Problems

1. A firm has a production function of Q = 0.25KL^0.5, the rental rate of capital is $100, and the wage rate is $25. In the short run, is fixed at 100 units.

   1. What is the short-run production function?

   2. What is the short-run demand for labor?

   3. What are the firm’s short-run total cost and short-run marginal cost?

2. Margarita Robotics has a daily production function given by Q = K^0.5L^0.5, where K is the monthly number of hours of use for a precision lathe (capital) and L is the monthly number of machinist hours (labor). Suppose that each unit of capital costs $40, and each unit of labor costs $10.

   1. In the short run, is fixed at 16,000 hours. What is the short-run demand for labor?

   2. Given that is fixed at 16,000 hours, what are total cost, average total cost, average variable cost, and marginal cost in the short run?

   3. What are the long-run demands for capital and labor?

   4. Derive total cost, average cost, and marginal cost in the long run.

   5. How do Margarita Robotics’ marginal and average costs change with increases in output? Explain.

3. A firm has a production function given by Q = 10K^0.25L^0.25. Suppose that each unit of capital costs R and each unit of labor costs W.

   1. Derive the long-run demands for capital and labor.

   2. Derive the total cost curve for this firm.

   3. Derive the long-run average and marginal cost curves.

   4. How do marginal and average costs change with increases in output? Explain.

   5. Confirm that the value of the Lagrange multiplier you get from the cost-minimization problem in part (a) is equal to the marginal cost curve you found in part (c).