2. Consider the following production function: y = f (21,12) = In (@1 + 1) + 21/r2. Assume the firm faces an output price p, and input prices (w1, w2). (a) Set up the long-run profit maximization problem. Solve for input de- mand functions, at (p, w1, w2) and r; (p, w1, w2) and the firm's sup- ply function, y* (p, w1, w2). (Hint: be careful about corner solutions). (b) Suppose the level of input 1 is fixed at some value z1 = $1. Solve for the short-run input demand function, 25, and the supply function, y*. Explain briefly why ; does not depend on the value of 1. (In other words, what is it about this production function that causes this?) (c) Set up the minimization problem and the corresponding Lagrange function you would use to solve for the long run cost function, c (w1, w2, y) (don't solve it out)