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1. Consider the profit maximization problem for the firm. Say the pro- duction function f(k, n) is strictly concave jointly in all arguments and once continuously differentiable (C), where k (resp, n) denotes the input of capital (resp, labor) (a) Assume a maximum exists for the firms profit maximization problem. What are the first order conditions (FOCs) to characterize all optimal solutions for factor demands for capital and labor in the general problem? (b) Say f(k, n) = (ak + (1-a)na))름 for α e (0,1) (i) What are the FOCs for the factor demands? (ii) Are these FOCs necessary for any optimal solution? Sufficient? Explain? MRTSk,)? when r → 0, f(k, n) = kani-a) problem). sary? Sufficient? (c) What is the version of the efficiency condition for this example (i.e. nds w/ Cobb Douglas production above (ie (e) What is the cost minimization problem for the firm? (i.e., state the (f) For the cost minimization problem, what are the FOCs? Are they neces- 2. Consider the utility maximization problem with n goods (n finite). (a) If the utility function u(c) is strictly concave, increasing, C, and as- suming interiority of the optimal solution, what is the problem the consumer is solving? What are the FOCs for this problem using an unconstrained ap- proach (i.e., variable substitution in primal problem)? (b) Do optimal solutions for all goods satisfy MRS-price ratio condition (i.e., MRSj(c) = Ps for all (V) iメj)? If so, explain why. If not, given a condition such that this will be case. (c) Now, for this problem, define a Lagrangian to solve this problem, and construct the FOCs for this problem. (d) Are the solutions obtained via the 2 methods for interior solutions (the primal method in part (a)-(b) vs the dual method in (c) are the same? If so, so explain why. If not, explain why 3. In problem 2, let n = 2 (2 goods), and say u(c)-c? (a) what is the problem the consumer is solving, and what are the optimal solutions?
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