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Long-Run EconomicGrowth

P A R TII

Long-run economic growth is the subject of this two-chapter part.

Chapter 3 covers the theory of growth, and Chapter 4 covers the world-

wide pattern of economic growth. Long-run economic growth is the

most important topic in macroeconomics. Standards of living in Canada

today are substantially better than what they were at the end of the

nineteenth century. Successful economic growth has meant that almost

all citizens of Canada today live better, and we hope happier, lives than

even the rich elite of a century ago. The study of long-run economicgrowth aims at understanding its sources and causes and at determin-

ing what government policies will promote or retard long-run economic

growth.

The study of long-run growth is also a separate module that is not very

closely connected to the study of business cycles, recessions, unem-

ployment, inflation, and stabilization policy that make up the bulk of

the subject matter of macroeconomics courses and of this book. Any

discussion of economic policy has to refer to long-run economic growth:

The effect of a policy on long-run growth is its most important element.

But the models used and the conclusions reached in Part II will by and

large not be used in subsequent parts. Starting in Chapter 5, our at-tention turns from growth to business cycles.

Why, then, include this two-chapter part on long-run growth? The prin-

cipal reason is that long-run economic growth is such an important topic

that it must be covered.

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QUESTIONS

What are the principal determinants of long-run economic growth?

What equilibrium condition is useful in analyzing long-run growth?

How quickly does an economy head for its steady-state growth path?

What effect does faster population growth have on long-run growth?

What effect does a higher savings rate have on long-run growth?

What government policies can enhance growth?

The Theory

of Economic Growth

C H A P T E R3

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3.1 SOURCES OF LONG-RUN GROWTHUltimately, long-run economic growth is the most important aspect of how the econ-omy performs. Material standards of living and levels of economic productivity inCanada today are about four times what they are today in, say, Mexico because of

favourable initial conditions and successful growth-promoting economic policiesover the past two centuries. Material standards of living and levels of economic pro-ductivity in Canada today are at least five times what they were at the end of thenineteenth century and more than ten times what they were at the founding of thecountry. Successful economic growth has meant that most citizens of Canada todaylive better, along almost every dimension of material life, than did even the rich elitein preindustrial times.

It is definitely possible for good and bad policies to accelerate or retard long-runeconomic growth. Argentines were richer than Swedes in the years before World

War I began in 1914, but Swedes today have perhaps four times the material stan-dard of living and the economic productivity level of Argentines. Almost all this dif-ference is due to differences in growth policies — good policies in the case of Swe-

den, bad policies in the case of Argentina — for there were few important differencesin initial conditions at the start of the twentieth century to give Sweden an edge.Policies and initial conditions work to accelerate or retard growth through two

principal channels. The first is their impact on the available level of technology thatmultiplies the efficiency of labour. The second is their impact on the capital inten-sity of the economy — the stock of machines, equipment, and buildings that the av-erage worker has at his or her disposal.

Better TechnologyThe overwhelming part of the answer to the question of why Canadians today aremore productive and better off than their predecessors of a century or two ago is“better technology.” We now know how to make electric motors, dope semiconduc-tors, transmit signals over fiber optics, fly jet airplanes, machine internal combus-tion engines, build tall and durable structures out of concrete and steel, record en-tertainment programs on magnetic tape, make hybrid seeds, fertilize crops withbetter mixtures of nutrients, organize an assembly line for mass production, and doa host of other things that our predecessors did not know how to do a century or soago. Perhaps more important, the Canadian economy is equipped to make use of allthese technological discoveries.

Better technology leads to a higher level of efficiency of labour — the skills andeducation of the labour force, the ability of the labour force to handle modern ma-chine technologies, and the efficiency with which the economy’s businesses andmarkets function. An economy in which the efficiency of labour is higher will be aricher and a more productive economy. This technology-driven overwhelming in-crease in the efficiency with which we work today is the major component of ourcurrent relative prosperity.

Thus it is somewhat awkward to admit that economists know relatively littleabout better technology. Economists are good at analyzing the consequences of ad-vanced technology, but they have less to say than they should about the sources of such technology. (We shall return to what economists do have to say about thesources of better technology toward the end of Chapter 4.)

3.1 Source of Long-Run Growth

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Solow proposed using a neoclassical production function for growth accounting.The production function tells us how much the economy can produce given theeconomy’s resources and the level of technology. We can write the production func-tion as

Y t = At × F(Kt, Lt), (3.1)

where Y t denotes real GDP, Lt denotes total labour used in the production process, Ktis total capital, and At denotes the current level of technology, more commonly re-ferred to as multi-factor productivity (MFP) or total factor productivity. F()is theproduction function, which shows how output is systematically related to capital,labour and the current level of technology. Often in economic modeling we assumethat the production function has a particularly simple form, referred to as the Cobb-Douglas function. With the Cobb-Douglas production function we have

Y t = At × Kt × Lt1– . (3.2)

The properties of the Cobb-Douglas production function will be further discussed inthe next section. At this stage, however, it will suffice to note that is a parameter

between 0 and 1 that measures the share of total income earned by capital, while 1- measures the share of total income earned by labour. These important results areshown in Box 3.1, and are used very effectively in growth accounting.

Using the rules for deriving proportional changes discussed in Box 2.4 of Chap-ter 2, it follows from equation (3.2) that a percentage change in output Y t /Y t willbe the sum of a percentage change in multi-factor productivity At / At, a fraction of a percentage change in capital Kt / Kt, and a fraction 1 – of the percentagechange in labour Lt / Lt. Thus, we will have

Y Y

t

t = ×

KK

t

t + (1 – ) ×

LL

t

t +

A A

t

t. (3.3)

3.2 Growth Accounting

ON THE PROPERTIES OF COBB-DOUGLAS PRODUCTION FUNCTION With the Cobb-Douglas production function (3.2) the parameter measures thefraction of total income earned by capital, while 1– measures the fraction of totalincome earned by labour. To see this, consider a change in output Y t brought aboutby a change in capital Kt. From equation (3.2), it is clear that Y t and Kt will berelated to each other as follows:

Y t = × At ×

Kt

K×

t

×KL

t

t1–

= × KY t

t × Kt.

Next, notice that we can re-write this equation as

=(Y t /

Y K

t

t) × Kt. (B3.1)

Finally, note that Y t / Kt is the marginal productivity of capital. It measures thechange in output due to a one unit increase in capital. With perfect competition, therental rate of capital is equal to its marginal productivity. Hence, Y t / Kt is incomeearned by each unit of capital, and the numerator of the right-hand side of equation(B3.1) is the income earned by capital. Clearly, then measures the fraction of totalincome earned by capital. Similarly, it can be shown that 1– is the fraction of totalincome earned by labour.

3.1

B O X

T O O L S

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3.3 The Neoclassical Growth Model

GROWTH ACCOUNTING

Using the neoclassical production function, one can decompose thegrowth rate of output into three components: (i) the contribution from thegrowth of capital, (ii) the contribution from the growth of labour, and (iii) thecontribution from the growth of multi-factor productivity. The rate of growth of multi-factor productivity in Canada during the 1980s and 1990s was substantiallylower than in the U.S., and it was moderate relative to other countries.

RECAP

FIGURE 3.1

Multi-Factor Growthfor OECD countries in1980s and 1990s

The rate of growth of

multi-factorproductivity in Canadaduring the 1980s wassubstantially lower thaits 1.2 percent averagefor the 1961–1999period. It was alsosubstantially lower thain the U.S., andmoderate relative toother countries.

P e r c e n t

0

1980–90 1990–99

I r e l a n d

F i n l a n d

N o r w a y

D e n m a r k

A u s t r a l i a

S w e d e n

C a n a d a

U n i t e d S t a t e s

N e w

Z e a l a n d

N e t h e r l a n d s

B e l g i u m

J a p a n

I t a l y

G e r m a n y

F r a n c e

U n i t e d K i n g d o m

S p a i n

1

2

3

4

5

Countries where trendMFP growth improved

Countries where trendMFP growth declined

Source: Adapted from The New Economy: Beyond the Hype, page 15. OECD copyright, 2001. The OECD Growth Project.

