Question

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?

Answer 1

1.a) The consumer is maximizing u(c) wrt the budget constraint. To make the problem unconstrained, we substitute in the utility function the value of c_n  from the budget constraint. Then u(c)=u(c1,c2,.....,y-p1c1-p2c2----pn-1cn-1/pn). Differentiating wrt each c separately, we get the FOC u_i(c)/u_j(c)=pi/pj. Or MRS=Price Ratio

b. As we have transformed a constrained problem into an unconstrained one, we won't get MRS=Price ratio for all goods. We will get them for all goods wrt the good which we have substituted for. In the above case, it will be in terms of the nth good.

c. L=u(c)+T(y-p1c1+p2c2+...+pncn) where T is the multiplier(Lambda). Foc will be u_i (c)-Tpi=0 for all i. Solving these, we get the same condition, MRS=Price ratio for al pairs.

d. The solutions obtained for the optimal consumption bundles will be the same due to duality of the optimization problem.

2. The consumer solves the problem to maximize the utility function wrt to the budget constraint.

The optimal solutions will be c1=a(Income)/P1. c2=1-a(Income)/P2.

Yes, the optimal solutions are increasing in the income.

Yes, c1 is increasing in the relative price as p will increase if p1 falls. If p1 falls the c1 rises. Similarly, c2 is decreasing in the price ratio

The definition of efficiency is the equality of MRS and the price ratio,

The Value function, or the indirect utility function is obtained by substituting the optimal consumption values in the utility function. So the value function is Income(a/p1)^a(1-a/p2)^(1-a)