Questions 2-4 are a modification to the cake-eating problem so that there is a return to storage. Let p > 0 be the return to storing cake from one period to the next. Assume that the law of motion for cake is, W++1 = pW: -C7 = Assume that W is known, but that the return on the cake takes place before the consumption decision so that if an agent consumed all of the cake in the first period, C1 = PW1. Agents are assumed to evaluate flow utility according to u(a) = ln(c) 2. t = 1, 2, ..., T: The T-period problem is to, T max -'u(cm) {ct, Wt+1} t=1 Subject to: W++1 = PW+ - C = W++1 > 0 (a) What is the value function and consumption for the T = 1 problem? (b) What is the value function and optimal consumption sequence for the T = 2 problem? (c) What is the value function and optimal consumption sequence for the T = 3 problem? (d) Write the Lagrangean function for the T-period problem. (e) Find the first order conditions. (f) Find the envelope condition. (g) Find the Euler-equation. (h) What is the difference between the Euler in this problem and from the standard cake-eating problem without storage return? Provide an economic interpretation. Questions 2-4 are a modification to the cake-eating problem so that there is a return to storage. Let p > 0 be the return to storing cake from one period to the next. Assume that the law of motion for cake is, W++1 = pW: -C7 = Assume that W is known, but that the return on the cake takes place before the consumption decision so that if an agent consumed all of the cake in the first period, C1 = PW1. Agents are assumed to evaluate flow utility according to u(a) = ln(c) 2. t = 1, 2, ..., T: The T-period problem is to, T max -'u(cm) {ct, Wt+1} t=1 Subject to: W++1 = PW+ - C = W++1 > 0 (a) What is the value function and consumption for the T = 1 problem? (b) What is the value function and optimal consumption sequence for the T = 2 problem? (c) What is the value function and optimal consumption sequence for the T = 3 problem? (d) Write the Lagrangean function for the T-period problem. (e) Find the first order conditions. (f) Find the envelope condition. (g) Find the Euler-equation. (h) What is the difference between the Euler in this problem and from the standard cake-eating problem without storage return? Provide an economic interpretation