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5. Consider an infinitely lived individual born in time zero, endowed with a cake of size xo. The cake is storable (without depreciation) and infinitely divisible. The agent derives contemporary utility from (cake) consumption through u(c,), and has as her ultimate objective to maximize her infinitely discounted sum of utility streams. Her discount rate is, as usual, given by B. (a) Formulate the mathematical problem corresponding to the above description (i.e formulate the Sequence Problem). (b) Provide a recursive formulation and show that its solution also solves the Se- quence Problem in (i) (i.e. Theorem 2). (c) Let u(c) = In(c). Derive both the value function, o(x), and the policy function, x' = g(x). N 6. Consider the following sequence problem V (ao, wo) = max EB'u(C.) (5) s.t do + >q' w = _q'c (6) 1=0 1=0 W/+1 = (1 -6)w + ow, (7) (a) For which value of & does (7) converge to a unique fixed point? What is the value of this fixed point? (b) Define a, Use this definition to show that the problem above has a recursive representa- tion. No formal proof is necessary, but explain the logic underlying Theorem 1 in this context. What interest rate do the "bonds" a, pay? (c) In a deterministic economy, such as the one above, it appears as if a one period bond is sufficient to allocate resources efficiently. Is this also true in a stochastic economy? Why or why not? 7. Consider a representative agent, exchange economy similar to that studied by Mehra and Prescott. Specifically, it is assumed that the endowment, x, grows stochastically as given by x/+1 = Ar+1x, where the growth rate, A, is assumed to be independently and identically distributed. Agents maximize lifetime expected utility: where B E (0, 1) and 0 5 / 0 affect the volatility of bond prices? Discuss the implications that these results have for the assumption of habit persistence to help resolve some asset pricing puzzles.1. Consider the following (real valued) sequence (x, ),2, defined by: xo = 1, x1 = 1+a, x2 = 1 + at a' , x, = 1 + a + a" + ... + a" and so on. Prove that if la| B'F(x1, X,+1) (3) s.t Xi+1 Er(x;) (4) Define x,, F(, ., .) and I(.) such that these two problems coincide exactly. 3. True or False: An implication of the Hansen indivisible labor RBC model is that consumption and labor productivity will have the same time series properties. Longer Answer Questions (Each question is worth 20 points.) 4. Consider an exchange economy populated by identical agents that trade equity shares, #1, defined as title to the endowment process. (That is, this is the same asset priced in the Lucas tree model.) Denote the price of equity as qr. Agents also trade one- period bonds which cost p, units of consumption in period t and return 1 unit of consumption in the following period. In addition to these assets, a one-period for- ward contract on bonds is traded. In this contract, agents agree at time ? to pay of units of consumption in period : + 1 for the promise of one unit of consumption to be received in period / + 2. The endowment, x,, is stochastic and varies over the interval (x, x); furthermore, x, is assumed to be independently and identically distributed. Given this environment, agents choose a sequence of consumption and assets in order to maximize (a) Formulate the agent's problem as a dynamic programming problem. Be ex- plicit in identifying the state and control variables. (b) Derive and interpret the necessary conditions which characterize the solution to this maximization problem. (c) Define a recursive competitive equilibrium in this economy. (d) Prove that equilibrium bond and equity prices are positively correlated with the endowment while the price of the forward contract is constant. Explain these results