3.3 THE NEOCLASSICAL GROWTH MODELEconomists begin to analyze long-run growth by building a simple, standard modelof economic growth — a growth model. This standard model is also called the Solowmodel, after Nobel Prize–winning MIT economist Robert Solow. Economists thenuse the model to look for an equilibrium — a point of balance, a condition of rest, a

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state of the system toward which the model will converge over time. Once you havefound the equilibrium position toward which the economy tends to move, you canuse it to understand how the model will behave. If you have built the right model,it will tell you in broad strokes how the economy will behave.

In economic growth economists look for the steady-state balanced-growth equilib-rium. In a steady-state balanced-growth equilibrium the capital intensity of the econ-omy is stable. The economy’s capital stock and its level of real GDP are growing atthe same proportional rate. And the capital-output ratio — the ratio of the econ-omy’s capital stock to annual real GDP — is constant.

The Production FunctionThe first component of the model is a technological relationship described by theproduction function. As discussed earlier, this relationship tells us how the produc-tive resources of the economy—the labour force, the capital stock, and the level of technology—can be used to determine the level of output in the economy.

In growth theory, economists have used different specifications for the level of tech-nology. We have already seen one specification in equation (3.1). Box 3.2 offers a de-

tailed discussion of different specifications for the levels of technology. Here, to sim-plify the analysis we assume that technological progress takes the form of increasingthe level of labour efficiency. Hence, the production function we use has the form:

Y = F(K, E × L),

84 Chapter 3 The Theory of Economic Growth

CLASSIFICATION OF TECHNOLOGICAL PROGRESS

Exogenous technological growth is the main engine for steady-state growth in theneoclassical model. Without technological progress all variables per worker will beconstant in the steady state. This is clearly an unrealistic feature of the model. Inmost industrialized counties the growth rates of consumption and output per

worker have been positive over long periods of time. In Canada, for example, dur-ing the historical period 1961 to 1999, the growth rate of output was 3.8 percent, of which 1.2 percent was due to exogenous productivity growth or technologicalprogress.

Economists John Hicks, Roy Harrod, and Robert Solow have used three alterna-tive means of introducing technological progress into the neoclassical growth model.

We state each definition in turn using a general production function with two in-puts, capital, K and labour, L. As we shall see, each definition is “neutral” with re-spect to a specific economic variable (i.e., it leaves that economic variable unaf-fected).

(a) Hick’s Neutral Technological Change

A technological innovation is “Hicks neutral” if it does not affect the ratio of themarginal product of labour (Y t / Lt) to the marginal product of capital (Y t / Kt) fora given capital-labour ratio. This property is satisfied if the production functiontakes the form

Y t =At × F(Kt, Lt),

where At is an index of the economy’s state of the technology.

D E T A I L S

3.2

B O X

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where capital K and labour L are defined as before, while E measures the level of labour efficiency or labour productivity, which is determined by the current level of

technology and the efficiency of business and market organization. We assume that F() is a constant returns to scale production function. Hence, if

K and E × L are increased by a factor of, say, 1\ L then output Y will also increase bythe same factor. Thus, we can write

Y L = FKL, E. (3.5)

Moreover, with constant returns to scale, the production function exhibits di-minishing marginal productivity in both its arguments. Hence, if one of the factorsof production is fixed, successive increments in the other factor will lead to succes-sively smaller increases in output. With this property, we say the production func-tion is concave in both its arguments.

It is often convenient to use the Cobb-Douglas specification for the productionfunction F():

Y = K × (E × L)1–,

or in per worker terms

Y L = KL

× E1–. (3.6)


(b) Harrod Neutral Technological ChangeA technological innovation is “Harrod neutral” if it leaves unaffected the ratio of thetotal income earned by capital to total income earned by labour. This definition isconsistent with a production function of the form

Y t =F(Kt, Et × Lt)

where Et is the technology index. This form of technological change is called labour-augmenting technological change because it increases output the same way as an in-crease in the stock of labour, L.

(c) Solow Neutral Technological Change

A technological innovation is “Solow neutral” if it leaves unaffected the ratio of thetotal income earned by labour to total income earned by capital. It can be shown thatthis definition implies a production function of the form

Y t =F(Bt × Kt, Lt)

where Bt is the technology index. This type of technological change is called capital

augmenting because it raises production the same way as an increase in the stock of capital, K.

It is important to note that in the neoclassical growth model only labour-aug-menting technological change turns out to be consistent with the existence of asteady state, where, as defined below, all relevant quantities grow at constant rates inthe long run. For this reason, we adopt this form of technological improvement inthis book in order to analyze the effects of technological growth on the economy.

D E T A I L S

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Hence, the economy’s output per worker Y/L is equal to the capital stock perworker K/L raised to the exponential power of some number and then multipliedby the current efficiency of labour E raised to the exponential power 1–.

The efficiency of labour E and the number are parameters of the model. The pa-rameter is always a number between 0 and 1. The best way to think of it is as theparameter that governs how fast diminishing returns to capital set in. We will referto it as the diminishing-returns-to-capital parameter. Figure 3.2 plots a typical Cobb-Douglas production function. So long as 0 <

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shift over time in any one country.) Thus we can calculate the growth of the labourforce between this year and the next with the formula

Lt+1 = (1 + n) × Lt (3.7)

Next year’s labour force will be n percent higher than this year’s labour force. Thusif this year’s labour force is 10 million and the growth rate parameter n is 2 percentper year, then next year’s labour force will be

We assume that the rate of growth of the labour force is constant not because webelieve that labour-force growth is unchanging but because the assumption makesthe analysis of the model simpler. The trade-off between realism in the model’s de-

Lt+1

= (1 + n) × Lt= (1 + 2%) × Lt= (1 + 0.02) × 10= 10.2 million


USING THE PRODUCTION FUNCTION

For given values of E (say, 10,000) and α (say, 0.3), this production function tells ushow the capital stock per worker is related to output per worker. If the capital stockper worker is $250,000, then output per worker will be

And if the capital stock per worker is $125,000, then output per worker will be

Note that the first $125,000 of capital boosted production from $0 to $21,334, andthat the second $125,000 of capital boosted production from $21,334 to $26,265,less than one-quarter as much. These substantial diminishing returns should not bea surprise: The value of α in this example — 0.3 — is low, and low values are sup-

posed to produce rapidly diminishing returns to capital accumulation.Nobody expects anyone to raise $250,000 to the 0.3 power in her or his head and

come up with 41.628. That is what calculators are for! This Cobb-Douglas form of the production function, with its fractional exponents, carries the drawback that wecannot expect students (or professors) to do problems in their heads or with justpencil and paper. However, this form of the production function also carries sub-stantial benefits: By varying just two numbers — the efficiency of labour E and thediminishing-returns-to-capital parameter α — we can consider and analyze a verybroad set of relationships between resources and the economy’s productive power.

In fact, this particular Cobb-Douglas form of the production function was devel-oped by Cobb and Douglas for precisely this purpose: By judicious choice of differ-ent values of E and α, it is simple to “tune” the function so that it can capture a largerange of different kinds of behaviour.

Y L

= $125,0000.3 × 10,0000.7

= $33.812 × 630.958= $21,334

Y L = $250,0000.3 × 10,0000.7

= $41.628 × 630.958= $26,265

3.3

B O X

E X A M P L E

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scription of the world and simplicity as a way to make the model easier to analyze isone that economists always face. They usually resolve it in favour of simplicity.

Savings and InvestmentLast, assume that a constant proportional share, equal to a parameter s, of real GDPis saved each year and invested. S is thus the economy’s savings rate. Such gross in-vestments add to the capital stock, so a higher amount of savings and investmentmeans faster growth for the capital stock. But the capital stock does not grow by thefull amount of gross investment. A fraction δ (the Greek lowercase letter delta, for“depreciation”) of the capital stock wears out or is scrapped each period. Thus theactual relationship between the capital stock now and the capital stock next year is

Kt+1 = Kt + (s × Y t) – (δ × Kt) (3.8)

The level of the capital stock next year will be equal to the capital stock this yearplus the savings rate s times this year’s level of real GDP minus the depreciation rateδ times this year’s capital stock, as Figure 3.3 shows.


This year’s capital stock: K t

Next year’s capital stock: K t +1

This year’s output level: Y t

Add: new investment (s Y )equal to a fraction s of thisyear’s output Y.

