A firm operates with the production function Q = K2 L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. The manager has been given a production target: to produce 8,000 units per day. She knows that the daily rental price of capital is $400 per unit and the wage rate is $200 day. a. What is the returns to scale of this production function? Show mathematically. b. Currently the firm employs 80 workers per day. What is the firm’s daily total cost if it rents just enough capital to produce at its target? c. Compare the marginal product per dollar spent on K and on L when the firm operates at the input choice in part (b). What does this suggest about the way the firm might change its choice of K and L if it wants to reduce the total cost in meeting its target? Explain your answer very clearly. d. In the long run, how much K and L should the firm choose if it wants to minimize the cost of producing 8,000 units of output a day? Make sure you show your work. What will the total daily cost of production be? e. Show your answers to parts (b) and (d) on a diagram.
A firm operates with the production function Q = K2 L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. The manager has been given a production target: to produce 8,000 units per day. She knows that the daily rental price of capital is $400 per unit and the wage rate is $200 day.
a. What is the returns to scale of this production function? Show mathematically.
b. Currently the firm employs 80 workers per day. What is the firm’s daily total cost if it rents just enough capital to produce at its target?
c. Compare the marginal product per dollar spent on K and on L when the firm operates at the input choice in part (b). What does this suggest about the way the firm might change its choice of K and L if it wants to reduce the total cost in meeting its target? Explain your answer very clearly.
d. In the long run, how much K and L should the firm choose if it wants to minimize the cost of producing 8,000 units of output a day? Make sure you show your work. What will the total daily cost of production be?
e. Show your answers to parts (b) and (d) on a diagram.
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