1.     Two economies, Hare and Tortoise, each start with a real GDP per person of $5000 in 1950. Real GDP per person grows 3% per year in Hare and 1% per year in Tortoise. In the year 2000, what will be real GDP per person in each economy? Make a guess first; then use a calculator to get the answer.

2.     For a particular economy, the following capital input K and labor input N were reported in four different years:

        Year                                    K                                     N

            1                                      200                                 1000

            2                                      250                                 1000

            3                                      250                                 1250     

            4                                      300                                 1200

The production function in this economy is

                                 Y = K^0.3N^0.7,

Where Y is the total output.

a.     Find the total output, the capital-labor ratio, and output per worker in each year. Compare year 1 with year 3 and year 2 with year 4. Can this production function be written in per worker form? If do, write algebraically the per-worker form of the production function.

b.      Repeat Part (a) but assume now that the production function is                 Y = K^0.3N^0.8.

3.     An economy has the per-worker production function

                             y_t = 3k_t^0.5

Where y_t is output per worker and k_t is the capital-labor ration. The depreciation rate is 0.1, and the population growth rate is 0.5. Saving is

                              S_t =0.3Y_t,

Where S_t is the total national saving and Y_t is total output.

a.     What are the steady-state values of the capital-labor ration, output per worker, and consumption per worker?

The rest of the problem shows the effects of changes in the three fundamental determinants of long-run living standards.

b.     Repeat part (a) for a saving rate of 0.4 instead of 0.3.

c.      Repeat part (a) for a population growth rate of 0.08 (with a saving rate of 0.3).

d.     Repeat part (a) for a production function of

                      y_t =4k_t^0.5.

Assume that the saving rate and population growth rate are at their original values.

4.     Consider a closed economy in which the population grows at the rate of 1% per year. The per-worker production function is y=6√k , where y is output per worker and k is capital per worker. The depreciation rate of capital is 14% per year.

a.     Households consume 90% of income and save the remaining 10% of income. There is no government. What are the steady-state values of capital per worker, output per worker, consumption per worker, and investment per worker?

b.     Suppose that the country wants to increase its steady-state value of output per worker. What steady-state value of the capital-labor ration is needed to double the steady-state value of output per capita? What fraction of income would households have to save to achieve a steady-state level of output per worker that is twice as high as in Part (a)?

5.     Both population and the work force grow at the rate of n = 1% per year in a closed economy. Consumption is C = 0.5(1-t)Y, where t is the tax rate on income and Y is the total output. The per-worker production function is y = 8√k , where y is output per worker and k is the capital-labor ration. The depreciation rate of capital is d = 9% per year. Suppose for now that there are no government purchases and the tax rate on income is t = 0

a.     Find expressions for national saving per worker and the steady-state level of investment per worker as functions of the capital-labor ratio, k. In the steady state, what are the values of the capital-labor ratio, output worker, consumption per worker, and investment per worker?

b.     Suppose that the government purchases goods each year and pays for these purchases using taxes on income. The government runs a balanced budget in each period and the tax rate on income is t = 0.5.

Repeat Part (a) and compare your results.

6.     According to Solow model, how would each of the following affect consumption per worker in the long run (that is, in the steady state)? Explain and illustrate your answers with graphs.

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).

REAL WORLD DATA

1.     This problem asks you to do your own growth accounting exercise. Using data since 1948, make a  table of annual growth rates of real GDP, the capital stock (private fixed asset from BEA web site, www.bea.gov, Table 6.1, or FRED, or Economagic), and civilian employment.

Assuming a_k = 0.3 and a_N = 0.7, find the productivity growth rate for each year.

a.     Graph the contributions to overall economic growth of capital growth, labor growth, and productivity growth for the period since 1948. Contrast the behavior of each of these variables in the post-1973 period to their behavior in the earlier period.

b.     Compare the post-1973 behavior of productivity growth with the graph of the relative price of energy, shown in Figure 3.12. To what extent do you think the productivity slow down can be blamed on high energy prices?

2.     Graph U.S. capital-labor ratio since 1948 (use private fixed assets from the BEA Web site, www.bea.gov, Table 6.1 as the measure of labor). Do you see evidence of convergence to a steady state during the postwar period? Now graph real output and real consumption per worker for the same period. According to the Solow model, what are the two basic explanations for the upward trends in these two variables? Can output per worker and consumption per worker continue to grow even if the capital-labor ratio stops rising?

3.     According to the Solow model, if countries differed primarily in temrs of their capital-labor ratios, with rich countries having high capital-labor ratios and poor countries that have a lower real GDP per capita income should grow faster than countries with a higher real GDP per capita. (This prediction of the Solow model assumes that countries have similar saving rates, population growth rate, and production functions.) You can test this idea using the Penn World Tables at pwt.econ.upenn.edu. Pick a group of ten countries and examine their initial levels of real GDP per capita in a year long ago, such as 1950. Then calculate the average growth rate of real GDP per capita since that initial year. Do your results suggest that countries that initially have lower real GDP per capita indeed grow faster than countries that initially have a higher real GDP per capita?