Subtract: new investment ( K )equal to a fraction of thisyear’s capital stock.

FIGURE 3.3

Changes in the CapitalStock

Gross investment addsto and depreciationsubtracts from thecapital stock.Depreciation is a shareα of the current capitalstock. Investment is ashare s of currentproduction.

The Mechanics of the Neoclassical Growth ModelIn growth theory the focus is on the determinants of output per worker Y/L, con-sumption per worker C/L, and capital per worker K/L, as these variables reflect thewell-being of the population. Let us first derive expressions for these variables. Wewill assume that the level of labour efficiency E is either fixed or it grows at an ex-ogenous rate. Hence, the level and its rate of growth will be assumed to have beendetermined outside the model. From the production function (3.5) it is clear that toderive output per worker one should first derive the capital-labour ratio. Once out-put per worker is known one can then derive consumption per worker,as C / L = (1 – s) × Y / L.

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We first derive the evolution of the capital per worker, which is given by

KLtt = L

K

t

t –

LK

t2t × Lt, (3.9)

where Kt = Kt+1 – Kt and Lt = Lt+1 – Lt.Next note that from equation (3.7) Lt = n × Lt while from equation (3.8) Kt =

s × Y t – δKt. Substituting these results into equation (3.9) and simplifying, we obtain

KLtt = s × Y Lt

t – (n + δ) ×

KLt

t. (3.10)

Equation (3.10) has a straightforward interpretation. s × Y t / Lt is total saving perworker in year t, which will be equal to actual gross investment per worker in yeart. On the other hand, (n + δ) × Kt / Lt is total gross investment per worker that isneeded in year t in order to maintain the capital per worker at a constant level. Witha fraction δ of K being worn out in any year, δ × Kt / Lt is the total investment perworker required to replace depreciated capital. With the population growing at therate of n, n × Kt / Lt is the amount by which capital per worker alive in year t shouldincrease in order for the capital per worker next year to be the same as it is this year.

Thus, according to equation (3.10), if actual gross investment per worker s × Kt / Lt islarger (conversely, smaller) than the investment per worker needed to keep capitalper worker at a constant level, then the capital per worker next year will be larger(conversely, smaller) than the capital per worker this year.

The Steady StateThe long-run equilibrium or steady state of the economy is determined by the re-quirement that all the endogenous variables in the model like output Y , capital K, andconsumption C grow constant rates over time, with their per worker counterparts,Y/L, K/L, and C/L either constant or grow at constant rates over time. The economiccondition that must be satisfied at the steady state is that total saving per workermust equal total required investment per worker.

Analytically, we define the steady state by the condition (K/L)= 0 and we useequation (3.10) to solve for the steady-state capital-labour ratio (K/L)* after substi-tuting the production function (3.5) for Y/L for a given level of labour efficiency E:

s × F((K / L)*, E) = (n + δ) × (K / L)*

Having computed (K/L)*, equation (3.1) can be used to find the steady-state level of output per worker as (Y/L)* = F((K/L)*, E). The steady-state level of consumption perworker can then be obtained from the relation (C / L)* = (1 – s) × (Y / L)*.

Figure 3.4 depicts the steady-state equilibrium as well as the adjustment of the cap-ital per worker to its steady-state level. Using the production function, saving perworker is s × F(Kt / Lt, E), which is the concave schedule s × (Y t / Lt) in Figure 3.4. This is

the production function per worker multiplied by the marginal propensity to save s.The schedule representing the investment per worker required to keep capital perworker constant is, on the other hand, a straight line through the origin with slope(n + δ). Henceforth, this schedule will be referred to as the required investment schedule.

Point SS gives the economy’s steady state, because at that point all the variablesgrow at constant rates. At point SS saving per worker is exactly equal to the invest-ment per worker that is required in order to keep capital per worker constant over

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time. With (K/L)* constant, it is clear from the production function (3.5) that out-put per worker will also be constant over time. With (Y/L)* constant over time, sav-ing per worker s × (Y / L)* and consumption per worker (1 – s) × (Y / L)* will also beconstant over time. It then follows that at point SS aggregate capital K, output Y , andconsumption C will be growing at the same rate as the labour force, at rate n.

The Adjustment Process to the Steady StateReferring to Figure 3.4, if capital per worker is initially at level (K / L)0 then savingper worker at point B, (i.e., gross actual investment per worker) will be larger thanthe required investment per worker at point D. As a result, capital per worker willbe increasing over time until the economy reaches the steady-state point SS. Simi-larly, if capital per worker is initially at level (K / L)1 then required investment perworker at point F is larger than saving per worker at point G and hence capital perworker will be decreasing over time until we reach point SS. Clearly, these argumentsshow that the neoclassical growth model possesses a unique and stable steady state.

An important implication of the adjustment process in the neoclassical growthmodel is that different economies with the same rate of saving and technology willtend to converge over time to the same steady-state level of per capita income. This


Capital per Worker

O u t p u t

p e r W o r k e r

S a v i n g p e r W o r k e r

I n v e s t m e n

t p e r W o r k e r Y t

L t

K t ×(n + δ)

L t

Y t s ×

L t H

F

D

SS

B

K L

0 K L

*

Y L

*

K L

K L

1

FIGURE 3.4

Steady-State Output, Savings, and Investment in the Neoclassical Growth Model

The saving per worker schedule is s times the output per worker schedule. The “required investmentschedule” is a straight line. The steady state of the economy is at point SS . If capital per worker is initiallybelow its steady-state level, then savings per worker will be larger than required investment per worker. As a

result, K /L will be increasing over time until the economy reaches the steady-state point SS . Similary, if capital per worker is initially larger than its steady-state level, then K /L will be decreasing over time towardthe point SS.

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prediction has received a lot of attention in recent literature on empirical growth,under the title of economic convergence across different economies. We will discussthe hypothesis of income convergence among different groups of countries in thenext chapter.

The Effects of an Increase in the Saving RateSuppose that, starting from an initial steady point SS0, with corresponding capital-labour ratio (K/L)* and output per person (Y/L)*, there is an increase in the savingrate s, from s0 to s1. As shown in Figure 3.5, this will shift up the saving schedule tothe new position s1 × (Y t / Lt). Hence, at the original capital per worker ratio (K/L)*

there will be an increase in saving per worker above that which is required to keepK/L constant. As a result, K/L will start to increase over time. The process will con-tinue until a new steady state is reached at point SS1, with higher levels of capital-labour ratio (K/L)** and output per person (Y/L)**.

Notice that during the adjustment period from point SS0 to point SS1, K/L in-creases because capital will be growing at a rate which is greater that the rate of growth of population. With K/L growing during the adjustment period, it is clearfrom the production function (3.5) that Y/L will also be growing until point SS1 isreached. Hence, during the adjustment period aggregate output Y will be growing ata higher rate than n.

Once the new steady state is reached, Y and K will be growing once again at therate of population growth n, and Y/L and K/L will be constant with a zero growthrate. Consequently, a key prediction of the neoclassical growth model is that aneconomy’s growth rate is independent of the saving rate in the steady state. Further,changes in the saving rate can have only a transitory effect on the growth rate of out-


FIGURE 3.5

An Increase in theSaving Rate in theNeoclassical GrowthModel

An increase in thesaving rate from s 0 tos 1 will shift up thesaving schedule.Hence, at the originalcapital per worker rati(K /L )* there will be anincrease in savings peworker above thatwhich is required tokeep K /L constant. Asa result, K /L willincrease until a newsteady state is reached

at point SS 1.

Capital per Worker

O u t p u t p e r W o r k e r


I n v e s t m e n t p e r W o r k e r

Y t L t

K t L

t

Y t s 0L t

Y t s 1L t

SS 0

SS 1

K

L

*

Y L

*

Y L

**

K

L

K

L

**

×

×

×

(n + δ)

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put per worker during the adjustment process to the new steady state. And the onlypermanent effect of an increase of the saving rate is to increase the steady state lev-els of the capital-labour ratio and the output per worker.

The Effects of an Increase in the Rate of Population

GrowthIt is not difficult to analyze the steady-state effects of population growth in the neo-classical growth model. In this model population growth has the same effect on theeconomy as does an increase in the depreciation rate of capital. The reason for this


*Note that MPK(K / L) gives us the additional output the country can have if K / L increases by one unit.

