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Review of the previous lecture 1. 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.

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Page 1: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Review of the previous lecture

1. 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.

Page 2: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Lecture 7

Economic Growth – 1(B)

Instructor: Prof. Dr.Qaisar Abbas

Page 3: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Lecture Contents

• Steady state

• Golden Rule

Page 4: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The equation of motion for k

• The Solow model’s central equation

• Determines behavior of capital over time…

• …which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g.,

income per person: y = f(k)

consump. per person: c = (1–s) f(k)

k = s f(k) – k

Page 5: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The steady state

If investment is just enough to cover depreciation

[sf(k) = k ],

then capital per worker will remain constant: k = 0.

This constant value, denoted k*, is called the steady state capital stock.

k = s f(k) – k

Page 6: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

Page 7: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

depreciation

k

k1

investment

Page 8: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k* k1

k = sf(k) k

k

k2

Page 9: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

k2

investment

depreciation

k

Page 10: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

k2

k

k3

Page 11: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

k3

Summary:As long as k < k*, investment will

exceed depreciation,

and k will continue to grow toward k*.

Summary:As long as k < k*, investment will

exceed depreciation,

and k will continue to grow toward k*.

Page 12: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

A numerical example

Production 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

Page 13: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

A numerical example, cont.

Assume:

• s = 0.3

= 0.1

• initial value of k = 4.0

1/ 2( )y f k k

Page 14: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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 4 4.584 2.141 1.499 0.642 0.458 0.184 … 10 5.602 2.367 1.657 0.710 0.560 0.150 … 25 7.351 2.706 1.894 0.812 0.732 0.080 … 100 8.962 2.994 2.096 0.898 0.896 0.002 … 9.000 3.000 2.100 0.900 0.900 0.000

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 4 4.584 2.141 1.499 0.642 0.458 0.184 … 10 5.602 2.367 1.657 0.710 0.560 0.150 … 25 7.351 2.706 1.894 0.812 0.732 0.080 … 100 8.962 2.994 2.096 0.898 0.896 0.002 … 9.000 3.000 2.100 0.900 0.900 0.000

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

Page 15: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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.

Page 16: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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

Page 17: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

An increase in the saving rate

Investment and

depreciation

k

k

s1 f(k)

*k1

An increase in the saving rate raises investment…

…causing the capital stock to grow toward a new steady state:

s2 f(k)

*k2

Page 18: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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.

Page 19: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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

Page 20: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The Golden Rule: introduction

• Different values of s lead to different steady states. How do we know which is the “best” steady state?

• Economic well-being depends on consumption, so the “best” steady state has the highest possible value of consumption per person:

c* = (1–s) f(k*)

• An increase in s

• leads to higher k* and y*, which may raise c*

• reduces consumption’s share of income (1–s), which may lower c*

• So, how do we find the s and k* that maximize c* ?

Page 21: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The Golden Rule Capital Stock

the Golden Rule level of capital, the steady state value of k that maximizes consumption.

*goldk

To find it, first express c* in terms of k*:

c* = y* i*

= f (k*) i*

= f (k*) k*

In general: i = k + k

In the steady state: i* = k* because k = 0.

Page 22: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Then, graph f(k*) and k*, and look for the point where the gap between them is biggest.

The Golden Rule Capital Stock

steady state output and

depreciation

steady-state capital per worker, k*

f(k*)

k*

*goldk

*goldc

* *gold goldi k

* *( )gold goldy f k

Page 23: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The Golden Rule Capital Stock

c* = f(k*) k*

is biggest where the slope of the production func. equals the slope of the depreciation line:

steady-state capital per worker, k*

f(k*)

k*

*goldk

*goldc

MPK =

Page 24: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Use calculus to find golden rule

We want to maximize: c* = f(k*) k*

From calculus, at the maximum we know the derivative equals zero.

Find derivative: dc*/dk*= MPK

Set equal to zero: MPK = 0 or MPK =

Page 25: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The transition to the Golden Rule Steady State

• The economy does NOT have a tendency to move toward the Golden Rule steady state.

• Achieving the Golden Rule requires that policymakers adjust s.

• This adjustment leads to a new steady state with higher consumption.

• But what happens to consumption during the transition to the Golden Rule?

Page 26: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Starting with too much capital

then increasing c* requires a fall in s.

In the transition to the Golden Rule, consumption is higher at all points in time.

* *If goldk k

timet0

c

i

y

Page 27: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Starting with too little capital

then increasing c* requires an increase in s.

Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption.

* *If goldk k

timet0

c

i

y

Page 28: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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.

Page 29: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving 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)

Page 30: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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

Page 31: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

The Solow Model diagram

Investment, break-even investment

Capital per worker, k

sf(k)

( + n ) k

k*

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

Page 32: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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.

Page 33: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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.

Page 34: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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

Page 35: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

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

In the Golden In the Golden Rule Steady State, Rule Steady State,

the marginal product of the marginal product of capital net of capital net of

depreciation equals the depreciation equals the population growth rate.population growth rate.

In the Golden In the Golden Rule Steady State, Rule Steady State,

the marginal product of the marginal product of capital net of capital net of

depreciation equals the depreciation equals the population growth rate.population growth rate.

Page 36: Review of the previous lecture 1.The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate

Summary

An increase in the saving rate leads to

higher output in the long run

faster growth temporarily

but not faster steady state growth.


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