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  3. 2 consider the utility maximization problem with n goods a...

Question: 2 consider the utility maximization problem with n goods a...

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2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function u(c) is strictly concave, increasing, C1, 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., MRSy(c) for all (V) i 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) - (a) what is the problem the consumer is solving, and what are the optimal solutions?
(b) Are the optimal solutions c* (s) increasing in w for fixed p. (c) Is ci (s) increasing in p (p the relative price of good 2 to good 1, where p >0) and c2(s) decreasing in p? (d) what is the definition of efficiency for this problem? (e) what is the value function? (f) is the value function Cl (i.e., once continuously differentiable?
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