A firm produces a single output using two variable inputs (denoted 1 and 2). The firmproduction function is given by = (1, 2) = (12) ^0.5. The firm can employ as much of either input it desires by incurring constant (and respectively denoted) per-unit input costs of 1 and 2. Assume throughout that all prices and quantities are positive and infinitely divisible. Finally, let 0 denote the target level of output that the firm envisions producing when deciding how much of the inputs to employ.

1. (a) Show that the firms production technology displays a diminishing (physical) marginal product with respect to either input.

2. (b) Write down the firms cost minimization problem and solve for the firms total cost function, (1, 2, 0). Show that the total cost function is homogeneous of degree one in input prices (i.e., (1, 2, 0) = (1, 2, 0) for > 0). Provide an interpretation of this result.

3. (c) Show that the firms average total cost and marginal cost functions are equal and constants (i.e., do not depend on the level of output produced by the firm). What does this imply about the firms production technology?

4. (d) Now consider another firm possessing a (short-run) total cost function given by () = 2 + + 100. Assume the firm takes the market price of its output, > 1, as given. At what market price will this firm earn zero economic profits in the short run?

5. (e) Define Hotellings Lemma and then use that result to derive the (short-run) supply function for the firm in part (d).