On the other hand, n + δ gives us the additional “required investment” that will be needed if K / L

increases by one unit. Hence, MPK(K / L) – (n + δ) gives us the increase in consumption resulting from a

one unit increase in K / L. As long as MPK(K / L) > (n + δ) consumption can be increased by increasing

K / L. Similary, as long as MPK(K / L) < (n + δ) consumption can be increased by decreasing K / L.

THE GOLDEN RULE LEVEL OF CONSUMPTION

The ultimate purpose of saving, investment, and growth is to increase the level of in-come per worker and to promote the living standards of a country’s residents. Oneindicator of well being that economists have used routinely is the level of consump-tion per worker. The higher the level of consumption per worker the higher is the

standard of living that the residents of a country can enjoy.In the neoclassical growth model the steady-state level of consumption will de-

pend on the savings rate s; and there is a savings rate that will maximize the steady-state level of consumption per worker. This maximum steady-state level of con-sumption per worker is known as the golden rule level of consumption per worker.Corresponding to the golden rule level of consumption per worker there is a goldenrule level of capital per worker. This is the steady-state level of capital per workerthat maximizes the steady-state level of consumption per worker. It can be shownthat the condition that guarantees the golden rule is given by the equation

MPK(K/L) = n + δ (B3.4)

where MPK(K/L) =

F(K/L, E) /)K is the marginal product of capital as a function of capital per worker, for a given level technology E.* The steady-state capital perworker ratio (K/L)gr , say, that satisfies this equation is the golden rule level of capi-tal per worker. Note that F(K/L, E) / K is the slope of the production function inFigure 3.6. Condition (B3.4) then says that at (K/L)gr the slope of the productionfunction is equal to the sum of the population growth rate, n, and the rate of capitaldepreciation, δ.

Once (K/L)gr is computed, we can use the production function to obtain thegolden rule level of income per worker, (Y/L)gr = F[(K/L)gr , E]. Further, denoting thecorresponding saving rate by sgr , we can obtain the golden rule level of consumptionper worker as (C/L)gr = (1 – sgr ) (Y/L)gr .

The golden rule steady-state analysis is useful in practice because it can be viewed

as a benchmark against which different government policies can be compared andevaluated. For instance, if the government follows policies that lead to a saving rates1 that is considerably less than sgr , then the economy will settle down to a steadystate at which capital per worker will be less than (K/L)gr , as shown in Figure 3.6.

D E T A I L S

3.4

B O X

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is that population growth depreciates the existing capital in the sense that it requiresnew investment in order to provide the new workers with the necessary capital andkeep the capital-labour ratio constant. With saving per worker fixed this implies areduction in K/L and Y/L in the new steady state.

To see this clearly, suppose that we start in a steady state at point SS0 in Figure 3.7and there is an increase in the population growth rate n, from n0 to n1. This will shift

up the schedule representing required investment per worker. With more people en-tering the economy but with saving per worker the same, this will start to reducecapital per worker. Over time, K/L will be falling until a new steady-state is reachedat point SS1, with a lower steady-state capital per worker (K/L)**. From the produc-tion function (not shown), it also follows that Y/L will fall to a new state level(Y/L)**.

Notice that during the adjustment period from point SS0 to point SS1, K/L falls be-cause capital will be growing at a rate that is smaller than the rate of growth of pop-


Capital per Worker


S a v i n g p e r W o r k

e r

I n v e s t m e n t p e r W o

r k e r

(n + δ)

slope = n + δ

K L

*

s gr

K L

K L gr

C L gr

Y L gr

Y L gr

s 1

s gr

Y t L t

Y t L t

×

Y t L t

×

K t L t ×

FIGURE 3.6The Golden Rule Levels of Captial and Consumption

The steady golden rule level of capital per worker (K /L )gr is determined by thecondition MPK (K /L ) = n + δ. The golden rule level of consumption per worker isgiven by (1 – s gr ) × (Y /L )gr . If the saving rate is at s 1, less than s gr , then thesteady-state consumption per worker is not maximized. In that case, policiesthat increase the saving rate towards s gr would be desirable.

Consequently, the steady-state consumption per worker will not be maximized andthe country’s steady-state standard of living will not be as high. In this case, govern-ment policies that induce people to increase their saving rate toward s gr will be de-sirable. Similar but opposite arguments apply when the saving rate is greater than sgr.

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ulation. With capital per worker K/L falling during the adjustment period, it is clearfrom the production function (3.5) that output per worker Y/L will also be fallinguntil point SS1 is reached. However, the growth rate of Y itself will be permanentlyhigher at the new level n1.

The Effects of an Exogenous Technological ImprovementNow we examine the effects of a once and for all improvement in the level of tech-nology. As discussed above, we have assumed that it is labour-augmenting , that is itcomes about through improvements in the efficiency of labour, which is representedby the letter E in the production function (3.5). Because we do not specify what de-termines E, we call it exogenous technological change. Indeed, this is a major weak-ness of the neoclassical growth model.

An increase in E causes the production function (3.5) to shift upward to a newposition, so that now more output per worker can be produced at each level of cap-ital per worker. As a result, saving per worker will also increase. Consequently, thesteady-state effects of exogenous technological change are similar to those that westudied above in the context of an increase in the economy’s saving rate.

Suppose that we start in a steady state at point SS0 in Figure 3.8, and there is anincrease in labour efficiency E, from E0 to E1. This will shift up the saving schedule,because it will increase output corresponding to each level of capital per worker.Hence, at the original capital per worker ratio (K/L)* there will be an increased sav-ing per worker above that which is required to keep K/L constant. As a result, K/Lwill start to increase over time. The process will continue until a new steady state isreached at point SS1, with steady state capital-labour ratio (K/L)**.


Capital per Worker



K t (n0 + δ) × L

t

K t (n1 + δ) × L t

Y t s L t

SS 0

SS 1

K L

**

K L

K L

*

×

FIGURE 3.7

An Increase inPopulation Growth inthe NeoclassicalGrowth Model

An increase in thepopulation growth ratefrom n0 to n1 will shiftup the requiredinvestment per workerschedule. As morepeople enter theeconomy and withsaving per worker thesame, there will be areduction in capital perworker. Over time, K /L will be falling until anew steady state isreached at point SS 1.

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Notice during the adjustment period the K/L increases because capital will begrowing at a rate that is greater that the rate of growth of population. With K/L grow-ing during the adjustment period, it is clear from the production function (3.5) thatY/L will also be growing until point SS1 is reached. Hence, during the adjustment pe-

riod aggregate output Y will be growing at a higher rate than n. What is the effect of the increase in labour efficiency on the rate of growth of out-put per worker Y/L? Using the tools discussed in Box 2.4 of Chapter 2, the rate of growth of Y/L is equal to the rate of growth of Y minus the rate of growth of L. Sinceduring the adjustment process the rate of growth of Y is larger than n, it follows thatY/L will be growing at a rate that is larger than zero. Nevertheless, the growth rate of Y/L will be declining over time until it reaches zero once the new steady is reached.Hence, the growth of Y/L eventually comes to a halt.

It is also important to keep in mind that here we analyzed only the effects of a onetime increase in E. Our results would be quite different if we had assumed insteadthat the exogenous technological improvement was occurring continuously everytime period at some given rate g. In the neoclassical growth presented in this chap-

ter continuous growth in output per capita requires continuous growth in E at theexogenous rate g. We discuss this case below in Section 3.5, Steady State Growth andthe Dynamics of the Growth Models. First, however, we will look at endogenousgrowth models, which are alternative ways in which economists have attempted toaccount for technological change in modern growth theory.


FIGURE 3.8

An Increase in LabourEfficiency in theNeoclassical GrowthModel

A once and for all increase in labourefficiency from E 0 to Ewill shift up the savingschedule. Hence, at thoriginal capital perworker ratio (K/L )*

there will be anincrease in saving perworker above thatwhich is required tokeep K/L constant.Thus, K/L will beincreasing over timetoward the new steadystate point SS 1.

