Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Economic Growth: The Solow Model

Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Economic Growth: The Solow Model

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-2

Chapter 6 Topics

• Economic growth facts

• Solow growth model

• Convergence

SOME EMPIRICAL FACTS



U.S. Per Capita Income Growth

In the United States, growth in per capita income has not strayed far from 2% per year (excepting the Great Depression and World War II) since 1900.


Figure 1 Natural Log of Real Per-Capita Income in the United States, 1869–2005


Real Per Capita Income and the Investment Rate

Across countries, real per capita income and the investment rate are positively correlated.


Figure 2 Real Income Per Capita vs. Investment Rate


Real per capita income and the rate of population growth

Across countries, real per capita income and the population growth rate are negatively correlated.


Figure 3 Real Income Per Capita vs. the Population Growth Rate


Real per capita income and per capita income growth

• There is no tendency for rich countries to grow faster than poor countries, and vice-versa.

• Rich countries are more alike in terms of rates of growth than are poor countries.


Figure 4 Growth Rate in Per Capita Income vs. Real Income Per Capita for the Countries of the World

All Key Facts Together

• US real p.c. income grew steady at 2% per year

• Across countries, real per capita income is:– positively correlated with investment rate– negatively correlated with population growth rate

• Rich countries don’t grow faster than poor, and vice-versa

• Rich countries are more alike w.r.t. growth rates than poor ones

6-12

SOLOW GROWTH MODEL



Solow Growth Model

• Want a model to capture and explain these facts Solow model

• This is a key model which is the basis for the modern theory of economic growth.

• A key prediction is that technological progress is necessary for sustained increases in standards of living.


Population growth

• Variables with ′ sign denote the value in the next period (N′ is population next year)

• In the Solow growth model, population is assumed to grow at a constant rate n.

where n > −1


Consumption-Savings Behavior

• Consumers are assumed to save a constant fraction s of their income, consuming the rest:

• If we allow for endogenous savings, model gets much harder, but the predictions do not change a lot

• Firms have a standard production function:

• Assume it has constant returns to scale

• Divide both sides by N to get:

• Will work with a per-capita P.F.:

6-17

Representative Firm’s Production Function (P.F.)

Example: Cobb-Douglas P.F.



Evolution of the Capital Stock

Future capital equals the capital remaining after depreciation, plus current investment.


Figure 5 The Per-Worker Production Function


Solving the Solow Model, 1

• Consumption market must clear:

• In equilibrium, future capital equals total savings S (= I ) plus what remains of current K.

since S = sY (rest gets consumed)



• Substitute for output from the P.F.:

• Rewrite in per-worker form:

where:



• Re-arrange to get the main equation:

• Determines the evolution of per-capita (or per worker) capital in the model

Example: Cobb-Douglas P.F.


Finding the Steady State

• Steady state – situation when per worker capital stock does not grow, i.e. k′ = k

• Think about steady state as the long-run equilibrium level

• Will discuss several ways of finding it– 2 graphical ways– and analytic way



Figure 6 Determination of the Steady State Quantity of Capital per Worker

Discuss adjustment dynamics to the steady state


Finding Steady State Level of Capital


Figure 7 Determination of the Steady State Quantity of Capital per Worker

Analyzing the Steady State

• Now, we found k*, what’s its growth rate?– Zero, by definition of steady state

• Hence y* = z f(k*) doesn’t grow either!

• So do we have a growth model that predicts growth will stop eventually?


Growth in the Steady State

• Of course not! Recall that k = K/N and

• Even though steady-state capital per worker k* doesn’t grow, total capital stock K= k N does!

• In fact, K and Y (and hence all other aggregate variables) grow at rate n in the steady state!


RUNNING EXPERIMENTSSolow Model:



Increase in the Savings Rate, s

• Having the steady state, can do experiments

• Suppose savings rate s goes up. Effects?– In the steady state, this increases capital per worker

and real output per capita– In the steady state, there is no effect on the growth

rates of aggregate variables– some intertemporal dynamics


Figure 8 Effect of an Increase in the Savings Rate on the Steady State Quantity of Capital per Worker


Figure 9 Effect of an Increase in the Savings Rate at Time T

Summary of Effects from Increase in s

• On per-capita variables (k, y)– New steady state level (higher)– No change in steady state growth rate (zero)

• On aggregate variables (K, Y)– No change in steady state growth rate (n)

• What about consumption? That’s what consumers usually care about!



Figure 10 Steady State Consumption per Worker

Golden Rule Level of Savings

• GR level of s is the one that maximizes consumption per worker in the steady state c*

– and hence aggregate consumption C *

• To find it, write down the expression for c* :

c* = (1 s) z f(k*)

• And maximize it


Finding the sGR, 1


Finding the sGR, 2



Increase in the Population Growth Rate, n

• Another experiment – increase the population growth rate n

• Results:– Capital per worker and output per worker decrease– Growth rates of aggregate variables go up (since

they are equal to n)


Figure 11 Steady State Effects of an Increase in the Labor Force Growth Rate


Limits to Growth

• So per capita income y will increase if:– savings rate s goes up– or if population growth rate n goes down (showed if

n goes up, y goes down)

• But there are limits to impact of these factors!

• How can we have sustained increases in per-capita income?


Increases in Total Factor Productivity, z

Sustained increases in z cause sustained increases in per capita income.


Figure 12 Increases in Total Factor Productivity in the Solow Growth Model

CONVERGENCE THEORYSolow Model:



Convergence in the Solow Growth Model

• Is there a tendency for poor countries to catch up with the rich ones?– The Solow growth model says “yes”!

• As always, need some assumptions:1.Two identical countries (same z, n, s)

2.The “rich” country initially has a higher level of capital per worker k. (Consequently, it also has a higher output per worker y)


Predictions of the Solow Model

If two countries are initially rich and poor, but identical in all other respects, they will converge in the long run to the same level and rate of growth of per-capita income.


Figure 13 Rich and Poor Countries and the Steady State


Figure 14 Convergence in Income per Worker Across Countries in the Solow Growth Model


Figure 15 Convergence in Aggregate Output Across Countries in the Solow Growth Model

Convergence in the Data

• Unfortunately, we don’t see evidence supporting the convergence theory in cross-country data:– while there seems to be some convergence among

the rich countries– there is nothing like this for the poor ones

• Here are the diagrams:


Figure 16 Hardly Any Convergence Across Countries

Want negative correlation, not sure it’s here6-52

Figure 17 Some Convergence Across Rich Countries

6-53Looks like negative correlation here

Figure 18 No Convergence Across Poor Countries

6-54Almost zero correlation here

Why No Convergence?

• If model predictions are wrong, usually this means some assumptions aren’t correct

• Here, assumed same z for all countries

• What if countries have access to different technologies?– say, due to some government policies that forbid

imports of technologies



Figure 19 Differences in Total Factor Productivity Can Explain Disparity in Income per Worker Across Countries

What Should Poor Countries Do?

• Some things could be done, it depends on the government to get things right– Promote greater competition among firms: Absence

of monopolies creates the incentive to innovate – Promote free trade

• This kind of policy seemed to have worked for Japan after WWII


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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Economic Growth: The Solow Model