Question

Problem 1. Consider the problem of an agent at date t, who marimizes the utility function に0 subject to a sequence of budget constraint, where B is the discount factor, c+k is consumption in period t+ k and 1/η is the intertemporal elasticity of substitution Let P+k be the price level prevailing at date t+k, i.e. Pt+k Dollar buy 1 unit of the consumption good in period t +k. Among the various assets that the agent can trade at date t are k-period discount bonds, who promise to pay 1 Dollar in period t +k and nothing else. Denote the date-t Dollar price of such a bond by Qtt+k. Assume that the agent can both buy and sell at that price. Let stt+k be the quantity of these k-period discount bonds, which the agent buys in period Some more notation. Let δ--log(β) be the discount rate of the agent. where here and everywhere else, log denotes the natural logarithm (not the log- arithm to the basis of 10). Let 9+log(u) - log(k-1) klog(c+k)-log(c+k-1) denote the grouth rate of consumption from t +k-1 to tk. Let be the corresponding inflation rate. Define ytt+k as t,t+k We assume throughout that Qtt+k is reasonably close to 1 and that gt+k and Tt+k are reasonably close to zero

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Answer 1