Aperfectly competitive firm produces output q with capital k and labor l according to the production function:

q = f(k, l) = 4k^1/4 l^ 1/4

the price of labor is w = 4, and the price of capital is r = 4. the firm has fixed costs equal to 8. the current market price is 8.

(a) prove that this production function has decreasing returns to scale.

(b) find the optimal (cost-minimizing) ratio of capital to labor inputs k/l for any level of output (use scale expansion path).

(c) for any q, and the cost minimizing inputs as functions of output: i. e., derive functions l(q) and k(q).

(d) find cost functions tc(q), atc(q), and mc(q), i. e. as functions of output. at what value of q does the minimum of atc occur?

(e) find total and marginal revenue as functions of output, i. e. tr(q) and mr(q). (f) find the pro t maximizing output q∗ and use that to and tr(q*), tc(q*), π(q*), l(q*), and k(q*).

(g) will other rms have an incentive to enter or exit this market? what will be the new market price that results from the entry/exit of firms? at the new price, what is the rm's new output? what is the rm's profit?