
11
ω
ω, , ∑
+=
≡s
miipd
12 θ
∑
∑
=
+=≡ n
ii
n
sii
d
1
13
ω
ω and
. Note that p is given by ∑+=
≡n
siipd
14 θ
)(/)(/
LlYLlYp
nn
mm
∂∂∂∂
= and stands for the relative
price of sub-sectors . nmi ,...1+=
If preferences are non-homothetic across sectors consumption expenditures on
agriculture and manufacturing ( ) do not grow at the same rate as
consumption expenditures on services ( ), when treating relative prices as
constants. Hence, I have to show that and do not grow at the same
rate when treating - as constants. It follows from equations (D.2) and (D.3)
that when treating - as constants the following equations are true
.. managrE +
.serE
.. managrE + .serE
1d 4d
1d 4d
(D.4) E
ddE
EEE
managr
managr
1
2..
..
1
1
+=
+
+&&
243
(D.5) E
ddE
EEE
ser
ser
3
4.
.
1
1
+=&&
which shows that and do not grow at the same rate when treating -
as constants, i.e., preferences are non-homothetic between the services sector
and the conglomerate of the agriculture sector. In the same way it can be shown
that preferences are non-homothetic between the manufacturing sector and the
agriculture sector. Q.E.D.
.. managrE + .serE 1d
4d
244
APPENDIX E The optimality condition (17) implies after some algebra that
(E.1) ipHh
iii ∀= ,ε
Hence,
(E.2) i
j
j
i
j
i
pp
hh
εε
= for sai ,...1+= and nsj ,...1+=
In equation (E.2) i stands for the manufacturing sector and j for the services sector.
Let us now take a look at an arbitrary producer of the manufacturing sector, e.g. the
producer i = 3, where a+1<3 < s. I rewrite equation (E.2) as follows to show the
viewpoint of “producer 3”:
(E.3) 3
33
pp
hh j
jj εε
= for nsj ,...1+=
From the view point of “producer 3” equation (E.3) determines the ratio between
the input of own intermediates (i.e. the amount of intermediates that is produced by
“producer 3” and used by “producer 3”) and input of services-sector-produced
intermediates (i.e. the amount of intermediates that is produced by “producer j”
from the services sector and used by “producer 3”). (Remember that and
enter the production function of “producer 3” via equations (1) and (7).) Hence, for
example, a decrease in
3h jh
jhh3 means that “producer 3” increases the input of
producer-j-intermediates relatively more strongly than the input of own
intermediates, i.e. “producer 3” substitutes own intermediate inputs by external
intermediate inputs, i.e. “producer 3” outsources additional intermediates
production to producer j. Therefore, I can conclude from equation (E.3) that
245
“producer 3” outsources more and more to “producer j” (i.e. jh
h3 decreases),
provided that 03
3 <−pp
pp
j
j && (i.e. provided that the price for the good j in terms of
the good 3 )(3p
p j decreases; or in other words: provided that the output of “producer
j” becomes cheaper and cheaper (or less and less expensive) in comparison to the
output of “producer 3”).
From this discussion and from equation (E.2) I can conclude the following:
manufacturing-sector-producers ( sai ,...1+= ) shift more and more intermediates
production to services-sector-producers ( nsj ,...1+= ), i.e. j
i
hh decreases, provided
that services-sector-production becomes cheaper and cheaper (or less and less
expensive) in comparison to manufacturing-production, i.e. provided that
0<−i
i
j
j
pp
pp &&
, and vice versa. Q.E.D.
Note that relative prices are determined by exogenous parameters. Hence, which
producers outsource and whether outsourcing from manufacturing to services
increases (or the other way around) depends on the parameterization of the model.
In general both cases are possible. By using optimality condition (17) the relative
prices can be calculated, so that I can reformulate equation (E.2) after some algebra
as follows
(E.4) μγμγϖνβ
ϑχα
εε −
−+−
⎟⎟⎠
⎞⎜⎜⎝
⎛= D
LK
lk
BA
hh
m
m
j
i
j
i
)(1 for mai ,...1+= and
nsj ,...1+=
246
(E.5) j
i
j
i
hh
εε
= for smi ,...1+= and nsj ,...1+=
where μν
γμ
χα
βν
χαϑ ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ,
γεμγενββϖ−−+
−−≡
)(1)( and
εμεγε
εμν
εχγαμ
χβαν
αχγ
−−−
= ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ∏
)1(11
1
n
ii
i
ABAD .
From equation (E.5) we can see that some of the manufacturing sector producers
(i.e. producers ) do not change their outsourcing behavior (i.e. these
producers keep their ratio of external to own intermediates production (
smi ,...1+=
j
i
hh
)
constant). Equation (E.4) implies that the rest of the manufacturing sector
producers (i.e. the producers mai ,...1+= ) change their outsourcing behavior.
Calculating the growth rate of equation (E.4) yields (remember Lemma 1):
(E.6) DDggg
hhhh
BAji
ji&&
)())((/
)/( * μγμγϖνβ −+−+−+−= for
and
mai ,...1+=
nsj ,...1+=
ϖ , and D are positive. I omit here a detailed discussion of *g D /&ji
ji
hhhh
/)/( &
, since
it is less relevant for my purposes. The only important thing is that ji
ji
hhhh
/)/( &
can be
positive (e.g. if 0,0 >−>− νβBA gg and 0>− μγ ) or negative (e.g. if
0,0 <−<− νβBA gg and 0<− μγ ) depending on the parameterization of the
model. Hence, the intermediates-production may be shifted from manufacturing to
services or the other way around, depending on the parameterization of the model.
Q.E.D.
247
APPENDIX F It is well known that balanced growth requires either labor-augmenting
technological progress (or production function(s) of type Cobb-Douglas.)
Furthermore, a standard assumption in macroeconomic models is that the
production function has constant returns to scale. (Later, we will see that the
aggregate production function has the same structure as the sectoral production
functions.) Since I want to reassess the standard growth theory I do not depart from
these assumptions. Therefore, I assume now that sectoral production functions are
given by:
(1)’ ( ) niLflBY iiiii ,...1 =∀Ω=
where
(26)’ niLBlKk
ii
ii ,...1 =∀≡Ω
iB stands for the level of sector-specific and labor augmenting technological
progress; is a sector-specific function of )( iif Ω iΩ ; it is the intensive form of a
“standard” constant returns to scale function, where in this appendix denotes
the capital-to-labor ratio in efficiency units in sector i.
iΩ
The sectoral growth rates of labor-augmenting technological progress ( ) are
constant, i.e. . The following equations remain the same as in the
previous discussion:
ig
igBB iii ∀=/&
(3)’ ∑ =i
ik 1
(3)’’ ∑ =i
il 1
(12)’ ∑≡i
iiYpY
248
I still assume that sector m is numéraire (m<n) (although I do not make here any
assumptions about which sector produces capital). Hence, equation (13) holds.
When labor and capital are mobile across sectors and markets are polypolistic the
following efficiency conditions must be true:
(17)’ jiLlYLlY
KkYKkY
pp
ii
jj
ii
jj
j
i ,)(/)(/
)(/)(/
∀∂∂
∂∂=
∂∂
∂∂=
(32)’ iKkYpr iii ∀∂∂=+ )(/δ
Note, that I do not make here any assumption about the household behavior. The
assumptions above are sufficient to derive Proposition 4.
The capital share of income in sector i (or: the elasticity of capital with respect to
output in sector i) is given by:
(F.1) ( ))()()(
ii
iii
i
iiiii f
fY
KkKkYΩΩ′
Ω=∂∂
≡Ωκ
where i
iiii
ffΩ∂Ω∂
≡Ω′)()( .
By inserting equations (1)’, (26)’ and (13) into equation (32)’ I obtain:
(F.2) )( mmfr κδ ′=+
Inserting first equations (1)’ and (26)’ into equation (17)’ and then inserting
equation (F.1) into this term yields:
(F.3) ( )( )
( )( ) i
lk
lk
mm
mm
m
m
ii
ii
i
i ∀ΩΩ−
=ΩΩ−
κκ
κκ 11
Solving this term for and inserting it into equation (3)’ yields (remember that ik
11
11
−−
=− ii
i
κκκ and ): ∑ =
iil 1
(F.4) ( )( ) ( )
1
11
1−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
Ω−=
ΩΩ− ∑
i ii
i
mm
mm
m
m llk
κκκ
Equations (13) and (17)’ imply:
249
(F.5) ( )( ) i
ffBffBp
iiiiii
mmmmmmi ∀
Ω′Ω−ΩΩ′Ω−Ω
=)()(
)()(
Inserting equations (1)’, (F.1) and (F.5) into equation (12)’ yields:
(F.6) ( )∑ −Ω−Ω=
i ii
immmmm
lLfBY)(1
)(1)(κκ
κ
Inserting equation (F.4) into equation (F.6) yields equation
(F.7) [ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛Ω−+Ω
ΩΩ
=m
mmtmmm
m
mm
lkf
KY )(1)()( κκ
Definition F.1: A PBGP is a growth path where KY and
YrK are constant.
Definition F.1 is consistent with Definition 1 (and with the Kaldor facts). In fact
both definitions yield the same equilibrium growth path (but Definition 1 is
stronger than necessary). However, now I use Definition F.1 in order to
demonstrate that the necessary condition for the PBGP is independent of the
numéraire. (Remember that, since KY and
YrK are ratios, they are always the same
irrespective of the choice of the numérarire.)
Lemma F.1: A necessary condition for the existence of a PBGP (according to
Definition F.1) is or equivalently ./ constkl mm = .)(1
constli ii
i =Ω−∑ κ
.
Proof: Definition F.1 requires that KY and
YrK are constant; hence r must be
constant; hence must be constant (due to equation (F.2)). Due to equation
(F.7), and
mΩ
.constm =Ω .constKY
= require ./ constkl mm = . and
require
./ constkl mm =
.constm =Ω .)(1
constli ii
i =Ω−∑ κ
(due to equation (F.4)). Note that
iii λκ =Ω− )(1 , since I assume that there are only two production factors capital
250
and labor. ( iλ stands for the output-elasticity of labor in sector i or equivalently for
the labor-income share in sector i.) Q.E.D.
251
LIST OF SYMBOLS of PART I of CHAPTER V * Denotes the PBGP-value of the corresponding variable.
A Parameter indicating technology/productivity level of
technology 1. (exogenous)
B Parameter indicating technology/productivity level of
technology 2. (exogenous)
iB Parameter indicating technology/productivity-level in sector i.
(exogenous)
iC Consumption of subsector-i-output; indicates how much of the
output of subsector i is consumed.
D Auxiliary parameter. (Function of exogenous model-
parameters.)
E Aggregate consumption expenditures; index of overall
consumption-expenditures of the representative household
E E in “efficiency units”.
.agrE Consumption-expenditures on agricultural goods.
.. managrE + Consumption-expenditures on agricultural and manufacturing
goods.
.manE Consumption-expenditures on manufacturing goods.
.serE Consumption-expenditures on services.
G Auxiliary parameter. (Function of exogenous model-
parameters.)
G~ Auxiliary variable. (Function of other model-variables.)
0G Level of G at the initial point of time of the model.
252
H Aggregate intermediate output; index of the value of all
intermediates produced in the economy
HAM Hamiltonian function.
MAH~ Maximum of the Hamiltonian function with respect to the
control-variables, given the state variable, co-state variable and
time.
NI Weighting factor between NEUTRAL and maximally non-
neutral structural change, i.e. it indicates whether structural
change was rather neutral or non-neutral.
J Determinant of the Jacobian matrix.
K Aggregate capital; i.e. the amount of capital that is used for
production in the whole economy.
K K in “efficiency units”.
0K Level of K at the initial point of time of the model.
L Aggregate labor; i.e. the “amount” of labor that is used for
production in the whole economy. (exogenous)
L Average labor supply. (exogenous)
0L Level of L at the initial point of time of the model. (exogenous)
LG Lagrange function.
1R Function restricting the economically meaningful space in the
phase-diagram.
2tR Function restricting the economically meaningful space in the
phase-diagram.
253
3tR Function restricting the economically meaningful space in the
phase-diagram.
T A point in the phase diagram.
U Life-time utility of the (representative) household.
ι Wealth/Assets of household W ι .
Y Aggregate output; index of economy-wide output-volume.
Y~ Auxiliary variable. (Function of other model-variables.)
Y Auxiliary variable. (Function of other model-variables.)
.agrY Output of the agricultural sector.
iY Output of (sub)sector i.
.manY Output of the manufacturing sector.
.serY Output of the services sector.
Z Index of intermediate production. Indicates how much
intermediate inputs are used in the whole economy.
a Auxiliary parameter used to assign the range of subsectors to
the sectors. Determines the upper range of the agricultural
sector and the lower range of the manufacturing sector.
(exogenous)
1c Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
2c Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
254
3c Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
4c Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
5c Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
6c Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
1d Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
2d Auxiliary variable. (Function of other model-variables.)
3d Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
4d Auxiliary variable. (Function of other model-variables.)
(.)if Sector-specific neoclassical production function (intensive
form).
*g Growth rate of aggregates along the PBGP.
Ag Growth rate of A. (exogenous)
Bg Growth rate of B. (exogenous)
Eg Growth rate of E.
Gg Growth rate of G.
Kg Growth rate of K.
Lg Growth rate of L . (exogenous)
255
ig Growth rate of labor-augmenting technological progress in
sector i.
ih Intermediates produced by subsector i; indicates how much
output of subsector i is used as intermediate in the whole
economy.
i Index denoting a sector or a subsector.
j Index denoting a sector or a subsector.
.agrk Capital-share of agriculture; indicates which share of aggregate
capital (K) is used in agriculture.
ik Capital-share of (sub)sector i; indicates which share of
aggregate capital (K) is used in (sub)sector i.
.mank Capital-share of manufacturing; indicates which share of
aggregate capital (K) is used in manufacturing.
.serk Capital-share of services; indicates which share of aggregate
capital (K) is used in services.
l Auxiliary variable. (Function of other model-variables.)
Indicates whether cross-capital-intensity structural change is
consistent with the PBGP (of the model with intermediates).
l~ Auxiliary variable. (Function of other model-variables.)
Indicates whether cross-capital-intensity structural change is
consistent with the PBGP (of the model without intermediates).
.agrl Employment-share of agriculture; indicates which share of
aggregate labor (L) is used in agriculture.
256
il Employment-share of (sub)sector i; indicates which share of
aggregate labor (L) is used in (sub)sector i.
.manl Employment-share of manufacturing; indicates which share of
aggregate labor (L) is used in manufacturing.
.serl Employment-share of services; indicates which share of
aggregate labor (L) is used in services.
1948il Employment share of sector i in 1948.
1987il Employment share of sector i in 1987.
max1987il Hypothetical employment share of sector i, which would result,
if the labor, which has been reallocated between 1948 and1987,
were reallocated in such a manner that the maximal decrease in
l~ was accomplished between 1948 and 1987.
lΔ Indicates the amount of labor that has been reallocated between
1948 and 1987.
actuall )~(Δ Measures the change in l~ that took place between 1948 and
1987.
max)~( lΔ Measures the maximal change in l~ , that would be
(hypothetically) possible with the given amount of cross-sector
factor reallocation between 1948 and 1987.
neutrall )~(Δ Measures the change in l~ that is caused by NEUTRAL
structural change.
m Index-number limiting the range of subsectors that use
technology 1.
n Number of subsectors.
257
p Relative price of goods nmi ,...1+= , where good m is
numéraire.
ip Relative price of good i, where good m is numéraire. Indicates
how many units of good m can be obtained for one unit of good
i on the market.
np Auxiliary variable. (Function of other model-variables.)
q Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
r Real rental rate of capital.
s Auxiliary parameter used to assign the range of subsectors to
the sectors. Determines the upper range of the manufacturing
sector and the lower range of the services sector.
t Index denoting time.
(.)u Instantaneous utility-function.
w Real wage-rate.
.agrz Intermediate-share of agriculture; indicates which share of
intermediate input-index (Z) is used in agriculture.
iz Intermediate-share of subsector i; indicates which share of
intermediate input-index (Z) is used in subsector i.
.manz Intermediate-share of manufacturing; indicates which share of
intermediate input-index (Z) is used in manufacturing.
.serz Intermediate-share of services; indicates which share of
intermediate input-index (Z) is used in services.
Γ Auxiliary variable. (Function of other model-variables.)
258
.agrΛ Auxiliary variable. (Function of other model-variables.)
.manΛ Auxiliary variable. (Function of other model-variables.)
.serΛ Auxiliary variable. (Function of other model-variables.)
iΩ Capital-to-labor-ratio “in efficiency units” in sector i.
α Parameter of the Cobb-Douglas production function; is equal to
output-elasticity of the corresponding input. (exogenous)
α Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
β Parameter of the Cobb-Douglas production function; is equal to
output-elasticity of the corresponding input. (exogenous)
β Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
γ Parameter of the Cobb-Douglas production function; is equal to
output-elasticity of the corresponding input. (exogenous)
δ Depreciation rate on capital (K). (exogenous)
ε Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
iε Parameter of the Cobb-Douglas-intermediate-index; indicates
the elasticity of Z with respect to . (exogenous) ih
iθ Parameter of the utility function; closely related to the utility of
. May be interpreted as minimum consumption regarding
good i (e.g. subsistence level), if positive. May be interpreted as
“natural” endowment of good i, if negative. (exogenous)
iC
259
ϑ Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
ι Index denoting a household.
ϖ Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
κ Auxiliary variable. (Function of other model-variables.)
κ Upper level of κ , which separates the phase-diagram into a
convergent and divergent section.
0κ Lower level of κ , which separates the phase-diagram into a
convergent and divergent section at the initial point of time.
0κ Level of κ at the initial point of time of the model.
.agrκ Income-share of capital in the agricultural sector.
.manκ Income-share of capital in the manufacturing sector.
poleκ Level of κ at the pole of the 0=κ& -locus.
.serκ Income-share of capital in the services sector.
λ Auxiliary variable. (Function of other model-variables.)
.agrλ Income-share of labor in the agricultural sector.
iλ Income-share of labor in sector i; output-elasticity of labor in
sector i in the model without intermediates.
.manλ Income-share of labor in the manufacturing sector.
.serλ Income-share of labor in the services sector.
1948iλ Labor-share of income in sector i in 1948.
1987iλ Labor-share of income in sector i in 1987.
260
μ Parameter of the Cobb-Douglas production function; is equal to
output-elasticity of the corresponding input. (exogenous)
ν Parameter of the Cobb-Douglas production function; is equal to
output-elasticity of the corresponding input. (exogenous)
ρ Time-preference rate. (exogenous)
σ Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
χ Parameter of the Cobb-Douglas production function; is equal to
output-elasticity of the corresponding input. (exogenous)
Hψ Co-state variable of the Hamiltonian function.
Lψ Lagrange multiplier.
ω Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
nω Auxiliary parameter. (Function of exogenous and constant
model-parameters.)
iω Parameter of the utility function; closely related to the utility of
. iC
261
262
PART II of CHAPTER V
A PBGP-Framework for Analyzing the Impacts of Offshoring on Structural Change and real GDP-
growth in the Dynamic Context
In the following I propose a framework for studying the effects of offshoring on
structural change and real GDP-growth. It is a multi-sector-growth-model where
(consumption-)sectors differ by TFP-growth and where capital accumulation
takes place. I argue that “standard offshoring theory” neglects important
productivity effects of offshoring by excluding capital accumulation and demand
patterns associated with Baumol’s “cost disease”. My model implies as well that
the transition from a manufacturing economy to a services economy, which takes
place in modern societies, is slowed down by offshoring in the long run.
Again, I modify the reference-model from Section 1 of Chapter III to adapt it to
the topic that is analyzed. The resulting model is a multi-sector Ramsey-Cass-
Koopmans model with some trade. I will discuss these modifications in detail
later; however, here are some short explanations: I simplify the preference-,
technology- and sector-structure for simplicity (homothetic preferences, identical
capital-intensity across sectors and only three sectors) and I add intermediate
structures; hence, the resulting model is similar to the model by Ngai and
Pissarides (2007). However, in contrast to Ngai and Pissarides (2007), I add
intermediate trade, introduce a measure of real GDP and focus on the impacts of
offshoring on real GDP-growth.
263
TABLE OF CONTENTS for PART II of CHAPTER V
1. Introduction .................................................................................................... 265
2. Model assumptions ......................................................................................... 270
3. Optimum and equilibrium .............................................................................. 277
4. Effects of offshoring on growth of aggregates ............................................... 280
4.1 The overall impact on aggregate growth .................................................. 281
4.2 Impact channels and their relative importance ......................................... 284
5. The effects of offshoring on structural change............................................... 290
6. Discussion and implications ........................................................................... 297
7. Concluding remarks........................................................................................ 301
APPENDIX A..................................................................................................... 306
APPENDIX B..................................................................................................... 307
APPENDIX C..................................................................................................... 309
APPENDIX D..................................................................................................... 313
APPENDIX E..................................................................................................... 315
APPENDIX F ..................................................................................................... 316
LIST OF SYMBOLS of PART II of CHAPTER V ........................................... 317
264
1. Introduction Let us first start with a definition of offshoring, since numerous definitions of the
term offshoring exist in the literature. Offshoring means here that firms shift
activities abroad (to unaffiliated firms or to own affiliates). Especially, a part of
the intermediate production is shifted abroad. The case where the whole
production process is shifted abroad is not analysed here; for this case traditional
(final-goods)trade theory may be more adequate.
Although a well known fact for a long time, offshoring has become one of the
most prominent terms in political debate in last years. For example, fears about
offshoring to the “new” EU-member countries in the Eastern Europe arose in
Western Europe; see e.g. Marin, 2006, for data on European offshoring. In the
United States of America fears about offshoring to India or China dominate the
political and scientific debate. These fears arose due to opening of international
borders, which has been caused by change in political regime/directives in (ex-
)communist countries, and due to new progress in information, communication
(but as well transportation) technologies, which made offshoring more profitable
and feasible. As a result, offshoring is “one of the most rapidly growing
components of trade” (Grossman and Helpman, 2005, p. 36) with the potential for
being the “next industrial revolution” (Blinder, 2007b).1
An overview of empirical and theoretical literature related to offshoring can be
found for example in Garner (2004), GAO (2005), Mankiw and Swagel (2006)
and Baldwin and Robert-Nicoud (2007). The two main discussion points in the
literature are: (1) whether offshoring leads to an increase in welfare (in the long
run in the industrialized countries) and (2) to what extent is offshoring associated
with unemployment. As discussed by Rodriguez-Clare (2007) there are two
1 For estimates of offshoring potential see Blinder (2007a), Jensen and Kletzer (2006) and Van Welsum and Vickery (2005).
265
opposing effects which are decisive for point (1): the positive productivity effect,
which refers to the increase in productivity of the domestic economy that is
caused by internalization of cross-country production-cost-differences, and the
negative terms-of-trade effect, which refers to a worsening of terms of trade due
to an increase in the supply of the domestic good on the world market. Regarding
point (2), e.g. Blinder (2005, 2007b) argues that offshoring can lead to a long
transition period with high unemployment, where the unemployment may be the
stronger, the more labor is reallocated across sectors that differ by skill type (not
skill level; e.g., the services sector requires relatively well developed soft-skills
whereas the manufacturing sector requires mathematical/engineering skills).
The aim of my paper is to contribute to both discussion-points (welfare and
unemployment) by studying (1) the productivity effect in detail (i.e. in addition to
the direct productivity effects of offshoring, which are in general studied in the
literature, I study the role of the productivity effects via structural change2
associated with Baumol’s cost disease3 and via capital accumulation), and (2) the
cross-sector reallocations (i.e. structural change) associated with offshoring. The
latter are not only important for assessing unemployment but are also relevant for
nearly every long run policy. Furthermore, I provide a dynamic framework for
studying the effects of offshoring (in contrast to the static frameworks studied by,
e.g., Bhagwati et al., 2004, and Samuelson et al. 2004). In this way I can analyze
the effects of offshoring on GDP-growth and on dynamic structural-change-
patterns between technologically distinct sectors. These dynamic effects are rather
2 Structural change means here reallocation of labor across sectors such as manufacturing, services and agriculture, where sectors differ by technology (e.g. by the growth rate of total-factor-productivity (TFP)). 3 Baumol (1967) shows that, if demand is price inelastic, there is a GDP-growth-slowdown in a multi-sector economy, since factors are reallocated to low-productivity-growth-sectors (hence average factor-productivity growth slows down). In the literature this slowdown is often referred to as the growth slowdown associated with Baumol’s “cost disease”. For some new evidence on the effects associated with Baumol’s cost disease see, e.g., Nordhaus (2008).
266
rarely analyzed in the literature4 (since they are difficult to study, i.e. there are no
balanced growth paths5).
