Consumer's Problem

Problem II. Consumer's Problem (25 points) Consider a consumer with a utility function on two consumption goods, u(c1, c2), and income /. Suppose that consumption prices are equal to (p1. p2). 1. Write down the constrained optimization problem. (5 points) 2. Write down the necessary conditions for optimality. (5 points) 3. Suppose that utility function is logarithmic, i.e. (c1, c2) = a In(c ) +bln(c2). Characterize the solution. Is there a unique solution? Why? (5 points) 4. Now, suppose that utility function is Cobb-Douglas, i.e. u(ci, (2) = cic;. Characterize the solution. Is there a pair of (o, ) such that the optimal consumption bundle be the same as the one with logarithmic utility under the same income and prices? Determine (a, ) in terms of (a, b). (10 points)