1. [Asset Pricing] Consider a version of Lucas (1978) where a representative agent chooses consumption and asset holdings to maximize the expected present value of utility: 162 00 170 E0213\" at , e (0,1), a>0 t:0 The real economy produces output gt that grows over time, yt+1 : 9t+134t7 f0]? all 75 Here gt+1 is the i.i.d. stochastic growth rate of output that can take one of 3 possible states: 9 6 {91,92, 93}. Let 7T3 denote the probability of state gs. The representative agent is endowed with property rights to the productive side of the economy. In terms of nancial markets, the agent can trade ownership shares in these property rights. Let pt denote the price of a share, and let at denote the agent's beginningofperiod t holdings of these shares. (i) Formulate the representative agent's optimization problem as a dynamic pro gramming problem. (ii) Dene a recursive competitive equilibrium (RCE) for this economy. (iii) Derive the Euler equation characterizing the optimal consumption path for the representative agent and impose the market clearing condition(s). (iv) Rewrite this pricing equation under the (correct) guess that the share price is proportional to the current dividend: i.e., Pt : 0y: for all t, where (9 > 0 is just a constant. Verify the guess by solving for 6 in terms of the parameters of the model. v ompu e e price 0 a oneperio r1s ee rea on . a is e r1s ee C tth ' f 'd'kfr ldeht'th 'kfr rate? (vi) A consol bought at date t is a riskfree claim to one unit of consumption in each future time period. What are the equilibrium price of a consol? What is the return on consols? Is it the same as the riskfree rate form part (v)