My framework is based on the model presented by Ngai and Pissarides (2007),
i.e. I do not rely on a (static) trade model but work with a growth model.6 The
aggregate structure of the model is similar to the Ramsey-model (for a discussion
of the Ramsey-model, see e.g. Barro and Sala-i-Martin, 2004). A part of the
intermediates production is taken over by the foreign country, i.e. offshoring takes
place. Differences in prices across countries come from differences in technology
(TFP-growth) across countries. Of course, since different technologies are used
across countries, I assume that domestic and foreign intermediates are not perfect
substitutes. Overall, I model an economy where capital is accumulated and where
the sectors differ by TFP-growth.7 The importance of analyzing offshoring in a
framework where capital is accumulated has been mentioned by Milberg et al.
(2006), pp. 6/7. Furthermore, the importance of analyzing offshoring in a
framework where technologically distinct sectors exist is implied by the previous
literature: As noted by Blinder (2005, 2007b) offshoring of high-productivity-
growth-activities (that became possible by progress in information and
communication technologies) could lead to a GDP-growth slowdown in the
economy, if the redundant factors are reallocated to sectors with lower
productivity growth (according to the framework of Baumol’s, 1967, “cost
4 There is some empirical literature on the productivity-growth-effects of offshoring at the industry and plant level, e.g. Amiti and Wei (2005, 2006), Mann (2004) and Girma and Gorg (2004). (A further overview of the literature can be found in Olsen, 2006.) However, no clear patterns as to how offshoring affects productivity can be concluded from this literature (see Olsen, 2006, p. 28). Furthermore, Rodriguez-Clare (2007) studies offshoring in a dynamic model. However, he omits cross sector differences in productivity growth and capital accumulation, which are crucial to our analysis. 5 On difficulties in the analysis of dynamic multi-sector models (no existence of balanced growth paths) see e.g. Kongsamut et al. (2001). 6 An overview of trade models dealing with offshoring can be found in Baldwin and Robert-Nicoud (2007). 7 Evidence on different TFP-growth rates across sectors is presented by, e.g., Baumol et al. (1985).
267
disease”). The findings by Fixler and Siegel (1999) imply that (domestic)
outsourcing has impacts on sector-productivity and on structural change in a
framework similar to that by Baumol (1967) (i.e. in a framework where
productivity growth differs across sectors). Therefore, one can expect that
offshoring (or international outsourcing) has similar effects, as well. Note that my
model may be regarded as a Ricardian model, since trade arises due to differences
in relative sector-productivities across countries.
My model postulates a chain of dynamic effects along which offshoring can
increase GDP-growth:
(1) Offshoring influences the (implicit) total-factor-productivity-growth of
intermediates-production. This effect implies that offshoring acts like a sort of
technological progress in intermediates-production by integrating the foreign
cost-advantages into domestic production. In fact, this effect is studied in classical
trade theories. (2) The productivity-changes in intermediates-production (effect
(1)) have an indirect effect on aggregate growth as well: they influence the rate of
capital-accumulation. This effect is similar to the effect of an increase in
technological progress in neoclassical growth models, e.g. in the Ramsey-model.
(3) The increase in capital accumulation leads to factor-reallocation from
consumption-industries to capital-industries. Since the structural change patterns
associated with Baumol’s cost disease arise in consumption industries (but not in
capital-industries), the withdrawal of factors from consumption-industries implies
that a smaller share of aggregate factor-use is exposed to Baumol’s structural
change patterns. This effect has a positive impact on GDP-growth, since
Baumol’s structural change patterns have a negative impact on real GDP-growth.
In general, models that exclude capital accumulation and structural change
associated with Baumol’s cost disease do not take channels (2) and (3) into
268
account. My model implies that therefore (in some cases) these models omit the
quantitatively more important part of the productivity effect: I show that the
growth effects via channel (1) are smaller than the other effects, provided that the
economy-wide output-elasticity of capital is higher than the economy-wide
output-elasticity of labor.
Furthermore, my results imply that offshoring of high-productivity-growth (hpg)
activities and offshoring of low-productivity-growth (lpg) activities are different
regarding the terms of trade development: Offshoring of lpg activities can be
beneficial for the home country even when the terms of trade worsen in the long
run. However, offshoring of hpg activities can be beneficial only if the terms of
trade improve in the long run. The reason for this fact is that lpg activities feature
increasing prices due to “Baumol’s cost disease”. Thus, even when the terms of
trade worsen in the long run, it can be “cheaper” importing intermediates than
producing them at home. This result may be of interest for the debate about
(future) hpg services offshoring.8
My results imply that structural change patterns are slowed down by offshoring
(and thus less unemployment due to inter-sectoral barriers may exist in reality) in
the long run equilibrium. (As explained above, the structural change slowdown
comes from the shift from consumption production to capital production.) These
results imply that offshoring can have different effects in comparison to domestic
outsourcing: While my results imply that offshoring leads to a structural change
slowdown, the results by Fixler and Siegel (1999) imply that domestic
outsourcing leads to an acceleration of structural change.
8 It is argued in the literature that, while offshoring of manufacturing activities included mainly lpg activities (see e.g. the discussion in Mankiw and Swagel (2006)), future services offshoring may include hpg activities; see e.g. Blinder (2005, 2007a, 2007b) and Irwin (2005).
269
Last but not least, my results imply that the economy, that offshores, first goes
through a turbulent phase before reaching the phase described above (i.e. the
long-run equilibrium where structural-change-smoothing occurs and higher
growth rates are achieved). During the turbulent phase there are strong
reallocations across sectors (thus, in reality high unemployment may be the case)
in order to adapt the production-structures to the effects of offshoring (in detail:
there are changes in exports, investment goods production and demand for
domestic intermediates). This result supports the argumentation by Blinder (2005,
2007b). All reallocations during the turbulent phase lead to a “manufacturing-
sector renaissance” in my model. That is, the manufacturing employment share
(which is normally decreasing in industrialized countries) increases during this
phase due to increased capital demand (and exports). This result may explain the
fact that offshoring is not associated with higher unemployment in the
manufacturing sector in empirical studies (e.g. by Amiti and Wei, 2005).
The rest of the paper is set up as follows: In the next section (section 2) I present
the model assumptions. Then I calculate the model-optimum and equilibrium
(section 3). Subsequently, I analyze the effects of offshoring on growth (section
4) and structural change (section 5). In section 6 I discuss the results/extensions of
my model, i.e. endogenous terms of trade, negative growth effects of offshoring,
the distribution of effects across phases, implications for unemployment,
“manufacturing renaissance” and “partial offshoring”. Finally, I summarize my
main results and suggest some topics for further research (section 7).
2. Model assumptions The model is a sort of disaggregated Ramsey-model with international trade. Due
to model-setting there exists an aggregate balanced growth path that features
270
balanced growth with respect to aggregates, but unbalanced growth (i.e. structural
change) with respect to disaggregated variables (for details see also Ngai and
Pissarides, 2007).
I assume that there are three types of goods and services (i). However, the model
could be extended for an arbitrary number of goods. To follow the recent
discussion about services offshoring9 I could assume, e.g., that the goods and
services are classified as follows: manufacturing goods (M), personal services (P)
and impersonal services I, i.e. IPMi ,,= . Blinder (2007a) suggests dividing the
service jobs into personal services, i.e. services that cannot be delivered
electronically from far away without degradation in quality (e.g. child care and
surgery), and impersonal services that can be delivered electronically from far
away without degradation in quality (e.g typesetting and programming).
Impersonal services are offshorable (or tradable), but personal services are not.
The representative household consumes all three types of goods and services. I
assume the lifetime utility function suggested by Ngai and Pissarides (2007).
They have proven that the lifetime utility (U) has to be a logarithmic function of
the consumption composite in order to allow for aggregate balanced growth. The
consumption composite itself is a CES-function of consumption ( ) of goods
and services ( ):
iC
IPMi ,,=
(1) IPMidte t
iiiCU ,, ,
0
ln)1/(
/)1( =−∫∞
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−
−∑= ρεε
εεω
(2) ii ∀><< 0 ,;10 ωρε
9 see e.g. Amiti and Wei (2005, 2006), Garner (2004) and Blinder (2005, 2007b) on the role of progress in communication and information technology for services offshoring and the future importance of services offshoring.
271
(3) ∑ =i
i 1ω
where t is the time index.
Since I assume here 1<ε , the goods are poor substitutes and relative demand is
price inelastic, which is necessary for analyzing the growth slowdown associated
with Baumol’s cost dissease. (For further explanations with respect to the features
of this utility function see Ngai and Pissarides, 2007.) These goods and services
are produced by the corresponding domestic sectors. Each sector produces its
output via a Cobb-Douglas production function by using labor, capital and
intermediate inputs. The amount of labor available is constant and normalized to
unity. However, exogenous population growth could be integrated into this model
easily. Total factor productivity (TFP) grows in each sector at a sector specific
rate : )( ig
(4) , βαβα )()(1 ZzKknAY iiiii−−= PIMi ,,=
0, >βα and 01 >−− βα
(5) ii
i gAA
=&
, PIMi ,,=
where is the output of sector i; iY K is the aggregate capital; Z is an index of the
total “amount” of intermediate inputs; , and represent respectively the
fraction of capital, labor and intermediate inputs devoted to the production of
sector i; is a sector-specific productivity parameter.
ik in iz
iA
I assume that sectors M and P do not produce intermediate inputs. In the autarky,
intermediate inputs are produced by sector I; hence,
272
(6a) IhZ =
where is the “amount” of domestic intermediate inputs produced by sector I.
When I allow for trade, I assume that the intermediate inputs index (Z) is a Cobb-
Douglas-function of domestic and foreign intermediate inputs:
Ih
(6b) , ϕϕ −= 1)( FI hhZ 10 << ϕ
where is the “amount” of foreign intermediate inputs. Note that this function
implies that foreign intermediates are not a perfect substitute for domestic
intermediates, since when using foreign intermediates domestic intermediates are
still necessary in production (i.e. there is only partial-offshoring). This may come
from the fact that foreign and domestic intermediates are produced by different
productions functions (which allows for price differences between abroad and
home). For example, if software is produced less expensively abroad, it may be
lower quality software (since foreign programmers have lower quality education);
hence, domestic software-programming may be necessary to repair software
failures which may show up later. Anyway some domestic services are always
necessary to integrate foreign services into domestic production (see also Blinder,
2007b). The assumption of equation (6b) is also consistent with the empirical
findings that domestic and foreign intermediate inputs complement each other in
the production of final goods (see e.g. Desai et al., 2005, or Hanson, et al., 2003).
Fh
Note that the parameter ϕ could overall be interpreted as a quality/tradability
parameter of foreign intermediates: The lower ϕ , the better substitutes are
foreign and domestic intermediates. ϕ can represent e.g. the quality of foreign
education, but also the quality of the transfer process (tradability) of services. For
273
example, if ϕ is very close to zero, the foreign services are nearly perfectly
tradable and their quality is comparable to the quality of domestic services, and
therefore intermediates will be produced nearly only abroad. Furthermore, ϕ
could also be interpreted as a measure of uncertainty regarding foreign
intermediates as modeled by Choi (2007).
Since the economy imports intermediate inputs, it has to “pay” for them by
exporting goods and services. Let sector P represent all goods and services that
cannot be exported. That is, only the output of sector M can be exported.
Alternatively, I could also assume that sector-P-output can be exported and
sector-M-output cannot. However, the key-model results would be the same. (I
assume here in accordance with the standard trade theory that the output of sector
I is not exported, i.e. the foreign country has some comparative advantage in I-
production. That is, I assume that goods and services that are exported are not
imported at the same time. However, the model could be modified easily to
include at the same time exports and imports of the same good.) I abstract from
any other trade not associated with offshoring in order to isolate the effects of
offshoring. Let denote the exports of sector M. Furthermore, let denote the
price of good i ( ). Thus, aggregate exports (E) are given by
Me ip
PIMi ,,=
(7) MM epE =:
I assume now that aggregate exports (E) are related to imports ( ) in the
following manner:
Fh
(8) FThE =
274
where T is the ratio of exports to imports associated with offshoring. It
determines how much the economy has to export in order to get one unit of
intermediates-imports (offshoring). Therefore, T is corresponding to the
(reciprocal of) “offshoring-terms of trade”. I assume that the offshoring-terms of
trade is changing at a constant rate ( ): Tg
(9) TgTT=
&
Later, in section 6, I will discuss the endogenous range of these terms of trade.
Capital (K) is produced only in sector M. (Therefore, this sector could also be
interpreted as the manufacturing sector10.) Capital depreciates at rate δ . Thus,
overall, sector-M-output is consumed ( ), exported and used as capital-input: MC
(10) MMM eKKCY +++= δ&
As explained above, the output of sector (I) is consumed ( ) and used as
intermediate input:
IC
(11) III hCY +=
The output of sector P is consumed ( ) only, as explained above: PC
(12) PP CY =
10 For empirical evidence that the manufacturing sector produces nearly all investment goods see e.g. Kongsamut et al. (1997).
275
I assume that capital, labor and intermediate inputs are mobile across sectors. All
capital, labor and intermediate inputs have to be used in domestic production, thus
(13) ∑ ==i
i PIMik ,, 1
(14) ∑ ==i
i PIMin ,, 1
(15) ∑ ==i
i PIMiz ,, 1
Furthermore, I define aggregate consumption expenditures (C) and aggregate
output (Y) as follows:
(16) ∑ ==i
ii PIMiCpC ,, :
(17) ∑ ==i
ii PIMiYpY ,, :
I choose the output of sector M as numéraire, thus:
(18) 1=Mp
Overall, we should keep in mind that the domestic country imports intermediate
inputs (that are substitutes for sector-I-output) and exports a part of the sector-M-
output. There is no labor mobility across countries and the households of the two
countries can invest their savings respectively only in the capital of their domestic
countries.
276
3. Optimum and equilibrium Now, the model is fully specified. Equations (1)-(18) (where we use equation (6b)
and not (6a)) can be optimized by using a Hamiltonian.11 Then, after some
algebra the following intertemporal and intratemporal optimality conditions can
be obtained for my model (I subdivide the equations describing the optimal
solution into aggregated and disaggregated level equations):
Aggregates
(19) ββ
βα
η −−= 11KAY M
(20) KCYK δβ −−−= )1(&
(21) ρδα −−=KY
CC&
(22) ββα
η −−= 11
1KZ
(23) YE βϕ)1( −=
where η is a function of exogenous model parameters growing at constant rate:
(24) ϕ
ϕϕ ϕϕβη−
− ⎟⎠⎞
⎜⎝⎛−=
11)1(
TpA I
I
Sectors
(25) iAAp
i
Mi ∀=
11 The optimality conditions which are obtained by the Hamiltonian, provided that there is free mobility of labor across sectors, are (u(.) denotes the instantaneous utility function from equation (1), i.e. u(.)=ln[…]):
iZzYZzY
KkYKkY
LlYLlY
CuCup
ii
MM
ii
MM
ii
MM
M
ii ∀
∂∂∂∂
=∂∂∂∂
=∂∂∂∂
=∂∂∂∂
= ,)(/)(/
)(/)(/
)(/)(/
/(.)/(.) ;
IM
MI h
ZZz
Yp∂∂
∂∂
=)(
;
FM
M
hZ
ZzYT
∂∂
∂∂
=)(
; ρδ −−∂∂=− )(/ KkYuu
MMM
M& ; where MM Cuu ∂∂= /(.) .
277
(26) YThhp FII βϕϕ=
−=
1
(27) izkn iii ∀==
(28) iXx
CCp iii ∀=
(29) YC
Xx
YYp
n PPPP ==
(30) ( ) βϕδ )1(* −+++==YKg
YC
Xx
YYp
n MMMM
(31) ϕβ+==YC
Xx
YYp
n IIII
where and X are time varying auxiliary variables and functions of exogenous
model parameters:
ix
(32) iAA
CCpx
i
M
M
i
M
iii ∀⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛==
−εε
ωω
1
:
(33) ∑==i
iM xCCX /:
*g denotes the equilibrium growth rate of aggregates in the equilibrium. (I will
discuss this growth rate and all the equations later in detail.)
Now, following Ngai and Pissarides (2007), I define a partially balanced
growth path (PBGP) such that aggregate consumption (C), aggregate output (Y)
and capital (K) grow at the same rate, thus, being consistent with the Kaldor facts.
This definition requires balanced growth with respect to aggregates. However, it
allows for unbalanced growth with respect to disaggregated variables such as
output shares, etc., i.e. structural change can take place.
278
Theorem 1a: A unique and globally saddle-path-stable PBGP exists in my model.
Proof: The equations describing the aggregate optimum (especially equations
(19)-(21)) resemble the ones from the “normal” one-sector Ramsey-model in all
relevant features. Thus, the model in aggregates behaves like a normal Ramsey
model. Therefore, we now that a unique and saddle-path-stable growth-path exists
in my model, where Y, K and C grow at a constant rate. For a discussion of the
normal one-sector Ramsey-model (or sometimes also named Ramsey-Cass-
Koopmans-model), see e.g. Barro and Sala-i-Martin (2004). Q.E.D.
Theorem 1b: Along the PBGP all aggregates (Y, K, C and E) grow at the
constant rate 12*g , which is given by
(34) βαββ η
−−
+−=
1)1(* gg
g M
where
(35) .)1(: constgppgg T
I
II =⎟⎟
⎠
⎞⎜⎜⎝
⎛−−+==
&&ϕ
ηη
η
and where is given by equation (25). II pp /&
Proof: Equation (19) implies that if K and Y grow at (the same) constant rate,
their growth rate is given by equation (34). Provided that K and Y grow at rate
, equations (20), (21) and (23) imply that aggregate consumption expenditures
(C) and aggregate exports (E) grow at rate as well. Q.E.D.
*g
*g
12 It follows from equation (22) that Z grows at a constant rate as well. However, its growth rate is different from . *g
279
Theorem 1c: Structural change takes place along the PBGP, i.e. sectoral factor-
input-shares ( ) and sectoral consumption-shares change over
time.
iii zkn ,, )/( CCp ii
Proof: Equations (27)-(33) imply that although K, C and Y grow at a constant
rate, and change over time. This result comes from the fact
that and X are functions of time-varying exogenous parameters. Hence, they
are not constant along the PBGP. Q.E.D.
iii zkn ,, CCp ii /
ix
Theorem 1d: The extent of offshoring changes along the PBGP, i.e.
changes.
FI hh /
Proof: This Theorem is implied by equations (25) and (26) as well as by Theorem
1b. Q.E.D.
Theorem 1e: is the (implicit) TFP-growth rate of intermediates-production
(Z).
ηg
Proof: see APPENDIX A.
4. Effects of offshoring on growth of aggregates First, I discuss the overall impact of offshoring on aggregate growth. Then in the
next subsection I will discuss the channels along which offshoring influences
GDP-growth and their relative importance. The analytical approach is as follows:
I compare the PBGP of an economy, which offshores, to the PBGP of an
economy, which is in autarky, ceteris paribus. I also discuss the factor
reallocations during the transition period in section 5.
280
4.1 The overall impact on aggregate growth Up to now I modeled an economy that offshores intermediate inputs. The
following Theorem implies how we can modify the model equations to describe
an economy without offshoring:
Theorem 2: The model describes an economy without offshoring if we set 1=ϕ
and in all model equations. In this case equation (6b) becomes (6a), i.e.
only the output of sector I is used as intermediate input.
0=Me
Proof: Note that in equation (24) 1=ϕ does not imply that 0=η , since .
The rest of the proof is trivial. Q.E.D.
100 =
Theorem 3a: Offshoring increases the growth rate of the economy ( ), provided
that the price of domestically produced intermediates ( ) grows at higher rate
than the price of imported intermediates (T), ceteris paribus. That is, the PBGP-
growth rate of an economy, which offshores, is higher in comparison to the
growth rate of an economy, which does not offshore, provided that
*g
Ip
0>− TI
I gpp& ,
ceteris paribus.
Proof: This Theorem is implied by equations (34) and (35) and by Theorem 2.
The economy, which offshores, features 1<ϕ ; the economy, which does not
offshore, features 1=ϕ . Note that I
I
pp& is a function of exogenous model
parameters; see equation (25). Q.E.D.
281
Theorem 3b: If 0<− TI
I gpp& , offshoring decreases the growth rate of the
economy ( ) in comparison to the state without offshoring, ceteris paribus. In
this case, the relative amount of offshoring ( ) decreases.
*g
IF hh /
Proof: The proof is analogous to the proof of Theorem 3a. Note that is
given by equation (26). Q.E.D.
IF hh /
Note that, in general, the negative outcome (Theorem 3b) will not occur, since, if
terms of trade develop unfavorably, the economy can return to the state of autarky
(i.e. equation (6a) becomes valid instead of equation (6b)). Hence, the growth rate
will never be lower than in autarky. However, in section 6 I will discuss some
extensions of the model where this negative case may occur.
Definition 1: High-productivity-growth-sectors are sectors where the TFP-
growth-rate is higher than the TFP-growth-rate of the capital-producing sector;
i.e. a sector i is named high-productivity-growth-sector, provided that
. Low-productivity-growth-sectors are sectors where the TFP-
growth-rate is lower than the TFP-growth-rate of the capital-producing sector;
i.e. a sector i is named low-productivity-growth sector, provided that
.
Migg Mi ≠> ,
Migg Mi ≠< ,
Theorem 4: Offshoring of high-productivity-growth-activities ( )
increases the growth rate of the economy ( ) only if the terms of trade improve
in the long run, i.e. only if
MI gg >
*g
0<Tg . On the other hand, offshoring of low-
productivity-growth-activities ( MI gg < ) can increase the growth rate even *g
282
when the terms of trade worsen in the long run, i.e. (provided that 0>Tg
0>− TI
I gpp& ).
Proof: Equation (25) implies that high-productivity-growth-activities have
decreasing prices, i.e. 0<i
i
pp&
for , and vice versa. The rest of the proof
is implied by equations (34) and (35) and Theorem 2. Remember: The economy,
which offshores, features
Mi gg >
1<ϕ ; the economy, which does not offshore, features
1=ϕ . Q.E.D.
Overall, Theorem 4 implies that even when the terms of trade worsen in the long
run, offshoring can increase the growth rate of aggregate output, provided that
low-productivity-growth-activities are offshored. The reason for this fact is that
these activities feature increasing prices due to the “cost disease” (see also
Baumol, 1967, and Ngai and Pissarides, 2007). Hence, even when the terms of
trade worsen (i.e. the price for foreign intermediates grows) it can be cheaper
using foreign intermediates instead of domestic intermediates (provided that the
price for domestic intermediates grows at higher rate than the price for foreign
intermediates). Overall, for positive growth effects of offshoring it is not merely
relevant whether the terms of trade improve or not, but rather how the terms of
trade develop in comparison to the price of the domestic sector (I) that is
competing with the foreign sector (Theorems 3 and 4).
283
4.2 Impact channels and their relative importance An interesting question within this model is along which channels offshoring
influences the growth rate of aggregate output (Y). It follows from equations (4),
(17), (22) and (27) that Y is determined as follows:
(19a) ∑−−=i
iii nApKY ββ
βα
η 11 .
This equation implies that there are five sources (or: channels) of growth within
this model (note that all variables with zero as exponent denote the initial value of
the corresponding variable):
(1) Intermediates production productivity (via term ββ
η −1 ). Remember that I
have shown in Theorem 1e that can be interpreted as the TFP-growth-
rate of intermediates-production (Z). An increase in
ηg
η increases Y, ceteris
paribus.
(2) Price-effect of structural change/technological cross-sector-disparity (via
). Cross-sector differences in technology (TFP-growth) cause
changes in relative prices (see equation (25)). In general, structural change
(reallocation of labor across sectors) leads to changes in relative prices as
well.
∑i
iii nAp 00
13 Changes in relative prices cause changes in aggregate output,
ceteris paribus.
(3) Quantity-effect of structural change (via ∑i
iii nAp 00 ). Labor-reallocation
across technologically distinct sectors (i.e. changes in ) leads to changes in
13 Note that, although in our model structural change does not directly lead to changes of relative prices, in general it does. This fact will be discussed later.
284
in aggregate output, ceteris paribus. In my model this effect is negative
due to the growth-slowdown associated with Baumol’s cost dissease: In
APPENDIX B I show that factors are reallocated from sectors with high
TFP-growth-rates to sectors with low-TFP-growth rates along the PBGP,
provided that demand is price-inelastic )1( <ε . That is labor, is
reallocated to low-productivity sectors; hence, average productivity of
labor decreases (see also the explanations by Ngai and Pissarides, 2007,
and Baumol, 1967 ).
(4) Technological progress (via ∑i
iii nAp 00 ). Changes in (exogenous)
technology parameter(s) ( ) lead to changes in aggregate output, ceteris
paribus.
iA
(5) Capital accumulation (via βα−1K ). An increase in capital increases
aggregate output, ceteris paribus.
