SGPE Summer School: Macroeconomics Lecture 5

SGPE Summer School:

Macroeconomics

Lecture 5

Recap: The natural levels of production and

interest rate

Y n =C Y,Y e, r,A( )+ I r,Y e,K( )

where Y n = F(K,E(1-un )L)

Capital stock

was taken as

exogenous

2

Recap: The natural rate of interest for given K

r

( ), , ,n eS Y Y r A

, ,eI r Y K

S,I

rn

3

Introduction

Presentation Outline:

• Capital accumulation and growth

• Wage determination and unemployment

• Money and inflation in the long run

4

Growth (Chapter 5)

Question:

• What factors determine the capital stock, production

and the real interest rate in the very long run?

Analysis of two cases:

• Constant population and technology

• Constant population growth and technical development

We also examine income differences between countries

5

Growth: Given population & technology

Optimal capital stock:

… but what determines the real interest rate?

In a closed economy, long-run equilibrium, without

growth in population or technology, consumption must

be constant, so we must have

MPK

1+m-d = r

u '(Ct)

u '(Ct+1

) / (1+r)=1+ r

t

r = r

6


Thus we have, in long-run equilibrium without growth in

population or technology:

We call long-run equilibrium ‘steady state’ (this term will

be explained later)

MPK

1+m-d = r Þ K*

7


8


What happens if we start with K<K*?

• MPK is high so companies want to invest and the real

interest is high:

• With high real interest, consumers want to consume

less today than in the future; they save and accumulate

assets (= capital in a closed economy)

• The capital stock grows until it reaches equilibrium

• As the equilibrium level is reached, MPK falls and

growth returns to zero

If we start out with K>K* the opposite occurs

9


We can also illustrate the adjustment in the diagram with

savings and investment.

In the long-run equilibrium (steady state) :

If we start out with a lower capital K’ then investments

are higher and income and savings are lower for a given

real interest rate, so the real interest has to be higher

S(Y *,Y *,r,A) = I(r,Y *,K) where Y * = F(K*,EN n )

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What happens if the consumer starts to care more about

the future (lower )?

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What happens if the consumer starts to care more about

the future (lower )?

13


Convergence

Using CRS we can express the model in terms of capital

and income per effective worker

• CRS means that for any z

• We can choose

• Production function:

F zK, zEN( ) = zY

Y

EN= F

K

EN,1

æ

èç

ö

ø÷

z =1

EN

14


• Let and

• Production function:

• Production per effective worker depends only upon

capital per effective worker, regardless of the size of the

population. This is a consequence of CRS.

• Marginal product of capital:

• Condition for long-run equilibrium:

y ºY

EN, k º

K

ENf (k)= f

K

EN

æ

èç

ö

ø÷ º F

K

EN,1

æ

èç

ö

ø÷

y = f k( )

MPK = FK k,1( ) = f ' k( )

f '(k*)

1+m-d = r

15


K*= k*EN

Y*= f k*( )EN

16


For given population and technology:

• The capital stock and production reach their long-run

equilibrium levels over time

• In long-run equilibrium we have no growth

• If different countries have the same technology and

same subjective discount rate, they will in the long run

have the same real GDP per capita – convergence!

17

Growth: Population growth and technical

development

• With given population and technology there is growth

only during the adjustment period if you start out from

low capital stock.

• To explain growth in the very long run we have to have

population growth and technical development.

Assume that the population is growing and technology is

improving at constant rates:

rDE

E= g

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development

Let us guess that real interest is constant on the long-term

growth path:

The optimal capital stock per effective worker is

determined by the same condition as before:

.r r=

f '(k*)

1+m-d = r

19


development

But what factors determine the real interest rate? To see this we assume a logarithmic utility function:

Real interest is determined by the subjective discount rate ( ) and the pace of technological development (g).

Consumers must be ‘bribed’ not to consume more today.

r » r+g

r

u Ct( ) = ln C

t( )

u '(Ct)

u '(Ct+1

) / (1+r)=1+ r

t

Þ u ' Ct( ) =1/C

t

Þ 1+ rt= 1+r( )

Ct+1

Ct

= 1+r( ) 1+ g( )

20


development

In steady state:

• K/EN and Y/EN are constant

• K and Y grow at the rate g+n

• K/N and Y/N grow at the rate g

K

EN= k * Þ K = k *EN Þ

DK

K=DE

E+DN

N= g +n

Y

EN= f k *( ) Þ Y = f k *( )EN Þ

DY

Y=DE

E+DN

N= g +n

21

Growth: ‘The golden rule’

• Does a larger capital stock always mean more

consumption?

• Which capital stock maximises consumption in steady

state?

