Alemayehu Gedaalemayehu.com/PhDMacro1Web/Lecture 1 Final Version... · 2014. 7. 12. · Romer, David th(2012). Advanced Macroeconomics.4 Edition Barro, Robert Jr. and Xavier Sala-i-Martin

Advanced and Contemporary Topics in

Macroeconomics I

Alemayehu Geda Email: [email protected]

Web Page: www.alemayehu.com

Lecture 1

Introduciton & The Solow Swan/Neoclassical

Model

Addis Ababa University

Departement of Economics

PhD Program

2014

Course Content/Outline 2014 • Lecture 0*: Review of Advance [Dynamics]

Mathematics [Optional]

• Lecture 1: The Solow-Swan/Neoclassical

Grwoth Model [Exogenous Saving]

• Lecture 2: The Ramsey-Cass-Koopmans

Growth Model [Endogenous Saving]

• Lecture 3: The Diamond/ OLG Model

• Lecture 4: The Endogenous Growth Models

Course Content….Cont’d

• Lecture 5: Real Business Cycle Models

• Lecture 6: The Political Economy of Grwoth

(in Ethiopia and Africa)

• Micro-Foundation

• Lecture 6: Consumption

• Lecture 7: Investment

• Lecture 8: The Labour Market

• Lecture 9*: Dynamic General Macroeconomic

Equilibrium Models & DSGE [Optional]

Literature For the Course

Main Readings/Books

Romer, David (2012). Advanced Macroeconomics.4th Edition

Barro, Robert Jr. and Xavier Sala-i-Martin (2004). Economic Growth.

Valdes, Benigno(1999).Economic Growth: Theory ,Empirics and Policy

Acemoglu, Daron (2008).Introduction to Economic Growth.

Nudulu et al (2008). The Political Economy of Grwoth in Africa, 2 volumes

(AERC and Cambridge University Press.

Alemayehu Geda (2008) The Political Economy of Grwoth in Ethiopia (In

the same book above/Nudulu et al, Vol 1, Ch 4)

• Chiang, Alpha C (1992). Elements of Dynamic Optimization. New York,

McGraw-Hill/Or

• Sydsaeter et al (2005). Further Mathematics for Econ Analysis (PTO)

• Taylor, Lance (2004). Reconstructing Macroeconomics. Princeton: Princeton

University Press.

• Weeks, John (2013). False Paradigm: The Irrelevance of Neoclassical

Macroeconomics. London: Edward Edgar

. .

Relevant Articles

• Solow, Robert M. (1956). “A Contribution to the Theory of Economic

Growth”, Quarterly Journal of Economics, 70:65-94.

• Baumol, William (1986). “Productivity Growth, Convergence, and

Welfare," American Economic Review, 76:1072-85

• DeLong, J. Bradford (1988). “Productivity Growth, Convergence, and

Welfare: Comment," American Economic Review, 78:1138-54.

• Mankiw, Gregory N., David Romer, and David N. Weil (1992). “A

Contribution to the Empirics of Economic Growth," Quarterly Journal

of Economics, 107:407-37..

Klenow, Peter J. and Andres Rodriguez-Clare (1997). “The

Neoclassical Revival in Growth Economics: Has It Gone Too Far?"

NBER Macroeconomics Annual, 12:73-103.

Hall, Robert E. and Charles I. Jones (1999). “Why Do Some Countries

Produce So Much More Output per Worker than Others," Quarterly

Journal of Economics,114:83-116

.

. Phelps, Edmund S., “The Golden Rule of Accumulation: A Fable for Growth men,” American Economic Review, September 1961, pp. 638-643

Oded Galor, Unified Growth Theory (Princeton, NJ: Princeton University Press,

2011).

Romer (1986) „Increasing Returns and Long run Growth‟ Journal of Political

Economy, 94:1002-1037.

Romer (1990) „Endogenous Technological Change’, Journal of Political

Economy, 98: 71-102.

Frankel (1962)

Domar, Evsey D. (1946), “Capital Expansion, Rate of Growth and Employment”,

Econometrica 14: 137-147.

Harrod, Roy (1939),“An Essay in Dynamic Theory”,Economic Journal 49: 14-33.

Kaldor, Nicholas (1963), “Capital Accumulation and Economic Growth”, In

Proceedings of a Conference Held by the International Economics Association,

Friedrich A. Lutz and Douglas C. Hague (editors). London: Macmillan.

Swan, Trevor W. (1956), “Economic Growth and Capital Accumulation”,

Economic Record 32: 334-361.

.

Class schedule:

Lucas, Robert E. Jr., “Why Doesn‟t Capital Flow from Rich to Poor

Countries?” American Economic Review, May 1990, pp. 92-96.

Uzawa, Hirofumi (1961), “Neutral Inventions and the Stability of the

Growth Equilibrium!”, Review of Economic Studies 28: 117-124

William A. Brock and M. Scott Taylor, “The Green Solow Model,”

Journal of Economic Growth, June 2010, pp. 125-153.

Aghion, P and P. Howitt (1992) „A Model of Growth through Creative

Destruction‟ Econometrica, 60: 323-351.