Effects (2) and (3) are well known form standard structural change theory. Effects
(4) and (5) are known from the neoclassical growth theory. In fact, my model is
neoclassical in the sense that changes in productivity (due to effects (1)-(4)) lead
to changes in capital accumulation and thus to even more aggregate output-
growth. This mechanism is the same as in the standard one-sector Ramsey-(Cass-
Koopmans-)model.
My model implies that offshoring has an impact on channels (1), (3) and (5):
• Equation (35) and Theorems 1e and 2 imply that offshoring
increases the productivity of intermediates production by
internalizing the price-difference between domestic and foreign
intermediates production. (Remember that only the case
285
0>− TI
I gpp& is relevant). Hence, aggregate output-growth is
increased by channel (1).
• Higher productivity in intermediates-production leads also to an
increase in capital accumulation and to more aggregate output-
growth via channel (5) (like in the normal Ramsey model); this fact
is implied by Theorems 1 and 3 and by equation (34).
• I show in the next section that this increase in capital accumulation
leads to a slowdown of structural change patterns in the long run;
i.e. due to offshoring the changes in over time along the PBGP
become smaller (see Theorem 5b). In other words, the structural
change patterns associated with Baumol’s cost disease are slowed
down by offshoring in the long run, which implies that less labor is
reallocated to low-productivity-growth-sectors. Hence, there is a
positive effect of offshoring on aggregate output-growth via
channel (3).
in
• This positive effect increases the rate of capital accumulation again
and thus aggregate output-growth and so on.
It should be noted that in my model channels (2) and (3) (and (4)) always “offset”
each other regarding Y -growth, since equations (14) and (25) imply that
Mi
iMi
i =ii
M
iiii AnAnA
AAnAp == ∑∑∑ . Hence, whether the changes in are
slowed down by offshoring or not, Y is always the same since equation (19a)
reduces to
in
(19b) MAKY ββ
βα
η −−= 11 .
286
However, this fact does not imply that offshoring has no impact on GDP-growth
via channel (3) in my model and in general. The reasons are the following:
I. Aggregate output (Y), as defined in my model, is not equivalent to the real
GDP, as measured in reality. Equations (17) and (18) imply that Y stands for
the aggregate output expressed in manufacturing terms (i.e. the manufacturing
sector is numéraire). In contrast, real GDP is not measured in manufacturing
terms but there is a compound numéraire. In APPENDIX C I show that real
GDP is given in my model by the following measure:
(19c) ∑−
=
i i
iM A
nA
YGDP )1( β
We can see from equation (19c) that changes in have an impact on real
GDP. I show in APPENDIX C that structural change has a negative impact on
real GDP-growth in my model, since factors are withdrawn from high-
productivity-sectors and reallocated to low-productivity-sectors. I show in the
next section (Theorem 5b) that this reallocation process is slowed down by
offshoring. Thus, offshoring has a positive impact on real GDP-growth.
in
II. In my model sectors are aggregated by using their current weights (i.e. by
using current (relative) prices) (see equation (17)). However, in reality, real
GDP is not calculated by using current weights, but by using fixed weights or
chain-weights. For example, Steindel (1995) discusses the usage of such fixed
weights and chain-weights in the GDP-growth calculations by the U.S.
Department of Commerce’s Bureau of Economic Analysis. The discrepancy
between aggregate output, as measured in my model, and real GDP as
287
measured in reality is mentioned as well by Ngai and Pissarides (2007) and
discussed in detail by Ngai and Pissarides (2004), pp. 21 ff.. (Due to this
discrepancy, some models use a fixed-weight definition of aggregate output,
e.g. Baumol, 1967, and Echevarria, 1997.) I show in APPENDIX D that
structural change determines the growth rate of real GDP, provided that real
GDP is calculated by the fixed-weights-method or the chain-weights-method.
Furthermore, I show in the next section that structural change is slowed down
by offshoring. Hence, offshoring has an impact on real GDP-growth via
channel (3), provided that real GDP is calculated by using the fixed-weights-
method or the chain-weights-method.
III. In general, the effects (2), (3) and (4) do not offset each other even regarding
Y -growth. This fact is well known from the structural change literature: More
complex assumptions, e.g. the assumption that output-elasticities of inputs
differ across sectors, would yield that in my model effects (2), (3) and (4) do
not offset each other regarding Y -growth. I omit here the explicit proof, since
it is well known in the literature (see e.g. the models by Acemoglu and
Guerrieri (2008) and Kongsamut et al. (1997)). Nevertheless, in APPENDIX
E I provide an example: I show that structural change has an impact on the
growth rate of Y, if sector differ by output-elasticites of inputs. I show in the
next section that structural change is slowed down by offshoring. Hence, we
know that in general offshoring has a “final” effect on Y-growth via channel
(3). (In my paper I do not use the more general assumption (equation (4a)
from APPENDIX E), but use equation (4), since in this way I can present my
results in the most comprehensible way: The assumption of equation (4a)
would complicate my analysis (i.e. a closed-form solution for the model could
not be derived, and I would have to rely on simulations), but would not
288
change my key result, namely the fact that offshoring has an impact on
channels (1), (3) and (5).)
Note that in models, where capital accumulation is excluded from analysis and
where structural change patterns associated with Baumol’s cost disease are not
taken into account, offshoring influences GDP-growth only via channel (1).
Hence, these models neglect the effects of offshoring via channels (3) and (5).
Therefore, it may be interesting to analyze what is the relative importance of
effect (1) in comparison to the other effects. Equation (19a) implies (remember
that ) Mi
iii AnAp =∑
(36) MgbaYY
++=&
where *
1ga
βα−
= and ηββ gb−
=1
.
Remember that offshoring acts in my model like an increase in . Hence, the
direct impact of offshoring on aggregate output-growth via channel (1) is covered
by b and the other effects are covered by a (and ). The impact of offshoring on
aggregate output-growth via channel (1) is given by:
ηg
Mg
ββ
η −=
∂∂
=1
)/(
.constag
YY& .
Furthermore, the impact of offshoring via the other channels is given by
βαβ
βα
βα
ηη −−−=
∂∂
−=
∂∂
=111
)/( *
.gg
gYY
constb
& (remember equation (34)). Hence,
we can be sure that the effect via channel (1) is weaker than the other effects
289
provided that βα
ββ
αβ
β−−−
<− 111
, which is equivalent to βαα −−> 1 .
Remember that my production functions imply that α is the economy-wide
output-elasticity of capital and βα −−1 is the economy-wide output-elasticity of
labor. Hence, we can be sure that the growth effects of offshoring via channel (1)
are smaller than the other effects, if output-elasticity of capital is higher than
output-elasticity of labor. Note, however, that these calculations do not take
channel (3) into account. That is, the effects via channel (1) are even less
significant.
5. The effects of offshoring on structural change In this model structural change is caused by differences in TFP-growth across
sectors. The differences in TFP-growth are reflected by prices (see equation (25)).
The representative household responds to the changes in prices by changing the
demand-ratios across goods. Hence, producers must adapt production to changing
demand, which leads to factor reallocations across sectors, i.e. structural change.
(For detailed discussion see Ngai and Pissarides, 2007.) I analyze now how
offshoring affects these structural change patterns.
Equations (29)-(31) are relevant for analyzing the effects of offshoring on
structural change. They represent the sectoral employment shares when the
economy offshores. Since labor is normalized to unity in my model, these
equations also represent the sectoral employment. I compare now the structural
change patterns in an economy that offshores with the structural change patterns
in an economy that does not offshore (see Theorem 2), ceteris paribus. I analyze
in this section the structural change patterns when 0>− TI
I gpp& . (As noted in the
290
previous section, the case 0<− TI
I gpp& in general cannot occur, since the country
could return the state of autarky if terms of trade develop unfavorably;
nevertheless, all the results for the case 0<− TI
I gpp& could be derived in the same
manner as in this section.)
We have to distinguish between “transitory effects” and “PBGP effects” of
offshoring with respect to structural change: Remember that we assume that in the
beginning the economy is in autarky and moves along the PBGP. Then opening of
borders occurs, which induces a transition to a new PBGP. The term “transitory
effects” denotes the factor reallocations which occur during this transition period
and which come to a halt when the economy is on the new PBGP. That is, as we
will see, some industry-employment-shares are constant in the old and in the new
PBGP; they only change during the transition period. The reallocations which are
associated with this change during the transition period are named transitory
effects. On the other hand, as we will see, when comparing the new and the old
PBGP the strength of the factor reallocation between some industries is not the
same in the old and the new PBGP. That is, offshoring induces a change in the
strength of structural change, when comparing the old and the new PBGP. This
effect of offshoring is named “PBGP effects”.
“Transitory effects” of offshoring: As just explained, this term stands for the
factor reallocations that are caused by offshoring and that take place only during
the transition period between two PBGPs. We have to distinguish between four
different “transitional effects”, which are explained in the following and
summarized in Theorem 5a:
291
Effect 1: Offshoring increases the exports-to-output ratio (E/Y), since the
economy has to “pay” for intermediate imports. This effect increases the
employment share of the exporting sector M (see equation (30); note that E/Y
is given by βϕ)1( − due to equation (23)). E/Y is constant along the PBGP
due to Theorem 1. Thus, the increase in exports is a transitional effect with
respect to structural change. That is, the changes in the employment share of
export-industries (which are a subsector of sector M) are accomplished during
the transition period.
Effect 2: Offshoring decreases the domestic-intermediates-to-output ratio
⎟⎠⎞
⎜⎝⎛
Yhp II , since some intermediates are imported from abroad. (Note that due
to equation (26) Yhp II is given by ϕβ when offshoring takes place; in the
case without offshoring, Yhp II is given by β (see Theorem 2 and equation
(26)). Thus, the domestic-intermediates-production-to-output ratio decreases
by βϕ)1( − due to offshoring.) This effect decreases the employment share of
sector I (via ϕβ ; see equation (31)), since intermediate industries are part of
sector I. Note again that this effect is transitional as well, since Yhp II is
constant along the PBGP (it is equal toϕβ ). That is, the decrease in the
domestic-intermediates-production-to-output ratio is accomplished during the
transition period.
Effect 3: It can be shown (see APPENDIX F) that the aggregate investment-
to-output ratio ( ) increases due to offshoring. This effect
increases the employment share of the sector M (see equation (30)), since
YKg /)( *+δ
292
capital-producing industries are a part of sector M. Since is
constant along the PBGP (see Theorem 1), this effect is transitional, i.e. the
accompanying reallocations are accomplished during the transition period.
The increase in the aggregate investment-to-output ratio occurs, because of
the higher aggregate productivity-growth (see also the previous section).
YKg /)( *+δ
Effect 4: The aggregate output of our economy is consumed, exported, used as
capital-input and used as intermediate input (see equations (7), (10)-(12) and
(16)-(17)). Thus, the following relation must be true:
Yhp
YE
YKg
YC II++
++=
)(1*δ . (This equation is implied by equations (20)
and (26).) My explanations of Effects 1 and 2 imply that E/Y increases due to
offshoring by the same amount as Yhp II decreases due to offshoring. Thus,
YKg
YC )( *++
δ must be constant when comparing the PBGP without
offshoring to the PBGP with offshoring. This fact implies that the aggregate
consumption-to-output ratio (C/Y) must decrease due to offshoring during the
transition period, since Effect 3 implies that increases due to
offshoring. Note that, as just explained, the decrease in C/Y is caused only by
the increase in the investment-to-output ratio ( ) (and not by
exports or by domestic intermediate goods production). What are the
implications of the decrease in C/Y for structural change? Theorem 1 implies
that C/Y is constant along the PBGP. Thus, equations (29)-(31) imply that the
decrease in C/Y is in part a transitional effect (the change in C/Y is
accomplished during the transition period) which reduces the employment
shares in all sectors during the transition period, since all sectors feature
YKg /)( *+δ
YKg /)( *+δ
293
consumption goods industries. However, ’s are not constant along the
PBGP (see equations (32)-(33)).
Xxi /
14 Hence, equations (29)-(31) imply that the
decrease in C/Y has also an PBGP-effect which will be discussed now.
“PBGP effects” of offshoring (for an explicit proof see Theorem 5b): The term
“PBGP effects” stands for the differences in strength of structural change when
comparing the old and the new PBGP. (Strength of structural change means the
amount of labor that is reallocated per unit of time. Hence, strength of structural
change can be measured by strength of changes in the employment shares.)
Hence, the PBGP-effects may the regarded as permanent or long-run effects of
offshoring on structural change. My discussion of Effects 1-4 above and
equations (29)-(31) imply that YC
Xxi ’s are the only terms, which determine the
strength of structural change along the PBGP. (YC
Xxi denotes the ratio of sectoral
consumption to aggregate output, since equation (28) implies that YCp
YC
Xx iii = .)
A decrease in C/Y (see Effect 4) decreases the strength of structural change (see
equations (29)-(31) and Theorem 5b below), which means that less labor is
reallocated across sectors over time. That is, offshoring causes a slowdown of
structural change (or in other words: structural change smoothing) in the long run.
(A discussion of the shape of the -curves can be found in APPENDIX B
and in the paper by Ngai and Pissarides, 2007.)
Xxi /
Now, let us summarize these results as follows:
14 Ie omit here the discussion of the shape of the -curves, since they are the same as in the model of Ngai and Pissarides (2007).
Xxi /
294
Theorem 5a: Offshoring leads to
• an increase in the exports-to-output ratio (E/Y) (i.e. offshoring has
a positive impact on the employment share of export industries
during the transition period)
• a decrease in the domestic-intermediates-production-to-output
ratio ⎟⎠⎞
⎜⎝⎛
Yhp II (i.e. offshoring has a negative impact on the
employment share of domestic intermediate industries during the
transition period)
• an increase in the investment-to-output ratio ( ) (i.e.
offshoring has a positive impact on the employment share on
investment-goods-industries during the transition period)
YKg /)( *+δ
• a decreases in the consumption-to-output ratio (C/Y) (i.e.
offshoring has a negative impact on the employment share of
consumption-goods-industries during the transition period).
Theorem 5b: Structural change along the PBGP is slowed down by offshoring.
That is, along the PBGP the changes in over time are weaker in an economy
that offshores in comparison to an economy that does not offshore, ceteris
paribus.
in
Proof: Since this Theorem is important, here is an explicit proof: Remember that
C, Y and K are constant along the PBGP. Hence, equations (29)-(31) imply that
the following relations are true along the PBGP:
(29a) dt
XxdYC
dtdn PP )/(
=
295
(30a) dt
XxdYC
dtdn MM )/(
=
(31a) dt
XxdYC
dtdn II )/(
=
where t stands for time. Theorem 5a implies that C/Y declines due to offshoring.
Hence, equations (29a)-(31a) imply that the changes in over time are weaker
due to offshoring. Hence, since structural change stands for changes in over
time, structural change is slowed down by offshoring. Q.E.D.
in
in
Theorem 5c: The structural-change-slow-down from Theorem 5b is caused by the
offshoring-induced decline in the consumption-to-output ratio.
Proof: This Theorem is implied by the proof of Theorem 5b. Q.E.D.
Corollary: Because of the positive productivity effects of offshoring, consumption
becomes a (quantitatively) less important part of aggregate output (shift from
consumption-goods-production to investment-goods-production). Hence, the
changes in consumption demand patterns (which are the only determinant
of structural change along the PBGP) become less relevant for the reallocation of
factors across sectors. Therefore, offshoring causes a structural change
slowdown.
Xxi /
All explanations regarding the development of sectoral employment shares are
also true for the sectoral capital shares , intermediate input shares (see
equation (27)) and sectoral output shares
in
ik iz
YYp ii (see equations (29)-(31)).
296
6. Discussion and implications In my analysis I have assumed that terms of trade are exogenous, since the
questions analyzed in my paper do not require endogenous terms of trade. As
mentioned in section 3 we know that, if offshoring takes place, terms of trade
must be such that 0>− TI
I gpp& , since otherwise the country would be better off
in autarky (i.e. offshoring would not take place). Furthermore, for 0>− TI
I gpp&
all my results are unambiguous. Hence, there is no need for endogenizing the
terms of trade. Nevertheless, it may be interesting to know the (endogenous)
terms of trade as function of deep parameters of the model. To derive the possible
range of the terms of trade I analyze the following example: I assume for the
moment that the foreign country (India) is the same as the domestic country
(USA) beside of the TFP in sector I. Let the TFP in sector I in India be .
Hence, due to equation (25) the price for the good I is given by
FIA
I
MI A
Ap = in the
USA and by FI
MFI A
Ap = in India. This implies that one of the countries will
offshore to the other country (and export M-goods) and that both countries will be
better off when trading, provided that and provided that T (i.e. the
reciprocal of the terms of trade) is somewhere between and (see also
Theorem 3). We know that T will be somewhere between and , since
otherwise both countries would be better off in autarky and in autarky both
countries would lose the gains from trade. (Remember that is not only an
indicator of technological differences between sectors, but also of the utility based
relative demand for the goods, see footnote 11).
FII AA ≠
Ip FIp
Ip FIp
ip
297
Although in the present model setting there seems to be no reason for negative
effects of offshoring for GDP-growth (i.e. terms of trade must always be such that
0>− TI
I gpp& ), it may be possible to construct cases where offshoring negatively
affects GDP-growth, i.e. 0<− TI
I gpp& . For example, assume that after the
departure from autarky the technology develops in such a manner that some
foreign intermediates, that cannot be produced at home, become essential for
state-of art production. In this case the state of art production process becomes
dependent on foreign resources (i.e. change from equation (6b) to (6a) becomes
impossible). The dependency on foreign resources would allow for unfavorable
terms of trade development (i.e. 0<− TI
I gpp& ) and hence for a GDP-growth
slowdown (see Theorem 3), since the foreign country has a better bargaining
position and can dictate the terms of trade. An intuitive example for this
argumentation may be mineral oil: Some industrialized countries do not have
(relevant) reserves of mineral oil at home. However, since they started importing
mineral oil in the early 20th century their technology developed such that mineral
oil is a key resource for the most products. Of course, the question here is whether
the development process in these countries would have been much slower, if they
never had started importing mineral oil (e.g. by researching right from the
beginning, i.e. in the early 20th century, for alternative non-oil-based
technologies). Hence, there is a trade-off between the losses from using
alternative technology (slower development) and the losses from offshoring-
dependency (weak bargaining position due to dependency). This trade-off would
have to be evaluated.
298
My distinction between “transitional effects” and “PBGP effects” of offshoring
on structural change in the previous section implies that only the “PBGP effects”
constitute a permanent effect on structural change in the long run. The
“transitional effects” can be regarded as transitory effects of offshoring with
respect to structural change.
A further interpretation of the distinction between “transitional effects” and
“PBGP effects” might be that the effects of offshoring will impact the economy in
two phases: In this case the “transitional effects” might be regarded as phase-1-
effects and “PBGP effects” might be regarded as phase-2-effects. That is, when
offshoring starts (e.g. due to technological progress or due to opening of
international borders) the economy must go through phase 1 first. The
reallocations during this phase are described by Effects 1-4 in the previous
section: employment in domestic intermediate-inputs-industries decreases,
employment in exports-industries increases, employment in capital-producing
industries increases and employment in consumption-goods-industries decreases.
Note that all these effects imply that labor is reallocated from all sectors to the
capital-producing sector in phase 1. That is, in phase 1 there is a kind of
“manufacturing sector renaissance” (provided that we interpret the capital-
producing sector as the manufacturing sector; see also p.271 and footnote 10).
After this phase is accomplished, phase 2 starts where smoother structural change
prevails, as explained in the previous section.
Thus, overall, this interpretation implies that the economy faces a turbulent phase
1 (where strong labor reallocations take place) due to offshoring. This result
supports Blinder (2005, 2007a, 2007b) who emphasizes the possible negative
(transitory) effects of offshoring. He argues that the reallocations during the
transitory phase can cause high unemployment, since they require that the labor
299
force changes its skill sets. This is especially true if the labor force has to be
reallocated across sectors rather than within sectors, since different skill sets are
required across sectors (for example, in the services sector soft skills are much
more important than in the manufacturing sector). My discussion of phase 1
implies that most of the labor force will have to be reallocated across sectors
during this phase, which implies that indeed high unemployment may arise in
reality. Furthermore, my discussion of phase 1 implies as well that unemployment
might be even higher than expected by now: Unemployment may not only arise in
the intermediates-industries, but also in the consumption-goods-industries (Effect
4).
A further interesting result from the previous section is that a sort of “partial
offshoring” occurs. That is, only a part of the intermediates-production is
offshored: the labor employed in the domestic intermediates production (ϕβ )
does not decrease in the long run, but is constant (see discussion of Effect 2), i.e.
the intermediates-production is not completely taken over by the foreign labor
force. This is consistent with the experience from manufacturing sector
offshoring: developed economies are still producing manufacturing goods.15 To
my knowledge, the only paper that models partial offshoring is the one by Choi
(2007), where partial offshoring occurs due to uncertainty. My results imply that
partial offshoring occurs even when there is no uncertainty, provided that foreign
intermediates are not perfect substitutes for domestic intermediates. In my model
this result comes from the fact that the relative extent of offshoring ( )
depends not only on price relations between domestic and foreign producers, but
also on quality of foreign products (indicated by
FI hh /
ϕ ); see also equation (26) and
discussion in section 2. 15 See, e.g. Blinder (2007b).
300
7. Concluding remarks Overall, my model implies that the inclusion of capital and structural change
associated with Baumol’s cost disease is crucial for an adequate assessment of the
productivity effect of offshoring. Standard trade theory in general does not
include these factors. Hence, the decision on the overall effect of offshoring (i.e.
whether the negative terms-of-trade effect is stronger or weaker than the positive
productivity effect) may be biased in this literature. In all my results capital plays
the key role: capital accumulation does not only create additional growth by itself,
but also makes the existence of growth effects via Baumol’s cost disease possible
(if there was no capital in my model, offshoring would not have any effects on
GDP-growth via Baumol’s cost disease).
These effects are based on the mechanism that an offshoring-induced increase in
domestic productivity accelerates domestic capital-production. Therefore, the
effects, which are shown in my model, are weaker, if some capital-goods are
imported from abroad. (In this case domestic capital production is less relevant
for domestic aggregate dynamics.) Nevertheless, we know that every country
produces some capital goods at home; thus, the effects that are modeled in my
paper exist in reality. Furthermore, in the discussion about North-South-
offshoring16 the assumption that the North produces the largest part of its capital-
goods on its own seems to be plausible, since the South does not posses the
technology to produce the high-tech capital-goods of the North.
Of course, it can happen that the North loses some of its comparative advantage
in the capital-producing sector and starts importing (more of) capital goods from
abroad. In this case the North would lose some of the positive effects of
offshoring, which are modeled in my paper. That is, even if the North gains
16 North-South-offshoring means here offshoring between industrialized countries and less developed countries, which is in the focus of the actual offshoring debate.
301
comparative advantage in some other sectors (which may ensure that the
offshoring-terms-of-trade develop favorably from the viewpoint of the North), the
loss of comparative advantage in the capital-producing sector (and allowing for
capital-goods-imports form abroad) means that the positive effects of offshoring
(via e.g. slowdown of Baumol’s cost disease, as modeled in my paper) do not
exist in the North any more. This argument is related to the debate about future
terms-of-trade development and the danger that the North loses some of its
comparative advantage (see e.g. Samuelson (2004)). However, my argument
brings another aspect to this debate. My results imply that it is important in which
sector the comparative advantage is lost: The manufacturing sector produces the
most of the capital goods. Hence, if the North loses its comparative advantage in
the manufacturing sector (and if it allows for capital-goods-imports), the North
loses its potential dynamic advantages from offshoring modeled in my paper.
That is, in this case, even if the offshoring-terms-of-trade are favourable (due to
gains in comparative advantage in other sectors), offshoring does not accelerate
domestic capital production (since capital is produced abroad) and, thus,
Baumol’s structural change patterns are not slowed down.
My results imply that offshoring has the potential to influence the long run
growth of industrialized economies (positively). Of course this influence persists
only as long as different countries use different technologies to produce similar
(not perfectly substitutable) goods.
My result regarding the (long-run) structural change slowdown associated with
offshoring implies that offshoring has an impact on the key feature of the modern
development process, namely the transition from a manufacturing economy to a
services economy. Hence, many topics associated with this transition (ranging
302
from labor market policy to education-system-design) are influenced by
offshoring. Note that structural change arises in my model due to cross-sector
differences in TFP-growth. I focused on this structural change determinant, since
it has important implications for aggregate growth (via Baumol’s cost disease).
However, this is not the only structural change determinant studied in the
literature; e.g. Kongsamut et al. (2001) show that the consumption demand
patterns associated with non-homothetic preferences can cause structural change
as well. My model results imply that these structural change patterns are slowed
down by offshoring as well, since in my model the structural change slowdown
comes from a decrease in the importance of consumption demand patterns for
factor allocation.
