5.- A new firm is producing Klinchis with the production function of ?(?,?,?)=?^1/3?^1/3?. (Yes, they don't use sugar, but some type of capital ?).It has an installed capital ?=4, and is considering possible changes of output in the short run, when changing ? and ? is perfectly possible, but changing ?, a fixed input in the short run, is not. The wage is ?=1, the price of ? is ?=4 and the price of capital is ?=15.

a) Obtain the short run cost function, and break it into variable cost and fixed cost. Also, obtain the SR average cost function, the SR average variable cost function and the SR average fixed cost function. Finally, obtain the marginal cost function. In the SR, does this firm operate with increasing, constant, or decreasing returns to scale?

b) Represent the SR average and marginal cost functions. Do this with care: study the derivative of each, and any important point, so that the graph is a good representation.

c) The firm is now considering the long run, and so deciding on all three inputs, including ?. Write, but don't solve, the cost minimization problem for the firm. Instead, considering the following approach: you have already obtained the SR cost function for ?=4. Repeat these derivations, but for an value of ?. Your derivations are good to obtain the SR cost function for ANY value of ?. Call this ?(?|?). Then, think about the following: the LR cost function, ?(?) is simply the minimum of ?(?|?) possible. That is, it is the result of minimizing ?(?|?) with respect to ?! Now obtain that level of ? that solves these minimization problem, and plug it back to obtain ?(?)