– To be worked out/ Flexible Modular

– 3 to 6 hours lecture per week

Additional articles are given on the Course CD that Contains the

lecture slides

Course Assesement Course Assessment

A) Each Candidate will write a review of literature in 4/5 topical areas of the work

based on 5 to 10 latest working papers/and Presentation. (20%)

B) There will be bi-weekly/Weekly drill on each of the lectures (to be handled by a

teaching assistant) (25%)

C) A Group paper on African Business Cycle (A group of 2 in each team) (20%)

D) Class Presentation and participation (10%)

E) Individual term paper on grwoth in Ethiopia or Africa on the following topics, 20

pages in a model of AER, Journal of African Economies etc (30%)

A) A Solow-Swan Model for Ethiopia OR

B) Human Capital Augmented Solow-Swan Model for Ethiopia

C) Ramesy-Cass-Koopman Model for Ethiopia

D) Endogenous Grwoth Model for Ethiopia

E) The Political Economy of Grwoth in Ethiopia: Political, Institutions and

Grwoth

F) A Dynamic Stochastic General Equilibrium Model for Ethiopia/Kenya

G) Modelling Investment (or Public Private Investment Interaction) in Ethiopia

H) Modelling Consumption/Saving in Ethiopia

.

Chapter One

The Solow –Swan Model

[The Neoclassical Model]

Lecture One contents I. Introduction and Review of Advanced Mathematics

II. The Solow Model

Assumptions

Inputs and outputs

Production function

Evolution of inputs into production

Solution

III. The Dynamics of the Model

The Dynamics of k

The Balanced Growth Path

IV. Comparative Dynamics: Impact of a Change in Savings

Rate

The Impact on Output

The Impact On Consumption

. V. Quantitative Implication Steady State: Quantitative Importance of Savings Rate in Affecting

Income Per Capita in the Long Run

Transition Dynamics: The speed of Convergence

VI. The Central Questions of Growth in the Solow Model

VII. Empirical Applications

Growth Accounting

Convergence

VIII. Conclusion

.

General Introduction about

Growth: Some basic facts about

economic growth

Method: Sherlock Holems’

Method!

• Sherlock Holems‟ 4 Rules of Scientific inquiry

Rule 1: Begin with the Data

Rule 2: Build a theory (or thoeries) capable of covering the facts that are know to u

Rule 3: Do not take for grants that your theory is correct because it cover all the facts (as new facts may come and other theories may cover that too)

Rule 4: If new evidence can not be accommodated with exiting theories reconsider your theories.

{See Valdes, Page 8-10)

Kaldor’s “Stylized Facts” about

Growth • In 1961 article he outlined empirical regularities: • “Capital Accumulation and Economic Growth” in F.A. Lutz and D.C. Hague

(eds). The Theory of Capital. New York: St Martin Press.

• SF1: Standard of living always increase from one generation to the next – Per capita income and labour productivity (y=Y/L) is increasing

(y grows at positive rate )

• SF2: The capital output ratio has no upward or downward trend (K/Y -no change in the long run )

• SF3: The functional distribution of income remains constant in the long run Π/K=profit, constant and hence Wage share is total income less profit share

• SF4: There are a variety of growth rates of percaptia income across the world

Kaldor’s “Stylized Facts”

• SFacts (1) and (2) imply the following:

• SFacts (2) and (3) together imply the following:

oo

oo

oo

ykL

Y

L

K

L

Y

Y

K

Y

K

Y

K

0

0/constant

/Y)( -1 W/Ydefinationby 1)W/Y/ (hence YWGiven

constant. be alsomust / capital physicl of share income

theS3by constant also is / and S2by constant is K/Y since

; ..

Y

Y

K

Y

K

YK

Y

YK

Some basic facts about economic growth

Standards of living in industrialized economies have

increased dramatically over the centuries

In the U.S and Western Europe real incomes is 10 – 30 times larger

than 100 years ago and 50 – 300 times larger than 200 years ago

Worldwide growth has not been constant

Average growth rates in industrialized countries were higher in the

twentieth century than in the nineteenth century, still growth rates

were higher in the nineteenth century than in the eighteenth

century…

Productivity growth has slowed down

In many industrialized countries annual growth in output per

person has slowed down since the 1970s

Standards of living vary enormously across countries

Real income is more than 20 times higher in the U.S than in

Bangladesh

Large variations in growth rates

Growth miracles, e.g. Japan, NICs

Growth disasters, e.g. Sub-Saharan Countries, Argentina

Over the modern era, cross-country income differences has

widened – enormous differences in human welfare across

different part of the world.

Some basic facts about economic growth:

(cont..)

An important quote fro Lucas(1988):

– When you think of the implication of a

solution to grwoth problem for mankind…..

“Once one start to think about economic

growth, it is hard to think about anything

else.”

(Robert E. Lucas, 1988)

.

Chapter One

The Solow –Swan Model

I. Introduction

As the first step, in order understand the role of

proximate causes (see nxt Slide) of economic

growth we develop a simple framework. We take

the Solow-Swan model as our starting point.

The model is named after Robert Solow and

Trevor Swan, who published two seminal papers

in the same year (1956).

Introduction.......cont’d

Robert Solow developed many applications of the

model, and was later awarded the Nobel prize in

economics.

This model has not only become the centerpiece of

growth theory but has also shaped the modern

macro theory.