I made several assumptions which are simplifying the model, but which are not
necessary for my key results: My subdivision of the economy into impersonal
services (which are offshored), personal services and manufacturing goods is not
necessary. Actually any sort of offshoring could be analyzed within my model.
Furthermore, manufacturing goods need not to be exported but other goods can be
exported. As discussed in the previous section, the terms-of-trade can be
endogenized in my model; however, endogenous terms of trade do not change my
key results.
If I used more complicated assumptions in my model (e.g. cross-sector
differences in output-elasticites of inputs), the effects, which are studied in my
model, would still exist, while the analysis would become much more
complicated (simulations would be necessary). However, I cannot exclude that
more complicated assumptions would yield additional growth-effects of
offshoring. Further research could focus on the study of the existence of such
effects.
303
Regarding further research the following points may be of interest as well:
First, as discussed in the previous section my model implies that offshoring could
theoretically have negative growth effects, provided that some kind of
dependency on foreign intermediates arises. Further research could try do develop
this argumentation further. However, as discussed in the previous section, this
case is difficult to analyze, since it requires comparing different paths of
technological development.
Second, as discussed in my paper, offshoring can increase GDP-growth via
structural change, since capital production becomes more important in
comparison to consumption goods production, where consumption goods
production (but not capital production) causes the growth slowdown associated
with Baumol’s cost disease. The question is whether the structural change
patterns associated with Baumol’s cost disease can also arise in the capital
producing sector. This would require, that different sorts of capital are produced
by different technology and that capital demand is price inelastic. If this is
possible, the offshoring-induced increase in capital production would also cause
by itself some growth effects via Baumol’s cost disease. Hence, the final effects
would depend on the relative strength of Baumol’s cost disease in the
consumption goods production in comparison to the capital goods production.
Third, unemployment could be explicitly analyzed in my model by assuming
some barriers to inter-sectoral reallocation of labor. This framework could be
used to analyze the effects of such barriers on the duration of the transition period
and on the growth rate of aggregates.
Fourth, there are of course several further potential impact channels of offshoring
on growth which I did not analyze in my model (e.g. offshoring could influence
304
the endogenous technological progress). All these topics are left for further
research.
305
APPENDIX A We know from equation (11) that is produced by sector I. Thus, the total factor
productivity (TFP) of -production is given by the TFP of sector I. It follows
from equation (4) that the TFP of sector I is given by . Thus, I can formulate
the following Theorem:
Ih
Ih
IA
Theorem A1: The TFP of the domestic intermediates-production ( ) is given by
.
Ih
IA
Equations (7), (8) and (18) imply
(A.1) Teh M
F =
We know from equation (10) that is produced by sector M. Thus, the TFP of
-production is given by the TFP of sector M. Thus, I can formulate the
following Theorem due to equation (4):
Me
Me
Theorem A2: The TFP of exports-production ( ) is given by . Me MA
It follows from equation (A.1) and Theorem A2:
Theorem A3: The implicit TFP of intermediates-imports ( ) is given by FhTAM .
Implicit TFP means here the TFP which is implied by the terms of trade and by
the TFP of the export sector.
Equation (6) and Theorems A1 and A3 imply that the implicit TFP of
intermediates-production (Z) is given by ϕ
ϕ−
⎟⎠⎞
⎜⎝⎛
1
TAA M
I . The growth-rate of this
term (i.e. the implicit TFP-growth-rate of Z-production) is given by
).)(1( TMI ggg −−+ ϕϕ This term is equal to , because of equations (25) and
(35). Q.E.D.
ηg
306
APPENDIX B In the proof of Theorem 5b I have shown that the dynamics of the employment
shares along the PBGP are given by the following equations:
(B.1) dt
XxdYC
dtdn PP )/(
=
(B.2) dt
XxdYC
dtdn MM )/(
=
(B.3) dt
XxdYC
dtdn II )/(
=
Remember that C/Y is constant along the PBGP, due to Theorem 1b. Hence,
equations (B.1)-(B.3) imply that the dynamics of the employment shares are
determined by the dynamics of s. Xxi / ‘
Equations (32) and (33) imply
(B.4)
1
111
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= −− εεεε
ωω
ωω
M
P
P
M
I
P
P
I
P
AA
AAX
x
(B.5)
1
111
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= −− εεεε
ωω
ωω
I
M
M
I
P
M
M
P
M
AA
AAX
x
(B.6)
1
111
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= −− εεεε
ωω
ωω
M
I
I
M
P
I
I
P
I
AA
AAX
x
I analyze here two cases: (1) high-productivity-goods are offshored and (2) low-
productivity-goods are offshored (see also Definition 1).
Case (1): In this case the sectoral productivity ranking is as follows:
(B.7) IMP AAA << , IMP ggg <<
307
Hence, the sector P is a low-productivity-sector and sector I is a high-
productivity-sector. (B.4), (B.6) and (B.7) imply that 0)/(>
dtXxd P and
0)/(<
dtXxd I , provided that 1<ε . Hence, equations (B.1) and (B.3) imply that
the employment share of sector P increases and the employment share of sector I
decreases, provided that 1<ε . That is, factors are reallocated from high-
productivity-sector(s) to low-productivity-sector(s), provided that demand is
price-inelastic.
Case (2): In this case the sectoral productivity ranking is as follows:
(B.8) , MIP AAA << MIP ggg <<
In this case sector M has the highest TFP-growth rate and sector P has the lowest
TFP-growth rate. (B.4), (B.5) and (B.8) imply that 0)/(>
dtXxd P and
0)/(<
dtXxd M , provided that 1<ε . Hence equations (B.1) and (B.2) imply that
0>dt
dnP and 0<dt
dnM , provided that 1<ε . That is, again factors are reallocated
from the sector with the highest-TFP-growth rate to the sector with the lowest-
TFP-growth rate, provided that demand is price-inelastic.
308
APPENDIX C In reality real GDP is calculated by using an average price as GDP-deflator; i.e. in
general a basket of all goods that have been produced is used as numéraire. (See
also Ngai and Pissarides (2007), p. 435, and Ngai and Pissarides (2004), p. 21.) In
my model I choose the manufacturing output as numéraire, since in this way I can
analyze equilibrium growth paths in the most convenient manner. Nevertheless, I
can calculate the real GDP by using an average price deflator as well. I use the
following compound deflator which may be regarded as the theoretical mirror
image of the deflators that are used to calculate real GDP in reality:
(C.1) ∑≡i
iN
Nii p
YYp
p
where and denote respectively the net-output of sector i and aggregate
net-output. are the prices of my model (where sector-M-output is numéraire).
“Net-output” means here gross-output minus real value of intermediates inputs
(which is equal to “real-value added”). Hence, is given by the following
relation:
NiY NY
ip
NiY
(C.2) HzYpYp iiiN
ii −≡
where H is the aggregate value of all intermediates that have been used for
production. Hence,
(C.3) FII ThhpH +≡
(Remember that, T is the price of foreign intermediates as e.g. defined in equation
(8)).
Hence,
(C.4) ∑≡i
NiiN YpY
309
I use “net output”, since in reality GDP does not include intermediates production
in order to avoid “double counting of intermediates production”. (See, e.g.,
Landefeld et al. (2008) on intermediate inputs and GDP.)
Overall, my GDP-deflator (definition (C.1)) is simply a weighted-average of
prices, where I used net-output-shares as weights (and where prices are in
manufacturing terms). Thus, if we divide our aggregate net-output (which is
expressed in manufacturing terms) by this deflator we have a GDP-measure that
is similar to that used in reality.
(C.5) p
YGDP N≡
Note that instead of definition (C.1) I could use the following definition:
(C.6) ∑≡i
iiiA p
YYp
p
where not net-output-shares are used as weights, but output-shares. However, it
does not matter, i.e. in my model p and Ap are the same. (I omit here the proof,
since its trivial.)
By using equations (C.1)-(C.5) and (15),(19), (22) and (25)-(27) it can be shown
that
(C.7) ∑−
=−
≡
i i
iM A
nA
Yp
YGDP )1()1( ββ Q.E.D.
Note that Y is given by equations (19) and (24). Furthermore the price index is
given by
(C.8) ∑=i i
iM A
nAp
310
We can see that the real GDP is determined by factor-allocation across sectors
(where sectors differ by the productivity parameters ). Hence, structural change
(i.e. changes ) affect the growth rate of GDP.
iA
in
Now, I show that the impact of structural change on real GDP is negative.
(C.7), (C.8), (19) and (24) imply that the impact of structural change on GDP
depends only on the development of p . If I show that reallocation of factors
across sectors (i.e. changes in ) increases in p , we know that structural change
has a negative impact on real GDP-growth. To do so I derive the following total
differential of p :
(C.9) ⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑
ii
iM dn
AApd 1
Due to equation (14) the following relation must be true
(C.10) ∑ =i
idn 0
where I have calculated the total differential of equation (14).
Like in APPENDIX B I distinguish between two cases: (1) high-productivity-
goods are offshored and (2) low-productivity-goods are offshored (see also
Definition 1).
Case (1): In this case the relations between sectoral total-factor-
productivities are as follows:
(C.11) IMP AAA << , IMP ggg <<
By using equation (C.10), equation (C.9) can be reformulated as follows:
(C.12) II
MP
P
M dnAA
dnAA
pd ⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−= 11
311
In APPENDIX B I have shown that in this case the employment share of sector P
increases and the employment share of sector I decreases; hence
(C.13) 0,0 <> IP dndn
(C.11)-(C.13) imply that 0>pd
Case (2): In this case the relations between sectoral total-factor-
productivities are as follows:
(C.14) MIP AAA << , MIP ggg <<
By using equation (C.10), equation (C.9) can be reformulated as follows:
(C.15) MI
MP
I
M
P
M dnAA
dnAA
AA
pd ⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−= 1
In APPENDIX B I have shown that in this case the employment share of sector P
increases and the employment share of sector M decreases; hence
(C.15) 0,0 <> MP dndn
(C.14)-(C.15) imply that 0>pd
Overall, I have shown that in both cases labor is reallocated from the low-
productivity-sector to the high-productivity sector. This reallocation process leads
to increases in p . Increases in p lead to decreases in real GDP (according to
equation (C.7)). Hence, structural change has a negative impact on real GDP-
growth. Furthermore, the stronger structural change is, the stronger is this
negative impact (where the strength of structural change is indicated by ’s).
Q.E.D.
idn
312
APPENDIX D Following Steindel (1995), GDP measured by the fixed-weights method is given
in my model by ∑∑ −−=≡i
iiii
iiF nApKYpGDP 0110 ββ
βα
η (where I used the prices in
manufacturing terms in as fixed sector-weights; however, any other fixed
weights could be used here). This equation implies that effects (2) and (3) (and
(4)) do not offset each other regarding GDP-growth (since
0=t
∑∑ =i i
iiM
iiii A
nAAnAp 000 ). That is, changes in have an impact on the growth rate
of . Hence, my discussion in section 4.2 implies that in my model
offshoring influences -growth via channel (3).
in
FGDP
FGDP
I study the impacts of offshoring on chain-weighted GDP-growth in my model by
using the following example: Assume that we want to calculate the growth rate of
chain-weighted GDP between two points in time (e.g. 1=t and ). Define 2=t
∑∑ −−=≡i
iiii
ii nApKYpY 11111 ββ
βα
η and ∑∑ −−=≡i
iiii
ii nApKYpY 21122 ββ
βα
η , where
and denote respectively the prices of goods i in 1ip 2
ip 1=t and . Following
Steindel (1995), the growth rate of chain-weighted GDP between and
is given by:
2=t
1=t 2=t
(D.1)
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
++−
+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛+≡
∑∑
∑∑
••
iiii
iiii
iiii
iiii
C
C
nAp
nAp
nAp
nApgg
YY
YY
GDPPDG
2
2
1
1
*2
2
1
1 )()(
21
1121
ηββ
βα&&&
Hence, we can see that factor reallocation (via ) has an impact on -
growth, i.e. the effects (2) and (3) (and (4)) do not offset each other regarding
in CGDP
313
CGDP -growth. Thus, we know from the discussion in section 4.2 that offshoring
influences -growth via channel (3). CGDP
314
APPENDIX E Assume that production functions are given by the following equation:
(4a) iZzKknAY iiiiiiiii ∀= −− ,)()()( 1 βαβα
This equation implies that input-elasticities of output differ across sectors.
Footnote 11 implies that the price of a good is given by:ii
MM
i
i
M
Mi Y
nnYp
βαβα
−−−−
=1
1 .
Hence, equations (14) and (17) imply that
(19b) ∑
∑
≠
−−−
−−=
Mii
i ii
i
MMM n
n
YY11
)1(βα
βα
Note that this equation can be restructured further (i.e. the growth path of
could be derived as function of exogenous parameters in the optimum); in this
respect see the paper by Kongsamut et al. (1997) and the paper by Acemoglu and
Guerrieri (2008) as well. However, for our purposes the function in (19b) is
sufficient: we can see now that the allocation of labor across sectors (via ) has
an impact on Y. That is, structural change has an impact o the growth-rate of Y,
i.e. the effects (2), (3) and (4) do not offset each other regarding Y-growth. Hence,
the discussion in section 4.2 implies that offshoring has a “final” impact on Y-
growth when output-elasticities differ across sectors.
MY
in
315
APPENDIX F Theorem 1 and equation (21) imply that the following relation is true along the
PBGP:
(F.1) ρδα −−=KYg *
This equation can be rearranged as follows:
(F.2) ρδαδδ
+++
=+ *
** )()(
gg
YKg
The first derivative of ρδαδ
+++
*
* )(g
g with respect to is given by *g
(F.3) 0
)(
**
*
*
>++
=∂
⎟⎟⎠
⎞⎜⎜⎝
⎛++
+∂
ρδαρρδ
αδ
ggg
g
Equations (F.2) and (F.3) and Theorem 3 imply that offshoring increases
YKg )( *+δ provided that 0>− T
I
I gpp& . Q.E.D.
316
LIST OF SYMBOLS of PART II of CHAPTER V iA Parameter indicating technology/productivity level of sector i.
(exogenous)
FIA Productivity parameter in sector I in the foreign country.
0iA Level of at initial point of time of the model. iA
C Aggregate consumption expenditures; index of overall consumption-
expenditures of the representative household
iC Consumption of subsector-i-output; indicates how much of the output of
subsector i is consumed.
E Aggregate exports.
GDP Real GDP.
CGDP Real GDP measured the chain-weights method.
FGDP Real GDP measured by the fixed weights method.
H Aggregate value of all intermediates that are used in production.
K Aggregate capital; i.e. the amount of capital that is used for production in
the whole economy.
T Ratio of exports to imports associated with offshoring.
U Life-time utility of the (representative) household.
X Auxiliary parameter. (Function of exogenous model-parameters.)
Y Aggregate output; index of economy-wide output-volume.
iY Output of sector i.
1Y Y at t =1
2Y Y at t = 2.
NY Aggregate net-output. Aggregate output minus aggregate value of
intermediates.
317
NiY Net-output of sector i. Value of (gross-)output of sector i minus value of
intermediate inputs that sector i uses.
Z Index of intermediate production. Indicates how much intermediate inputs
are used in the whole economy.
a Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
b Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
Me Export of sector-M-goods.
*g Growth rate of aggregates along the PBGP.
ig Growth rate of total factor-productivity in sector i.
Tg Growth rate of T.
ηg Growth rate of η .
Fh Intermediates produced abroad.
Ih Intermediates produced by sector I in the home country.
i Index denoting a sector.
ik Capital-share of sector i; indicates which share of aggregate capital (K) is
used in sector i.
in Employment-share of sector i; indicates which share of aggregate labor is
used in sector i.
0in Level of at initial point of time of the model. in
p Price-index (“deflator”).
318
ip Relative price of good i, where good M is numéraire. Indicates how many
units of good M can be obtained for one unit of good i on the market.
0ip Level of at initial point of time of the model. ip
1ip at t = 1. ip
2ip at t = 2. ip
FIp (relative) price of I in the foreign country.
t Index denoting time.
(.)u Instantaneous utility-function.
ix Auxiliary parameter. (Function of exogenous model-parameters; grows at
constant rate.)
iz Intermediate-share of subsector i; indicates which share of intermediate
input-index (Z) is used in sector i.
α Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
β Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
δ Depreciation rate on capital (K). (exogenous)
ε (relative) price elasticity of demand. (exogenous)
ρ Time-preference rate. (exogenous)
iω Parameter of the utility function; closely related to the utility of . iC
ϕ Parameter of the Cobb-Douglas-intermediate-index; indicates the elasticity
of Z with respect to . (exogenous) Ih
319
320
321
PART III of CHAPTER V
A PBGP-Framework for Analyzing the Impacts of Ageing on Structural Change and real GDP-growth
Ageing (i.e. an increase in the old-to-young-ratio) leads to demand-shifts (and
thus to factor-reallocation) across sectors, since demand-patterns differ across the
“old” and the “young”. As a result, ageing has an impact on GDP-growth and thus
on old-age-pension-to-output-ratios, depending on the technological attributes of
these sectors. This relationship is neglected by standard (one-sector-)frameworks.
I show in a theoretical model that a rich portfolio of parameters determines the
non-monotonous dynamics of this relationship, implying that the future
challenges by ageing may vary significantly across countries and across time.
Moreover, my model provides potential policy-channels for reducing the negative
impacts of ageing.
Again, I modify the reference-model from Section 1 of Chapter III to adapt it to
the topic that is analyzed. The resulting model is a multi-sector Ramsey-Cass-
Koopmans model. I will discuss these modifications in detail later; however, here
are some short explanations: For simplicity, I restrict the number of technologies
to only two. I introduce another preference-structure; especially, I introduce some
exogenous and income-independent demand shifts to model ageing-related
demand-shifts. Furthermore, I introduce some intermediate structures and use a
GDP-measure similar to the one used in essay on offshoring (PART II of Chapter
V). Note that I present here two models: In contrast to the complex version of my
ageing-model, the simpler version features identical capital-intensity across
sectors. This makes the model much simpler and maybe easier to understand.
However, the more complex version has much richer dynamics and implications.
322
TABLE OF CONTENTS for PART III of CHAPTER V
1. Introduction .................................................................................................... 323
2. Model assumptions ......................................................................................... 330
2.1 Utility........................................................................................................ 330
2.2 Production................................................................................................. 333
2.3 Numéraire ................................................................................................. 335
2.4 Aggregates and sectors ............................................................................. 337
3. Model equilibrium .......................................................................................... 338
3.1 Optimality conditions ............................................................................... 338
3.2 Aggregates ................................................................................................ 339
3.3 Sectors ...................................................................................................... 342
4. Effects of ageing............................................................................................. 343
4.1 Partially Balanced Growth Path (PBGP) without ageing......................... 344
4.2 Ageing and cross-sector differences in TFP-growth ................................ 347
4.3 Ageing and cross-sector differences in input-elasticities ......................... 352
4.3.1 Productivity effect: Impacts and channels......................................... 353
4.3.2 Additional impacts on GDP: The price-effect................................... 359
4.3.2.1 Transitional effects of ageing on GDP ....................................... 359
4.3.2.2 PBGP-effects of ageing .............................................................. 366
4.3.3 Dynamic aspects ................................................................................ 367
5. Concluding remarks........................................................................................ 368
APPENDIX A..................................................................................................... 373
APPENDIX B..................................................................................................... 379
APPENDIX C..................................................................................................... 380
APPENDIX D..................................................................................................... 390
LIST OF SYMBOLS of PART III of CHAPTER V.......................................... 391
323
1. Introduction Ageing, a term which in general refers to an increasing life span of an average
member of a society, is a recurrent topic on the economic agenda. It has already
had some major impacts on the economic environment (e.g. there have been some
major changes in the pension and health systems of some industrialized
economies) and it is regarded as one of the trends which (will) have major
impacts on economic and social structures in (industrialized) economies in
present and in the future. Hence, it is not surprising that there is a large body of
literature dealing with several aspects of ageing. (For an overview of models
dealing with ageing and economic growth see, e.g., Gruescu (2007); for a shorter
discussion of these growth-effects see, e.g., Mc Morrow and Röger (2003); an
overview of related empirical studies is provided by, e.g., Groezen et al. (2005).)
In this paper I focus on an impact channel of ageing which seems to be rarely
studied in this literature (at least there seems to be a shortage of theoretical
models that analyze the following relationship): I analyze the impacts of ageing-
induced demand-shifts on factor-allocation across technologically distinct sectors
and their consequences for GDP-growth and old-age-pension-to-output ratios.
The working hypothesis is the following: An increase in the relative share of the
“old” in an economy changes the structure of aggregate demand, since the “old”
have another structure of demand in comparison to the “young”. If there are some
differences in technologies between sectors that produce the goods for the old and
sectors that produce the goods for the young, there may arise some effects on
aggregate productivity growth and thus on GDP-growth and pension-to-output-
ratios. (I name this whole line of arguments “factor-allocation-effects of ageing”).
In other words, ageing-induced changes in aggregate demand may cause some
cross-technology factor-reallocation, hence causing changes in aggregate (or:
324
average) productivity growth. Thus, the increasing old-age pension payments
(due to the increasing number of recipients) are confronted with changes in the
growth rate of the tax-base, which may require changes in the old-age-pension
system.
This line of arguments seems to be quite obvious, especially when thinking of
services, like health care services and geriatric nursing services: in general, the
“old” demand more of such services in comparison to the “young”; furthermore,
the “production process” of these services is regarded to be technologically
distinct (i.e. relatively labor-intensive) in comparison to e.g. manufacturing goods
(see also IMF (2004), chapter 3, and especially p. 159). However, there are also
some other differences in demand between the old and the young, e.g. the young
have a relatively larger demand for commodities and investment goods (e.g.
housing, car and furniture, i.e. things that the old may already have). Furthermore,
in general, the old seem to spend a larger share of budget on services (see
Groezen et al. (2005)).
Empirical evidence on such differences in demand patterns between the old and
the young and their growing importance (not only for factor reallocation across
sectors) has been presented by, e.g., Börsch-Supan (1993, 2003), Fuchs (1998)
and Fougère et al. (2007). Furthermore, empirical evidence implies that there are
strong differences in technology across products/sectors (e.g. when comparing
services and manufactured products or health care services and commodities
production): Evidence on differences in TFP-growth across sectors/products has
been presented by, e.g., Baumol et al. (1985) and Bernard and Jones (1996).
Evidence on differences in capital intensities across sectors has been presented
by, e.g., Close and Schulenburger (1971), Kongsamut et al. (1997), Gollin (2002),
Acemoglu and Guerrieri (2008) and Valentinyi and Herrendorf (2008). Nordhaus
325
(2008) presents some evidence on the relevance of cross-sector reallocations for
aggregate growth. Overall, this (partly indirect) evidence on factor-allocation-
effects of ageing seems to provide sufficient incentive to take a look at their
relevance from a theoretical perspective.
My model is related to the theoretical literature that postulates the importance of
cross-sector technology-differences for GDP-growth, e.g. Baumol (1967) and
Acemoglu and Guerrieri (2008). Baumol (1967) claims that cross-sector
differences in (labor-)productivity-growth can cause (by themselves) a GDP-
growth-slowdown via relative price changes (“Baumol’s cost disease”). However,
Baumol (1967) does not analyze (ageing-induced) demand-shifts, and he makes
as well some simplifying assumptions (e.g. he excludes capital accumulation)
which may be not accurate for my goals as we will see later. Acemoglu and
Guerrieri (2008) show that cross-sector differences in capital-intensities have an
impact on aggregate growth. However, they as well do not include (ageing-
induced) demand-shifts into their analysis. Furthermore, Rausch (2006) provides
a two-sector Heckscher-Ohlin model with ageing, where ageing leads to an
increase in the savings rate, since the old have relatively larger amounts of assets.
He argues that ageing leads to changes in the relative sector-size (and thus in
GDP-growth), provided that sectors differ by capital intensity (see Rausch (2006),
pp. 20 ff.). He as well does not take account of the impacts of ageing-induced
demand-shifts.
To my knowledge, the model by Groezen et al. (2005) is the only one that
explicitly includes ageing-induced demand-shifts into analysis, where ageing is
incorporated into a two-sector overlapping-generations model. The old consume
the output of a sector that uses only labor as input. Furthermore, there is no
productivity growth in this sector. The young consume the output of a progressive
326
sector that uses capital and labor as input factors. Furthermore, this sector
“produces” capital and endogenous technological progress which increases its
productivity with time. Groezen et al. (2005) show the importance of the elasticity
of substitution between capital and labor in the progressive sector. If this
elasticity is equal to unity, ageing has no impacts on growth in their model.
However, if this elasticity is greater (smaller) than unity, ageing has a negative
(positive) impact on growth.
