Venture capitalist Sarah purchases two firms to produce widgets. Each firm produces identical products and each has a production function given by
qi = √ Ki ∙ Li
Where
i = 1, 2
The firms differ, however, in the amount of capital equipment each has. In particular, firm 1 has K1 = 25, whereas firm 2 has K2 = 100. The marginal product of labor is MPL = 5 = / (2√L) for firm 1, and MPL = 5= √L for firm 2. Rental rates for K and L are given by w = v = $1.
a. If Sarah wishes to minimize short-run total costs of widget production, how would output be allocated between the two firms?
b. Given that output is optimally allocated between the two firms, calculate the short-run total and average cost curves. What is the marginal cost of the 100th widget? The 125th widget? The 200th widget?
c. How should Sarah allocate widget production between the two firms in the long run? Calculate the long-run total and average cost curves for widget production.
d. How would your answer to part c change if both firms exhibited diminishing returns to scale?