Introduction.......cont’d

The central model of macroeconomics before the

Solow model came along was the Harrod-Domar

model, which was named after Roy Harrod and

Evsey Domar (Harrod (1939) and Domar (1946)).

The Harrod-Domar model focused on

unemployment and growth.

The distinguishing feature of the Solow model is

the neoclassical aggregate production function.

II. The model

The Solow model focuses on four variables: –Output (Y)

–Capital (K)

–Labour (L)

–Effectiveness of labour (A)

The Production function takes the form:

where t denotes time, which enters indirectly into the production function through K, L and A.

)1.1.(.........., tLtAtKFtY

Some properties of the production function:

Output changes over time only if input changes over

time

The amount of output obtained from given quantities

of capital and labour rises over time only if there are

technological progress, i.e. the effectiveness of labour

increases over time.

A and L enter multiplicatively into the model such that

the term AL is referred to as „effective labour‟ meaning

that technological progress is labour-augmenting

(Harrod-neutral).

Some critical assumptions regarding the production

function:

CRS – doubling input doubles output (A held constant):

• Implicitly assumes that the economy is sufficiently large that

any gains from specialization has been exhausted

• Implicitly assumes that other production factors (e.g. land) is

relatively unimportant

)2.1.........(..........0,, cALKcFcALcKF

With CRS, an intensive form of the production

function is easily specified:

Set

Hence (1.3) can be rewritten in intensive form production

function as:

ALc /1

)3.1........(..........,11,AL

YALKF

ALAL

KF

AL

KkWhere

kfyAL

KFALKF

ALAL

Y

)4.1......().........()1,(),(1

The intensive form production function is assumed

to satisfy the following conditions:

0)('lim

)('lim

0)(''

0)('

00

0

kf

kf

kf

kf

f

k

k

f(k)

k

. Since it follows that the

marginal product of capital,

{see equation 1.3}

Thus the assumptions that is positive and

is negative imply that the marginal product of

capital is positive ,but that it declines as capital

(per unit of effective labor) rises.

)(),(AL

KALfALKF

).()1

)((),(

kfALAL

KfAL

K

ALKF

)(kf )(kf

. These conditions (which are stronger than needed

for the model‟s central results) states that the

marginal product of capital is very large when the

capital stock is sufficiently small and that it

becomes very small as the capital stock becomes

larger;

their role is to ensure that the path of the economy

does not diverge.

Example:

Cobb-Douglas production function:

First order condition:

Second order condition:

kkfy

kAL

K

AL

ALK

AL

Y

ALKY

)(

)(

)5.1 ...(........................................ 10,)(

1

1

0)1(

1)(''

2

2

k

kkf

0)()( 1

kkfk

k

k

kf

The Evolution of the Inputs into Production

The remaining assumptions of the model concern how

the stock of labour, knowledge, and capital change over

time.

The model is set in continuous time(i.e., the variables

are defined every point in time)

Labour and knowledge (technology) is assumed to

grow at a constant rate over time:

where n and g are exogenously given constant growth

rates and a dot over a variable denotes a derivative w.r.t

time

)9.1(..............................).........(

)8.1.(..............................).........(

tAgA

tLnL

)/)()(( dttdLtL

. The growth rate of a variable refers to its

proportional rate of change .(i.e., the phrase the

growth rate of X refers to the quantity .

Thus equation (1.8) implies that the growth rate of

L is constant and equal n , and equation (1.9)

implies that A‟s growth rate is constant and equal

to g .

)(

)(

tX

tX

. A key fact about growth rates is that the growth

rate of a variable equals the rate of change of its

natural log. i.e.,

To see this: since lnX is a function of X and X is a

function of t, using the chain rule to write:

.)(ln

)(

)(

dt

tXd

tX

tX

)10.1.(....................).........()(

1)(

)(

)(ln)(lntX

tXdt

tdX

tdX

tXd

dt

tXd

. Applying the result that a variable‟s growth rate

equals the rate of change of its log to (1.8) and

(1.9) tells us that the rate of change of the logs of

L and A are constant and that they equal n and g,

respectively. Thus,

)12.1.........(....................,.........)]0([ln)(ln

)11.1.........(....................),........)]0([ln)(ln

gtAtA

ntLtL

. • Where L(0) and A(0) are the values of L and A at

time 0. Exponentiating (or taking the ant-log) both

sides of these equations gives us

Thus our assumption is that L and A each grow

exponentially.

)14.1.(..................................................)0()(

)13.1.(..................................................)0()(

gt

nt

eAtA

eLtL

. Coming to our model: Output is used to either consumption or investment (saving).

• The saving rate (s) is assumed to be constant and

exogenously given.

– For simplicity one can assume that one unit of

investment is equal to 1 unit of new capital.

• Existing capital depreciates at a rate δ.

These assumptions imply that the capital stock

grows according to:

)15.1..(..........).........()()( tKtsYtK

. Although no restrictions are placed

,their sum is assumed to be positive .i.e.,

This completes the model.

The Solow model is grossly simplified in a

number of ways.

There is only one good

No government

Fluctuation in employment are ignored

, andgn

0 gn

. Production is described an aggregate production

function with just three inputs

The rate of saving, deprecation, population growth

and technological progress are constant

Etc….

