1. Optimization: From Dixit's excellent book \"Optimization in Economic Theory\" (35 points) There is a xed total Y = 1 of goods at the disposal of society. There are two con sumers who envy each other. If consumer 1 gets Y1 and consumer 2 gets Y2, their utilities are U1 = r1 psi/22; U2 = Y2 krf, where k is a positive constant. The allocation must satisfy the resource constraint: Y1 + Y2 g Y. You are the social planner who is trying to find the allocation that maximizes the utilitarian welfare function of U1 + 1.12. Setup the social planner's optimization problem. (5 points) . What are the optimal allocations when k = l? (10 points) . What are the optimal allocations when k = 2? (10 points) Dar-30"?) . Show that if Y > %, the resource constraint will be slack at the optimum. Inter- pret this result. (10 points)