Problem 3-1

Suppose there are two inputs in the production function, labor and capital, and these two inputs are perfect substitutes. The existing technology permits 1 machine to do the work of 3 workers. The firm wants to produce 100 units of output. Suppose the price of capital is $750 per machine per week. What combination of inputs will the firm use if the weekly salary of each worker is $300? What combination of inputs will the firm use if the weekly salary of each worker is $225? What is the elasticity of labor demand as the wage falls from $300 to $225?

Problem 3-3

Firm would hire 20,000 workers if the wage rate is $12 but will hire 10,000 workers if the wage rate is $15. Firm B will hire 30,000 workers if the wage is $20 but will hire 33,000 workers if the wage is $15. The workers in which firm are more likely to organize and form a union?

Problem 3-4

Consider a firm for which production depends on two normal inputs, labor and capital, with prices \(w\) and \(r\), respectively. Initially the firm faces market prices of \(w = 6\) and \(r = 4\). These prices then shift to \(w = 4\) and \(r = 2\).

1. In which direction will the substitution effect change the firm’s employment and capital stock?
2. In which direction will the scale effect change the firm’s employment and capital stock?
3. Can we say conclusively whether the firm will use more or less labor? More or less capital?

Problem 3-5

What happens to employment in a competitive firm that experiences a technology shock such that at every level of employment its output is 200 units/hour greater than before?

Problem 3-6

Consider each of the following, and explain why it is or is not a valid instrument for estimating labor supply elasticity and/or labor demand elasticity in the United States.

1. Variation in state income tax rates.
   
2. Variation in state corporate tax rates.
   
3. Changes in federal income tax rates over time.

Problem 3-7

Suppose a firm purchases labor in a competitive labor market and sells its product in a competitive product market. The firm’s elasticity of demand for labor is \(-0.4\). Suppose the wage increases by 5 percent. What will happen to the amount of labor hired by the firm? What will happen to the marginal productivity of the last worker hired by the firm?

Problem 3-8

A firm’s technology requires it to combine 5 person-hours of labor with 3 machine-hours to produce 1 unit of output. The firm has 15 machines in place when the wage rate rises from $10 per hour to $20 per hour. What is the firm’s short-run elasticity of labor demand?

Problem 3-9

In a particular industry, labor supply is \(ES = 10 + w\) and labor demand is \(ED = 40 \times 4w\), where \(E\) is the level of employment and \(w\) is the hourly wage.

1. What is the equilibrium wage and employment if the labor market is competitive? What is the unemployment rate?
2. Suppose the government sets a minimum hourly wage of $8. How many workers would lose their jobs? How many additional workers would want a job at the minimum wage? What is the unemployment rate?

Problem 3-10

Suppose the hourly wage is $10 and the price of each unit of capital is $25. The price of output is constant at $50 per unit. The production function is \[f(E,K) = E^{\frac{1}{2}}K^{\frac{1}{2}}\] so that the marginal product of labor is \[MPE = \frac{1}{2}\left(\frac{K}{E}\right)^{\frac{1}{2}}\] If the current capital stock is fixed at 1,600 units, how much labor should the firm employ in the short run? How much profit will the firm earn?

Problem 3-12

How does the amount of unemployment created by an increase in the minimum wage depend on the elasticity of labor demand? Do you think an increase in the minimum wage will have a greater unemployment effect in the fast food industry or in the lawn care/landscaping industry?

Problem 3-13

Which one of Marshall’s rules suggests why labor demand should be relatively inelastic for public school teachers and nurses? Explain.

3-15. Consider a production model with two inputs–domestic labor \((EDom)\) and foreign labor \((EFor)\). The market is originally in equilibrium in that \[\frac{MP_{Edom}}{w_{dom}}=\frac{MP_{Efor}}{w_{for}}\]

1. Suppose a shock occurs that increases the marginal product of foreign labor. Assuming no changes in domestic or foreign wages, explain what will happen to domestic and foreign labor in order to restore the above condition.
2. In the years following the shock, what are three (significantly different) policies that the domestic country could employ if it wanted to reverse the outflow of labor?