Question
Suppose a firm faces a ‘Cobb-Douglas' production function: Q = 0.5K0.5 L 0.5 where Q is output, and K and L are the amounts of capital and labour used in the production process. The marginal productivities associated with this production function are MPL = 0.25L -0.5K 0.5 and MPK = 0.25L 0.5K -0.5 . Labour costs (W) are €10 per hour and capital costs (R) are also €10 per hour.
a. If the firm is operating efficiently, what is the cost of producing 100 units of output? How many units of labour and of capital will be employed?
b. If, in the short run, the amount of input K is fixed at K = 75, how much labour would be needed to produce an output of 100?
c. When K is fixed at 75, and 100 units of output are produced, by how much do short-run costs exceeds costs in the long run?
d. When K is free to vary, what inputs will the firm use if the rental rate doubles to €20, W is unchanged and if the firm still wishes to produce 100 units of output?