Economics 202A, Problem Set 4Maurice Obstfeld1. Interest rates and consumption. An individual has the exponentialperiod utility functionu(C) = eC=( > 0) and maximizesu(Ct) + u(Ct+1)(0 < < 1) subject to the budget constraintCt+ RCt+1= Yt+ RYt+1 Wt[where R = (1 + r)1; so that a fall in the real interest rate r means arise in the market discount factor R].(a) Solve for Ct+1as a function of Ct, R, and using the consumer’sintertemporal Euler equation.(b) What is the optimal level of Ct, given Wt, R, and ? [In otherwords, solve for the date t consumption function.](c) By di¤erentiating your consumption function (including Wt) withrespect to R, show that:dCtdR= Ct1 + R+Yt+11 + R+1 + R[1 log(=R)] :[Hint: Your consumption function has the form C = f(W; R)=(1 +R):Therefore,dCdR= C1 + R+11 + R@f@WdWdR+@f@R;which gives you half the answer.](d) What is the intertemporal substitution elasticity for the exponentialutility function? [Calculate this elasticity at an allocation where Ct=Ct+1= C. It is a function (C) of C, not a constant.]1(e) Show that the derivative calculated in part (c) above can be ex-pressed asdCtdR=(Ct+1)Ct+11 + R+Yt+1 Ct+11 + R:(f) Explain intuitively why, for someone with Yt+1> Ct+1; a rise inR (that is, a fall in the real interest rate r), unambiguously raisesconsumption on date t:2. Optimal consumption with incomplete markets. A consumer has thequadratic period utility function u(C) = C (=2) C2and maximizesu(Ct) + u(Ct+1) subject to the constraintsAt+1= (1 + r)At+ Yt Ct; Atgiven,Ct+1= (1 + r)At+1+ Yt+1(s); s 2 f1; 2; :::; Sg :Here, r is the real rate of interest (so that 1=(1 + r) is the current priceof a unit of output delivered next period with probability 1). Let (s)be the probability of state of nature s from the perspective of date tand assume that = 1= (1 + r).(a) Ignore for the moment the constraint that date t + 1 consumptionbe nonnegative. Compute and interpret the optimal level of Ct.(b) Suppose the consumer has an in…nite horizon and there is uncer-tainty over output on all future dates. Use your answer to (a) to guessthe consumption function and use the “random walk” result to provethat your guess is correct.(c) Let’s return to the 2-period case in part (a) but now take seriouslythe constraint thatCt+1(s) 0; 8s:Renumber the states of nature s (if necessary) so thatYt+1(1) = minsfYt+1(s)g :Show the following: If(1 + r)At+ Yt EtfYt+1g > (2 + r)Yt+1(1)=(1 + r);2then the result in (a) still holds. (Why, intuitively?) Otherwise, theconsumption function is:Ct= (1 + r)At+ Yt+ Yt+1(1)=(1 + r):(d) Still thinking about the 2=period case, suppose the consumer facescomplete asset markets such that p(s); the price in terms of sure datet + 1 output of a state-s contingent unit of date t + 1 output, equals(s). Compute the optimal value of Ct. Do we gave to worry nowabout the non-negativity constraint on date t + 1
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