Economics The Cost Minimizing Input Proportions

MC Economics The Cost Minimizing Input Proportions Question

Labor Economics

There are three problems for a total of 100 points. Please label your graphs and if you mark a point on the graph, indicate the corresponding values on both axes. If you make additional assumptions, state them precisely and discuss briefly why they are necessary.

 1. (20 points) Suppose there are two inputs in the production function, labor and capital, and these two inputs are perfect substitutes. The existing technology permits 5 machines to do the work of 2 workers. So the production function is F (E, K) = 2K + 5E. The firm wants to produce q units of output, where q > 0 is some number. Suppose the price of capital is $10 per machine per hour. What combination of inputs will the firm use if the wage rate is $15 or $25 or $35 per hour?

2. (20 points) A firm’s technology requires it to combine 4 labor hours with 3 machine hours to produce 1 unit of output. So capital and labor are perfect complements. So the production function is f (E, K) = min(E/4, K/3). The firm wants to produce q units of output, where q > 0 is some number. Suppose the price of capital is $10 per machine per hour. What combination of inputs will the firm use if the wage rate is $10 per hour? What combination of inputs will the firm use if the wage rate is $15 per hour? What if one of the inputs (but not both) is free?

3. (60 points) In this problem, you can assume all solutions are interior. Suppose the hourly wage is $20 and the price of each unit of capital is $20. The price of output is constant at $60 per unit. The production function is: F (E, K) = E 1/9 K 1/3

 (a) If the current capital stock is fixed at 64 units, how much labor should the firm use in the short run? How much profit will the firm earn?

(b) How much labor and capital should the firm use in the long run? How much profit will the firm earn?

(c) Now, assume w = $40. Repeat (b).

(d) Find substitution and scale effects from (b) to (c). Include a table like the one at the end of this document. Not just with the signs but the actual values of substitution and scale effects!

 (e) On a graph below, draw isoquant and isocost curves for (b) and (c) and show the allocations that you find in (b), (c) and (d).

(f) Are the substitution and scale effects that you found in (d) consistent with the theory? Why?

 (g) Calculate short-run and long-run labor demand elasticities based on your answers in (a) and (b), respectively. What does the theory say about the relationship between them? Is it consistent with your results? Why or why not?

 E Substitution Effect Scale Effect Total Effect K