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MPN for each level of employment

1. The following data give real GDP, Y, capital, K, and labor, N, for the U.S. economy in various years.

Year Y K N
1960 2377 2606 65.8
1970 3578 3850 78.7
1980 4901 5569 99.3
1990 6708 7419 118.8
2000 9191 9849 135.2

Units and sources are the same as in Table 3.1. Assume that the production function is Y = A*(K^0.3)*(N^0.7).

a. By what percentage did U.S. total factor productivity row between 1960 and 1970? Between 1970 and 1980? Between 1980 and 1990? Between 1990 and 2000?
b. What happened to the marginal product of labor between 1960 and 2000? Calculate the marginal product numerical as the extra output gained by adding 1 million workers in each of the two years. (The data for employment, N, are measured in millions of workers, so an increase of 1 million workers is an increase of 1.0).

2. An economy has the production function Y = 0.2(K + N^1/2).

In the current period, K = 100 and N = 100.
a. Graph the relationship between output and capital, holding labor constant at its current value. What is the MPK? Does the marginal productivity of capital diminish?
b. Graph the relationship between output and labor, holding capital constant at its current value. Find the MPN for an increase of labor form 100 to 110. Compare this result with the MPN for an increase in labor form 110 and 120. Does the marginal productivity of labor diminish?

3. Acme Widget, Inc., has the following production function.

Number of Workers Number of Widgets Produced
0 0
1 8
2 15
3 21
4 26
5 30
6 33

a. Find the MPN for each level of employment.
b. Acme can get $5 for each widget it produces. How many workers will it hire if the nominal wage is $38? If it is $27? If it is $22?
c. Graph the relationship between Acme's labor demand and the nominal wage. How does this graph differ from a labor demand curve? Graph Acme's labor demand curve.
d. With the nominal wage fixed at $38, the price of widgets doubles from $5 each to $10 each. What happens to Acme's labor demand and production?
e. With the nominal wage fixed at $38 and the price of widgets fixed at $5, the introduction of a new automatic widget maker doubles the number of widgets that the same number of workers can produce. What happens to labor demand and production?
f. What is the relationship between your answers to part (d) and part (e)? Explain.

4. The marginal product of labor (measured in units of output) for a certain firm is
MPN = A(100- N), where A measures productivity and N is the number of labor hours used in production. The price of output is $2.00 per unit.

a. If A = 1.0, what will be the demand for labor if the nominal wage if $10? If it is $20? Graph the demand curve for labor. What is the equilibrium real wage if the supply of labor is fixed at 95?
b. Repeat part (a) for A = 2.0.

5. Consider an economy in which the marginal product of labor MPN is MPN = 309 - 2N, where N is the amount of labor used. The amount of labor supplied, NS, is given by NS = 22 + 12w + 2T, where w is the real wage and T is a lump-sum tax levied on individuals.

a. Use the concepts of income effect and substitution effect to explain why an increase in lump-sum taxes will increase the amount of labor supplied.
b. Suppose that T = 35. What are the equilibrium values of employment and the real wage?
c. With T remaining equal to 35, the government passes minimum-wage legislation that requires firms to pay a real wage greater than or equal to 7. What are the resulting values of employment and the real wage?

6. Suppose that the production function is Y = 9* (K^0.5) * (N^0.5). With this production function, the marginal product of labor is MPN = 4.5 * (K^0.5) * (N^-0.5). The capital stock is K = 25. The labor supply curve is NS = 100[(1 - t)w]^2, where w is the real wage rate, t is the tax rate on labor income, and hence (1 - t)w is the after-tax real wage rate.

a. Assume that the tax rate on labor income, t, equals zero. Find the equation of the labor demand curve. Calculate the equilibrium levels of the real wage and employment, the level of full-employment output, and the total after-tax wage income of workers.
b. Repeat part (a) under the assumption that the tax rate on labor income, t, equals 0.6.
c. Suppose that a minimum wage of w = 2 is imposed. If the tax rate on labor income, t, equals zero, what are the resulting values of employment and the real wage? Does the introduction of the minimum wage increase the total income of workers, taken as a group?

7. Consider an economy with 500 people in the labor force. At the beginning of every month, 5 people lose their jobs and remain unemployed for exactly one month; one month later, they find new jobs and become employed. In addition, on January 1 each of year, 20 people lost their jobs and remain unemployed for six months before finding new jobs. Finally, on July 1 of each year, 20 people lose their jobs and remain unemployed for six months before finding new jobs.

a. What is the unemployment rate in this economy in a typical month?
b. What fraction of unemployment spells lasts for one month? What fraction lasts for six months?
c. What is the average duration of an unemployment spell?
d. On any particular date, what fraction of the unemployed are suffering a long spell (six months) of unemployment?

8.

Not Labor Force
74 million

Unemployed
8.5 million

------ 2 %→
←13% ---------

2%↓ ↑3% 1%↑ ↓22%

Employed
137.4 million

Use the data above to calculate how many people become unemployed during a typical month. How many become employed? How many leave the labor force?

9. You are give the following data on the unemployment rate and output.

Year 1 2 3 4
Unemployment rate 8% 6% 7% 5%
Output 950 1030 1033.5 1127.5
- - -
a. Assume that the natural rate of unemployment is 6% and that (Y - Y) / Y = 2 (u - u ). Find the full-employment level of output in each year.
b. Calculate the growth rate of full-employment output in years 2, 3, and 4 two different ways. First, calculate the growth rates by using the values for full-employment output that you found in part (a). Then, calculate the growth rate of full-employment output by using only the change in the unemployment rate, the growth rate of output, and the growth rate version of Okun's law (∆Y/Y = 3 - 2∆u). Compare your answers for the two sets of calculations.

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Calculate the growth rate of full-employment output.

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