Capital per Worker



K t , E 1s × F L t

K t (n + δ) ×

L t

SS 1

SS 0

K

L

*

K t , E 0s × F L t

K

L

K

L

**

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3.4 ENDOGENOUS GROWTH MODELSPaul Romer of Stanford University and Robert Lucas of the University of Chicagopointed out in the 1980s that there were some empirical regularities that contradictthe predictions of the neoclassical growth model. First, the model suggests thatcountries with similar rates of growth of population should eventually converge tothe same steady-state point in Figure 3.4. This cannot account for the disparities weobserve in standards of living around the world, where poor countries seem to havestandards of living that are permanently below those of the rich countries.

Another empirical regularity that contradicts the predictions of the neoclassicalgrowth model is with regards to the growth rate of output per worker. Recall in theneoclassical model that after a technological improvement there will be an increasein the growth rate of output per worker, but this growth rate will gradually declineto its original level in the new steady state, which is zero (see also Section 3.5 below

for more details). Romer, considering growth rates over centuries, discovered thatover such long periods growth seems to have accelerated over time, and in recentdecades it has been relatively stable.

Another shortcoming of the neoclassical growth model is that it relies on exoge-nous factors to generate ongoing growth in the economy (see Box 3.5 below). Ide-ally, economists would like to explain the effects of policy changes on growth, andrecommend policies that would foster growth. We would like to explain, for exam-ple, whether free trade and economic integration would have any effect on growth.

These empirical considerations and theoretical issues have led to a new wave of research in growth theory that aims at constructing models in which growth is gen-erated by endogenous factors that could be influenced by government policies. In thissection we will present simplified versions of three popular endogenous growth

models.

Romer’s Growth ModelRomer argued that capital K should not be regarded as physical capital alone, but itshould also include knowledge. Hence, when measuring the amount of capital avail-able in a company we are interested in the amount of knowledge that is incorporatedin, for example, the computer software as much as the machinery that is available tothem.


THE NEOCLASSICAL GROWTH MODEL

In growth theory the focus is on the determinants of output per workerY/L, consumption per worker C/L, and capital per worker K/L, as these variablesreflect the well-being of the population. In the neoclassical growth model thestate of technology is exogenous. In the steady-state aggregate capital, output,and consumption will be growing at the same rate as the labour force. An increasein the savings rate will increase the steady-state level of K/L, Y/L, and C/L. An in-crease in the rate of growth of population (the labour force) will reduce thesteady-state levels of K/L, Y/L, and C/L. An exogenous increase in the level of labour efficiency will increase the steady-state levels of K/L, Y/L, and C/L.

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3.4 Endogenous Growth Models

Once K incorporates physical capital and knowledge, then one can extend themodel to incorporate learning-by-doing. The idea here is that the employees of thecompany can learn and improve upon the knowledge (e.g., the software) that is in-corporated in the machineries used in the company. With the knowledge availableto them the employees are better able to foresee the next steps involved in improv-ing the machinery, or the production process. This idea of learning-by-doing andhow it can be related to K can be captured by assuming that labour efficiency E isproportional to capital:

Et = ρ × Kt where 0 ≤ ρ ≤ 1. (3.11)

Substituting for Et from equation (3.11) into the Cobb-Douglas production func-tion (3.6), we obtain a simple expression for the production function:

Y Lt

t = KLt

t

× (ρ × Kt)1–.

Now multiplying the right-hand side of this equation by Lt / Lt (i.e., unity), we canre-write it as

Y Lt

t = (ρ × Kt)1–

× KLt

t. (3.12)

Assume there is no labour force growth, and Lt = L for all t. Then we can re-writethe production function (3.12) as

Y Lt

t = A ×

KL

t where A = (ρ × L)1– (3.13)

In Figure 3.9 the production function (3.13) will be a straight line through theorigin with slope A, and saving per worker will be a straight line through the originwith slope s × A. With no population growth (n = 0), the required investment sched-ule will have a slope of δ.

As should be clear from Figure 3.9, in this model saving per worker will always

be above the investment per worker that is needed in order to keep K/L at a constantlevel. Hence, growth will never come to a halt. The amount by which K/L will in-crease over time will be saving per worker (s × A × Kt /L) minus the investment perworker that is needed to keep K/L at a constant level (δ × Kt /L). Hence, capital perworker will grow regardless of how rich the country is; and its growth rate will beconstant. As a result, a poor country will never be able to catch up with a rich coun-try in terms of its capital per worker.

What is the prediction of this model regarding the effects of free trade and eco-nomic integration on growth? Free trade and market integration will give people inthe country access to knowledge in other countries. Hence, with free trade labour ef-ficiency will increase. Thus, by increasing labour efficiency, free trade will fostergrowth.

Barro’s Growth ModelRobert Barro of Harvard University has proposed an endogenous growth model thatexplicitly incorporates government expenditures, and can be used to discuss theissue of the optimum size of government. Barro regards government expenditures onthe economy’s infrastructure as entering the production function as another factor of production. This is a very reasonable assumption to make, as government expendi-

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tures on health services, education, law and order, communication systems andother infrastructures do increase productivity in the private sector.

In terms of the model we have developed above, in Barro’s growth model labourefficiency in period t can be regarded as depending on government expenditures Gt.Let us assume

Et = ρ × Gt where 0 ≤ ρ ≤ 1. (3.14)

Substituting for Et from equation (3.14) into the Cobb-Douglas production func-tion (3.6), we obtain the production function proposed by Barro:

Y Lt

t = KL

t

t

× (ρ × Gt)1– (3.15)

To simplify the analysis, we assume that government expenditures are financedby proportional taxes on capital. Hence, for simplicity let us assume that

Gt = τ × Kt, (3.16)

where τ is the tax rate.1

Substituting for Gt from equation (3.16) into (3.15), and assuming that popula-tion growth is zero (i.e., Lt = L for all t), we obtain

Y L

t = KL

t

× (ρ × τ × Kt)1–.


Capital per Worker




s = s × A ×

K t δ ×

L t

Y t L

K t L

= A ×Y t L

K t L

K

L

FIGURE 3.9

Output, Savings, andInvestment in Romer’sGrowth Model

In Romer’s model the

production functionper worker and thesavings function perworker are straightlines through theorigin. As a result,saving per worker willalways be above theinvestment per workerthat is needed in orderto keep K/L at aconstant level, Hence,growth will never cometo a halt.

1In fact, Barro assumes that government expenditures are financed by a proportional tax on income: Gt = τ × Y t . It can be shownthat the essence of Barro’s results remain unchanged with the alternative assumption that government expenditures arefinanced with proportional taxes on capital.

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3.4 Endogenous Growth Models

Now multiplying the right-hand side of this equation by L/L (i.e., unity), we canre-write it as

Y L

t = A × τ1– ×

KL

t where A = (ρ × L)1– (3.17)

Equation (3.17) implies that in Figure 3.10 the production function will be a

straight line through the origin with slope A × τ1–

.Because households pay a tax to the government, disposable income per worker

is (Y t /L) – τ × (Kt / L) or, upon using equation (3.17), A × τ1– × (Kt / L) – τ × (Kt / L).Saving per worker is a fraction s of disposable income. Hence, saving per worker iss × ( A × τ1– – τ) × (Kt / L). The saving per worker schedule will be a straight linethrough the origin with slope s × ( A × τ1– – τ). The slope of the saving function willalso be equal to saving per K/L. Let us call this average saving or average actual in-vestment.

On the other hand, the required investment schedule will be as before. It will bea straight line through the origin with slope δ. The slope of the required investmentschedule will also be equal to required investment per K/L. Let us call it average re-quired investment. In order to maximize growth the government should maximize the

difference between average saving (or average actual investment) and average re-quired investment. As average required investment n + δ cannot be affected by thegovernment, to maximize growth the government should choose a tax that will max-imize average saving.

In Figure 3.11 we plot average saving s × ( A × τ1– – τ) for different values of τ.Average saving is increasing in τ when τ is small. It reaches a maximum at τ , andthen it is decreasing for values of τ above τ . The reason for this result is that an in-crease in τ has two competing influences on income, and saving. First, an increasein τ has the direct effect of reducing the disposable income of the households, which

FIGURE 3.10

Output, Savings, andInvestment in Barro’sGrowth Model

In Barro’s growthmodel the schedulesrepresentingproduction per workerand saving per workerare straight linesthrough the origin.Hence, as in Romer’smodel, growth does

not come to a halt. InBarro’s model labourefficiency depends ongovernmentexpenditures. Hence,governmentexpenditures can affecthe slopes of theseschedules.