I focus here on the pure factor-allocation-effects of ageing, i.e. I analyze the
impacts of ageing on GDP-growth and on the pension-to-output-ratios via factor
reallocation, ceteris paribus. That is, I do not try to analyze which endogenous
growth effects alternative allocations of factors may have (this question has
already been analyzed in part by Groezen et al. (2005)); hence, I keep the TFP-
growth-rates exogenous. Furthermore, I keep the elasticity of substitution
between capital and labor equal to unity, since, as just mentioned, the effects of
non-unitary substitution-elasticity have already been analyzed adequately by
Groezen et al. (2005). Moreover, I do not use an overlapping-generations model
but a Ramsey-model, since the effects of ageing in overlapping generations
models are well known from the previous literature; especially the reallocation
effects of ageing in an overlapping generations model are known from Groezen et
al. (2005). The Ramsey-model is analytically much more convenient regarding
the questions which are in my focus. In some respect, these simplifying
assumptions allow us to elaborate more exactly which (further/new) parameters
and technological specificities are relevant for the reallocation-effects of ageing,
i.e. I can model the sector that produces the goods for the old in a more realistic
way: its productivity increases due to technological progress and it uses capital,
labor and intermediates as inputs. My model is a sort of disaggregated Ramsey-
327
model1 where the representative household(s) consume(s) two groups of goods:
“senior-goods” (i.e. goods that are primarily consumed by “older” people) and
“junior-goods” (i.e. goods that are primarily consumed by younger people).
Ageing (i.e. an increasing ratio of old-to-young) yields an increasing weight of
senior-needs in the aggregate utility function, hence leading to a demand-shift in
direction of senior-goods. I assume that the production of senior-goods and the
production of junior-goods differ by TFP-growth and by capital-intensity (i.e.
output-elasticity of inputs), according to the empirical evidence discussed above.
Moreover, I include intermediates production into the model; this allows for
linkages between senior- and junior-goods-production which have been stated to
be important by Fougère et al. (2007) and by Kuhn (2004).
In my model ageing has three effects regarding GDP-growth:
(1) The ageing-induced demand-shift alters the factor allocation across
technologically distinct sectors, which yields to a direct productivity effect
(average factor-productivities change).
(2) This productivity effect has also an impact on GDP-growth via capital
accumulation (change in the savings rate). This effect is similar to the effect of a
productivity-increase in the standard one-sector Ramsey-model: a change in
productivity leads to a change in the opportunity costs of consumption, since
return on savings depends on productivity of capital (remember that savings are
invested in capital). Note that the change in the savings rate in my model is not
the same as the ageing-induced savings-change in other ageing-models (e.g. in the
model by Rausch (2006)). In these models the effect of ageing on the savings rate
is modeled as a direct effect (“more saving for retirement”).
1 For discussion of the standard (one-sector) Ramsey-model, see e.g. Barro and Sala-i-Martin (2004).
328
(3) Since the ageing-induced demand-shift leads to changes of sectoral output-
shares, the average price-index (which is the weighted average of sector prices)
changes as well. Hence, ageing leads to changes in the GDP-deflator (average
price index), which has an impact on the (real-)GDP as well.
If the change in GDP-growth is negative, there is additional upward-pressure on
the old-age-pension-to-output ratio. (Remember that upward pressure on the old-
age-pension-to-output-ratio comes from an increasing number of pension-
recipients as well).
I show that the strength and the direction of the factor-allocation-effects of ageing
depend on the combination of several parameters, including
• technology parameters, e.g. sectoral TFP-growth-rates and (initial) TFP-levels
as well as input-elasticities of sectoral production functions (including
intermediates-elasticities, which supports Fougère et al. (2007) and Kuhn
(2004)),
• parameters determining the savings rate (due to effect (2)), e.g. the time-
preference rate and
• population parameters (the old-to-young-ratio and the growth rate of working-
population).
Hence, the strength and direction of factor-allocation effects of ageing may vary
across countries according to their values of these parameters. Therefore, the fact
that empirical studies were not able to identify an unambiguous effect of ageing
on growth (see Groezen et al. (2005)) may come from the neglect of the
importance of cross-country differences in these parameters. Overall, my model
implies that the fact that capital intensity in the senior-sectors is relatively low is
not sufficient to constitute negative effects of ageing.
329
In contrast to Groezen et al. (2005) I show that reallocation-effects of ageing
arise, even if the elasticity of substitution between capital and labor is equal to
unity and that the exact constellation of the parameters listed above determines
the direction of this effect. My results imply as well that the impact of ageing via
the savings rate, which is modeled by previous literature (e.g. by Rausch (2006)),
may be weakened or strengthened depending on the parameters listed above.
Furthermore, in contrast to the previous literature I show that the impacts of
ageing on GDP-growth may change over time, i.e. in the beginning ageing may
have a positive (negative) impact on GDP-growth and later the effect of ageing
may be negative (positive). Finally, my results imply that projections of future
GDP-growth and of future pension-system-challenges may be too optimistic. For
example, the paper by the Economic Policy Committee of the EU Commission
(2003) based on Mc Morrow and Röger (2003) (see there especially pp. 12 ff.)
does not include factor-reallocation-effects in its ageing-related projections.
Hence, the negative impacts of ageing may be stronger than expected by now and
the reforms of old-age-pension systems (and health-systems) may be too weak.
The rest of the paper is set up as follows: In sections 2 and 3 I present the
assumptions and the solution of my model. In section 4 I analyze the impacts of
ageing: first I describe the dynamics of the equilibrium without ageing (section
4.1); subsequently, I compare this equilibrium to the equilibrium with ageing,
where I present a simpler version of the model in section 4.2 (where only cross-
sector-differences in TFP-growth exist) and the more sophisticated version of the
model in section 4.3. Finally, I make some concluding remarks in section 5.
330
2. Model assumptions
2.1 Utility I assume an economy where two groups of goods exist: “junior-goods” (goods
mi ,...1= ) and “senior-goods” (goods ),...1 nmi += . The representative household
consumes a mix of these goods and maximizes the following life-time utility
function (in the following I omit the time-indices):
(1) ∫∞
−=0
dtueU tρ
where
(2) SJ uNLu
NLu )1( −+=
(3) ⎥⎦
⎤⎢⎣
⎡−= ∏
=
m
iiiJ
iCu1
)(ln ωθ
(4) ⎥⎦
⎤⎢⎣
⎡−= ∏
+=
n
miiiS
iCu1
)(ln ωθ
(5a) ∑ ∑= +=
==m
i
n
miii
1 10,0 θθ
(5b) ∑ ∑= +=
==m
i
n
miii
1 1
1,1 ωω
(6) LNi gLLg
NNi ==∀<<
&&,,10 ω
where iC stands for consumption of good i and ρ is the time-preference rate. N
is an index of overall-population (including the young and the old) growing at
constant exogenous rate; L is an index of the young (working) population
growing at constant exogenous rate. Hence, the ratio NL / is an index of the
331
share of the young as part of overall population, and a decreasing NL / can be
interpreted as ageing.
The utility function is based on the Stone-Geary-preferences, where the iθ s can
be respectively interpreted as the subsistence levels (if iθ > 0) or as levels of
home-production (if iθ < 0). The income-elasticity of demand differs across
goods; the price-elasticity of demand differs across goods as well and is not equal
to unity. (See also Kongsamut et al (1997, 2001) for the discussion of a similar
utility function.)
In fact this utility function introduces ageing induced demand shifts in the most
simple way. The utility function implies that ageing (a decreasing NL / ) makes
the consumption of senior-goods relatively more contributing to aggregate utility,
and, as we will see later, this leads to a shift of demand towards senior-goods. In
order to focus on the effects of ageing I introduce the restrictions (5a) and (5b). In
this way I ensure that there are no other shifts in demand between the junior and
senior sector, beside of those induced by ageing (a decreasing NL / ): Provided
that NL / is constant (no ageing), the demand for senior-goods and the demand
for junior-goods grow at the same rate, yielding no factor reallocations between
the senior- and the junior-sector. (Nevertheless, there are still demand shifts and
reallocations within these sectors, due to the iθ s.)
Alternatively, the functions Ju and Su could be assumed to be of type Cobb-
Douglas or CES. I chose Stone-Geary-preferences, since in this way I can add
additional sources of demand-shifts (others than ageing) by omitting the
restriction (5a) and (5b). This will be of importance later.
Note that there is a difference between demand-shifts, which are modeled in
standard structural change theory (e.g. in the paper by Kongsamut et al. (2001)),
332
and ageing-induced demand shifts, which are modeled in my paper. In standard
structural change theory demand shifts are caused by differences in income-
elasticity of demand across goods. Hence, some repercussions arise: changes in
income -> demand shifts -> productivity impacts-> changes in income and so on.
This repercussion does not arise in my model. In my model the chain of impacts
is rather only in one direction: (income–independent) exogenous change in old-to-
young ratio -> demand shifts -> productivity impacts -> change in income. Of
course, one could postulate that changes in income are associated with changes in
old-to-young ratio to some extent (e.g. due to improvement in medicine or do to
some change in socio-cultural parameters which are associated with increasing
income). This would imply that changes in the old-to-young ratio are endogenous.
Although I believe that this is an interesting topic in general, a model with
endogenous old-to-young ratio would yield very similar results as the standard
structural change theory. The only difference would be that there is a further link
in the chain of impacts: income-change -> change in old-to-young ratio ->
demand shifts -> productivity impacts -> income-change and so on.
Therefore, we can summarize this discussion as follows: Ageing seems to cause
productivity-impacts via demand-shifts in two ways: On the one hand, it acts
similar like income-elasticity-differences across goods. This sort of impact is
modeled implicitly in standard structural change theory. On the other hand,
ageing acts like an exogenous shift in demand, which is income-independent.
This sort of impact is modeled in my paper. Hence, in my model I assume that
ageing arises due to some (from economist’s point of view) exogenous changes.
For example, some socio-cultural parameters change (e.g. change in religiosity,
emancipation) and/or some progress in medicine occurs independently of income
333
level. Of course both factors depend on the income of a country to some extent;
however, they must have some income-independent timely component.
Note that I am not the only one that models ageing as exogenous shifts in
demand. For example, Groezen et al. (2005) models it in this way too. Overall,
there seems to be a research gap in this field, which may be interesting to fill.
2.2 Production According to the evidence discussed above, the senior-goods are not produced by
the same technology as junior-goods; the technologies differ by TFP-growth and
by output-elasticities of inputs (i.e. capital intensities differ between the senior-
and the junior-sector). Furthermore, I assume that only the young supply labor on
the market; hence L (and not N) is input in production:
(7) miZzKkLlAY iiii ,...1,)()()( == γβα
(8) nmiZzKkLlBY iiii ,...1,)()()( +== μνχ
(9) BA gBBg
AA
==&&
,
(10) 1;1;1,,,,,0 =++=++<< μνχγβαμνχγβα
where iY denotes the output of sector i; K denotes the aggregate stock of capital;
Z is an index of intermediate inputs; ii kl , and iz denote respectively the fraction
of labor, capital and intermediates devoted to sector i; A and B are exogenous
technology parameters, where I assume that TFP-growth differs between the
junior- and the senior-sector.
334
I assume that each sector’s output is consumed and used as intermediate input
)( ih ; only sector-m-output is used as capital:
(11) mihCY iii ≠∀+= ,
(12) KKhCY mmm δ+++= &
where δ is the depreciation rate of capital. Provided that it is assumed that
senior-goods are rather services, the assumption that only the junior-sector
produces capital seems to be consistent with empirical evidence which states that
nearly all capital goods are produced by the manufacturing sector (see e.g.
Kongsamut et al. (1997, 2001)).
The intermediate-inputs-index )(Z is a Cobb-Douglas function of sectoral
intermediate outputs ( ih ):
(13) ∑∏==
=∀<<=n
iii
n
ii ihZ i
11
1,10,)( εεε
Note that it is important to assume intermediates production within this model. In
general we can assume that the old and the young consume many goods that are
nearly the same (however, the manner of consumption is quite different). For
example, the young and the old consume food. However, while probably many
young cook the food by themselves, some very old consume the food by being
served in retirement homes or hospitals. Hence, although the old and the young
eat similar things, the share of services is larger in the consumption of the old. If
we did not assume some intermediate linkages between the junior and senior
335
consumption goods we would not take account for the fact that the old are the
same human beings as the young (i.e. having the same basic needs). For example,
the assumption that the old and the young consume different goods (that have no
intermediate linkages) would e.g. imply that the old do not eat food. It would not
be necessary to take account for these facts if intermediates production were
irrelevant for the ageing-effects. However, as we will see, the output-elasticites of
intermediate inputs determine among others the strength of the ageing impacts via
structural change. Hence, we have to include intermediate linkages between
senior-goods and junior-goods into my model.
All labor, capital and intermediate inputs are used in production, i.e.
(14) ∑∑∑===
===n
ii
n
ii
n
ii zkl
1111;1;1
2.3 Numéraire Let ip denote the price of good i. I choose the output of sector m as numéraire.
Hence,
(15a) 1=mp
It should be noted here that in reality real GDP is calculated by using an average
price as GDP-deflator; i.e. in general a basket of all goods that have been
produced is used as numéraire. (See also Ngai and Pissarides (2007), p. 435, and
Ngai and Pissarides (2004), p. 21.) I choose the manufacturing output as
numéraire, since in this way I can analyze the equilibrium growth paths in the
most convenient manner. Nevertheless, I will always calculate the GDP by using
336
an average price deflator as well. I use the following compound deflator which
may be regarded as the theoretical mirror image of the deflators that are used to
calculate real GDP in reality:
(15b) ∑=
≡n
ii
N
Nii p
YYpp
1
where NiY and NY denote respectively the net-output of sector i and aggregate
net-output. “Net-output” means here gross-output minus real value of
intermediates inputs (which is equal to “real-value added”). Hence, NiY is given
by the following relation:
(15c) HzYpYp iiiN
ii −=
where H is the aggregate value of all intermediates that have been produced (see
later equation (18) as well). I use “net output”, since in reality GDP does not
include intermediates production in order to avoid “double counting of
intermediates production”. (See, e.g., Landefeld et al. (2008) on intermediate
inputs and GDP. Furthermore, the relationship between gross-output and net-
output in my model can be seen in equation (A.25) from APPENDIX A and
equations (16).)
Overall, my GDP-deflator (equation (15b)) is simply a weighted-average of
prices, where I used net-outputs as weights. If we divide our aggregate net-output
(expressed in manufacturing terms) by this deflator we have a GDP-measure that
is similar to that used in reality. However, all the issues regarding the choice of
the numéraire are irrelevant when looking at shares or ratios (since the changes
337
in the numéraire of the numerator offset the changes in the (same) numéraire of
the denominator). For example, the capital-to-output ratio ( NYK / ) is the same
irrespective of the numéraire. (See also Ngai and Pissarides (2007), p. 435 and
Ngai and Pissarides (2004), p. 21.)
2.4 Aggregates and sectors I define aggregate (gross-)output (Y ), aggregate net-output ( NY ), real GDP
(GDP ), aggregate consumption expenditures (E) and aggregate value of
intermediate inputs ( H ) as follows:
(16a) ∑=
≡n
iiiYpY
1
(16b) HYYN −≡
(16c) p
YGDP N≡
(17) ∑=
≡n
iiiCpE
1
(18) ∑=
≡n
iiihpH
1
Throughout the paper I use aggregate net-output instead of aggregate (gross-
)output (Y), since in general GDP does not include intermediates. (In my model Y
is equal to the sum of investment, consumption and intermediates-value (H); see
equation (A.25) in APPENDIX A.)
The aggregate labor share of the junior-sector )( Jl and the aggregate labor share
of the senior-sector )( Sl are given by:
338
(19) ∑∑+==
≡≡n
miiS
m
iiJ llll
11
The aggregate consumption expenditures on junior-goods ( JE ) and senior-goods
)( SE are given by:
(20) ∑∑+==
≡≡n
miiiS
m
iiiJ CpECpE
11
SE could also be interpreted as the budget devoted to the old. Throughout the
paper I assume that the aggregate budget is distributed across the old and the
young according to the representative household utility function (social welfare
function). That is, budgets are such to maximize social welfare.
3. Model equilibrium
3.1 Optimality conditions The model, as specified in the previous section, can be solved by maximizing life-
time utility (equations (1)-(6)) subject to equations (7)-(15a), e.g. by using a
Hamiltonian function. The intra- and intertemporal optimality conditions are
(where I assume that there is free mobility of factors across sectors):
(21) ihZ
ZzY
ZzYZzY
KkYKkY
LlYLlYp
im
m
ii
mm
ii
mm
ii
mmi ∀
∂∂
∂∂
=∂∂∂∂
=∂∂∂∂
=∂∂∂∂
= ,)()(/
)(/)(/)(/
)(/)(/
(22) iCuCup
m
ii ∀
∂∂∂∂
= ,/(.)/(.)
(23) ρδ −−∂∂
=−)( Kk
Yuu
m
m
m
m&
339
where mm Cuu ∂∂≡ /(.) . The proof that these conditions are necessary and
sufficient conditions for an optimum is analogous to the corresponding proof in
the Kuznets-Kaldor-Puzzle essay (see there APPENDIX A).
By using equations (1)-(20) these optimality conditions (equations (21)-(23)) can
be transformed into the following equations (sections 3.2 and 3.3) describing the
aggregate and sectoral behavior of the economy (for a proof of these equations
see APPENDIX A):
3.2 Aggregates
(24) Kc
ggEKK G
Lc
mc
mˆ)
1(ˆˆ)(ˆ
−++−−+= − δβλαλ&
(25) c
ggKEE G
Lcc
m −−−−−= −−
1ˆ
ˆˆ
11 ρδβλ&
(26a) ανχβ
λαχββαλ−
−+−= − mc
mc vKY )()(ˆˆ
(26b) )(ˆˆm
cm
cN KY βλαλ += −
(26c)
αχγεμε
λαβανχβ
βνγεμε
λSS
cm
cSS
mK
ENL
+−
⎟⎠⎞
⎜⎝⎛ −
−−+−
=−
1
ˆˆ
11
(26d) ανχβ
λαμχγβνγβμαλ−
−+−= − mc
mcKH )()(ˆˆ
(27) 1
2 )1()1()1()(ˆˆ−
−⎥⎦
⎤⎢⎣
⎡−
−−
−−++= Smmmc
mc pKPDG
ανχβμαβλβλαβλαλ
340
where 1)1(1
)1(0 <
−−−+−
≡<μεεγγνεμεβ
SS
SSc , m
mm k
l≡λ , ∑
+=
≡n
miiS
1
εε , GGgG
&≡
SS
i
S
S
S n
iiBAG
μεεγγ
ε
εμνχεγ
με
εγγμ
βν
αχ
−−−
=
−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛≡ ∏
)1(1
1
1
, and
[ ] )(1
)(1)()(
1
ˆαμγχεα
α
χεμγγαμγχεαανχβ
μαγχε
αμαναχ
λεγ
μγ
νβ
χα
−+−+−
−+−−
= ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ∏
S
S
S
i
BAKp
m
n
iiS .
Definition 1: NYandPDGYEK ˆˆ,ˆ,ˆ,ˆ stand for NYandGDPYEK ,,, expressed in
“labor-efficiency-units, i.e. cLG
YY−
≡1
1ˆ ,
cLG
KK−
≡1
1ˆ ,
cLG
EE−
≡1
1ˆ ,
cLG
GDPPDG−
≡1
1ˆ
and c
NN
LG
YY−
≡1
1ˆ .
Note that this definition of variables in efficiency units makes my discussion
about the equilibrium growth path easier later.
Proposition 1: Sp stands for the price of senior goods. Sp is given by:
[ ] )(1
)(1)()(
1
ˆαμγχεα
α
χεμγγαμγχεαανχβ
μαγχε
αμαναχ
λεγ
μγ
νβ
χα
−+−+−
−+−−
= ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ∏
S
S
S
i
BAKp
m
n
iiS
Proof: Remember that senior goods are produced by the same production
functions; hence each senior good has the same price, Sp . The rest of the proof is
given in APPENDIX A. Q.E.D.
341
We can see that (beside of the GDP-measure) the optimum aggregate structure of
this economy is quite similar to the optimum structure of the standard Ramsey-
model (or sometimes also named “Ramsey-Cass-Koopmans model”). (For
discussion of the standard Ramsey-model, see e.g. Barro and Sala-i-Martin
(2004).) For a given mλ , equations (24)-(26b) determine the equilibrium savings
rate of the model (optimal intertemporal allocation of factors), like in the normal
Ramsey-model. (In fact, for mλ = 1 equations (24)-(26b) are the same as the
corresponding equations of the standard Ramsey-model). Equation (26c)
determines mλ as function of cross-sectors demand patterns (see also later
equations (30) and (31)). mλ can be regarded as a productivity indicator of
aggregate production: it captures the changes in aggregate productivity that are
caused by factor-reallocations across technologically distinct sectors (junior and
senior sector), since mλ depends only on the allocation of labor across the junior
and senior sectors: ⎟⎟⎠
⎞⎜⎜⎝
⎛+= SJm llβχανλ .2 Furthermore, equation (26c) determines
the mλ that is consistent with the efficient (intratemporal) allocation of factors
across sectors, since equation (26c) can be derived from equations (14) and (21)
(among others); see as well the derivations in APPENDIX A. (Equations (14)
state that all factors must be used in production, i.e. no factors are wasted;
equation (26c) requires that marginal rates of technical substitution are equal
across sectors, i.e. factors are efficiently allocated across sectors.)
2 This equation can be derived by using equation (A.23) from APPENDIX A and equations (14) and (19).
342
3.3 Sectors
(28) )(
)()(ανχβ
λαχβνβαδε−
−+−⎟⎟⎠
⎞⎜⎜⎝
⎛ +++= m
JJ
J YKK
YH
YEl
&
(29) αχ
ανχβλαχβνβαε
)()()(
−−+−
⎟⎠⎞
⎜⎝⎛ += m
SS
S YH
YEl
(30) NLEEJ =
(31) ⎟⎠⎞
⎜⎝⎛ −=
NLEES 1
(32a) m
m
i
i
lk
lk
χβαν
= for nmi ,...1+=
(32b) im
mi l
lk
k = for mi ,...1=
(33a) m
m
i
i
lz
lz
χγαμ
= for nmi ,...1+=
(33b) im
mi l
lz
z = for mi ,...1=
where ∑=
≡m
iiJ
1
εε .
For a proof of these equations see APPENDIX B.
We can see that ageing (i.e. changes in L/N) leads to demand shifts between the
junior- and the senior-sector (equations (30) and (31)). These demand shifts lead
to changes in factor allocation between these two sectors (here shown by changes
in employment shares; see equations (28) and (29)). Further factor-reallocation
between the senior- and the junior-sector is caused by changes in aggregate
capital demand (since only the junior-sector produces capital) and by changes in
aggregate intermediates demand.
343
Proposition 2: Capital intensity (intermediates-intensity) is lower in the senior
sector in comparison to the capital-intensity (intermediates intensity) in the junior
sector, provided that χβαν < ( )χγαμ < .
Proof: Since capital intensity in a subsector i is given by iLlKk
i
i ∀, , and
intermediates intensity in a subsector i is given by iLlZz
i
i ∀, , equations (32) and
(33) imply this proposition. Q.E.D.
4. Effects of ageing To study the effects of ageing I compare the economy without ageing (L/N =
constant) to the economy with ageing (L/N decreases), ceteris paribus. In all the
following argumentation, ageing (i.e. a change in L/N or a change in N
LN − )
means that Ng changes and not Lg . That is, I assume that L is independent of
ageing (i.e. it grows at constant rate Lg irrespective of whether ageing takes place
or not). In this way I can clearly distinguish between growth-effects of ageing via
factor-reallocation (which are in the focus of my paper) and growth effects of
changes in labor supply (i.e. changes in the growth rate of L). The latter are well
known from standard (one-sector) models.
(Working)Definition 2: Ageing stands for an increase in N/L, where Lg is
constant.
344
In the next section (4.1) I discuss the equilibrium without ageing. In section 4.2, I
analyze the effects of ageing in a simpler version of my model, where only cross-
sector-differences in TFP-growth are allowed for. In section 4.3 the effects of
ageing are analyzed in the general version of the model, where it is allowed for
cross-sector differences in input-elasticities as well.
4.1 Partially Balanced Growth Path (PBGP) without ageing In this subsection I assume that there is no ageing, i.e. L/N = constant.
Definition 3: A “partially balanced growth path” (“PBGP”) is an equilibrium
growth path where NYYEK ˆ ,ˆ,ˆ,ˆ and mλ are constant.