The model omits many obvious features of the real

world which are important for growth

But the purpose of a model is not to be realistic

(NB Positivist Method!). After all, we already

possess a model that is completely realistic-the

world itself (see my 2nd Trade book Ch4

Appendix)

III. The Dynamics of the Model

Here we want to determine the behavior of the

economy we have just described in the previous

slides.

The evolution of two of the three inputs into

production, labor and knowledge, is exogenous.

Thus to characterize the behavior of the economy,

we must analyze the behavior of the third input,

Capital.

a) The Dynamics of k

Because the economy may grow over time, it is much easier

to focus on the capital stock per unit of effective labour, k,

than on the unadjusted capital stock K.

Since k(t)=K(t)/A(t)L(t), i.e. a function of K, L and A, which

are all functions of t, the chain rule applies and we can find

the intensive form of the capital growth equation from

.

kAL

Kwhere

tA

tA

tLtA

tK

tL

tL

tLtA

tK

tLtA

tK

tLtAtLtAtLtA

tK

tLtA

tK

dtdKtKtAtA

tktL

tL

tktK

tK

tktk

t

tLtA

tK

tk

simply is

)(

)(

)()(

)(

)(

)(

)()(

)(

)()(

)(

(1.16)

)()()()([))()((

)(

)()(

)(

etc...../)( NB )()(

)()(

)(

)()(

)(

)()(

)(

)()(

)(

)(

2

Since from (1.8) and (1.9) ,

respectively

Substituting these facts into (1.16) &using (1.15)

yields:

ntL

tL

)(

)(g

tA

tA

)(

)(

)18.1...(........................................).........()(

))(()()(

)(fact that the UsingAnd

)()()()()(

)(

(1.17)

)()()()(

)()()()(

)()(

)()(

tkgntksf

tkftLtA

tY

tgktnktktLtA

tYs

gtkntktLtA

tKtsYgtkntk

tLtA

tKtk

Break-even investment Total investment

Equation (1.18) is the key equation in the Solow model.

It states that, the rate of change in the capital stock per unit of

effective labour is the difference between total investment

per effective labour unit and the amount of investment

needed to keep the capital-to-effective labour-ratio constant.

The first term, sf(k), is actual investment per unit of effective

labor

The second term, (n + g + )k is break-even investment, i.e.

the amount of investment that must be done to keep k at its

existing level. Investment is needed to prevent k from falling

because the existing capital deprecates at rate , which is

captured by k term, and also because the quantity of effective

labor is also growing at rate (n+g), which is captured by the

term (n+g)k.

.

Therefore, when the actual investment per unit of

effective labor exceeds the break-even investment, k

rises, and vice-versa.

Moreover, when the two are equal, k is constant.

This is depicted in figure 2 below.

• The figure plots the two terms of the right-hand side of

the fundamental law of motion as functions of k. Since

F(0,L,A) = 0 it implies that f(0) = 0, and therefore

actual investment and break-even investment are equal

at k = 0.

. Furthermore, the Inada conditions imply that as k goes to zero f’(k) becomes very large. Therefore

sf(k) is steeper than break even investment line

around k = 0; and actual investment is larger than

break-even investment.

• The Inada conditions also imply that f’(k) falls to

zero as k becomes very large. As a result, at some

point the slope of he actual investment curve

becomes less than the slope of the break-even

investment line, implying that the two lines must

eventually cross

Figure 2: Actual investment and break-even

investment

. Finally, due to diminishing returns, fkk < 0, the two lines intersect only once. Let k* be the value

where actual investment equals break-even

investment, or in other words there is no change in

capital per unit of effective labor, .

This value of k is also called the steady state value

of k.

The next figure shows that the economy converges

to k regardless of where it starts.

0k

Figure 3: Steady State

When k < k* actual investment exceeds break-

even investment as a result .

When k > k* the actual investment is less than

break-even investment and therefore .

Finally for k =k*, .

0k

0k

0k

b) The Balanced Growth Path

How are output, capital and consumption growing

in this economy when k = k*.

We know that L and A are growing at

exogenously given rates - n and g, respectively.

The capital stock K = ALk, and since k is constant

at k*, the aggregate capital stock of the economy

is growing at the rate(n+g).

. With both capital and effective labor growing at the same rate (n+g), the assumption of constant returns to scale

implies that aggregate output is also growing at the rate (n

+ g). Since consumption is (1 − s)Y , where s is constant,

consumption also grows at the same rate as output.

Finally, capital per unit of labor, K/L, and output per unit

of labor, Y/L, grow at the rate g.[Nb:! at (n+g)-n, n for the

percpita handling!)

Thus, the Solow model implies that regardless of its

starting point, the economy converges to a balanced

growth path, where each variable grows at a constant rate

. At this point we also need to discuss our assumption that technological change is labor augmenting.

This is a restriction that is required for the existence of a

balanced growth path.

NB: Other types of technological change - Hicks-neutral

(unbiased technological change) and capital augmenting-

technical change - are not consistent with a balanced

growth path. For a proof look at Uzawa (1961).

Notice also that off the balanced growth path technological

change is no longer required to be labor-augmenting. (Your

reading assignment begin with Valdez)

. The idea of balanced growth though seemingly abstract has a parallel in the data.

