Consider the Arrow-Debreu economy we have been studying in class (slide page numbers 56-62) with a general endowment process (as in "Economy II" on p. 72). Assume the period utility function, u, is given by u (c) = ln c. For the questions that follow, please make sure you present all the intermediate steps of your derivations. 1. Solve agent i's optimization problem and show that the resulting consumption functions are as given on p. 76. Since these functions depend on R, please denote them ci t (R), for i 2 {1, 2} and t 2 {0, 1}. 2. Use the consumption functions ! c1 1 (R), c2 1 (R) " and the market-clearing condition for period t = 1 to find the equilibrium interest rate, R. Verify that you obtain the same solution reported on p. 77, where the procedure we followed was to instead use the consumption functions ! c1 0 (R), c2 0 (R) " and the market-clearing condition for period t = 0. 3. Define the following XD 0 : R ! R as XD 0 (R) P2 i=1 ci 0 (R) P2 i=1 ei 0, where ci 0 (R) denotes the consumption function obtained in the first part of this question. Show that the function XD 0 () has the following properties: XD 0 () is continuous on the domain (0,1). XD 0 () is strictly decreasing. limR!0 XD 0 (R)=+1. limR!1 XD 0 (R) < 0.