The name “partially balanced growth path” reflects the fact that along the PBGP
some variables ( NYEKY and ,, ) behave as if they were on a balanced growth
path (steady state), while the other variables (e.g. GDP) do not behave in this
manner, i.e. they grow at non-constant rates, as we will see soon. (This concept is
similar to the concept of “aggregate balanced growth”, which is used by Ngai and
Pissarides (2007).)
Lemma 1: There exists a unique PBGP of the dynamic equation system (24)-(26),
provided that L/N is constant.
Proof: It can be seen at first sight that equations (24)-(26) imply that there is an
equilibrium growth path where NYYEK ˆ ,ˆ,ˆ,ˆ and mλ are constant, provided that
L/N is constant. Q.E.D.
345
Lemma 2: Along the PBGP, the growth rate of the variables NYEKY and ,, is
given by
(34) .)1(
)1(* constgggg LSS
BSAS =++−+−
=χγεαμε
γεμε
(where L/N is constant).
Proof: This lemma is implied by Definitions 1 and 3. Q.E.D.
Lemma 3: Along the PBGP, factors are not shifted between the senior- and the
junior-sector, i.e. Jl and Sl are constant, (where L/N is constant).
Proof: This lemma is implied by equations (28)-(31), by Definitions 1 and 3 and
by Lemma 2. Q.E.D.
Definition 4: An asterisk (*) denotes the PBGP-value of the corresponding
variable.
Now, I derive the PBGP-values of variables as functions of exogenous
parameters:
Lemma 4: Along the PBGP, the variables PDGYYEK Nˆ,ˆ ,ˆ,ˆ,ˆ and mλ are given
by the following functions of exogenous model parameters (where L/N is
constant)
(35a) *11
*ˆm
csK λ−=
(35b) *11
1*ˆm
ccc
ssE λρα −− +=
(35c) ανχβ
λαχββα−
−+−= −
*1* )()(ˆ mc
c vsY
346
(35d) )(ˆ *1*m
cc
N sY βλα += −
(35e) s
NLN
NLN
S
S
m
βρανχβεμαχγα
ανχβεμβνγβ
βαλ
−−+−+
−−−−+
=)()(
)()(*
(35f) 1
***2*1* )1()1()1()(ˆ−
−⎥⎦
⎤⎢⎣
⎡−
−−
−−++= Smmmc
c
psPDGανχβμαβλβλαβλα
where
(35g)
[ ] )(1
)(1)()(
11
1
*
αμγχεα
α
χεμγγαμγχεαανχβ
μαγχε
αμαναχ
εγμγ
νβ
χα
−+−+−
−+−
−
−
= ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛= ∏
S
S
S
i
BAsp c
n
iiS
(35h)
cg
gs
GL −+++
≡
1ρδ
β .
Proof: To determine the PBGP-values of NYYEK ˆ ,ˆ,ˆ,ˆ and mλ we have to set
0ˆ =K& and 0ˆ =E& (because of Definition 3). Then equations (24)-(27) imply
Lemma 4. Remember that in this section L/N is constant. Q.E.D.
Lemma 5: The young-to-old ratio (NL ) has an impact on the PBGP-levels of
aggregate variables ***** ˆ,ˆ ,ˆ,ˆ,ˆ PDGYYEK N and *mλ (where L/N is constant).
Proof: This lemma is implied by equations (35). Q.E.D..
Lemma 6: *ˆPDG does not grow at constant rate along the PBGP (even when N/L
is constant).
347
Proof: This lemma is implied by (35f). Note that equation (35g) implies that *Sp
is not constant along the PBGP. Q.E.D.
Lemma 6 shows a quite convenient feature of my model: we can study the rich
dynamics of the GDP (where the reallocation-effects of ageing cause unbalanced-
growth of GDP) while the other variables are on a (partially) balanced growth
path (partial steady state). This fact makes it possible to analyze the impacts of
ageing without simulations.
Lemma 7a: A saddle-path, along which the economy converges to the PBGP,
exists in the neighbourhood of the PBGP of the dynamic equation system (24)-
(26).
Lemma 7b: If intermediates are omitted (i.e. if 0== μγ ), the PBGP of the
dynamic equation system (24)-(26) is locally stable.
Proof: see APPENDIX C.
Corollary 1: Even if the initial capital level is not given by equation (35a), the
economy (which is described by the aggregate equation system (24)-(26))
converges to the PBGP, provided that L/N is constant.
Proof: This corollary follows from Lemmas 1, 4 and 7. Q.E.D.
4.2 Ageing and cross-sector differences in TFP-growth In this subsection I provide a simpler version of my model, which is helpful to
understand the general mechanism which leads to the reallocation effects of
348
ageing. I assume now that input-elasticities are equal across sectors, i.e.
νβχα == , and thus μγ = . Furthermore, I assume that ageing takes place.
Lemma 8: If νβχα == , and thus μγ = , equations (24)-(35) become:
(24)’ Kc
ggEKK G
Lc ˆ)
1(ˆˆ)(ˆ
−++−−+= δβα&
(25)’ c
ggK
EE G
Lc
−−−−−= −
1ˆ
ˆˆ
1 ρδβ&
(26a)’ cKY ˆˆ =
(26b)’ )(ˆˆ βα += cN KY
(26c)’ 1=mλ
(26d)’ YYYH Nˆˆˆˆ γ=−=
(27)’ 1
1)(ˆˆ−
⎥⎦⎤
⎢⎣⎡ −++=
BBAlKPDG S
c βα
(28)’ ⎟⎟⎠
⎞⎜⎜⎝
⎛ +++=
YKK
YH
NL
YEl JJ
δε&
(29)' ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛ −= SS Y
HNL
YEl ε1
(34)' LBSAS g
ggg +
+−=
αγεγε )1(*
(35a)’ csK −= 11
*ˆ
(35b)’ ccc
ssE −− += 11
1*ˆ ρα
(35c)’ cc
sY −= 1*ˆ
(35d)' )(ˆ 1* βα += −cc
N sY
349
(35f)' 1
1* 1)(1)(ˆ−
−
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛ −+
−++= S
cc
NLs
BBAsPDG γεραβα
where 10 <+
=<βα
βc , βαβα
εε
βα εγ+−−
=
+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎦⎤
⎢⎣⎡≡ ∏
1
1
1 n
ii
i
S
ABAG .
Proof: The proof is quite straight-forward. Therefore, I omit it here. Note that
following steps are necessary to obtain equation (27)’: By inserting equation
(26c) into equation (27). the following equation can be obtained:
[ ] 1
2 )1)(1(/1
)/ˆ/(ˆ/)()(ˆˆ−
−⎥⎦
⎤⎢⎣
⎡−−
+−−−
−++= SSS
cmS
mmc
mc pKENLNKPDG μ
αχγεμελγεβλαβλαλ
This term can be reformulated by using the other equations to obtain
[ ][ ] 1)/1(ˆ/ˆ/)(1)(ˆˆ −
−−+−+= BAYENLNKPDG Sc γεβα . Then, by using
equations (26d)’ and (29)’, equation (27)’ can be derived. Q.E.D..
Lemma 9: If input-elasticities are equal across sectors, there exists a unique
PBGP, irrespective of whether ageing takes place or not, and irrespective of the
rate of ageing.
Proof: Lemma 8 implies that equations (24)’-(26)’ apply here. The proof of
Lemma 9 can be seen directly from equations (24)’-(26c)’, which are nearly the
same as in the standard one-sector Ramsey model. Since equations (24)’-(26)’
are not dependent on L/N, the existence of the PBGP is not affected by changes in
L/N. Q.E.D.
Lemma 10: If input-elasticities are equal across sectors, the growth rate of the
variables NYEKY and ,,, is given by equation (34)’ along the PBGP.
350
Proof: This lemma is implied by Lemma 8 and Definitions 1 and 3. Q.E.D.
Lemma 11: If input elasticities are equal across sectors, the PBGP is globally
saddle-path stable, irrespective of whether ageing takes place or not.
Proof: Lemma 8 implies that equations (24)’-(26)’ apply. Equations (24)’ and
(25)’ are the same as in the standard Ramsey-model regarding all relevant
features. Therefore, the aggregate system of my model behaves like the standard
Ramsey-model, i.e. it is globally saddle-path stable. (See also Ngai and Pissarides
(2007) on the stability of such frameworks.). Since equations (24)’-(26)’ are
independent of L/N, ageing has no impact on the stability of the PBGP. Q.E.D.
Corollary 2: When input-elasticities are equal across sectors, ageing is irrelevant
regarding the development of the variables NYandEKY ,,, in my model: Neither
the PBGP-growth rate *g nor the PBGP-levels **** ˆ andˆ,ˆ,ˆNYYEK are affected
by (the level or the growth rate of) L/N. A change in L/N does not induce a
deviation from the (initial) PBGP with respect to NYandEKY ,,, .
Proof: This corollary is implied by Lemmas 8-10 and equations (35). Q.E.D.
Now we take a look at the disaggregated variables of the economy.
Theorem 1: If input-elasticities are equal across sectors, ageing shifts demand
from the junior-sectors to the senior-sectors along the PBGP. That is, decreases
in L/N lead to decreases in EEJ / and increases in EES / .
351
Proof: This theorem is implied by equations (30) and (31). Remember that, as
argued in section 2, the choice of the numéraire is irrelevant when looking at
shares or ratios. Q.E.D.
Theorem 2: If input-elasticities are equal across sectors, ageing reallocates
factors from the junior-sectors to the senior-sectors along the PBGP; i.e.
decreases in L/N lead to decreases in Jl and increases in Sl .
Proof: This theorem is implied by Lemma 8 and equations (28)’ and (29)’.
Q.E.D.
Theorem 3: If input elasticities are equal across sectors, ageing reduces the
growth rate of GDP along the PBGP, provided that the TFP-growth rate (and the
TFP-level) is lower in the senior sector in comparison to the junior sector. That
is, a decreasing L/N causes a reduction of the GDP-growth rate, provided that
A>B and BA gg > .
Proof: This theorem is implied by Lemma 8 and equation (35f)’. Q.E.D.
Corollary 3: If input-elasticities are equal across sectors, ageing shifts demand
from the junior-sectors to the senior-sector. These demand shift cause factor
reallocation from the junior-sector to the senior-sector. This reallocation process
reduces the growth rate of GDP provided that the senior-sector has a relatively
low TFP(growth-rate) in comparison to the senior sector.
Proof: This corollary is implied by Theorems 1-3. Q.E.D.
352
Hence, whether ageing increases or decreases the GDP-growth-rate depends only
on the TFP-relation between the junior and senior sectors. The factors, which
determine the strength of the ageing-impact, are analyzed in the next section.
As argued in section 2, the choice of the numéraire is irrelevant when looking at
shares or ratios. Hence, we can analyze the senior-goods-consumption-to-output
ratio ( NS YE / ) without worrying about numéraire choice. The share of senior-
budget in aggregate output ( NS YE / ) increases at the same rate as the old-to-
young ratio (see equation (31) and remember that along the PBGP E and NY grow
at the same rate).
All the results from this section are valid for the case that the budget devoted to
seniors (e.g. old age pensions) develops according to the social welfare function
(representative household utility function). If however political issues led to a
reduction of old age pensions, the ageing-impacts would be weaker. We will
discuss this case later.
4.3 Ageing and cross-sector differences in input-elasticities Now let us assume that input-elasticities differ across sectors, i.e. νβχα ≠≠ ,
and μγ ≠ . (The TFP-growth rates differ across sectors as well.) Furthermore, in
this paper I analyze only the case where the capital intensity in the senior sector is
lower in comparison to the junior sector (i.e. χβαν < ), since this case is in
general assumed in the literature (see also Proposition 2). I assume that initially
the economy is in the equilibrium described in section 4.1 with L/N = constant. In
sections 4.3.1 and 4.3.2 I analyze what happens if there is a one time decrease in
353
L/N (according to Definition 2). (After this decrease L/N is constant again.) In
section 4.3.3 I generalize my results to the case where L/N increases
consecutively. Furthermore, in section 4.3.1 I analyze the effects of ageing on net-
output and on the pension-to-output ratio and I derive the impact channels,
whereas in section 4.3.2 I look at the differences in this analysis when my GDP-
measure is taken into account.
4.3.1 Productivity effect: Impacts and channels In this subsection the term “aggregates” refers only to EYY N ,, and K but not to
GDP.
Lemma 12: A one time decrease in L/N leads to a change of the PBGP. That is,
the economy leaves the old PBGP and there is a transition period where the
economy converges to the new PBGP. The growth rate of aggregates ( *g ) is the
same along the old and the new PBGP.
Proof: Remember that I assume here again that the input-elasticities differ across
sectors; hence equations (24)-(35) apply here. Equations (35) imply that there
must be a transition period, since the old and the new PBGP require different
equilibrium capital levels; i.e. *K depends on L/N. (That is, the capital level that
exists when the decrease in L/N occurs, is not the same as the capital level that
brings the economy directly on the new PBGP; we know from the discussion of
the standard one-sector Ramsey-model that this induces a transition period, where
the economy is converging to the new PBGP.) Furthermore, Lemma 7 and
Corollary 1 imply that the economy will converge to the new PBGP (provided
that the decrease in L/N is not too strong). Equation (34) implies that the growth
354
rate of aggregates ( *g ) is the same along the old and the new PBGP, since *g
does not depend on L/N. Q.E.D.
Lemma 13: A one-time decrease in L/N reduces the growth rate of aggregates
during the transition period between the old and the new PBGP. That is, the
growth-rate of aggregates ( NYandEK ,, ) during the transition period is lower
in comparison to the growth rate of aggregates along the (old and new) PBGP
( *g ).
Proof: To prove this lemma we need the following derivatives of equations (35).
(Note that my key results would not change, if I calculated here elasticities
instead of derivatives.)
(36a) 0ˆ
11
*
*
>=∂∂ −c
m
sKλ
(36b) 0ˆ
11
*
*
>=∂∂ −c
m
sE ρλ
(36c) ανχβαχβ
λ −−
=∂∂ − )(ˆ
1*
*c
c
m
sY
(36d) 0ˆ
1*
*
>=∂∂ −c
c
m
N sY βλ
(36e) 0
)()(
)()()( 2
*
<
⎥⎦
⎤⎢⎣
⎡ −−+−+
−++−+−−=
⎟⎠⎞
⎜⎝⎛ −∂
∂
sN
LN
ss
NLN
S
SSm
βρανχβεμαχγα
βρεμβγνρεμαχγα
βαανχβ
λ
355
From these equations we can see that a one-time decrease in L/N leads to a
decrease in *mλ . The decrease in *
mλ leads to a decrease in *** ˆ and ,ˆ,ˆNYEK .3
Hence, the values of *** ˆ and ,ˆ,ˆNYEK along the new PBGP are lower in
comparison to those of the old PBGP. Therefore, we can conclude that
*** ˆ and ,ˆ,ˆNYEK decrease during the transition period. Hence, the growth rate of
aggregates ( NYEK and , ) during the transition period is lower than *g .
(Remember that Definitions 1 and 3 and Lemma 2 imply the following: if
*** ˆ and ˆ,ˆNYEK are constant, aggregates ( NYEK and , ) grow at the constant rate
*g ; hence, if *** ˆ and ˆ,ˆNYEK decrease, the growth rate of NYEK and , is lower
than *g . Note that this argumentation works, since “efficiency units” are the same
along the old and the new PBGP: I express the variables in efficiency units (see
Definition 1) as follows: e.g. cLG
YY−
≡1
1ˆ ; since cLG −1
1
does not change due to
ageing, efficiency units are the same along the old and the new PBGP.) Q.E.D.
I will discuss the intuition behind this lemma soon; at first I postulate two lemmas
which are helpful to understand Lemma 13.
Lemma 14: A one-time decrease in L/N shifts demand from the junior-sector to
the senior-sector. That is, along the new PBGP EES is higher (
EEJ is lower) in
comparison to the EES (
EEJ ) of the old PBGP.
3 Note that the effects of ageing on aggregate gross-output Y may be positive or negative depending on the sign of the term αχ − . This reflects the fact that depending on the input-elasticities ageing can lead to an increase in intermediates production that is stronger than the decrease in net-output and vice versa.
356
Proof: This lemma is implied by equations (30) and (31). Q.E.D.
Lemma 15: A one-time decrease in L/N leads to factor reallocation from the
junior-sector to the senior-sector. That is, along the new PBGP Sl is higher in
comparison to the Sl of the old PBGP.
Proof: By using equation (A.23) from APPENDIX A; it can be shown that the
employment share along the PBGP is given by ανχβ
χβλ−
−= )1( **mSl . Since
equation (36e) implies that 01
*
<⎟⎠⎞
⎜⎝⎛ −∂
∂
NL
mλ , a decrease in L/N leads to a decrease in
*
mλ . Therefore, *Sl increases due to a decrease in L/N. (Remember that I assume
that 0>−ανχβ .) Q.E.D.
Now, I discuss the intuition behind Lemma 13: We know that output is produced
by using labor and capital. Ageing shifts demand (and thus production factors)
towards senior-sectors (as implied by Lemmas 14 and 15). The key feature of the
senior-sectors is that capital is less productivity-enhancing in comparison to the
(junior-sectors). This is reflected by the fact that optimal capital intensity in the
senior-sector is lower in comparison to the junior-sector (see Proposition 2).
Hence, the ageing-induced (one-time) demand-shift implies that aggregate capital
becomes less productive when looking at the economy-wide-averages. Therefore,
at the aggregate level a one-time decrease in L/N acts similarly like a negative
357
productivity-shock (a decrease in the productivity of capital).4 This leads to the
negative impacts on aggregate net-output-growth, aggregate capital-growth and
aggregate consumption-expenditures-growth (and of course, the savings rate
decreases, since savings which are invested in capital become less rentable, i.e.
the opportunity costs of consumption decrease). This adjustment-process occurs
during the transition period. Since a one-time productivity-level-shock has no
impacts on productivity-growth rates, the economy converges to a growth path
(PBGP) where the growth rate is the same as before. (Remember that steady state
growth rates are determined only by productivity-growth and not by productivity-
levels within the standard (one-sector) Ramsey-model; in this respect my
aggregate model is the same as the standard Ramsey model.)
A further interesting question is about the effects of ageing on the senior-budget-
to-the-net-output-ratio ( NS YE / ). Equations (31), (35b) and (35d) imply the
following derivative (consider also equation (36e)):
(37) 01)(ˆ
ˆ)ˆ/ˆ(2*
*
*
***
>⎟⎠⎞
⎜⎝⎛
−++
+−
⎟⎠⎞
⎜⎝⎛ −∂
∂−=
⎟⎠⎞
⎜⎝⎛ −∂
∂c
ggsN
LN
NLNY
E
NLN
YE GL
m
m
N
NS δβλααλ
Hence, we can see that ageing increases the senior-budget-to-output-ratio. The
first term on the right-hand-side of equation (37) may be regarded as the direct
effect of ageing (i.e. an increase in the old-to-young ratio, increases the share of
the seniors in the overall consumption-to-output-share).5 The second term may be
4 This fact is reflected by the ageing-induced decrease in *
mλ , which is implied by equation (36e);
as discussed in section three *mλ can be interpreted as a productivity indicator, which reflects the
aggregate impacts of cross-sector factor-reallocation. 5 since **** /*/)(/ NNS YENLNYE −= .
358
regarded as an indirect effect: the increase in the old-to-young ratio leads to an
increase in the overall consumption-to-output-share6 (i.e. as noted above the
savings-rate decreases due to lower productivity level).
We can see from equations (36) and (37) that a rich “portfolio” of parameters
determines the strength of the impact of ageing. (Note that this portfolio would
not change, if I calculated elasticities instead of derivatives in equations (36) and
(37).) These parameters are:
a) technology parameters: input-elasticities of sectoral production functions
(including labor, capital and intermediates elasticities), TFP-growth-rates (via
Gg ) and the depreciation rate
b) time preference rate
c) old-to-young ratio and the growth rate of labor.
The reason for the fact that so many parameters determine the impact of ageing is
the following: The demand-shift across technologically distinct sectors makes it
necessary to change the (average) aggregate structure of the economy, especially
the ratios between aggregate capital, labor and aggregate intermediates. The
sectoral technology parameters (especially the input-elasticities) determine how
strong this change has to be. Furthermore, since changes in capital in general
require an adjustment of the savings rate (*
*
ˆˆ
1YE
− ), all the variables which
determine the savings rate come into account, especially the parameters captured
by the auxiliary variable s; see equations (35), e.g. the time-preference rate. The
parameters, which determine the savings rate, contain all those variables which
6 since ( ) ( ) 01)(/)(/)(
)ˆ/ˆ(2*
***
>⎟⎠⎞
⎜⎝⎛
−++
+−∂∂
−=−∂
∂c
ggs
NLNNLNYE G
Lm
mN δβλααλ
.
359
are already known to determine the savings-rate of the standard Ramsey-model
(see the auxiliary variable s). However, my model provides a sector foundation of
those parameters: especially, Gg and c are assumed to be exogenous in the
standard Ramsey-model, while in my model these two variables are functions of
sectoral parameters.
4.3.2 Additional impacts on GDP: The price-effect Remember that I have shown in the previous section that a one-time increase in
L/N leads to a transition from the old PBGP to a new PBGP. Due to this fact, the
effect of ageing on real GDP-growth can be divided into transitional effects and
PBGP-effects. Transitional effects have an impact on the real GDP-growth-rate
during the transition between two PBGP’s, while PBGP-effects of ageing have an
impact on the growth rate along the new (PBGP). In fact I have shown that the
effects from the previous section are transitional. In this section I will introduce a
new effect, which affects real GDP-growth (but not the growth rate of other
aggregate variables). I name this effect price effect, and I show that this effect is
not only transitional.
4.3.2.1 Transitional effects of ageing on GDP To show these facts we have to calculate the derivative of equation (35f):
(38)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−−
−−−⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
+′
=⎟⎠⎞
⎜⎝⎛ −∂
∂
++−
−
4444 84444 76444 8444 7644 844 76 )(
**
)(
*
)(
*2*
*1* 1)1()1()()(
)(ˆ
χανχβγμ
χανββα
βλαλβ S
SSm
mc
c
lplp
s
NLN
PDG
360
where ⎟⎠⎞
⎜⎝⎛ −∂
∂≡′
NLN
mm
** )( λλ , )1(1)1(1 *
*
**
Sm
m pp −+−
−−
−=βλαλ
ανχβμαβ and
ανχβχβλ−
−= )1( **mSl .
and where *Sp is given by equation (35g).
(For an explicit proof see APPENDIX D.) A “(+)” (a “(-)”) above a term denotes
that this term is positive (negative).
Theorem 4: A one-time decrease in L/N has a negative impact on the growth rate
of GDP during the transition between the old and the new PBGP, provided that
senior goods are “more expensive” in comparison to junior-goods; i.e. provided
that 1* >Sp , where *Sp is given by equation (35g).
Proof: This theorem is implied by equation (38). If 1* >Sp , an increase in the
old-to-young-ratio has a negative impact on the *ˆPDG -level (and hence a
negative impact on the GDP-growth-rate during the transition period; see also the
argumentation in the proof of Lemma 13). Note that *Sp is always positive and
determined by exogenous parameters. Furthermore, note that the relative price of
senior goods is given by *Sp (see proposition 1) and the price of junior goods is
given by 1. The latter comes from the fact that sector m is numéraire (see
equation (15a)) and belongs to the junior-sector and all junior sub-sectors have
identical production functions (see also equations (A.5) and (A.6) in APPENDIX
A). Q.E.D.
361
If 1* <Sp , the effect of an increase in the old-to-young ratio may be positive or
negative, depending on the parameter constellation, where the effect can be
positive provided that *Sp is relatively small (i.e. relatively close to zero). To
isolate a set of parameter-values, which ensures that the GDP-effect of ageing is
positive when *Sp is relatively close to zero, we have to calculate the limit-value
of the term within the squared brackets of equation (38), i.e.
(39)
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−−
+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
−−
−−−⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
→
*
***
0
)(2)()(
1)1)(1(lim
S
SSSp
l
lplS
ανχβαναβχβα
χανχβγμ
χανββα
where ανχβ
χβλ−
−= )1( **mSl .
If (39) is negative, equation (38) implies that for small values of *Sp the effect of
ageing is positive regarding GDP-growth. Equation (39) implies that, e.g., αβ <
is a stronger than necessary condition for this. (Remember that I assume that
χβαν < .)
Now, the question is which parameter constellations ensure that *Sp is relatively
small.
Lemma 16: In the limit 1* >Sp ( )1* <Sp , provided that the growth rate of labor-
augmenting technological progress in the junior-sector is higher (lower) in
comparison to the growth rate of labor-augmenting technological progress in the
senior-sector, i.e. provided that BA ggχα11
> ( BA ggχα11
< ).
