lecture 12 economic growth. slide 1 economic growth explain improvements in standards of living (gdp...

Lecture 12Economic Growth

slide 2

Economic Growth

Explain improvements in standards of living (GDP per capital) along time

Explain differences across countrieslearn how our own growth rate is

affected by shocks and our government’s policies

Solow Growth Model

slide 3

some statistics

In Uganda, 96% of people live on less than $2/day. (data link)

2.8 billion people live on less than $2/day (1.1 billion under $1/day)

GDP per capita Chad in 1960: $1212, in 2000: $908 Venezuela in 1960: $7840, in 2000: $6420 Korea in 1960: $1495, in 2000: $15875 H.K. in 1960: $3090, in 2000: $26698

slide 4

Huge effects from tiny differences

1,081.4%243.7%85.4%

624.5%169.2%64.0%

2.5%

2.0%

…100 years…50 years…25 years

percentage increase in standard of living after…

annual growth rate of income per capita

slide 5

Long term growth effect

Rule of 72: 1% growth rate, approximately takes 72 years to double GDP

What will happen if China keeps 10% growth rate and US keeps 3% growth rate (US per capita GDP $42,000 China $6800)

slide 6

World Distribution of Income

6

slide 7

World Income Map

7

slide 8

South vs. North

8

slide 99

slide 10

Real GDP per capita, 1975–2003

10

slide 11

Life Expectancy and Income (Preston, 1976)

11

slide 1212

slide 13

Heights of Males and Females in China1

551

601

65

1920 1940 1960 19801920 1940 1960 1980

1 2

95% CI Fitted values

cohort

Graphs by 1=URBAN SITE(U) 2=RURAL SITE(R)13

slide 14

Happiness and Income

14

slide 15

The Solow Model due to Robert Solow,

won Nobel Prize for contributions to the study of economic growth

a major paradigm:widely used in policy makingbenchmark against which most

recent growth theories are compared

looks at the determinants of economic growth and the standard of living in the long run

slide 16

How Solow model is different from Chapter 3’s model

1. K is no longer fixed:investment causes it to grow, depreciation causes it to shrink.

2. L is no longer fixed:population growth causes it to grow.

3. The consumption function is simpler.

4. No G and T

slide 17

Production

Initially assume constant population (L) and no technology change

Production of goods and services:

Constant Returns to Scale:

),( LKFY

),( zLzKFzY

slide 18

Production

Letting z = 1/L, we get the production function in per capita terms:

y = Y/L = output per workerk = K/L = capital per worker

Constant Returns to Scale size of the economy does not affect the relationship between capital per worker and output per worker

)()1,/(/ kfyLKFLY

slide 19

Production

Decreasing MPK:

This implies the following shape for the production function:

)(/ kfKFMPK

0)( kf

slide 20

Production

MPK is the slope of this curve.k

yf(k)Low MPK

High MPK

slide 21

Production

Cobb-Douglas case:

1LAKY

AkLKALAKLLAK

LYy

)/(

1

Aky

slide 22

Demand

Assume a closed economy with no government: NX = G = 0

Assume that people save a fraction s of their income (and therefore consume 1 – s),

LILCLYICY /// icy

yscYsC )1()1( 10 s

slide 23

Demand

Substituting:

In equilibrium:

iysicy )1(

syi

)(ksfi

)(kfy

slide 24

Capital Accumulation

Two elements determine how the capital stock changes over time:

Investment: addition of new plants and equipment (makes capital stock rise)

Depreciation: wearing out of existing capital stock (makes capital stock fall)

slide 25

Capital Accumulation

In other words:

Stock Capitalin Change Investment onDepreciati

k )(ksf k

slide 26

Capital Accumulation

k

f(k)

sf(k)c

iy

slide 27

Capital Accumulation

Investment higher than depreciation capital stock increases

Depreciation higher than investment capital stock increases

0)( kkksf

0)( kkksf

slide 28

Capital Accumulation

Steady-state capital stock (k*):

Steady state output, consumption, investment:

**)( kksf

*)(***)()1(*)1(*

*)(*

ksfsyikfsysc

kfy

slide 29

Determining the capital–labor ratio in the steady state

29

slide 30

Capital Accumulation

Low k high MPK high returns from investment capital stock grows

High k low MPK low returns from investment capital stock decreases

In both cases, the economy converges to the steady state (long-run equilibrium)

0)(*111

kkksfkk

0)(*222

kkksfkk

slide 31

Capital Accumulation

Cobb-Douglas:

In steady state:

Akkf )(

**)(*)( kkAksf

)1/(1)1/(11 **)(

sAsA

ksA

k

slide 32

Increase in Savings Rate

k

s1f(k)

k

s2f(k)

*1

k *2

k

slide 33

Increase in Savings Rate

Higher s means that more resources will be dedicated to investment higher capital stock in steady state

Therefore, output per capita will be also higher

slide 34

Golden Rule

What is the relationship between steady-state consumption and savings rate?