The Kaldor‟s stylized facts, Kaldor (1963), show

that while output per capita grew, the capital-

output ratio (K(t)/Y (t)), the interest rate(r(t)), and

the distribution of income between labor

(w(t)L(t)/Y (t)) and capital (R(t)K(t)/Y (t)) remain

roughly constant.

The figure below shows the factor shares for the

US since 1929.

Figure 4: Labor and capital shares in value added in the U.S.

IV. Comparative Dynamics: Impact of a Change in

Savings Rate

The parameter of the Solow model that policy is

most likely to affect is the savings rate.

• What is the effect of (unanticipated) change in the

savings rate s?

• Consider an economy that is on a balanced growth

path, and suppose that there is a permanent

increase in s. The increase in s shifts the actual

investment curve upwards, thereby resulting in an

increase in k*.

A. The Impact on Output

• Initially, when s increases and the curve shifts up,

at the initial steady-state value of k the actual

investment exceeds break-even investment.

• Thus is positive resulting in an accumulation of

k, which continues till it reaches the new steady-

state value of k.

• This is depicted in the figure below.

k

Figure 5: Effect of change in savings rate on investment

.

• Y/L, we concluded grew at rate g when k = k*.

• However, when k is increasing, as the economy moves

from one steady-state to another, Y/L grows at a rate

higher than g (because the source of growth is not

only the growth rate of A[=g], but also growth rate

of k (which was constant when k=k* hence y*=f(k*)

before s increased to s*).

• Once k reaches its new steady-state value, growth rate

of Y/L falls back to g.

• Thus a permanent increase in s produces temporary

increase in in the growth rate of output per worker, k

rises for some time but eventually it reaches a level at

which additional savings are devoted to maintaining

the higher level of k.

Figure 6: Effect of an increase in savings rate

Effect of increase in saving, 2nd diagram explained

• After the transition period of “T” the new steady

state log y will follow the trajectory {At+T=log

y*t+T, At+T+1=log y*t+T+1,…}

– This is above &parallel to the previous steady state

– In the transition period (T) the slop of log y [which is

grwoth rate] is greater than both stead states. (ie

during transition percapita income growth at a rate

greater than “g”

– Note that y=(Y/L) is one-to-one linked to y~= (Y/AL)

through A, y=A(y~), Thus as y~ growth at growth

rate of A=g (since it is f(k*) in which k is constant at

steady state), so is y will be growth (at “g‟ rate).

– The reason behind this dynamics is the diminishing MPK

. • At the end of the day a change in s has a level effect but not a growth rate effect:

– ie. it changes the balanced growth path of the economy

and its level of output per worker, but it does not affect

the growth rate of output of per worker on the new

balanced growth path.

• In fact in the Solow model only changes in the rate

of technological progress have growth effects of

percaipita income; all other changes have level

effects.

B. The Impact on consumption

• Since consumption per unit of effective labor c =

(1 − s)f(k), an increase in s at the initial steady-

state level of k results in an initial decrease in c

and then as k rises to its new level c also rises.

• Whether or not c exceeds its original level can be

seen by writing down the expression for

consumption per unit of effective labor.

Steady-state consumption is given by:

• An increase in s raises k. Thus, c will rise in

response to an increase in s if the marginal product

of capital, fk, is greater than (n + g + δ).

• Intuitively when k rises investment must increase

by (n + g + δ) times k in order to sustain the new

level of k.

)20.1.(..........),,,(*)]()),,,(*([*)19.1...(..................................................*,........)(*)(*

s

gnskgngnskf

s

c

kgnkfc

k

. • If fk is less than (n + g +δ ), then the additional output from a higher k is not enough to support the higher

level of k.

• As a result c must decline in the long run to maintain

the stock of capital.

• On the other hand, if fk exceeds (n+g+δ) there is more

than enough output to support the higher level of k,

and therefore c increases in the long run.

• However, if the steady-state value of k to start with is

changed duet to s in such a way that fk = (n + g + δ)

then a marginal change in s does not change c.

. • This value of k is called the golden rule level of the capital stock (& the associate saving “saving gold”).

• At the golden rule level of capital, consumption is at

its maximum level {see the next 2 slides).

• Since s is exogenous in the Solow model, there is no

guarantee that k will be at its golden rule level.

• This cases are depicted in the figures below.

– See the implications of this for the choice of either

current or future consumption (hence Ramsey

model) & OLG models!! see Barro &Sala-i-

Martin, p. 34 on dynamic inefficiency [also briefly

in few slides ahead]

Figure 7: Output, investment and consumption on the

balanced growth path

V. Quantitative Implications of the Model

• A) Effect of saving on per capita income

• B) A little bit about Dynamic inefficiency

• C) Transition dynamics – the speed of

convergence from y to y* /or from k to k*

A. Steady State: Quantitative Importance of Savings

Rate in Affecting Income Per Capita in the Long Run

• In our discussions we saw that income per capita varies significantly across countries. Can the savings

rate have a quantitatively important impact on income

per capita so as to explain such large income

differences?

• The long run effect of a change in savings rate on

output per unit of effective labor is given by: (see

derivation of 1.21, next slide)

)()(

)()(

)21.1.....(........................................),,,(

)(

*

**

**

*

kfsgn

kfkf

s

gnskkf

s

y

. • Where is the level of output per unit of effective

labor on the balanced growth path.