362
Proof: We know from equation (35g) that the actual level of *Sp is determined
by a time-variant term )/( αχ BA and by a constant term
( ανχβμαγχ
εαμαναχ
εγμγ
νβ
χα −
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛ ∏ sn
ii
i
1
). αχ BA / approaches infinity (zero),
provided that BA ggχα11
> ( BA ggχα11
< ). Thus, in the limit *Sp approaches
infinity (zero) as well, i.e. *Sp becomes larger (smaller) than 1. Q.E.D.
Hence, depending on the parameter setting, several cases can exist: (1) *Sp can
be relatively close to zero in the beginning, but approach to infinity with time; (2)
*Sp can be relatively close to zero in the beginning and approach to zero with
time; (3) *Sp can be relatively large in the beginning but approach to zero with
time; (4) *Sp can be relatively large in the beginning and approach to infinity
with time.
These cases and the discussion above (about equations (35g) and (38)) imply that
ageing may have positive and negative impacts on GDP-growth (during the
transition period) depending on the exact constellation of parameters from
equations (35g) and (38). Moreover, the effect of ageing may change with time
(in cases (1) and (3)), i.e. in the beginning the effect on GDP-growth may be
positive (negative) but later negative (positive).
Nevertheless, in the limit only the term αχ BA / (together with equation (38))
determines whether a future increase in the old-to-young ratio leads to an increase
or to a decrease in GDP(-growth). Hence, from the today’s point of view the
growth rate of this term (namely BA gg αχ − ) is deciding for the question about
363
the (distant) future impacts of ageing: If 0>− BA gg αχ (or: BA ggχα11
> ) we
know that *Sp approaches infinity. Hence, we know that sooner or later ageing
will have negative (transitional) effects on GDP-growth. Otherwise, if
0<− BA gg αχ (or: BA ggχα11
< ) we know that sooner or later ageing could
have positive (transitional) effects on GDP-growth. This seems to be a quite
convenient rule of thumb. Especially, since in this way the effects of ageing are
related to two quite comprehensible and estimable parameters: in fact the
production functions, which I assumed, imply that Agα1 and Bg
χ1 are the
growth rates or labor-augmenting technological progress in the senior sector and
junior sector respectively. Nevertheless, this is only a rule of thumb, since the
other variables from equation (35g) may be dominant for a long period of time, if
BA gg αχ − is not very large (i.e. if αχ BA / changes slowly).
Theorem 5: In the limit, a one-time decrease in L/N has a negative impact on the
growth rate of GDP during the transition between the old and the new PBGP,
provided that the growth rate of labor-augmenting technological progress in the
junior-sector is higher in comparison to the growth rate of labor-augmenting
technological progress in the senior-sector, i.e. provided that BA ggχα11
> .
Proof: This theorem is implied by Theorem 4 and Lemma 16. Q.E.D.
To understand why it is important for the GDP-effects of ageing whether *Sp <1
or >1, we have to remember that I have shown in the proof of Theorem 4 that *Sp
364
is the price of senior-sector-goods and that the price of junior-sector-goods is
equal to unity. Hence, *Sp <1 (>1) means that senior-goods are less (more)
expensive than junior-goods. Furthermore, with respect to GDP-growth ageing
has two types of effects:
a) The “productivity effect” has already been discussed in section 4.3.1. I stated
there that ageing acts like a negative productivity shock, i.e. it leads to a
decrease in net-output ( NY ), provided that capital intensity in the senior sector
is lower in comparison to the junior-sector. This effect affects the GDP
measure, since p
YGDP N≡ (see equation (16c)).
b) “Price effect”: Remember that we divide our net-output ( NY ) by the price-
index ( )p to obtain GDP. Hence, the changes in p have an impact on GDP
as well. Ageing leads to changes in p , since the ageing-induced demand-shift
leads to changes in output-shares, which have been used to weight the prices
of the price index (see equation (15b)). Hence, if the price of the senior sector
is lower (higher) in comparison to the price of the junior-sector, ageing
induced demand-shifts lead to a decrease (increase) of p (since the relatively
inexpensive senior-goods become a stronger weight in p ). The price effect
increases (decreases) GDP, provided that the senior-sector price ( *Sp ) is
lower (higher) in comparison to the junior-sector price ( = 1).
Hence, if *Sp >1, both effects (the productivity effect and the price effect) point to
the same direction, i.e. GDP-growth decreases. On the other hand, if *Sp <1, the
productivity effect has a negative impact on GDP-growth, but the price effect
increases GDP-growth. Hence, it is deciding which of those effects is stronger.
365
Summary: If the model parameters (from equation (35g)) are such that the price
of senior-sector-goods is relatively low, ageing may have positive transitional
impacts on GDP. For example, if parameters from equation (35g) are such that
*Sp is close to zero and if αβ < , ageing has a (temporary) positive effect on
GDP-growth, since in this case the positive price effect is stronger than the
negative productivity effect. However, whether the transitional effects of a future
decrease in L/N will be positive depends on the growth rate of αχ BA / (which
determines the growth rate of the senior-goods-price, and hence the price effect).
On the other hand, if the model parameters are such that the price of the senior
sector ( *Sp ) is higher than the price of the junior sector ( =1), ageing has a
negative transitional impact on GDP-growth, since the productivity effect and the
price effect point to the same direction. Whether future ageing will have negative
(transitional) effects in this case depends on the development of the term αχ BA /
(and on the parameters of equation (35g)). Equations (35g) and (38) imply that
the parameter-portfolio, which determines the strength and direction of the
ageing-impact, comprises:
a) sectoral labor-, capital- and intermediates-elasticities of output
( iεμνχγβα ,,,,,, )
b) the parameters, which determine in neoclassical growth models the steady-
state savings rate (e.g. the time-preference rate, depreciation rate) (via parameter
“s” in equation (35g))
c) the relative level and growth rate of labor-augmenting technological
progress in junior-sector in comparison to the senior sector (via the term αχ BA / )
366
d) population-parameters (the old-to-young ratio (N
LN −) via *
Sl and the
growth-rate of labor ( Lg ) via parameter “s”).
4.3.2.2 PBGP-effects of ageing In this subsection I show that ageing has not only transitional effects on GDP, but
it affects the growth rate of GDP along the PBGP.
Theorem 6: A one-time decrease in L/N reduces (increases) the PBGP-growth
rate of GDP, provided that the growth rate of labor-augmenting technological
progress in the junior-sector is higher (lower) in comparison to the growth rate of
labor-augmenting technological progress in the senior-sector, i.e. provided that
BA ggχα11
> ( BA ggχα11
< ). That is, the growth rate of GDP along the new
PBGP is lower (higher) in comparison to the growth rate of GDP along the old
PBGP, provided that BA ggχα11
> ( BA ggχα11
< ).
Proof: PDG ˆ along the PBGP is given by equation (35f). Along the PBGP all
terms of equation (35f) are constant beside of *Sp , which is given by equation
(35g). Therefore, we obtain the following growth-rate:
)1()1()1(
)1()1(
ˆˆ
***
**
*
*
Smm
Sm
p
p
PDGPDG
−−−
−−+
−−
−−=
ανχβμαβλβλα
ανχβμαβλ &&
Calculating the derivative of this growth rate implies:
367
(40) 2***
***
*
)1()1()1(
)1()/(
ˆˆ
⎥⎦
⎤⎢⎣
⎡−
−−
−−+
+−−
⎟⎠⎞
⎜⎝⎛∂
∂=
∂
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛∂
Smm
Sm
pp
NLNL
PDGPDG
ανχβμαβλβλα
βαανχβμαβλ
&
&
Equation (36e) implies that ⎟⎠⎞
⎜⎝⎛∂
∂
NLm
*λ >0. Furthermore, remember that I assume
0>−ανχβ .
Equation (35g) implies *Sp& > 0 , if BA gg
χα11
> , and *Sp& < 0, if BA gg
χα11
< .
Hence, equation (40) is positive (negative), if BA ggχα11
> ( BA ggχα11
< ). That
is, a decrease in L/N has a negative (positive) impact on the GDP-growth rate
along the PBGP, provided that BA ggχα11
> ( BA ggχα11
< ). Q.E.D.
4.3.3 Dynamic aspects By now, in this section I have analyzed the impacts of a one-time increase in the
old-to-young ratio. If ageing is not regarded as a one-time increase but as a
sequence of (discrete) increases in the old-to-young ratio, my results still remain
applicable: Since I have shown in Lemma 7 (Corollary 1) that the PBGP is
saddle-path-stable, the economy will be on the converging path. The qualitative
results remain the same. The overall magnitude of the change in the macro-
variables (e.g. in GDP) is determined by the sum of the changes in the old-to-
young-ratio (overall-change in the old-to-young ratio). Only the period of change
(the transition period) is more prolonged, since the overall change is dispersed
over a sequence (i.e. the economy cannot reach the “final” PBGP before the
sequence is finished).
368
5. Concluding remarks In this paper I have specified how ageing affects the GDP-growth rate and the
pension-to-GDP-ratio via factor-allocation effects. In comparison to the previous
literature I have shown that ageing affects these variables along additional
channels, for example:
• in contrast to Groezen et al. (2005), ageing-impacts arise in my model
despite of the fact that the elasticity of substitution between capital and labour is
equal to unity;
• in section 4.3.2 I introduced the “price effect”.
Moreover, my model adds new determinants of the ageing-impact (e.g. the fact
that capital-intensity in the senior sectors is lower in comparison to that of the
junior-sector does not necessarily lead to negative impacts of ageing, but a rich
portfolio of technology and preference parameters determines these ageing-
impacts in my model). Furthermore, in contrast to the previous literature, I have
shown (in section 4.3.2) that the ageing-effects may be non-monotonous over
time. Hence, the strength of the ageing impact (and the necessary reforms of
pension systems) may vary widely across countries and across time, depending on
the parameters derived in my model.
This non-monotonousity comes from the fact that although the senior sector has
relatively low capital intensity, the price of senior-goods needs not necessarily
being higher than the price of junior-goods. Hence, the positive “price effect” of
ageing can overweight the negative “productivity effect” of ageing; however, if
the growth rate of labor augmenting technological progress is relatively high in
the junior sectors, the senior goods must become more expensive than junior
369
goods at some point of time, i.e. the price effect becomes negative as well. (See
the following explanations and the explanations on pp.28-31.)
As shown in section 4.3.2, despite of its “complexity” my model provides quite
easily interpretable results which can be used in empirical research and policy
making:
(1) The present and past effects of ageing via structural change can be analyzed
by assessing the (market) prices of senior-goods and junior-goods. In fact my
model implies that ageing has a negative impact on real GDP-growth via
structural change, if senior goods are “more expensive” than junior goods.
(Otherwise, if senior goods are cheaper, the effects of ageing can be positive or
negative, depending on the exact model parameters which will be discussed in the
following.) In contrast to Groezen et al. (2005), my model does not imply that
low capital-intensity in the senior sector is sufficient to constitute negative effects
of ageing. That is, lower capital-intensity in the senior sector does not necessarily
imply that senior goods are more expensive in comparison to junior goods. The
reason is that TFP-plays a role for the effects of ageing as well: for example high
TFP may offset the negative effects of low capital intensity.
(2) For the discussion about the future effects of ageing we have to know which
parameters determine the development of relative prices of senior and junior
sector. Only in this way we can asses whether it makes sense to assume that in
future ageing will have negative impacts on real GDP-growth via structural
change. My results imply that in the limit (or: in the very long run) ageing has
negative impacts on real GDP-growth, provided that the growth rate of labor-
augmenting technological progress is lower in the senior sector in comparison to
the junior-sector. In this case sooner or later senior goods become more expensive
in comparison to junior goods. Hence, when discussing the future effects of
370
ageing it is important to know about the development of labor-augmenting
technological progress in the senior and junior sector. However, as mentioned in
section 4.3.2, this is only a rule of thumb, since “sooner or later” is a quite vague
concept. That is, the exact parameter restrictions (which were derived in section
4.3.2) may be the key determinant of the ageing impact for a very long period of
time.
(3) The portfolio of parameters, which determine the effects of ageing via
structural change, which has been derived in section 4.3.2, provides a range of
policies to counteract the negative impacts of ageing on real GDP-growth. My
model can help to isolate more or less efficient policies: For example, equation
(40) implies that the relative price of senior goods can be influenced by policies
that have an impact on
• the savings rate (via s in equation (40)),
• the sectoral output-elasticities of inputs,
• the growth rate of working population (via s),
• and the sectoral levels and rates of labor-augmenting technological
progress.
For example, policies, which influence the savings rate, seem to be not effective
in the very long run, since (as just explained) in the very long run the impacts of
labor-augmenting technological progress are dominant regarding the development
of the relative price of senior goods price. On the other hand, e.g., policies that
increase the birth rate seem to be relatively effective: they do not only decrease
the old-to-young ratio directly (hence reducing the rate of ageing) but they also
reduce the relative price of senior goods (via Lg from s in equation (40)).
371
Throughout the paper I assumed that parameter restrictions (5a,b) hold. In fact,
these restrictions ensure that there are no other sources of demand-shifts between
the senior and junior sector beside of that caused by ageing. However, the impact
of social welfare parameters (especially, the question how the utility of the old
and the young is weighted in a society) may be captured by deviation from these
restrictions. The question is whether pension systems (and private “savings-for-
retirement”-behavior) change systematically (i.e. whether the weight is shifted to
the old or to the young) with an increasing income. The answer to this question
has an impact on the strength of the ageing-impacts. This fact could be modeled
by a departure from the restrictions (5a,b). Furthermore, it should be mentioned
here that if fewer budgets were devoted to the old (i.e. if SE was restricted
exogenously, e.g. by an “inefficient” pension system), the ageing impacts would
be weaker. However, from the social welfare point of view this would be
suboptimal (i.e. the social welfare would be suboptimal). These facts may as well
have some explanatory power regarding differences in the strength of ageing
impacts across countries.
In section 4.3 I modeled ageing like a shock (or series of shocks) and not like a
smooth and perfectly foresighted process. The difference between these two
approaches is that the latter is more difficult to model (I could not rely on the
PBGP-results) and I would have to use simulations. Furthermore, if perfectly
foresighted the effects of ageing would be smoother, i.e. dispersed over a longer
period (i.e. even before the increase in the old-to-young ratio the effects of ageing
would show up), which would affect my results quantitatively but not
qualitatively (i.e. the impact channels would be the same). Furthermore, it should
be questioned whether it makes sense to model ageing like a smooth perfectly
foresighted process, especially when taking into account irrational or bounded
372
rational behavior of households in reality. Last but not least, since my model is
rather aimed to postulate some crude qualitative macroeconomic relationships
between economic parameters and variables, the question whether ageing is
modeled as a series of shocks or as a perfectly foresighted smooth process seems
to be less relevant.
Overall, my results imply that for assessing (the future) growth-impacts of ageing
some further empirical research is necessary to estimate the exact technological
properties of the junior and senior sector. Especially, it seems to be necessary to
link the data on demand-differences across the old and the young with the data on
technological properties of the sectors. Only in this way a general conclusion can
be drawn about the past and future strength of the ageing impacts. Furthermore,
the discussion about whether the senior sector needs necessarily having a lower
growth rate of labor-augmenting technological progress in future seems to be
interesting regarding the effects of ageing.
Last not least, as mentioned in Section 2.1, it could be interesting to endogenize
the population development (e.g. old-to-young ratio as a function of income) and
to analyze what factor reallocation impacts arise in such a model.
Analyzing these questions is left for further research.
373
APPENDIX A Inserting equations (7) and (8) into equation (21) yields:
(A.1) im
mi l
lk
k = and im
mi l
lz
z = , for mi ,...1=
(A.2) im
mi l
lk
kβν
χα
= and im
mi l
lz
zγμ
χα
= , for nmi ,...1+=
Inserting equations (A.1) and (A.2) into equation (7) yields
(A.3) miLAlY mmii ,...1for, == γβζκ
(A.4) nmiLBlY mmii ,...1for, +== μνζϑκ
where LlZz
m
mm ≡ζ ,
μν
γμ
χα
βν
χαϑ ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≡ and
LlKk
m
mm ≡κ .
Inserting equations (A.3) and (A.4) into equation (21) yields:
(A.5) mipi ,...1for ,1 ==
(A.6) nmipBAp Smmi ,...1for,1
+=≡= −− μγνβ ζκϑχ
α
It follows from equations (15a) and (21) that ihZhZ
pm
ii ∀
∂∂∂∂
= ,//
, which implies
that
(A.7) ip
hh
im
mii ∀= ,1ε
ε
Inserting equation (A.7) into equation (18) yields:
(A.8) m
mhH
ε=
Inserting equations (A.7), (A.5) and (A.6) into equation (13) yields:
(A.9) ∏=
−− ⎟⎠⎞
⎜⎝⎛=
n
immi
S
i
ABHZ
1
εγμβνε ζϑκ
αχε
374
where ∑+=
≡n
miiS
1εε
Inserting equations (A.3) and (A.5) into equation (21) yields for mi =
(A.10) HZ
ZzLAl
m
mmmγβζκ
γ=1
Solving this equation for H yields:
(A.11) γβζκγ mmm
m ALzl
H =
Inserting equation (A.9) into equation (A.10) yields
(A.12) ψκζ mm D=
where γεμγ
ενββψ
−−+−−
≡S
S
)(1)(
and
SS
i
Sn
iiA
BADμεεγ
ε
εμν
εχγαμ
χβαν
αχγ
−−−
= ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ∏
)1(11
1
.
Inserting equations (A.1) and (A.2) into equation (14) yields (remember: it
follows from equation (14) that ∑∑+==
−=n
mii
m
ii ll
111 ):
(A.13) 1
111
−
+=⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=∑ β
νχαn
mi m
mi k
ll
(A.14) ∑+=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
n
mii
m
m lzl
111
γμ
χα
It follows from equations (1)-(6) and (22) that
(A.15a) ( ) miforCC immm
ii ,...1=+−= θθ
ωω
(A.15b) nmiforp
CN
LNC ii
mm
m
ii ,...1+=+
−−= θ
θωω
Inserting equations (A.15), (A.5) and (A.6) into equation (17) yields
375
(A.16) m
mmCLNE
ωθ−
=
Inserting equations (2)-(6) and (A.3) into equation (23) yields due to equation
(A.16):
(A.17) ρδζκβ γβ −−= −mmA
EE 1&
Inserting equations (A.3)-(A.6) into equation (16a) yields (remember:
∑∑+==
−=n
mii
m
ii ll
111 ):
(A.18) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−= ∑
+=
n
miimm lALY
111
χαζκ γβ
Inserting equation (A.13) into equation (A.18) yields:
(A.19) ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
m
mmm k
lbaALY γβζκ
where
χβανχα
−
−−≡
1
11a and
χβανχα
−
−≡
1
1b
Inserting first equation (A.14) and then equation (A.13) into equation (A.11)
yields:
(A.20) ⎥⎦
⎤⎢⎣
⎡+=
m
mmm k
lddALH 21γβζκγ
where
χβανχγαμ
−
−−≡
1
111d and
χβανχγαμ
−
−≡
1
12d .
It follows from equation (A.4) that
(A.21) μν ζϑκ mm
ii BL
Yl = for nmi ,...1+= .
376
Inserting first equation (11), then equations (A.15) and (A.6) and finally equation
(A.19) into equation (A.21) yields
(A.22) Yph
BLYC
NLNl ii
mm
imm
m
ii ~~ α
χζϑκ
θθωω
αχ
μν ++−−
= for nmi ,...1+=
where
m
m
klba
YY+
≡~ .
It follows from equation (A.13) that
(A.23) ∑+=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
n
miim l
111
χβανλ
where m
mm k
l≡λ
Inserting first equation (A.22), then equations (A.16) and (A.7) and finally
equation (A.8) into equation (A.23) yields after some algebra (remember that
∑+=
=n
mii
10θ ):
(A.24) ⎟⎠⎞
⎜⎝⎛ +
−⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−−= S
m
mm Y
HYE
NLN
klba ε
αχ
χβανλ 11
where m
mm k
l≡λ .
Equations (11), (12), (16a), (17), (18) and (15a) imply:
(A.25) HEKKY +++= δ&
Inserting equation (A.12) into equation (A.19) yields
(A.26) ( ) ccm
cm KLbaGY −− += 1λλ
where μεεγγνεμεβ
SS
SSc−−−+−
≡)1(1
)1( ,
377
SS
i
Sn
iiA
BAAGμεεγ
γ
ε
εμν
εχγαμ
χβαν
αχγ
−−−
= ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ∏
)1(1
1
, and m
mm k
l≡λ .
Inserting equation (A.12) into equation (A.17) yields
(A.27) ρδλβ −−= −−− cccm LKG
EE 111&
Inserting equation (A.19) into equation (A.20) yields equation
(A.28)
m
m
m
m
klba
kldd
YH+
+=
21
γ
Equations (A.24)-(A.28) can be transformed into equations (24), (25),
(26a,b,d) and (26c). Q.E.D.
Now I only have to derive equation (27). By using equations (14), (15c), (16a),
(A.5) and (A.6) equation (15b) can be transformed as follows:
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−−−= ∑ ∑
+= +=
n
mi
n
miiSiSN
N
zHpYpYY
p1 1
)1(1
where Sp is given by equation (A.6). Now, inserting equations (A.2), (A.4) and
(A.6) yields:
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−−=
m
mmmSSN
N lzHLAlpY
Yp
ß
γμζκ
χα γ)1(1 .
where Sl is given by equation (19). Inserting equations (16b), (A.14), (A.19) and
(A.20) yields
( )⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−−
+−
+−−= μ
χγμαχγλ
λχα
S
m
mNSS
l
ddbaY
Ylpp1
1)1(1 21
378
where
χβανχα
−
−−≡
1
11a ,
χβανχα
−
−≡
1
1b ,
χβανχγαμ
−
−−≡
1
111d and
χβανχγαμ
−
−≡
1
12d . Now,
inserting equations (26a,b) and (A.23) yields
(A.29)
( )
m
Smm pp
βλαανχβμαβλβλα
+−−
−−−+=
1)1)(1(
where Sp is given by equation (A.6). Inserting equation (A.12) into (A.6) yields
after some algebra:
(A.30)
[ ] )(1
)(1)()(
1
ˆαμγχεα
α
χεμγγαμγχεαανχβ
μαγχε
αμαναχ
λεγ
μγ
νβ
χα
−+−+−
−+−−
= ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛≡ ∏
S
S
S
i
BAKp
m
n
iiS
(Hint: the following equations may be useful for obtaining (A.30): 1=++ γβα
and 1=++ μνχ ; these equations imply among others that
ανβχβμγννβ −=−+− and μαγχγνμβμγ −=−+− .)
The rest of the proof is quite simple: equation (27) can be obtained by
dividing the net-output (equation (26b)) by the price-index (equation (A.29)),
where Sp is given by equation (A.30). Q.E.D.
379
APPENDIX B Equations (A.7), (A.8), (A.16) and (A.22) from APPENDIX A and equations
(5a,b) and (19) imply equation (29).
Equation (28) can be derived in the same way as equation (29).
Equations (A.5), (A.6), (A.15a,b), (A.16) from APPENDIX A and equations
(5a,b) and (20) imply equations (30) and (31).
For a proof of equations (32) and (33) see APPENDIX A equations (A.1) and
(A.2).
380
APPENDIX C First, I show by using linear approximation that the saddle-path-feature of the
PBGP is given (Lemma 7a). Then I prove local stability by using a phase diagram
(Lemma 7b).
Existence of a saddle-path (Lemma 7a)
The study of local stability of the PBGP is analogous to the proof by Acemoglu
and Guerrieri (2008) (see there for details and see also Acemoglu (2009), pp. 269-
273, 926).
First, I have to show that the determinant of the Jacobian of the differential
equation system (24)-(25) (where mλ is given by equation (26c)) is different from
zero when evaluated at the PBGP (i.e. for *** ,ˆ,ˆmEK λ from equations (35a,b,e)).