Two conflicting forces:Higher s higher output higher the

amount of resources available for consumption c*

Higher s lower the proportion of income allocated to consumption c*

*)()1(* kfsc

slide 35

Golden Rule

For low values of s, c* increases with s

For high values of s, c* decreases with s

Golden Rule: capital stock implied by the savings rate such that c* is maximized

slide 36

Golden Rule

More formally:

But in steady state:

*)(*)(*)()1(* ksfkfkfsc

**)( kksf

**)(* kkfc

slide 37

Golden Rule

Golden Rule: find k* such that c* is maximized

**)(*max*

kkfck

)*(0)*(*d*d

gg kfkfkc

MPK

slide 38

Golden Rule

k

k

f(k)

*gk

MPK =

slide 39

Golden Rule

sg is the savings rate that implies kg*:

k

k

f(k)

*gk

MPK =

sgf(k)

slide 40

The relationship of consumption per worker to the capital–labor ratio in the steady state

40

slide 41

Golden Rule

Cobb-Douglas case:

Golden Rule: MPK =

Akkf )(

1)( kAkfMPK

)1/(1)1/(1

1 *)*(

AA

kkA gg

slide 42

Golden Rule

In steady state: *)*( ggg kkAs

A

AA

AAk

s gg

1)1/(11)*(

gs

slide 43

Transition to Golden Rule

Case 1: s > sg, i.e., steady-state capital too high. Decrease s in order to reach sg

k

sgf(k)

s2f(k)

*gk *k

k

slide 44

Transition to Golden Rule

t

t t

k y

c

k*

kg*

y*

yg*

c*cg*

i*

ig*

ti

slide 45

Transition to Golden Rule

Case 2: s < sg, i.e., steady-state capital too low. Increase s in order to reach sg

k

sf(k)

sgf(k)

*k *gk

k

slide 46

Transition to Golden Rule

t

t

k y

c

k*

kg* y*

yg*

c*

cg*

i*

ig*i

t

t

slide 47

Transition to Golden Rule

If the economy begins above the golden rule (s too high), consumption increases in all future periods decrease in s leads to welfare improvement

If the economy begins below the golden rule (s too low), consumption falls during transition there is a tradeoff between consuming today or in the future

slide 48

Egypt

Chad

Pakistan

Indonesia

ZimbabweKenya

India

CameroonUganda

Mexico

IvoryCoast

Brazil

Peru

U.K.

U.S.Canada

FranceIsrael

GermanyDenmark

ItalySingapore

Japan

Finland

100,000

10,000

1,000

100

Income per person in 1992(logarithmic scale)

0 5 10 15Investment as percentage of output (average 1960–1992)

20 25 30 35 40

International Evidence on Investment Rates and Income per Person

slide 49

Population Growth

Assume that the population--and labor force-- grow at rate n. (n is exogenous)

Ln

L

EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02).

Then L = n L = 0.02 1000 = 20,so L = 1020 in year 2.

slide 50

Break-even investment

( + n)k = break-even investment, the amount of investment necessary to keep k constant.

Break-even investment includes: k to replace capital as it wears out

n k to equip new workers with capital(otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers)

slide 51

The equation of motion for k

With population growth, the equation of motion for k is

k = s f(k) ( + n) k

break-even

investment

actual investme

nt

slide 52

The Solow Model diagram

Investment, break-even investment

Capital per worker, k

sf(k)

( + n ) k

k*

k = s f(k) ( +n)k

slide 53

The impact of population growth

Investment, break-even investment

Capital per worker, k

sf(k)

( +n1) k

k1*

( +n2) k

k2*

An increase in n causes an increase in break-even investment,leading to a lower steady-state level of k.

slide 54

Prediction:

Higher n lower k*.

And since y = f(k) , lower k* lower y* .

Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.

slide 55

Chad

Kenya

Zimbabwe

Cameroon

Pakistan

Uganda

India

Indonesia

IsraelMexico

Brazil

Peru

Egypt

Singapore

U.S.

U.K.