22

Growth: ‘The golden rule’ (golden rule capital

stock)

• Investments in steady state where :

• Consumption per effective worker:

• To get the capital stock that maximises consumption,

take the derivative with respect to k:

I = DK +dK =DK

KK +dK = n+ g( )K +dK = n+ g +d( )K

DY

Y=DK

K= n+ g

Y

EN-I

EN= f k( )- n+ g +d( )k

f ' k( ) = n+ g +d

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Growth: ‘The golden rule’ (golden rule capital

stock)

Assume K=2Y and, for the sake of simplicity,

Marginal return on capital is higher than

Conclusion: the capital stock is lower than theconsumption-maximising capital stockIncreased saving would increase consumption in the longrun. The problem is that we are impatient ...

f ' k( ) = n+ g +d

f ' k( ) =MPK =aKa-1N1-a =aKaN1-a

K=a

Y

K»

1

3×1

2» 0.17

0 0.03 0.07 0.10n g

n+ g +d

m = 0

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Growth: The Solow model

Central assumptions:

• Constant savings rate (s) (no optimising)

• Closed economy: Investments = savings: I = sF(K,EN )

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Growth: The Solow model

• Change in capital stock: DK = sF(K,EN )-dK

sF(K) >dK Þ K grows

sF(K) <dK Þ K falls

sF(K*) =dK* steady-state

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Why are some countries richer than others?

GDP per capita in 2010 (PPP, 2005 constant prices, USD)

• USA 41 365

• Sweden 36 132

• China 7 130

• India 3 477

• Zimbabwe 319

27


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Our theory: convergence

• Two countries with the same

should converge to the same income per capita in the

long run

• If one of the countries is poorer, growth should be

higher and over time the country should catch up with

the richer country

• Obviously, this is not happening – at least not very

quickly. Income differentials are large and persistent

E,a,m,un and d

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Why are some countries richer than others? Capital

stocks

What does ‘capital’ do in the model?

• Contributes to the production of goods and services• Is produced and increases through savings/investments• Decreases through capital consumption

Different kinds of capital are complements in production:

• Private capital: machinery, buildings• Public capital: infrastructure such as roads, airports,

telecommunication, schools, courts, admin, buildings• Human capital: education, experience

Differences in policies, taxes, corruption can lead to different capital stocks per worker

30


Can differences in physical capital explain income

differences?

Compare two counties:

Differences in physical capital explain only part of the

differences in income

Y

N=K aE 1-aN 1-a

N= KaE 1-aN -a =

Ka

NaE1-a =

K

N

æ

èç

ö

ø÷

a

E 1-a

YIndia

/ NIndia

YUS

/ NUS

=KIndia

/ NIndia

KUS

/ NUS

æ

èçç

ö

ø÷÷

a

EIndia

EUS

æ

èçç

ö

ø÷÷

1-a

0.08

1.00=

0.07

1.8

æ

èç

ö

ø÷

1/3

EIndia

EUS

æ

èçç

ö

ø÷÷

2/3

= 0.34 ×EIndia

EUS

æ

èçç

ö

ø÷÷

2/3

31


32


Can differences in human capital explain incomedifferences?

• Hard to measure

• One measure is the average number of years the adultpopulation went to school

• To see how much it can explain we need to measurethe effect on productivity of one more year ofschooling

• Empirical estimates: Between 10 and 30% of thedifferences in income can be explained by differences inhuman capital

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Conclusion:

• At most 50 % of the income differences can be

explained by differences in physical capital and

schooling

• At least 50% of the differences in income must be

explained by other factors than the differences in

physical capital and schooling

36


What other factors can explain differences in income?

• Inadequate access to technology – different E

• Overpopulation, lack of natural resources? No!

• Inadequate institutions (‘social infrastructure’)

– Corruption and lawlessness

– Lack of competition

– Poor public infrastructure

– Taxes and regulations

– Poorly functioning labour markets

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Does income per capita converge?

• Some convergence in Europe but no general convergence between rich and poor countries

• The growth funnel: – in rich countries, GDP per capita grows by about 2 per cent

per year– some poorer countries catch up, others fall behind

• Geographical differences: many Asian countries have had rapid growth while many African countries have fallen behind

40

What determines technical development?

• In the long run it is technology that drives growth in

GDP per capita

• We have treated E as exogenous – our theory did not

explain what determines technical development

• Endogenous growth theory tries to explain what drives

technical development

• These models often contain an explicit sector for

research and development

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What determines technical development?

Production function for knowledge

where is the number of staff in R&D

Assume

– Growth depends on how many people work in R&D

– Growth depends on the amount of existing knowledge

that influences the development of new knowledge:

growth eventually slows down

growth goes on forever

‘We stand on the shoulders of giants’

DE = aNR

nEq

NR= lN Þ g =

DE

E= a(lN )n Eq-1

RN

q =1: g =DE

E= a(lN )n

q <1:

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SGPE Summer School: Macroeconomics Lecture 5

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