• Thus to find , we need to find

• To do this, note that k* is defined by the condition

; thus k* satisfies

• Equation (1.22) holds for all values of s(and of n, g, and δ)

*)(* kfy

s

y

*

s

k

*

0k

)22.1......(..........).........,,,(*)()),,,(*( gnskgngnsksf

. • Thus the derivatives of the two sides with respect to s are equal:

Where the arguments of k* are omitted for simplicity.

• Rearranging (1.23) gives :

)23.1..(..............................*

)(*)(*

*)(s

kgnkf

s

kkfs

)25.1..(..............................*)()(

*)(*)(**

*

*

obtained be could (1.21) rulechain and ) (1.24 using

)24.1..(........................................*)()(

*)(*

kfsgn

kfkf

s

y

s

k

k

y

kfsgn

kf

s

k

. Two changes help in interpreting this expression.

• First convert (1.25) into an elasticity by

multiplying both sides by s/y*

• Second use the fact that to substitute for s .

*)(*)( kgnksf

Making these changes results:

*)](/*)(*[1

*)(/*)(*

(1.26)

*)](/*)(*)()*)[((

*)(*)(*))(

*)(

*)('

*)()*)((

*)(

*)()(

*)(*)(

*)(

*

*

kfkfk

kfkfk

kfkfkgngnkf

kfkfkgn

kf

kf

kfsgnkf

ksf

kfsgn

kfkf

kf

s

s

y

y

s

. • is the elasticity of output with

respect to capital at k=k*.Denoting this by

• If markets are competitive and there are no externalities,

capital earns its marginal product . In this case, the total

amount received by capital (per unit of effective labor) as

the share of output on the balanced growth path is

*)(/*)('* kfkfk

have we*),(kk

)27.1....(..................................................*)(1

*)(*

* k

k

s

y

y

s

k

k

*).( *),(/*)('* korkfkfk k

• The elasticity of output w.r.t. saving depend on capital share of

income, . ie – With competitive markets and no externalities capital earns its marginal

product.

– & On the balanced growth path the share of income attributed to capital

must be

• Capital share of income is found to be about 1/3 in most

countries, which implies that (from eqn 1.27 above) the

elasticity of output w.r.t. saving is about 0.5:

– Thus, a 10% increase in savings rate increases per worker output in the

long run by 5% relative to the path it would have followed. For a 50%

change in savings rate y rises by only 25%.

– . Thus, big changes in savings rate have only a moderate effect on the

level of output on the balanced growth path

*)(kK

*)(*)(/*)('* kkfkfk K

5.03/11

3/1

*)(1

*)(*

*

k

k

s

y

y

s

k

k

B. Saving, Dynamic inefficiency and the prelude

to Ramesy-Cas-Koopman Model

• A little bit about Dynamic inefficiency here:

– If higher saving leads to higher level of income (higher

steady state) [though no higher growth rate], the best saving

rate will be 100% - but this is tautological [ie higher s,

higher k, higher Y etc….]

– Specifically in SS model a saving rate above the golden rule

reduces per capita consumption at steady state [called

dynamically inefficient/over saving – hence the need to

reduce it from s2 (the need to increase it form s1 in the next

diagram)

– From a societies welfare perspective high saving is not

necessarily good and low saving necessarily bad when once

see it dynamically either -partly because higher saving

today means lower consumption today (may be higher

consumption in the future!).

– .

Saving, Dynamic inefficiently and the prelude to

Ramsy-Cas-Koopman Model

– If we follow Keynes: “consumption is the sole end and

objective of all economic activity” so we can use it for

societal welfare indications

– However the welfare depends on hh valuation of future

versus current consumption (or themseleves versus future

generation).

– Thus, the optimal saving of a nation (hence investment)

should be inferred from the optimal level of

consumption/utility it generates over time

– Thus the importance of inter-temporal view and hence the

Ramesy-Kass-Koopman model: !!how much should a

nation save?!!Ramesy

– See Barro and Saal-i-Martin, p. 34, & nxt slide

Saving, Dynamic inefficiently and the prelude to

Ramsy-Cas-Koopman Model

• A little bit about Dynamic efficiently here:

– See Barro and Saal-i-Martin, p. 34, nxt slide

C. Transition Dynamics: Speed of Convergence

How rapidly does k approach k* when s changes?

– The growth of capital per unit of effective labour depends on

the size of the capital-to-effective labour ratio:

– At the point where we know that .

– These conditions imply that a first order Taylor-series

approximation of around the point give:

)(kkk

*kk 0k

)(kk *kk

*/)(*)()( kkkkkktktk

In the neighbourhood of the balanced growth path, k converges to

k* at a speed that is approximately proportional to its distance from

k*. Hence, the gap between k(t) and k* narrows at a rate that is

approximately constant and equal to :

To find an expression for we differentiate

w.r.t. k and evaluate the resulting equation at the point k=k*:

*)0(*)( kkektk t

)()()( tkgntksftk

gnk

kf

kfkgngn

gnksfk

kk

K

kk

*)(1

*)(

)('*

*)(')(

*

Using the expression for we can conclude that the speed of convergence of the capital stock is:

A reasonable assumption is that . With we have .

k moves 4% of the remaining distance toward k* per year

It takes approximately 17 years for k to get half the way to its balanced growth path if k(t) is in the vicinity of k* to begin with.