Canada

FranceFinlandJapan

Denmark

IvoryCoast

Germany

Italy

100,000

10,000

1,000

1001 2 3 40

Income per person in 1992(logarithmic scale)

Population growth (percent per year) (average 1960–1992)

International Evidence on Population Growth and Income per Person

slide 5656

slide 57

Clark 2005, p1308 Fig 1

slide 58

The Golden Rule with Population Growth

To find the Golden Rule capital stock, we again express c* in terms of k*:

c* = y* i*

= f (k* ) ( + n) k*

c* is maximized when MPK = + n

or equivalently, MPK = n

slide 59

Technology Progress

Rewrite the production function to incorporate technology change:

E = efficiency of labor

E L = effective workers

Assume: Technological progress is labor-augmenting: it increases labor efficiency at the exogenous rate g:

),( ELKFY

slide 6060

Technology Progress

Assume that E grows at rate g

Therefore E L grows at rate n + g

Redefine all variables in terms of effective workers:

k = K/EL = capital per effective worker

slide 6161

Technology Progress

Then y = Y/EL (= output per effective worker) is given by:

Similarly for consumption and investment:

)()1,/(/ kfyELKFELY

icyELIELCELYICY ///

yscELYsELCYsC )1(/)1(/)1( syiELsYELIsYI //

slide 6262

Technology Progress

Therefore, the equations are the same as before

The only change is in the law of motion for k. Capital per effective worker: Increases with investment Decreases with physical depreciation Also decreases because there are more

effective workers to share the existing capital (higher L and E)

slide 6363

Technology Progress

Then:

In steady-state, capital per effective worker is fixed:

kgnksfkgnik )()()(

0k*)(*)( kgnksf

slide 6464

Technology Progress

k

sf(k)

( +n+g)k

k*k1 k2

slide 6565

Technology Progress

In steady state, income, consumption and investment per effective worker are also constant over time:

*)(***)()1(*)1(*

*)(*

ksfsyikfsysc

kfy

slide 6666

Technology Progress

Therefore capital, income, consumption and investment per worker grow at the rate g in steady-state:

*/*/**

*/*/**

*/*/**

*/*/**

EiLIELIi

EcLCELCc

EyLYELYy

EkLKELKk

slide 67

slide 67

Steady-State Growth Rates in the Solow Model with Tech. Progress

n + gY = y E L Total output

g(Y/ L ) = y E Output per worker

0y = Y/ (L E )Output per effective worker

0k = K/ (L E )Capital per effective worker

Steady-state growth rate

SymbolVariable

slide 6868

Technology Progress

This follows since steady-state variables are constant and E is growing at the rate g

Therefore, the inclusion of technology progress in the Solow model can generate sustained long-run growth

slide 6969

Technology Progress

Moreover, total capital, output, consumption and investment grow at the rate n+g in steady state:

Given that steady-state variables are constant and EL is growing at the rate n+g

** ,**** *,*

iELIcELCyELYkELK

slide 7070

Consumption per effective worker in steady state:

Golden Rule: find k* s.t. c* is maximized:

Golden Rule

*)(*)(*)(*)(*)()1(* kgnkfksfkfkfsc

*)(*)(*max*

kgnkfck

gnMPKkfgnkfkc

gg )*(0)()*(*d*d

slide 71

Government Policies to raise the rate of productivity growth

Improving infrastructure Would increased infrastructure spending increase

productivity?• There might be reverse causation: Richer countries

with higher productivity spend more on infrastructure, rather than vice versa

• Infrastructure investments by government may be inefficient, since politics, not economic efficiency, is often the main determinant

slide 72

Government Policies to raise the rate of productivity growth

Building human capital• There’s a strong connection between productivity and

human capital• Government can encourage human capital formation

through educational policies, worker training and relocation programs, and health programs

• Another form of human capital is entrepreneurial skill• Government could help by removing barriers like red tape

Encouraging research and development• Government can encourage R and D through direct aid to

research

slide 73

Why is technological breakthroughs progress so unequal across countries?

What determined whether/when new technology adopted?Geography view: importance of ecology,

climate, disease environment, geography, in short, factors outside human control.

Institutions view: importance of man-made factors; especially organization of society that provide incentives to individuals and firms.

History’s accidents: some countries are unlucky and trapped in underdevelopment.

73

slide 74

The Geography Factor

74

slide 75

The Institutions Factor

75

slide 76

Institutions and Economic Performances

76

slide 77

Institutions and Economic Performances

77

slide 78

But institutions are complicated: identification problem

Good institutions are correlated with many other good things. Theories about institutions are thus very difficult to test.

The study of the causal role of institutions on economic growth is therefore complicated by concerns about endogeneity.

For example, the United States is rich; it has good institutions; it has high levels of education; it has a common law heritage; it has a temperate climate.

Good institutions are difficult to pin down precisely. We want to be very careful to disentangle different causal effects and isolate the effect of interest.

78

slide 79

But institutions are also endogenous

Institutions could vary because underlying factors differ across countries: Geography, ecology, climate

Montesquieu’s story:

– Geography determines “human attitudes”

– Human attitudes determine both economic performance and political system.