Capital per Worker

O

u t p u t p e r W o r k e r


I n v

e s t m e n t p e r W o r k e r

s ×

K t δ ×

L t

Y t L t

= A × τ ×Y t L t

K t L

K L

1 – α

= s × A × τ ×K t L

1 – α

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tends to reduce saving. Second, the taxes that are collected by the government areused to provide productive government expenditures on infrastructure, which tendsto increase production, income, and saving. For values of τ smaller than τ the sec-ond effect dominates, while for values of τ larger than τ the first effect dominates.

Average saving is maximized when τ = τ . This then maximizes average actual in-vestment over required investment in any period, which in turn maximizes thegrowth rate. Barro’s model, then, is a useful model for studying the optimum size of government, and it has received attention in policy discussions.

Rebelo’s Growth ModelAs can be seen from equations (3.13) and (3.17) in the Romer and Barro models out-put is a linear function of capital. This implies that with these models there is no di-minishing marginal productivity of capital. As we shall see shortly, this is the mainingredient of endogenous growth models that distinguishes them from the neoclas-sical model. Recognizing this, Sergio Rebelo of Northwestern University has sug-gested using the production function

Y t = A × Kt,

where A is treated as a fixed parameter. Rebelo’s AK model is particularly simple, andis often used to discuss issues when other endogenous growth models are cumber-some.


Tax Rate

A v e r a g e S a v i n g s (A τ – τ)

1 – α

ττ

× ×

FIGU RE 3.11

The GrowthMaximizing Tax Rate

in Barro’s GrowthModel

An increase in τ hastwo competing effectson income and savings.First, it reducesdisposable incomedirectly, Second, thetaxes collected by thegovernment provideproductive governmentexpenditures oninfrastructure, whichtends to increase

production. The growthmaximizing tax rate isτ . For values of τsmaller than τ thesecond effectdominates, while forvalues of τ larger than τ the first effectdominates.

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3.5 Steady-State Growth and Dynamics of the Growth Models

3.5 STEADY-STATE GROWTH AND DYNAMICSOF THE GROWTH MODELS

In this section we study in more detail the steady-state growth properties of the neo-classical and endogenous growth models, and compare and contrast their properties.As the subsequent analysis will make clear, without exogenous technological growththere will be no steady-state growth in per worker variables in the neoclassicalgrowth model. Growth comes to a halt in the steady state due to diminishing returnsto capital. On the other hand, in the endogenous growth model returns to capital areconstant and thus positive growth can go on forever in the steady state.

Steady-State Growth and Dynamics in the NeoclassicalGrowth ModelIn Figures 3.4 to 3.8 the intersection of the saving schedule and the required invest-ment schedule is the steady state of the economy. At this point all variables grow atconstant rates. Hence, sometimes the steady state is also referred to as the balancedgrowth equilibrium. It turns out that a substantial amount can be said about theproperties of the steady state. In Box 3.5 we derive the detailed properties of thesteady-state when labour efficiency grows at an exogenous rate of g. Here we treatthe level of labour efficiency as constant, and show that in that case there will be nosteady-state growth in per worker variables in the neoclassical model.

In order to gain more insight into the source of the last result it is useful to ana-lyze the transition dynamics of the neoclassical growth model, that is, the model’sadjustment path to its steady-state equilibrium. To this end, consider again equation(3.10) divided though by K/L so that the left-hand variable is the growth rate of cap-ital per worker, gk say:

gk =s × F

K(K / L / L, E) – (n + δ) (3.18)

where gk =(

KK / L / L).

Equation (3.18) shows that gk is the difference of two terms, the first being theproduct of the saving rate s multiplied by the average product of capital per worker

s × FK(K / L / L, E) and the second is (n + δ). Figure 3.12 plots both terms against K/L.

ENDOGENOUS GROWTH MODELS

Romer extends the neoclassical growth model by assuming that capitalincorporates both physical capital and knowledge; and labour efficiency is pro-portional to capital. Barro extends the neoclassical growth model by assumingthat labour efficiency depends on government expenditures. The models devel-oped by Romer and Barro generate growth that goes on forever without having torely on exogenous technological progress or population growth. Hence, they arecalled endogenous growth models. This is in contrast to the neoclassical growthmodel that relies on exogenous increases in population or productivity for ongo-ing growth.

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STEADY-STATE GROWTH

In the neoclassical growth model the intersection of the savings and required in-vestment schedules is the steady state of the economy. At this point all variablesgrow at constant rates. A substantial amount can be said about the properties of thesteady state. This is important because in extensions of the neoclassical model thefocus is sometimes on the properties of the steady state, as the model is sometimesso complicated that not much can be said about the properties of the model outsidethe steady state.

Here we derive the growth rates of output and capital when population L growsat the rate of n and labour efficiency E grows at the rate of g. To this end, first notethat the equality of savings and investment, equation (3.8), implies

s × Y t = Kt – δ × Kt.

Dividing both sides of this equation by Kt, we obtain

s × KY t

t =

KK

t

t – δ. (B3.5.1)

Next, note that by definition all variables grow at constant rates in the steady

state. Hence, in equation (B3.5.1)

KK

t

t must be a constant in the steady state.

With

KK

t

t constant, it follows from (B3.5.1) that in the steady state the ratio of

output to capital must also be constant. Hence, in the steady state, output and cap-ital must be growing at the same rate. That is, in the steady state

Y Y

t

t =

KK

t

t. (B3.5.2)

Furthermore, note that we can substitute for Y t from the production function(3.4) into (B3.5.1) to obtain

s ×

F(Kt,KE

t

t × Lt)

=KKt

t

– δ. (B3.5.3)

Upon using the fact that the production function is homogeneous of degree one inKt and Et × Lt, equation (B3.5.3) reduces to

s × F1, Et × Lt /Kt = KK

t

t – δ. (B3.5.4)

As in the steady state

KK

t

t is constant, it follows from (B3.5.4) that in the steady

stateEt

K×

t

Lt must also be constant. Hence, in the steady state the rate of growth of

labour efficiency Et (that is,

EE

t

t or g) plus the rate of growth of population (that is,

L

L

t

t or n) must be equal to the rate of growth of capital. Thus, in the steady state the

rates of growth of output and capital must be equal to g + n:

Y Y

t

t =

KK

t

t = g + n

The rate of growth of output per person Y/L or capital per person K / L will be g.This is the rate of growth of output or capital (that is, g + n) minus the rate of growthof population (that is, n).

D E T A I L S

3.5

B O X

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The first term is downward sloping and approaches zero as K/L tends to infinity, andthe second term is a horizontal line at (n + δ). The vertical distance between the twoschedules is gk, the growth rate of capital per worker. The intersection of the twoschedules determines the steady-state value of K/L where gk = 0. To the left of thesteady state gk is positive and thus K/L rises over time. But as K/L rises the distancebetween the two schedules decreases and hence gk approaches zero as K/L ap-proaches (K/L)*. Similar but opposite arguments can be made for levels of K/L to theright of the steady state. These arguments show formally that the neoclassical growthmodel has a unique steady-state equilibrium, and that steady-state growth of capitalper worker comes to an end in an economy with no technological change (i.e., g =0).

The source of these findings is the diminishing returns to capital per worker. Dueto the concavity of the production function, as K/L increases the average product of

capital per workerF(K

K / L / L

, E) falls causing a reduction in the saving per worker

and in the growth rate of the economy. Eventually, growth of K/L comes to a halt,when saving per worker is just equal to required investment per worker. With K/Lnot growing in the steady state, and with E fixed, it is clear from the production

function (3.5) that output per worker Y/L will also not be growing in the steadystate. With consumption per worker C/L equal to (1–s)Y/L, it is clear that C/L willalso not be growing in the steady state.

Obviously, if diminishing returns to capital could be avoided, we could have aneconomy with perpetual positive growth in K/L even in the steady state. This is in-deed the case with the endogenous growth models that we study next.

