Problem 7.1: Country A and B has the production function:

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

A. Does this production function have constant returns to scale?

B. What is the per-worker production function, Y/L=f(K/L)

C. Assume that neither country experiences population growth or technological progress and

that 5 percent of capital depreciates each year. Assume further that country A saves 10

percent of output each year, and country B saves 20 percent of output each year. Find the

steady state level of capital per worker, the steady-state level of income per worker and

consumption per worker.

D. Suppose that both countries start off with a capital stock per worker of 2. What are the

levels of income per worker and consumption per worker? Remembering that the change in

the capital stock is gross investment minus depreciation, calculate capital stock per worker,

income per worker, and consumption per worker over time. How many years will it be before

consumption per worker in Country B is higher than the level of consumption per worker in

country A.

Problem 7.2:

In the discussion of German and Japanese postwar growth, the text describes what happens

when part of the capital stock is destroyed in a war. By contrast, suppose that a war does not

affect the capital stock, but that casualties reduce the labor force.

A. What is the immediate impact on total output and on output per person?

B. Assuming that the saving rate is unchanged and that the economy was in a steady state

before the war, what happens subsequently to output per worker in the postwar economy? Is

the growth rate of output per worker after the war smaller or greater than normal?

Problem 7.3: Consider an economy described by the production function:

0.3 0.7 Y F K L K L = = ( , )

A. What is the per-worker production function?

B. Assuming no population growth or technological progress, find the steady-state capital

stock per worker, output per worker, and consumption per worker as a function of the

saving rate and the depreciation rate.

C. Assume that the depreciation rate is 10 percent per year. Make a table showing steadystate capital per worker, output per worker, and consumption per worker for saving

rates of 0 percent, 10 percent, 20 percent, and 30 percent and so on. What saving rate

maximizes output per worker? What saving rate maximizes consumption per worker?

D. Use calculus to find the marginal product of capital. Add to your table the marginal

product of capital net of depreciation for each of the saving rates.

Problem 7.4. "Devoting a larger share of national output to investment would help restore

rapid productivity growth and rising living standards". Do you agree with this claim? Explain.

Problem 7.5: (somewhat hard) One view of the consumption function is that workers have

high propensities to consume and capitalists have low propensities to consume. To explore the

implications of this view, suppose that an economy consumes all wage income and saves all

capital income. Show that if the factors of production earn their marginal product, this

economy reaches the Golden rule level of capital. (Hint: Begin with the identity that saving

equals investment. Then use the steady-state condition that investment is just enough to keep

up with depreciation and population growth, and the fact that saving equals capital income in

this economy.)

Problem 7.6: Many demographers predict that the US will have zero population growth in the

21st century, in contrast to an average population growth of about 1 percent per year in the

20th century. Use the Solow model to forecast the effect of this slowdown in population

growth on the growth of total output and the growth of output per person. Consider the effects

both in the steady state and in the transition between steady states.

Problem 7.9: Empirically, as an indicator of the saving rate is investment/GDP. In a closed

economy, saving equals investment. And in an open economy saving may be close to

investment. No question here.

Problem 7.10:

Assume the Solow growth model with population growth and with no technological progress:

Derive steady-state expressions for capital per worker, output per worker, the real wage and

the real return to capital.

1

Y F K L A K L ( , )

= =

As functions of the exogenous variables:

What happens to the real wage and the real return to capital if the saving rate increases?