ANALYTICAL PROBLEMS

1. According to the Solow model, how would each of the following affect consumption per worker in the long run (that is, in the steady state)? Explain.

a. The destruction of a portion of the nation's capital stock in a war.

b. A permanent increase in the rate of immigration (which raises the overall population growth rate).

c. A permanent increase in energy prices.

d. A temporary rise in the saving rate.

e. A permanent increase in the fraction of the popu­lation in the labor force (the population growth rate is unchanged).

2. An economy is in a steady state with no productivity change. Because of an increase in acid rain, the rate of capital depreciation rises permanently.

a. According to the Solow model, what are the effects on steady-state capital per worker, output per worker, consumption per worker, and the long-run growth rate of the total capital stock?

b. In an endogenous growth model, what are the effects on the growth rates of output, capital, and consumption of an increase in the depreciation rate of capital?

3. This problem adds the government to the Solow model. Suppose that a government purchases goods in the amount of g per worker every year; with N_t workers in year t, total government purchases are gN_t. The government has a balanced budget so that its tax revenue in year t, T_t, equals total government purchases. Total national saving, S_t, is

St = s(Y_t - Tt),

where Y_t is total output and s is the saving rate.

a. Graphically show the steady state for the initial level of government purchases per worker.

b. Suppose that the government permanently increases its purchases per worker. What are the effects on the steady-state levels of capital per worker, output per worker, and consumption per worker? Does your result imply that the optimal level of government purchases is zero?

4.

S_t = sY_t - hK_t.

The extra term, -hK_t, reflects the idea that when wealth (as measured by the capital stock) is higher, saving is lower. (Wealthier people have less need to save for the future.)

Find the steady-state values of per-worker capi­tal, output, and consumption. What is the effect on the steady state of an increase in h ?

5. Two countries are identical in every way except that one has a much higher capital-labor ratio than the other. According to the Solow model, which country's total output will grow more quickly? Does your answer depend on whether one country or the other is in a steady state? In general terms, how will your answer be affected if the two countries are allowed to trade with each other?

6. Suppose that total capital and labor both increase by the same percentage amount, so that the amount of capital per worker, k, doesn't change. Writing the production function in per-worker terms, y = f (k), requires that this increase in capital and labor must not change the amount of output produced per worker, y. Use the growth accounting equation to show that equal percentage increases in capital and labor will leave output per worker unaffected only if

7. An economy has a per-capita production function

where A and a are fixed parameters, y is per-worker output, k is the capital-labor ratio, and h is human capital per worker, a measure of the skills and training of the average worker. The production function implies that, for a given capital-labor ratio, increases in average human capital raise output per worker.

The economy's saving rate is s, and all saving is used to create physical capital, which depreciates at rate d. Workers acquire skills on the job by working with capital; the more capital with which they have to work, the more skills they acquire. We capture this idea by assuming that human capital per worker is always proportional to the amount of physical capital per worker, or h = Bk, where B is a fixed parameter.

Find the long-run growth rates of physical capi­tal, human capital, and output in this economy.

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