Economic Growth I

Economics 331

J. F. O’Connor

"A world where some live in comfort and plenty, while half of the human race lives on less than

$2 a day, is neither just, nor stable."

Quote of the Day, NYT of Wednesday, July 18, 2001

Output per Capita, 0-1995

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Logarithmic scale

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Output per Capita by Major Region1820-2000

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1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

W. Eur E. Eur W. OffshootLat. Amer Japan Asia (ex. Japan)

Model of Capital Accumulation

• Y = F(K, L) where F displays constant returns to scale and diminishing marginal products

• Y = C + I• C = (1-s)Y where s is the saving rateis the change in the

capital stock and is the depreciation rate of capital

Per Unit of Labor Version

• Y/L = F(K/L,1) = f(k) • Y/L = C/L + I/L • C/L = (1-s)Y/L LLL

• Note:• y - c = i• y - c = sy =sf(k)

• y = f(k)• y = c + i• c = (1-s)y k i - k

• sf(k) = i k sf(k) - k

Key Relationship

k sf(k) - k• The change in capital per worker, k=K/L, is

equal to saving (investment) per worker less depreciation of capital per worker. Capital per worker increases if saving per worker is greater than the depreciation in capital per worker, and vice versa.

Growth and the Steady State

• From above, the equation governing the growth in capital per worker is:

• k i - k = sf(k) - k

• In the steady state equilibrium, the change in the capital per worker is zero, k =0. Investment = depreciation.

The Steady State Equilibrium

• Given the equilibrium capital-labor ratio, k*, we can find output, f(k*) and consumption per capita, (1-s)f(k*).

• Note that the economy converges in the long run to a steady state, where there is no growth! It arrives at a stationary state and stays there.

Saving and Growth• Note experience of Germany and Japan

after WWII• See how a change in the saving rate affects

growth. It causes the capital to grow towards a new steady state. But once that is reached, there is no more growth.

• So, what does a high rate of saving and investment do for you? It gives you a higher per capita income in the steady state.

Prediction:

• Higher s higher k*.

• And since y = f(k) , higher k* higher y* .

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

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

A numerical exampleProduction function (aggregate):

1/ 2 1/ 2( , )Y F K L K L K L

1/ 21/ 2 1/ 2Y K L KL L L

1/ 2( )y f k k

To derive the per-worker production function, divide through by L:

Then substitute y = Y/L and k = K/L to get

A numerical example, cont.

Assume:

• s = 0.3

= 0.1

• initial value of k = 4.0

Approaching the Steady State: A Numerical Example

Year k y c i k k 1 4.000 2.000 1.400 0.600 0.400

0.200 2 4.200 2.049 1.435 0.615 0.420

0.195 3 4.395 2.096 1.467 0.629 0.440

0.189

Year k y c i k k 1 4.000 2.000 1.400 0.600 0.400

0.200 2 4.200 2.049 1.435 0.615 0.420

0.195 3 4.395 2.096 1.467 0.629 0.440

0.189

Assumptions: ; 0.3; 0.1; initial 4.0y k s k

Exercise: solve for the steady state

Continue to assume s = 0.3, = 0.1, and y = k 1/2

Use the equation of motion k = s f(k) k

to solve for the steady-state values of k, y, and c.

Solution to exercise:0 def. of steady statek

and * * 3y k

( *) * eq'n of motion with 0s f k k k

0.3 * 0.1 * using assumed valuesk k

*3 *

*

kk

k

Solve to get: * 9k

Finally, * (1 ) * 0.7 3 2.1c s y

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.

Introducing Population Growth• Recall that k = K/L and therefore,

• % k = % K - % L

• k/k = K/K - L/L

• = ( I - K )/K - n

• where n is the rate of population growth

• Multiply both sides by k = K/L to get

k = I/L - k - nk

• where I/L = i

Growth and the Steady State

• From above, the equation governing the growth in capital per worker is now :

• k i - k - nk = sf(k) - (n)k

• In the steady state equilibrium, the change in the capital stock is zero, k =0. Investment per capita = the depreciation rate + population growth rate.

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)

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

The Solow Model diagram

Investment, break-even investment

Capital per worker, k

sf(k)

( + n ) k

k*

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

The impact of population growthInvestment, 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.

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.

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

Chapter Summary1. The Solow growth model shows that, in the long

run, a country’s standard of living depends positively on its saving rate.

negatively on its population growth rate.

2. An increase in the saving rate leads to higher output in the long run

faster growth temporarily

but not faster steady state growth.