There are two types of agents, doctors (D) and athletes (A), and they live for two periods. For each type i ?{D,A}, given an initial endowmentxi 1,xi 2,we consider the following problem:

max

xi 1?0,xi 2?0

logxi 1 + ? logxi 2 s.t. xi 2 = xi 2 + (1 + r)xi 1 ?xi 1

where xi 1 is the consumption of type i in period 1, xi 2 is the consumption of type i in period 2, ? ? (0,1) is the discount factor, and r > 0 is the real interest rate.

There are two types of agents, doctors (D) and athletes (A), and they live for two periods. For each type ic {D, A}, given an initial endowment (1, 12), we consider the following problem: ax x, 20,2;20 log r; + Bloga; s.t. x2 = 12 + (1 +r) (z; - z;) where r; is the consumption of type i in period 1, x, is the consumption of type i in period 2, B E (0, 1) is the discount factor, and r > 0 is the real interest rate. 1. Derive the demand function of each type. 2. Derive optimal individual savings of each type. 3. Suppose that (IP, 2?) = (0, 1) and (71, 14) = (1,0). Denote the num- bers of agents of types D and A by N" and NA, respectively. Assuming the existence of equilibrium, derive the equilibrium value of r, that is, the value of r such that the total optimal savings are zero