Since output per effective worker only depend on k, k and y converges to k* and y* at the same speed.

The impact of a change in the saving rate on output is both quite small and fairly slow!

*)0(*)( *)(1 kkektk tgnkK

%6 gn 3/1K

%41 gnK

VI. The Central Questions of Solow model the

growth theory

– Why are some economies much richer than others?

– Are income levels converging across nations?

The Solow model identifies two potential source to why output per

worker varies across countries and over time:

1. Differences in capital per worker (K / L)

2. Differences in the effectiveness of labour (A)

Due to convergence of k to k* changes in the effectiveness of labour is

the only factor that lead to permanent changes in the growth rate

• There are two ways to see that the Solow model implies that differences in

capital accumulation cannot account for large differences in income:

1. Required differences in output per worker**

• Differences in incomes between rich and poor countries roughly

corresponds to a factor of 10. This implies that the capital stocks in rich

and poor countries differs by a factor of .

In the real world one observes that the capital-to-labour ratio in rich

countries are 20 to 30 times larger than in the poor countries. Moreover,

capital-to-output ratios tend to be fairly constant over time.

2. Required differences in the rate of return to capital

• If markets are competitive the rate of return to capital equals its marginal

product minus depreciation.

– A tenfold difference in capital per worker implies a difference in the rate of

return to capital by a factor of 100.

– MPK in poor countries would be so high that there would be no reasonable

answer to why not all capital in the world relocates to the poor countries.

)3/1(100010 /1 KK

VII. Empirical Application

A. Growth Accounting

In the Solow model ,long run growth of output per worker

depends only on technological progress.

But short run growth can result from either technological

progress or accumulation.

Thus the model implies that determining the sources of

short run growth is an empirical issue

Thus the growth accounting relates to an empirical

extension that allows to distinguish between different

sources of growth.

. Growth accounting is pioneered by Abramovitz(1956) and

Solow(1957)

To see how it works, consider the production function

given as

).()(

)()(

)(

)()(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

yeilds side handright on the terms therewriting and )(by sides bothe Dividing

.ly respective,][ and ][ denote and where

)()(

)()(

)(

)()(

)(

)()(

is derivative totalits implies, This).()(),(()(

tRtL

tLt

tK

tKt

tA

tA

tA

tY

tY

tA

tL

tL

tL

tY

tY

tL

tK

tK

tK

tY

tY

tK

tY

tY

tY

L(AL)

YA

(AL)

Y

A

Y

L

Y

tAtA

tYtL

tL

tYtK

tK

tYtY

tLtAtKFtY

LK

. Note that

is the elasticity of output with respect to labor

at time t,

is the elasticity of output with respect to

capital at time t and

.

)(tL

)(tK

. Subtracting from both sides and using the fact that results an expression for the

growth of output per worker as:

Note that:

The growth rate of Y,K and L are straight forward

to measure

If capital earns its marginal product, can be

measured using data on the share of income that

goes to capital

)(

)(

tL

tL

1)()( tt KL

)35.1..(..........).........(])(

)(

)(

)()[(

)(

)(

)(

)(tR

tL

tL

tK

tKt

tL

tL

tY

tYK

K

. R(t) then can be measured as the residual in

equation (1.35) above.

Thus equation (1.35) provides a way of

decomposing the growth of output per worker into

the contribution of growth of capital per worker

and a remaining term, the Solow Residual

The Solow residual –some times interpreted as a

measure of the contribution of technological

progress (TFP).

But as derivation shows, it reflects all sources of

growth other than the contribution of capital

accumulation via its private return

. Growth Accounting has been applied to many

issues:

Yong(1995)-used detailed growth accounting to argue that

the higher growth in the newly industrialized countries of

East Asian than the rest of the world is mainly Due to

rising investment, increasing labor force participation, and

improving labor quality(in terms of education),and not to

rapid technological progress other forces affecting the

Solow residual

See:Hsieh(1998a); Denison(1985); Bailyand

Gordon(1988); Griliches(1988);and Jorgeson(1988) for a

classic application of growth accounting

Alemayehu (2008; 2013) used it to examine Ethiopia‟s

growth “Miracle” 2000-2013.

Table 2.3: A Growth Accounting Exercise for Ethiopia in the Last Decade (2000-2010)

Note: The growth accounting is based on the result of econometric estimation reported in Alemayehu and Befekadu (2005) and Alemayehu et al (2008). Using various models (both macro and micro) these

studies have come up with the capital share coefficient (β) that ranges from 0.28 to 0.36. We have used

0.30. The capital stock is generated using the perpetual method (Geda, 2013/14).

Year

GDP Growth The Contribution of

Capital

The Contribution of

Labour

The Contribution of Total

Factor Productivity (TFP)

2000/2001 7.4 0.6 2.6 4.2

2001/2002 1.6 0.8 2.7 -1.9

2002/2003 -2.1 1.0 2.6 -5.7

2003/2004 11.7 0.7 2.7 8.3

2004/2005 12.6 1.2 2.6 8.8

2005/2006 11.5 1.1 2.7 7.7

2006/2007 11.8 1.5 2.2 8.1

2007/2008 11.2 2.1 2.2 6.9

2008/2009 9.9 1.8 2.3 5.9

2009/2010 10.4 2.7 2.2 5.5

Average(2003/04-2009/10) 11.3 1.6 2.4 7.3

B. Convergence

Are poor countries growing faster than the rich ones, i.e. is

there convergence?