This implies that this differential equation system is hyperbolic and can be
linearly approximated around *** ,ˆ,ˆmEK λ (Grobman-Hartman-Theorem; see as
well Acemoglu (2009), p. 926, and Acemoglu and Guerrieri (2008)). The
determinant of the Jacobian is given by:
(C.1) EK
KE
EE
KK
EE
KE
EK
KK
J ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
∂∂
∂∂
−∂∂
∂∂
=
∂∂
∂∂
∂∂
∂∂
=&&&&
&&
&&
The derivatives of equations (24)-(25) are given by:
381
(C.2)
EcKE
cggK
EE
KKcKcE
KE
EccK
EK
cgg
KccKKc
KK
mcm
cGL
ccm
mcm
ccm
c
mcm
cm
c
GL
mcm
cm
ccm
cm
c
ˆ)1(ˆˆ1
ˆˆˆ
ˆˆ)1(ˆ)1(ˆ
ˆˆ
1ˆ))1((ˆˆˆ
1ˆ))1((ˆ)(ˆˆˆ
111
112
1
111
∂∂
−+⎟⎠⎞
⎜⎝⎛
−−−−−=
∂∂
⎟⎠⎞
⎜⎝⎛
∂∂
−+−=∂∂
−∂∂
−+−=∂∂
⎟⎠⎞
⎜⎝⎛
−++−
∂∂
−+−++=∂∂
−−−−
−−−−
−−−
−−−−−−
λλβρδβλ
λλλβ
λβλαλ
δλβλαλβλαλ
&
&
&
&
where the derivatives of equation (26c) are given by
(C.3)
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎠⎞
⎜⎝⎛ −++⎟
⎠⎞
⎜⎝⎛ −+
−−
=∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎠⎞
⎜⎝⎛ −++⎟
⎠⎞
⎜⎝⎛ −+
−−
−=∂∂
−
−
−+
−
−
cm
cSS
cm
cm
cm
cSS
c
cm
m
KEc
NLN
KEc
NLN
K
KEc
NLN
KNLN
E
1
1
1
1
1
ˆˆ
111
ˆˆ
ˆ
ˆˆ
111
ˆˆ
λαβανχβ
ααμχγε
ααμχγε
λαβανχβ
λ
λαβανχβ
ααμχγε
ααμχγε
λαβ
ανχβλ
Inserting the derivatives (C.2) and (C.3) into (C.1) and inserting the PBGP-values
from equations (35a,b,e) yields after some algebra the following value of the
determinant of the Jacobian evaluated at the PBGP:
(C.4)
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎠⎞
⎜⎝⎛ −++⎟
⎠⎞
⎜⎝⎛ −+
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
−+++⎟
⎠⎞
⎜⎝⎛ −+
−−+
−−−
−=
−
−
cm
cSS
GLS
KEc
NLN
cgg
LNN
KE
NLNc
J
1**
*1
*
*
*
)()ˆ(
ˆ111
11ˆ
ˆ)1(
λαβανχβ
ααμχγε
ααμχγε
ρδααμχγε
ανχβαρ
αανχβ
We can see that the determinant evaluated at PBGP is different form zero. Hence,
the PBGP is hyperbolic. Furthermore, we can be sure that 0* <J , provided that
382
0>−ανχβ . Since I assume in my paper that the capital intensity in the senior-
sector is lower in comparison to the junior sector, the relation 0>−ανχβ holds
(see section 4.3).
Our differential equation system consists of two differential equations ((24) and
(25)) and of two variables ( E and K ), where we have one state and one control-
variable. Hence, saddle-path-stability of the PBGP requires that there exist one
negative (and one positive) eigenvalue of the differential equation system when
evaluated at PBGP (see also Acemoglu and Guerrieri (2008) and Acemoglu
(2009), pp. 269-273). Since 0* <J we can be sure that this is the case. ( 0* <J
can exist only if one eigenvalue is positive and the other eigenvalue is negative. If
both eigenvalues were negative or if both eigenvalues were positive, the
determinant *J would be positive.) Therefore, in the neighborhood of the PBGP
there is a saddle-path, along which the economy converges to the PBGP. Q.E.D.
Local stability (Lemma 7b)
In the following, I omit intermediates for simplicity, i.e. I set 0== μγ .
Furthermore, as noted above I study here only the case 0>−ανχβ (see also
section 4.3). Since αχανχβ −=− if 0== μγ , I can say as well that I study
here only the case 0>−αχ . Note, however, that the qualitative stability results
for the other case (i.e. 0<−αχ ) are the same.
To show the stability-features of the PBGP, the three-dimensional system (C.1)-
(C.3) has to be transformed into a two dimensional system, in order to allow me
using a phase-diagram. By defining the variable m
Kλ
κˆ
≡ , the system (24)-(25)-
(26c) can be reformulated as follows (after some algebra):
383
(C.5) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
+++−= −
βρδβκ β
1ˆˆ
1 GL
ggEE&
(C.6)
β
ββ
κβχα
ρκαβ
χακβ
δκ
κκ
EN
LNN
LNEgg GL
ˆ1
1ˆ
)1
( 11
−−+
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−−
−++−
=
−−
&
I can focus attention on showing that the stationary point of this differential
equation system is stable: The discussion in section 4.1 (Definition 3 and Lemmas
1-4) implies that κ and E are jointly in steady state only if K , E and mλ are
jointly in steady state and that K , E and mλ are jointly in steady state only if κ
and E are jointly in steady state. Therefore, the proof of stability of the stationary
point of system (C.5)-(C.6) implies stability of the stationary point of system
(24)-(25)-(26c). Hence, in the following I will prove stability of the stationary
point of system (C.5)-(C.6).
It follows from equations (C.5) and (C.6) that the steady-state-loci of the two
variables are given by
(C.5a)
β
βρδ
βκ
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−+++
==
11
*
1
:0ˆˆ
GL
ggEE&
(C.6a) κκω
αβχαρ
βδκ
κκ
β
β
κ−
−
= −−
−++−
==1
1
0
1
)1
(ˆ:0
n
GL
ggE &
&
Now, I could depict the differential equation system (C.5)-(C.6) in the phase
space ( κ,E ). Before doing so, I show that not the whole phase space ( κ,E ) is
economically meaningful. The economically meaningful phase-space is restricted
384
by three curves ( 321 ,, tt RRR ), as shown in the following figure and as derived
below:
Figure C.1: Relevant space of the phase diagram
Only the space below the 1R -line is economically meaningful, since the
employment-share of at least one sub-sector i is negative in the space above the
1R -line. This can be seen from the following fact:
As shown in APPENDIX A (see there equation (A.23)), the following relation is
true
(C.7) ∑+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
n
miim l
11
χβανχβλ
Note that αχανχβ −=− when 0== μγ .
Since, il cannot be negative (hence , 101
≤≤ ∑+=
n
miil ) this equation implies that
(C.8) χβαν
<m
m
kl
κ
E
1R
30=tR
20=tR
385
Inserting equation (26c) into this relation yields
(C.9) βκχα
LNNER−
<ˆ:1 (remember that 0== μγ ).
Hence, the space above 1R is not feasible. When the economy reaches a point on
1R , no labor is used in sub-sectors i=1,…m. If I impose Inada-conditions on the
production functions, as usual, this means that the output of sub-sectors i=1,…m
is equal to zero, which means that the consumption of these sectors is equal to
zero. Our utility function implies that life-time utility is infinitely negative in this
case. Hence, the household prefers not to be at the 1R -curve.
Now I turn to the 2tR and 3
tR -curves. I have to take account of the non-negativity-
constraints on consumption ( iCi ∀> 0 ), since our Stone-Geary-type utility
function can give rise to negative consumption. By using equations (A.6), (A.15)
and (A.16) from APPENDIX A and Definition 1 the non-negativity-constraints
( iCi ∀> 0 ) can be transformed as follows (remember that I assume here
0== μγ ):
(C.10) miLA
LNE
i
i ,...11ˆ1 =
−>
αωθ
(C.11) nmiLBA
LNNE v
i
i ,...111ˆ2 +=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−> −− β
αβν
χ
κνβ
χβαν
ωθ
This set of constraints implies that at any point of time only two constraints are
binding, namely those with respectively the largest i
i
ωθ− . Hence, the set (C.10),
(C.11) can be reduced to the following set:
(C.12) αω
θ1
2 1ˆ:LA
LNER
j
jt
−>
386
where mii
i
j
j ,...1=−>
−ωθ
ωθ
and mj ≤≤1 .
(C.13) βαβν
χ
κνβ
χβαν
ωθ
−−⎟⎟⎠
⎞⎜⎜⎝
⎛−
−> v
x
xt
LBALN
NER 11ˆ: 23
where nmii
i
x
x ,...1+=−
>−
ωθ
ωθ
and nxm ≤≤+1
These constraints are time-dependent. It depends upon the parameter setting
whether 2tR or whether 3
tR is binding at a point of time. In Figure C.1 I have
depicted examples for these constraints for the initial state of the system. Only the
space above the constraints is economically meaningful, since below the
constraints the consumption of at least one good is negative. Last not least, note
that equations (C.12)/(C.13) imply that the 2tR -curve and the 3
tR -curve converge
to the axes of the phase-diagram as time approaches infinity.
Now, I depict the differential equation system (C.5)-(C.6) in the phase space
( κ,E ).
387
Figure C.2: The differential equation system (C.5)-(C.6) in the phase-space for
0>−=− αχανχβ
Note that I have depicted here only the relevant (or: binding) parts of the
restriction-set of Figure C.1 as a bold line R.
The phase diagram implies that there must be a saddle-path along which the
system converges to the stationary point S (where S is actually the PBGP). The
length of the saddle-path is restricted by the restrictions of the meaningful space
321 ,, tt RRR (bold line). In other words, only if the initial κ ( 0κ ) is somewhere
between 0κ and κ , the economy can be on the saddle-path. Therefore, the
system can be only locally saddle-path stable. Now, I have to show that the
system will be on the saddle-path if κκκ << 00 . Furthermore, I have to discuss
what happens if 0κ is not within this range.
Every trajectory, which starts above the saddle-path or left from 0κ , reaches the
1R -curve in finite time. As discussed above, the life-time utility becomes
infinitely negative if the household reaches the 1R -curve. These arguments imply
κ
E
*κ
0=κ&
0ˆ =E&
S
saddle-path
κ0κ
R
388
that the representative household will never choose to start above the saddle path
if κκκ << 00 , since all the trajectories above the saddle-path lead to a state
where life-time-utility is infinitely negative.
Furthermore, all initial points that are situated below the saddle-path or right from
κ converge to the point T. If the system reaches one of the constraints ( 32 , tt RR )
during this convergence process, it moves along the binding constraint towards T.
However, the transversality condition is violated in T. Therefore, T is not an
equilibrium. To see that the transversality condition is violated in T consider the
following facts: The transversality condition in my model requires that
01
lim 1 >−
−−−−
∞→ βδβκ β G
Lt
gg , which is equivalent to:
β
βδ
βκ
−
∞→
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−++
<
11
1
limG
Lt gg
. However, equation (C.6a) implies that in point T in
Figure C.2
β
βδ
κ
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−++
=
11
1
1G
Lgg
. Hence, the transversality condition is
violated if the system converges to point T.
Overall, we know that, if κκκ << 00 , the household always decides to be on the
saddle-path. Hence, we know that for κκκ << 00 the economy converges to the
PBGP. In this sense, the PBGP is locally stable (within the range κκκ << 00 ).
If the initial capital is to small ( 00 κκ < ), the economy converges to a state
where some existence minima are not satisfied (curve 1R ) and thus utility
becomes infinitely negative. This may be interpreted as a development trap. On
the other hand, if initial capital-level is too large ( κκ >0 ), all trajectories violate
389
the transversality condition. Therefore, in this case, the representative
household must waste a part of its initial capital to come into the feasible area
( κκκ << 00 ).
Q.E.D.
390
APPENDIX D Due to equation (16c) we know that
(D.1) *
**
**
*
p
Y
NLN
pp
NLN
Y
NLN
GDP
NN
⎟⎠⎞
⎜⎝⎛ −∂
∂−
⎟⎠⎞
⎜⎝⎛ −∂
∂
=⎟⎠⎞
⎜⎝⎛ −∂
∂
Equation (38) can be obtained by inserting equation (A.29) from APPENDIX A
and equation (35d) into equation (D.1). Hints: Equation (A.30) from APPENDIX
A and equation (35a) imply that 0*
=⎟⎠⎞
⎜⎝⎛ −∂
∂
NLN
pS . Furthermore, I used equation
(A.23) to transform *mλ into *
Sl .
391
LIST OF SYMBOLS of PART III of CHAPTER V * Denotes the PBGP-value of the corresponding variable.
^ Implies that the variable is expressed in “labor-efficiency units” (see
Definition 1).
A Parameter indicating technology/productivity level of junior-goods.
(exogenous)
B Parameter indicating technology/productivity level of senior-goods.
(exogenous)
iC Consumption of sector-i-output; indicates how much of the output of
sector i is consumed.
D Auxiliary parameter. (Function of exogenous model-parameters; growing
at constant rate.)
E Aggregate consumption expenditures; index of overall consumption-
expenditures of the representative household.
JE Consumption expenditures on junior goods.
SE Consumption expenditures on senior goods.
G Auxiliary parameter. (Function of exogenous model-parameters; growing
at constant rate.)
GDP Real GDP.
H Aggregate intermediate output; index of the value of all intermediates
produced in the economy
J Determinant of the Jacobian matrix.
K Aggregate capital; i.e. the amount of capital that is used for production in
the whole economy.
L Index of working population (economy-wide labor-input). (exogenous)
392
N Index of overall-population. (exogenous)
1R Function restricting the economically meaningful space in the phase-
diagram.
2tR Function restricting the economically meaningful space in the phase-
diagram.
3tR Function restricting the economically meaningful space in the phase-
diagram.
R Set of binding parts of functions restricting the meaningful space in the
phase-diagram
U Life-time utility of the (representative) household.
Y Aggregate output; index of economy-wide output-volume.
Y~ Auxiliary variable. (Function of other model-variables.)
iY Output of sector i.
NY Aggregate net-output. Aggregate output minus aggregate value of
intermediates.
NiY Net-output of sector i. Value of (gross-)output of sector i minus value of
intermediate inputs that sector i uses.
Z Index of intermediate production. Indicates how much intermediate inputs
are used in the whole economy.
a Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
b Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
393
c Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
1d Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
2d Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
*g Growth rate of aggregates along the PBGP.
Ag Growth rate of A. (exogenous)
Bg Growth rate of B. (exogenous)
Gg Growth rate of G.
Lg Growth rate of L. (exogenous).
Ng Growth rate of N. (exogenous)
ih Intermediates produced by subsector i; indicates how much output of
subsector i is used as intermediate in the whole economy.
i Index denoting a sector.
ik Capital-share of sector i; indicates which share of aggregate capital (K) is
used in sector i.
il Employment-share of sector i; indicates which share of aggregate labor (L)
is used in sector i.
Jl Employment share of the junior sectors.
Sl Employment share of the senior sectors.
m Index-number limiting the range of sectors that belong to the junior-sector.
n Number of sectors.
p Price-index (“deflator”).
394
ip Relative price of sector i. (Sector m is numéraire.)
Sp Relative price of senior goods. (Sector m is numéraire.)
s Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
t Index denoting time.
u Instantaneous utility-function.
Ju Instantaneous utility index, closely related to the utility of junior goods.
mu First derivative of u with respect to mC .
Su Instantaneous utility index, closely related to the utility of senior goods.
iz Intermediate-share of sector i; indicates which share of intermediate input-
index (Z) is used in sector i.
α Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
β Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
γ Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
δ Depreciation rate on capital (K). (exogenous)
iε Parameter of the Cobb-Douglas-intermediate-index; indicates the elasticity
of Z with respect to ih . (exogenous)
Jε Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
395
Sε Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
iθ Parameter of the utility function; closely related to the utility of iC . May
be interpreted as minimum consumption regarding good i (e.g. subsistence
level), if positive. May be interpreted as “natural” endowment of good i, if
negative. (exogenous)
ϑ Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
κ Auxiliary variable. (Function of other model-variables.)
κ Upper level of κ , which separates the phase-diagram into a convergent
and divergent section.
0κ Lower level of κ , which separates the phase-diagram into a convergent
and divergent section at the initial point of time.
0κ Level of κ at the initial point of time of the model.
mκ Capital-intensity in sector m.
mλ Auxiliary variable. (Function of other model-variables.)
μ Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
ν Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
ρ Time-preference rate. (exogenous)
mζ Intermediate-intensity in sector m.
χ Parameter of the Cobb-Douglas production function; is equal to output-
elasticity of the corresponding input. (exogenous)
396
ψ Auxiliary parameter. (Function of exogenous and constant model-
parameters.)
iω Parameter of the utility function; closely related to the utility of iC .
(exogenous)
CHAPTER VI
Summary
My work is about using the methods of the PBGP-school for modelling structural
change and economic growth. Broadly speaking, this new school of structural
change can be characterized upon two attributes (a mathematical one and a
theoretical one): (1) The concept of “partially balanced growth” is used to study
the differential-equation-systems of the theoretical models. (2) The modelling
framework may be regarded as “neoclassical” in many ways.
I have explained in Chapter IV that there is a large body of literature on
structural change consisting of several schools, dealing with several aspects of
structural change and using different approaches and methods. After elaborating
the features of these schools in Chapter IV, I have decided to focus on the New
School of Structural Change (or: PBGP-School of Structural Change).
I decided for this school, since it seemed to me very promising: It allows me to
study structural change in analytically-solvable frameworks, while including
some important aspects of the neoclassical growth-school (e.g. capital
accumulation). Furthermore, since the PBGP-school is very familiar with
neoclassical mainstream assumptions, I hoped that I can draw references to the
mainstream.
After elaborating the mathematical and modelling prerequisites of the PBGP-
method in Chapters II and III, I searched for several open questions in growth
theory associated with structural change. From all the open questions that I have
found, three questions seemed important and feasible to me: the Kuznets-Kaldor-
Puzzle, dynamic effects of offshoring associated with structural change and the
397
growth effects of ageing via structural change. I have studied these topics by
using the PBGP-method in Chapter V.
The elaboration of the mathematical prerequisites of the PBGP-school
(Chapter II) has shown that from the mathematical point view there are three
main challenges in using the PBGP-method:
1.) A partially balanced growth path exists only in very “rare” cases. Therefore, it
is difficult to find (meaningful) economic assumptions that yield differential
equation systems where partially balanced growth paths exist.
2.) Since the assumptions of the PBGP-school are based on neoclassical growth
theory, the solution of PBGP-models requires the solution of optimal control
problems. In continuous time these problems can be solved by using the
Hamiltonian. However, due to the high dimension of multi-sector differential
equation systems, proving the sufficiency of Hamiltonian optimality conditions
becomes very difficult, especially calculating the determinant of the Hessian for
proving concavity of the Hamiltonian.
3.) The proof of the stability of a partially balanced growth path as well as the
study of transitional dynamics are relatively difficult in the PBGP-school. The
problem is that the differential equation systems of the PBGP-school are (at least)
three-dimensional. Therefore, it is necessary to find (adequate) transformations of
the three-dimensional differential equation systems into two-dimensional ones to
be able to use a phase diagram. This is often difficult (at least for me).
Note that these problems do not arise in other structural change-schools, since
other structural change schools do not require proving the existence and stability
of a PBGP.
398
Nevertheless, I believe that the models of Chapter V have shown that the three
mathematical problems of the PBGP-school can be solved sometimes: at least I
have managed it to solve them in my models of Chapter V.
The elaboration of the modelling-foundations of the PBGP-school in Chapter
III has shown that from the theoretical point of view there is always one
challenge to the PBGP-school: The existence of a PBGP requires usage of knife-
edge parameter restrictions. It is quite difficult to find a theoretical rationale for
these knife-edge restrictions. In Part I of Chapter V, I have searched for such a
theoretical rationale. Exactly speaking, I have argued that independency between
preferences and technologies may explain (in part) these knife-edge conditions.
Furthermore, the discussion in Chapter IV implies that structural-change-theorists
(and growth theorists) have always used some knife-edge conditions to simplify
the analysis.
Now, the last question is about the general conclusion from my work: What is
the general value of the PBGP-method? I have discussed this question
throughout my work.
In Sections 6, 7 and 8 of Chapter I, I have explained that the PBGP-method
seems to be very useful for studying structural-change-topics that require the
inclusion of capital into analysis. In contrast to the PBGP-method, the traditional
methods of structural-change-modelling require simulations to include capital (or
even they omit capital). The price that we have to pay for this advantage of the
PBGP-method is: PBGP-models may not be regarded as descriptive, i.e. they
depict only some of the many channels along which structural change affects
aggregate growth. This is due to the necessity to use knife-edge conditions to
generate PBGPs. However, the discussion in Chapter IV (especially Section 3)
399
has shown that structural-change-theoreticians (and growth-theoreticians in
general) have always used some knife-edge conditions to increase the
“understandability” of their models.
The discussion in Sections 2 and 3 of Chapter II has shown that it is quite
difficult to analyse differential equation systems of multi-sector models in
general. In Chapter III, I have provided an example, explaining the difficulties
of understanding the dynamics of a multi-sector model (when capital is included
into analysis). Furthermore, I have demonstrated there how the PBGP-method
resolves these challenges by using knife-edge conditions.
Last not least, as I hope, the essays from Chapter V have shown as well that the
PBGP-method is valuable. Although the essays seem complicated, they are
actually not: Remember that all the topics/channels are in general that
complicated that only few intuitive results could be obtained without using the
PBGP-method. Normally, simulations would be necessary to get a notion of
ruling dynamics. Hence, the PBGP-method seems to be a valuable method for
deriving impact-channels and theories of structural change, i.e. for
“understanding” structural change.
In fact, my evaluation of the PBGP-method must depend on the topic-related
insights that are provided by my Chapter-V-models as well. I believe that there
are relevant/interesting insights (however, you have your own opinion on this, I
guess):
In the essay on the Kuznets-Kaldor-puzzle I have shown by using the PBGP-
method that the assumption of independency between preferences and
technologies may solve the Kuznets-Kaldor-puzzle. The Kuznets-Kaldor-puzzle
is one of the most important/essential empirical observations regarding the
relationship between structural change and aggregate development. I have also
400
tried to develop a method for assessing the degree of independency in reality and
applied it to the data of the United States.
In the essay on Offshoring I used the PBGP-method to point to the existence of an
impact channel of offshoring on real GDP-growth, which has been neglected in
the literature. Especially, I argued that offshoring slows structural change down
(in the long run) and thus reduces the negative impacts of structural change on
real-GDP-growth (via Baumol’s cost disease).
The essay on population ageing shows that the factor-reallocation effects of
ageing can be analysed by using the PBGP-method in a quite convenient way. I
hope this essay has improved the insights into structural change, which is induced
by ageing, and into the parameters, which are of importance for the relationship
between ageing and real-GDP-growth.
It seems to me that there are always new topics that could be studied by using the
PBGP-method. In fact, the most topics that are associated with structural change
could be analysed by using (some extension of) the models from Chapter V.
Last not least, as mentioned in the beginning of Chapter V, I believe that the
mathematical models, which I have presented in my work, are quite valuable on
their own, since they can be reinterpreted to analyse some new topics associated
with structural change. Remember that I have discussed in Chapters II and III that
it is very difficult to find assumptions that ensure the existence of a PBGP and
that it is even more difficult to find assumptions that additionally make the proof
of sufficiency of Hamiltonian conditions and global stability of the PBGP feasible
(when all structural change determinants are included into analysis). In fact, I
have done all this work and my models are “convenient” mathematical a-priori-
solutions, which can be adapted/reintepreted to analyse some new economic
401
questions associated with structural change, without having trouble with proving
sufficiency and stability of optimal solutions.
To conclude: I believe that my research has shown that the PBGP-method is a
valuable tool for the “modern structural-change-theorist”. Of course, the PBGP-
method is only one tool among several tools for structural change modelling. That
is, the PBGP-method completes the set of traditional methods (traditional
structural change theory and simulation-based models).
I am very grateful to Prof. Wagner for supervising this work.
402
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Curriculum Vitae (Lebenslauf) March 2011 (März 2011)
Name (Name): Denis Stijepic Date of Birth (Geburtsdatum): 27th November1981 (27. November 1981) Place of Birth (Geburtsort): Doboj, Jugoslavia (Doboj, Jugoslawien) Education (Bildungsweg): 2001
Higher education entrance qualification, academic high school in Wallduern, Germany; Grade: 1.2 (Allgemeine Hochschulreife, wirtschaftswissenschaftliches Gymnasium in Walldürn; Notendurchschnitt: 1,2)
2006 Master degree in economics, J.W. Goethe-University in Frankfurt am Main, Germany; Grade: 1.5 (Diplom Volkswirt, J.W. Goethe-Universitaet in Frankfurt am Main; Gesamtnote: sehr gut (1,5))
Occupation (Beruflicher Werdegang): Since 2007 research assistant, Chair of Macroeconomics, University in
Hagen (Seit 2007 wissenschaftlicher Mitarbeiter am Lehrstuhl für
Makroökonomie, FernUniversität in Hagen)
Erklärung laut §6(8) der Promotionsordnung
Hiermit versichere ich, dass ich die vorliegende Dissertation selbständig und ohne unerlaubte Hilfe angefertigt und andere als die in der Dissertation angegebenen Hilfsmittel nicht benutzt habe. Insbesondere habe ich nicht die Hilfe einer Promotionsberaterin/eines Promotionsberaters in Anspruch genommen. Alle Stellen, die wörtlich oder sinngemäß aus veröffentlichten oder nicht veröffentlichten Schriften entnommen sind, habe ich als solche kenntlich gemacht.
Denis Stijepic
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