FIGURE 3.12

Growth Dynamics inthe NeoclassicalGrowth Model

The growth rate, gk , oK/L is given by thevertical distancebetween the savingschedules × F (K/L , E )/(K/L ) andthe “depreciation” linen + δ. If K/L < (K/L )*

the growth rate of K/Lis positive and K/L increases toward (K/L )

If K/L > (K/L )*

thegrowth rate of K/L isnegative and K/L declines toward (K/L )*.

Capital per Worker

S a v i n g p e r U n i t o f K / L

I n v e s t m e n t p e r U n i t o f K / L

K L

gk > 0gk = 0

gk < 0

s ×

K L

0 K L

* K L

1

, E F K L

K L

n + δ

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Steady-State Growth and Dynamics in the EndogenousGrowth ModelTo simplify the analysis we consider Rebelo’s AK model:

Y = A × K (3.19)

SOLVING FOR THE STEADY STATE

Here is a numerical example to show how to solve for the steady state in the neo-classical and in Rebelo’s endogenous growth models. For simplicity, we consider aneconomy with no exogenous technological growth, and assume that the economy’ssaving rate s is 10 percent (i.e., s = .10), the capital’s depreciation rate d equals 4 per-cent per year (i.e., δ = .04), and the labour force grows at the rate n equal to 1 per-cent per year (i.e., n = .01).

(a) Neoclassical Growth Model

Assume that the production function is Cobb-Douglas and has the form

Y =F(K, E × L) = K1/2 × L1/2

where we have set E =1 without loss of generality. Dividing both sides of this equa-tion with L, we can write production function as

Y/L = (K/L)1/2.

Using this information, the steady-state condition (SS) in the text becomes

.10(K/L)1/2 = (.01 + .04) × (K/L).

Solving this equation we obtain the steady-state level of capital per worker for thiseconomy (K/L)* = 4. Substituting this into the production function gives us thesteady-state output per worker (Y/L)* = 2. Steady-state consumption per worker is(C/L)* = (1–.10) × (Y/L)* = 1.8.

Since, there is no exogenous technological growth in this economy, the steady-state growth rate of output per worker is zero, while output itself grows at the rateof growth of population n, which we have assumed to be 1 percent per year.

(b) The Rebelo Endogenous Growth ModelNow assume that the production function is

Y = A × K = 1 × K,

where we set A equal to 1 for simplicity. In per worker terms, the production func-tion has the form

Y/L = 1 × (K/L).

Recall that there is no unique steady-state capital-labour ratio in this model.

There is an infinity of K/L ratios that are consistent with this economy’s steadygrowth rate. This steady-state growth rate is

gk = s × A – (n + δ) = .10 × (1) – (.01 + .04) = .06.

Thus, in contrast to the neoclassical economy, the endogenous growth economygrows at the rate of 6 percent per year, even though we have assumed that there isno exogenous technological growth. As an exercise, we encourage the reader to pro-vide the graphical representation of these results.

3.6

B O X

E X A M P L E

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Also, assume that labour L grows at rate n and that capital K depreciates at rate δ.In order to analyze the growth properties of this economy, notice first that equa-

tion (3.19) implies that Y and K grow at the same rate. It is easy to derive this rate.Using equation (3.8), which equates saving to investment, together with equation(3.19) we get

K = s × Y – δ × K = s × A × K – δ × K (3.20)which upon divided through by K gives

KK = s × A – δ. (3.21)

Therefore output and capital grow at the rate s × A – δ over time. Similarly withlabour growing at rate n, it follows immediately that output per worker Y/L and cap-ital per worker K/L grow at the rate given by

gk = s × A – δ – n. (3.22)

The growth rate of K/L is now the difference between the two terms sA and (n + δ),both of which are straight lines as shown in Figure 3.13. The figure is drawn under

the assumption that s × A is greater than (n – δ). As long as this is the case, gk willbe positive for all values of K/L .The economy will be growing at the steady-state rateforever, without having to rely on exogenous technological progress. With Y/L =

A × K/L and C/L = (1 – s) × K / L, it is clear that output per worker and consump-tion per worker will also be growing at the rate of gk.

This startling result is a direct consequence of the nature of the production func-tion (3.19), which is associated with a constant average and marginal product of cap-ital equal to the positive parameter A for any level of K. Hence, with non-diminish-ing returns to capital economic growth can go on forever.

Comparison of the Neoclassical and Endogenous Growth

Models We are now in a position to summarize and compare the findings of the neoclassicaland endogenous growth models.

1. In the neoclassical growth model with no exogenous technological growth, thesteady-state growth rate output per worker is zero and independent of theeconomy’s saving rate. An increase in the saving rate can have only atemporary effect on the economy steady-state growth rate of capital perworker, output per worker, or consumption per worker. Under the sameconditions, the endogenous growth model predicts a positive steady-stategrowth of capital per worker, output per worker, and consumption per worker.Moreover, with endogenous growth these growth rates are greater the larger isthe economy’s saving rate (see equation (3.22)).

2. Population growth does not affect long-run growth of capital per worker,output per worker, and consumption per worker in the neoclassical growthmodel, but it reduces steady-state growth of these variables in the endogenousgrowth model. The same comments apply for an increase in the depreciationrate δ.

3. In the neoclassical growth model, positive steady-state growth of capital perworker, output per worker, and consumption per worker can occur only at therate of exogenous technological change g. In the endogenous growth model,steady-state growth can occur even when g = 0.

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4. The reason for the result in (3) above is that without exogenous technicalprogress (i.e., with g = 0) growth in the neoclassical model comes to an enddue to the diminishing returns to capital given the concavity of theneoclassical production function. In the endogenous growth model returns tocapital are constant and this is the fundamental cause for continuous steady-state growth in that model.


STEADY-STATE GROWTH AND DYNAMICS OF THEGROWTH MODELS

In the steady state, per worker variables such as capital per worker oroutput per worker are either constant or grow at constant rates. In the neoclassi-cal model with no technological change there is no steady-state growth in theeconomy. Growth comes to a halt due to diminishing returns to capital. With pos-itive technological progress, output per worker in the steady state grows at therate of growth of the exogenous technological progress. On the other hand, in theendogenous growth model returns to capital are constant and thus positivegrowth can go on forever in the steady state, even without any exogenous tech-nological progress.

RECAP

Capital per Worker

S a v i n g p e r U n i t o f K / L

I n v e s t m e n t p e r U n i t o f K / L

K

L

K L

gk > 0 for all

n + δ

s × A

FIGU RE 3.13

Growth Dynamics withEndogenous Growth

If the productionfunction is AK then the

saving schedule is astraight line at s × A,while the requiredinvestment schedule isa straight line at n + δ.Given that s × Aexceeds n + δ, there iscontinuous growth inthe steady state, evenwithout exogenoustechnological change.

3.6 ECONOMIC CONVERGENCE ANDDIVERGENCE

An important issue in the recent literature on empirical growth is whether differentcountries across the globe tend to converge or diverge over time in terms of their per

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3.6 Economic Convergence and Divergence

capita income levels and therefore their standards of living. If growth rates in realper capita output converge over time, then poor countries tend to grow faster thanrich ones and catch up with them eventually. This has given rise to the convergencehypothesis of whether poor regions or countries have the tendency to grow fasterthan rich ones.

Two types of convergence have been analyzed in the literature, absolute conver-gence and conditional convergence. The hypothesis of absolute convergence states thatpoor economies tend to grow faster per capita than rich ones. This concept assumesthat countries are similar in other characteristics regarding saving rates, populationgrowth rates, production technologies, education and human capital, governmentpolicies and the like. On the other hand, conditional convergence takes into accountthe fact that countries may differ in these important characteristics. Hence, condi-tional convergence states that poor economies tend to grow faster in terms of percapita income than rich ones, provided that we take into account differences incountry characteristics.

Economic theory is not entirely supportive of either type of convergence. Whereas the standard neoclassical growth model predicts economic convergence,the more recent endogenous growth models reject convergence and predict eco-nomic divergence across countries in general. We now use the earlier analysis to gainmore insight on these ideas.

