1. Answer questions (a)and (b) regarding a long-run profit maximizing, perfectly competitive firm with the following production function y = f(K, L) = K$ + L (a) Solve the first-order conditions, and find expressions for the input demand functions L* = L' (w, v, p) and K* = K*(w, v, p), where w = price of capital, v = price of labour and p = price of production good. (b) Find comparative statics expressions: OL' /Ow, OL' /Ov, OL* /8p.2. Determine whether the following functions are homogeneous. If so, of what degree? (a) f(z, y) = Vay- (b) g(x, y, 2) = x3= + 2y'xz3. Let Q = Q(K, L. E) be the output function of three inputs Q (K, L, E) = AKOLEY (a) Is this function homogeneous? (b) Under what conditions would there be constant returns to scale?4. Answer the following questions about an individual with utility function U(x1 , 12, 13) = = Inc1 + - In 12 + 12 who maximizes utility subject to the standard budget constraint: Pic1 + pace + pars = M. (a) Set up the Lagrangian and solve the first-order conditions for the regular demand func- tions of (p1, p2. p3, M), x3(P1, p2. p3, M) and zy(p1, p2, p3, M) (b) Are these demand functions homogeneous of any degree? (Show.) (c) Find an expression for the optimal value of the Lagrange multiplier A"(p1, p2, M)- (d) Find all the comparative statics terms for i = 1, 2, 3. OM5. Find the general solution of the following difference equations: To+2 - Oct1 + 83: =06. Let us consider the second order non-homogeneous difference equation which is a mathematical representation of a version of the Samuelson model Y- c(1+v)Witch-2=1 0