A perfectly competitive rm produces output q with capital K and labor L according to the production function: q = f(K; L) = 4K 1 4L 1 4 The price of labor is w = 4, and the price of capital is r = 4. The rm has xed costs equal to 8. The current market price is 8. (a) Prove that this production function has decreasing returns to scale.

(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, nd 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 prot maximizing output q and use that to nd TR(q), TC(q), (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 rms? At the new price, what is the rm's new output? What is the rm's prot?