3. (15 points)(Competitive Firm) Suppose a competitive firm has the production function of f(x1, X2) = X] T2 . The output and input markets are competitive, and their prices are denoted by p, w1, and w2. Moreover, there are no fixed costs. (a) Mathematically express the firm's profit maximization problem. " (b) Derive the firm's supply function y(p, w1, w2). Does this firm produce more if p increases? Does this firm produce more if wi and w2 decrease? Are they intuitive? (c) Derive the firm's factor demand functions for both inputs (i.e., 1 (p, w1, w2) and x2(p, W1, w2)). (d) Mathematically express the first stage of the firm's cost minimization problem. (e) Derive the firm's conditional factor demand functions for both inputs (i.e., xi(y, w1, W2) and x2(y, W1, W2)). (f) Construct the firm's cost function (i.e., c(w1, w2, y)). (g) Mathematically express the second stage of the firm's cost minimization problem. Then, derive the firm's supply function. Is it equivalent to the one from (b)? (h) Answer (a) and (d) again if the government imposes the firm an ad-valorem tax t (on input 1) when the firm buys input 1 (just (a) and (d), nothing else). Hint: Do not use the Lagrangian optimization method, but think about how these firms would behave if they want to produce y units of output. For example, 1) if this firm is to produce 10 units of output, how many of each input does it need? 2) How much do these inputs cost? 3) Then, try to change 10 to y. "For example, write down something like: max pf (z, y) - Pzz - Pyy