Suppose that a representative consumer receives a piece of cake, whose size wo > 0, at...

 

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Suppose that a representative consumer receives a piece of cake, whose size wo > 0, at time 0. The consumer can live upto time T > 1. Since the cake is the only food that the consumer has during her entire life (say the cake is really, really big!), she should decide how to eat the cake throughout her life. In particular, she solves the following lifetime utility maximization problem: subject to T ΣB¹u(c) max (C₁,w₁+1)=0 t=0 (1) W₁+1=W₁-C for t=0,1,...,T (2) ct ≥ 0, we+1 ≥ 0 where we+1 is the cake left for the next period and 3 € (0,1). Assume that utility function satisfies all nice properties; it is strictly increasing, strictly concave, continuous, and differentiable in consumption. Furthermore, Inada condition is assumed to hold. 1. Set up the Lagrangian for this problem and obtain the set of conditions that describe the consumer's optimal choice. 2. Derive the lifetime budget constraint for this problem. If you use this constraint instead of the periodic budget constraint in the Lagrangian method, do optimality conditions change from 17 3. Assume that u(c) = Inc. Obtain the closed-form solution for c. What happens to optimal consumption plan when the consumer becomes more patient (i.e. 3 becomes higher)? 4. Assume that u(c) = c; we now assume that utility function is weakly concave and hence Inada condition does not hold. Obtain the closed-form solution for c, and explain why the optimal consumption plan is same or different from 3. 5. Keep the assumption on the utility function that u(c) = c. Assume for this question that 3 = 1 instead of B € (0,1). Obtain the solution(s) for this question and compare it with your answer for 4. Explain why your answers are same (or different). Suppose that a representative consumer receives a piece of cake, whose size wo > 0, at time 0. The consumer can live upto time T > 1. Since the cake is the only food that the consumer has during her entire life (say the cake is really, really big!), she should decide how to eat the cake throughout her life. In particular, she solves the following lifetime utility maximization problem: subject to T ΣB¹u(c) max (C₁,w₁+1)=0 t=0 (1) W₁+1=W₁-C for t=0,1,...,T (2) ct ≥ 0, we+1 ≥ 0 where we+1 is the cake left for the next period and 3 € (0,1). Assume that utility function satisfies all nice properties; it is strictly increasing, strictly concave, continuous, and differentiable in consumption. Furthermore, Inada condition is assumed to hold. 1. Set up the Lagrangian for this problem and obtain the set of conditions that describe the consumer's optimal choice. 2. Derive the lifetime budget constraint for this problem. If you use this constraint instead of the periodic budget constraint in the Lagrangian method, do optimality conditions change from 17 3. Assume that u(c) = Inc. Obtain the closed-form solution for c. What happens to optimal consumption plan when the consumer becomes more patient (i.e. 3 becomes higher)? 4. Assume that u(c) = c; we now assume that utility function is weakly concave and hence Inada condition does not hold. Obtain the closed-form solution for c, and explain why the optimal consumption plan is same or different from 3. 5. Keep the assumption on the utility function that u(c) = c. Assume for this question that 3 = 1 instead of B € (0,1). Obtain the solution(s) for this question and compare it with your answer for 4. Explain why your answers are same (or different).