Are income levels converging across nations?

• Solow model suggests three reasons why countries are

expected to converge: 1. The model predicts that countries converge to their balanced growth

paths – to the extent that differences in output per worker depend on countries being at different stages relative to their balanced growth path, cross-country income differentials are expected to decrease.

2. The model predicts that the rate of return to capital is lower in countries with more capital per worker, which induce capital to flow from rich to poor countries and result in convergence.

3. Lags in technology diffusion can account for income differentials between countries and to the extent that poor countries gain access to state-of-the art technologies poor countries would catch up on rich countries.

.

• For classic application on the convergence

hypothesis see Baumol(1986), De

Long(1988); also Findely, AER, 1996. The

first two are discussed below

• Reading Assignment on Beta and Sigma

convergence/ Presentation - Seminar (form

the convergence folder)

Convergence… Cont‟d


• De Long (1988) noted Baumol‟s Finding is

largely spurious. Cause problems of:

– (a) sample selection (countries that have long data

series are those that are the most industrialized).

Countries not rich 100 years ago and in the sample

must be there if they grow fast

– He included countries as rich as the 2nd poorest in

Baumol‟s sample, Finland. This led to add more

countries (see Diagram below)


• De Long (1988) 2nd problem:

– (b) measurement error: estimates of 1870 are

imprecise. Measurement errors bias result

towards convergence [if 1870 is overstated,

growth 1870-79 is understated by an equal

amount and vice versa (growth tends to be low

where measured initial income is high even if no

relation between initial income actual growth).

– He then estimated the following model (see

Diagram below)

– This reduced the estimate of “b” to -0.566


• For example: if we find measured growth is

negatively related to initial income this could be

– Either measurement error is not important and there

is convergence OR

– Measurement is important and there is no

convergence

– This is what is called “model identification problem”

• However, De Long argues we can have an idea

of the accuracy of the 1870 data

Convergence… Cont‟d • In term of standard value (SD), an SD of 0.01

implies, we measured initial income with 1%

error [-implausibly low] and SD=0.50 is 50%

error [-Implausibly high]

• He found even moderate SD has substantial

impact on the result.

– For unbiased sample an SD of 0.15 makes the estimate

of “b” reaches 0 (no convergence); AND an SD of

0.20 makes it 1 (convergence big time!)

• Thus, a moderate measurement error could take

most of the remaining Baumol‟s estimate of

convergence (as we noted was already-0.566)

C. Human capital Augmented Solow-Swan Model

• Attributed to Mankiew et al (1992)….

• Good summary in Heijira (2009) P.411

The Augmented Solow Swan (Human

Capital) Neoclassical Model

1) The Puzzle :

Emperical estimates of the speed of adjustment shows that values around

(this is actually for the US & say 2% of the gap from the steady state is closed each year (i.e., the gap between

However ,having this values and bench mark value for US and other DC‟s

0226.0

tt yy and

estimate share capital theis 68.0)(

of valuecomputable the,04.0 and 032.0

gn

gnα

ygn

The augmented Solow Swan (Human

Capital) Neoclassical Model ….. Cont’d

However, the capital share for most DC‟s is about 30% .The question is what is happening. Why the SS model gives α=68% [see our Diagram bfr]

2) Mankiw, Romer and Weil(1992) suggested a simple solution to this problem/ puzzle

→we are not thinking about K in a right way. We take it as physical capital but it also incorporates human capital (skill)-this is because all output not consumed is not used to increase physical capital only; but also skill (health education, etc)

Human capital ….Cont’d • Key idea: add human capital to the model.

• Technology:

Y(t)= 𝐾(𝑡)αk+𝐻(𝑡)αH+[𝑍 𝑡 𝐿 𝑡 ]1−αk−α𝐻

( 0 < αK+ αH < 1)

where H(t) is the stock of human capital and αK and αH are

the efficiency parameters of the two types of capital

(0 < αK, α H < 1).

• In close accordance with the Solow-Swan model, productivity

and population growth are both exponential (𝑍 (t)/Z(t) = nZ

and 𝐿 (t)/L(t) = nL.

• The accumulation equations for the two types of capital can

be written in effective labour units as:

Human capital ….Cont’d

• Stability: The phase diagram is give next


• In Summary the model is given by:


Humanĸ capital ….Cont’d


k

VIII. Conclusion

Central conclusion from the Solow model:

Only differences in the productivity of labour

can account for vast differences in wealth across

space and time (see slide 84&85 in this lecture).

The „technology factor‟ is, however, exogenous to the Solow model – the model makes no prediction of what this

factor really is, how it behaves or how it grows.

END… END…. END

Alemayehu Gedaalemayehu.com/PhDMacro1Web/Lecture 1 Final Version... · 2014. 7. 12. · Romer, David th(2012). Advanced Macroeconomics.4 Edition Barro, Robert Jr. and Xavier Sala-i-Martin

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