– Institutions potentially influenced by the determinants of income

79

slide 8080

Factor Prices

So far, we solved the model without any reference to wages and rental rates (factor prices)

We just focused on how income is generated, but not on how it is distributed

Assume that a competitive firm hires capital and labor to generate output

slide 8181

Factor Prices

Assuming Cobb-Douglas technology:

Then the problem for this firm is given by:

KRLwELPAKLK

..)(max 1,

1)(),( ELAKELKFY

slide 8282

Factor Prices

First-order condition for K implies that:

In steady-state, the real rental rate is fixed (since k is fixed)

11

11

)/(/

)(//

kAELKAPR

ELKAKFMPKPR

slide 8383

Factor Prices

First-order condition for L implies that:

In steady-state, the real wages increase at the rate g (since k is fixed and E grows at the rate g)

EkAEELKAPw

EELKALFMPLPw

)1()/)(1(/

)(//

slide 8484

Factor Prices

Assume that capital is initially below the steady-state. Then k will evolve according to the following path:

t

k

k*

slide 8585

Factor Prices

Rental rate:initially high (low k implies high MPK)decreases over time as capital

accumulates and MPK decreases

t

R/P

slide 8686

Factor Prices

Define wage in terms of efficiency units as:

Then : initially low (low k implies low MPL labor

abundant relative to capital) increases over time as capital accumulates

and MPL increases constant in steady state

kAE

Pww )1(/~

w~

slide 8787

Factor Prices

This means that real wages (w/P):Grow faster than g during the

transitionGrow at the rate g in steady-state

t

w~

slide 88

Growth Accounting

Want to be able to explain why and how countries grow

There are many sources of growth First step is to decompose aggregate

growth into its components:Growth in the labor forceGrowth in capitalGrowth in productivity

88

slide 8989

Sources of Economic Growth

Assume Cobb-Douglas Production Function

Take log and differentiating

1Y zK L

(1 )dY dz dK dL

Y z K L

slide 90

Computing TFP

z: Total Factor Productivity (TFP) or “Solow Residual”

Y is GDP, K is aggregate capital, N is number of workers

Need to know α

90

slide 9191

What is “z”

Human Capital (Education) Technological Progress Externality: environmental Issues Institutional Effect

Firm OrganizationPatent ProtectionCorruptions

slide 92

Labor Share in the Cobb-Douglas Production Function

Firm optimization:

First-order condition with respect to N:

Labor share is wN/Y. Here:

92

1max( )zK N wN rK

(1 )w zK N

1/ (1 ) / 1wN Y zK N Y

slide 93

Result

Can use average labor share as measure of Labor share (total wages divided by

GDP) in the U.S. is about 64%Estimate α to be 0.36

Can now compute TFP as:

93

1

.36 .64/ ( )z Y K N

slide 94

Total Factor Productivity in the U.S.

94

slide 95

Decomposing Growth Rates

Taking logs of production function:

The same applies to log differences:

Log differences are approximately equal to percentage changes:

95

log log 0.36log 0.64logY z K N

log log 0.36 log 0.64 logY z K N

% % 0.36% 0.64%Y z K N

slide 96

Growth Decomposition for the U.S.

96

slide 97

Growth Decomposition for the Asian Tigers

97

slide 98

Growth Accounting for China

slide 99

Human Capital in China

99

slide 100

Human Capital in China

100

slide 101

Innovation

New Goods

Patents

Technology in China

101

slide 102

102

02

46

mea

n o

f p

ate

ntt

ot

1 1.5 2 2.5 3 3.5 4 4.5

Management and ProductivityP

aten

ts

1996

-200

4

Management ScoreNote: European firms only as uses the European Patent Office database

slide 103

103

Policies to promote growth

Saving Rate Human capital investment Encouraging technological progress Right Institutions

slide 104

104

Growth empirics: Confronting the Solow model with the facts

Solow model’s steady state exhibits balanced growth - many variables grow at the same rate. Solow model predicts Y/L and K/L grow at

same rate (g), so that K/Y should be constant.

This is true in the real world.

Solow model predicts real wage grows at same rate as Y/L, while real rental price is constant.

Also true in the real world.

slide 105

105

Convergence Solow model predicts that, other things

equal, “poor” countries (with lower Y/L and K/L ) should grow faster than “rich” ones.

If true, then the income gap between rich & poor countries would shrink over time, and living standards “converge.”

In real world, many poor countries do NOT grow faster than rich ones. Does this mean the Solow model fails?

slide 106

106

Convergence

No, because “other things” aren’t equal. In samples of countries with similar savings

& pop. growth rates, income gaps shrink about 2%/year.

In larger samples, if one controls for differences in saving, population growth, and human capital, incomes converge by about 2%/year.

slide 107

107

Convergence

What the Solow model really predicts is conditional convergence - countries converge to their own steady states, which are determined by saving, population growth, and education. And this prediction comes true in the real world.