Consider first a situation where all countries have identical preferences and tech-nologies, but they differ only in terms of their initial capital-labour ratios, or equiv-alently, their initial levels on per capita incomes, with poor countries having lowerinitial per capita incomes than rich ones. Under these conditions, the growth dy-namics of the neoclassical growth model in Figure 3.12 make it clear both that allthe economies will have the same steady-state capital-labour ratio or per capita in-come, and that the poor economies will grow faster than the rich ones (i.e., giventhat (K / L)0 poor < (K / L)0rich it follows that gk,poor > gk,rich). Hence, if all the economiesare similar in terms of their saving rates, population growth, depreciation rates, andproduction technologies, but differ only in their starting per capita incomes (becauseof, say, exogenous events such as wars or transitory shocks to their production func-tions), then poor countries will catch up to the rich ones eventually, and in the longrun living standards around the world will be more or less equalized. That is, in thiscase the neoclassical model predicts absolute convergence.

Alternatively assume that, in addition, two or more economies differ in terms of some important characteristic such as saving rates. Specifically, suppose that a groupof economies have a low saving rate, and another group a high saving rate. Then,using Figure 3.12, it is easy to see that the first group of economies will have asteady-state capital-labour ratio or per capita income, which is less than that of thesecond group of economies. Further, within each group, the poor countries will tendto grow faster than the rich ones and thus converge over time to their commonsteady state. However, across the two groups, the economies will converge to twodifferent steady states. That is, in this case the neoclassical model predicts condi-tional convergence. This means that living standards will converge only within eachgroup of similar economies, not across all the economies. For example, under con-ditional convergence, a poor country with a low saving rate will catch up in the fu-ture to a rich country with also a low saving rate, but it will never achieve the livingstandards of a rich country with a high saving rate.

Considering the endogenous growth models it is easy to see that they predict eco-nomic divergence. Consider for simplicity Rebelo’s AK model and assume again

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there are two groups of countries, one group with a low saving rate and the otherwith a high saving rate. Suppose also that all the other parameters are the sameacross all the countries. Then, it is clear from Figure 3.13 that the low-saving coun-tries will have a growth rate that is permanently lower than that of the high-savingcountries, regardless of initial conditions. Hence, the two groups of economies willnever converge in their per capita incomes and living standards will be permanentlydifferent across the world. Thus, this endogenous growth model predicts economicdivergence among different groups of countries.

Notice that for convergence to hold we must observe an inverse relationship be-tween per capita growth rates and initial per capita incomes in a cross section of countries. This is the same thing as saying that countries with low starting per capitaincomes have a higher growth rate than countries with high starting per capita in-comes. Also, the speed of convergence to the steady state, depends on the gap be-tween initial per capita income and steady state per capita income. The larger is thisgap, the faster will be the convergence to the steady state.

To examine the hypothesis of convergence empirically we collected annual dataon 17 countries in the Asia-Pacific region2, and computed their real per capita GDPgrowth rates for the period 1960–1990. Figure 3.14 plots the average annual growthrate of real per capita GDP against the log of real per capita GDP at the start of theperiod, 1960, for the 17 countries. It is evident from this figure that the initiallypoorer East Asian economies experienced significantly higher per capita growthrates than the more developed economies of Australia, Canada, New Zealand, andthe U.S. Clearly, the “best fitting” line through these data points is one with a nega-tive slope that highlights the inverse relationship between growth rates and initialper capita incomes (draw this line as an exercise). In other words, this sample sup-ports the hypothesis of absolute convergence among these countries.

A large number of recent studies have used actual economic data to analyze thehypothesis of economic convergence across different countries, and also differentprovinces/states within countries.3 The empirical findings has been mixed with somestudies supporting convergence and other studies rejecting convergence in favour of divergence. Serge Coulombe of the University of Ottawa and Frank Lee of the De-partment of Finance analyzed economic convergence among the Canadianprovinces.4 These authors found evidence in favour of convergence among the Cana-dian provinces for different measurements of per capita income and output for theperiod 1961–1991. We examine the issue of convergence and other issues of empir-ical growth in greater detail in the next chapter.

2The countries are Australia, Canada, Chile, China, Hong Kong, Indonesia, Japan, Korea, Malaysia, Mexico, New Zealand, PapuaNew Guinea, Philippines, Singapore, Taiwan, Thailand, and the U.S.

3For a summary of this literature see Barro and Sala-i-Martin, Economic Growth, McGraw Hill (1995).

4See Serge Coulombe and Frank Lee, “Convergence Across Canadian Provinces, 1961 to 1991, “Canadian Journal of Economics , 1991, pp. 886–98.

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3.6 Economic Convergence and Divergence

ECONOMIC CONVERGENCE AND DIVERGENCE

Economic convergence among a set of countries or regions exists if theirgrowth rates of real per capita output converge over time. Otherwise the

economies are said to diverge. Two notions of convergence have been analyzed inthe literature: absolute convergence and conditional convergence. Absolute conver-gence states that poor economies tend to grow faster than rich ones and eventu-ally catch up with them. Conditional convergence states that poor economiestend to grow faster than rich ones, provided that we take into account differencesin country characteristics. The neoclassical growth model is supportive of bothtypes of convergence, whereas the endogenous growth model predicts economicdivergence in general. The empirical evidence is mixed. Some studies supportconvergence and some indicate divergence.

RECAP

FIGURE 3.14

Asia-Pacific Region:GDP Growth vs InitialGDP

The average annual

growth rate of real percapita GDP for theperiod 1960–1990 isplotted against the logof real per capita GDPin 1960 for 17countries of the Asia-Pacific region. Initiallythe poorer East Asianeconomies grew fasterthan the developedeconomies of AustraliaCanada, New Zealand,and the U.S. Thisevidence supports thehypothesis of absoluteconvergence amongthese countries.Real Per Capita GDP (1960)

G D P G r o w t h ( % ) ( 1 9 6 0 –

1 9 9 0 )

0

5

10

15

20

25

0 2000 4000 6000 8000 10000 12000

Sgp

Kor HK

Twn

Jpn

Mex

Chl

Can

Aus

NZUSA

ThaMys

Ind

Chn

PNG

Phl

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Chapter Summary1. Growth accounting attempts to measure the sources of

growth. It decomposes the sources of growth into threeparts: growth in labour; growth in capital; and growthin multifactor productivity.

2. One principal force driving long-run growth in outputper worker is the set of improvements in the efficiencyof labour springing from technological progress.

3. A second principal force driving long-run growth inoutput per worker is the increases in the capital stockwhich the average worker has at his or her disposal andwhich further multiplies productivity.

4. An economy undergoing long-run growth convergestoward and settles onto an equilibrium steady-state

growth path, in which the economy’s capital-outputratio is constant.

5. The steady-state level of the capital-output ratio is equalto the economy’s savings rate divided by the sum of itslabour-force growth rate, labour efficiency growth rate,and depreciation rate.

6. In endogenous growth models growth is driven by en-dogenous factors that can be influenced by governmentpolicies. Endogenous growth models predict a wider di-versity of living standards among different countriesthan the neoclassical growth model.

Key Terms and Equationscapital (p. 84)

capital intensity (p. 79)

capital-output ratio (p. 84)

convergence (p. 107)

depreciation (p. 88)

efficiency of labour (p. 79)

labour force (p. 86)

output per worker (p. 86)

production function (p. 84)

savings rate (p. 88)

steady-state growth path (p. 84)

A A

t

t =

Y Y

t

t – ×

KK

t

t – (1 – ) ×

LL

t

t (3.4)

KLtt = s × Y Lt

t – (δ + n) ×

KLt

t (3.10)

Y Lt

t = A ×

KL

t where A = (ρ × L)1– (3.13)

Y L

t = A × τ1– ×

KL

t where A = (ρ × L)1– (3.17)

gk =s × F

K(K / L / L, E) – (n + δ) (3.18)

Y = A × K (3.19)

gk = s × A – δ – n (3.22)

Solow residuals:

Evolution of K/L:

Romer’s production function:

Barro’s production function:

Steady-state growth rate of K/Lin the neoclassical model:

Rebelo's production function:

Growth rate of K/L in theEndogenous growth model:

www.mcgrawhill.ca/college/delong

Analytical Exercises1. Explain the effects of an increase in the rate of depreci-

ation of capital in the neoclassical growth model, and inRomer’s model.

2. Suppose the economy’s aggregate production function

is given by

Y t = (Kt)0.5 × (E × Lt)0.5

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Delong Ch 03

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