Economics 202A, Problem Set 5Maurice Obstfeld1. Consumer durables. Our consumption analysis implicitly assumed thatconsumption is perishable. But if some consumer goods instead were durable(washing machines, autos, etc.), spending in one period would secure an itemthat yields utility in a number of subsequent periods. This question asks thatyou analyze consumer behavior when some goods are durable.Before going further, here is a digression on a solution method, themethod of lag and lead operators, that should be in your toolkit. For anytime series fxtg, de…ne the lag operator L byLxt= xt1:De…ne the lead operator L1byL1xt= xt+1:(Obviously, LL1xt= L1Lxt= xt, which fact inspires the notation.) Youneed to know two facts about these operators, both derived from the standardformula for summing geometric series.Fact #1. Let yt= (1  L)xt, where jj < 1: Thenxt= (1  L)1yt;where (note the formal similarity to the usual formula)(1  L)1= 1 + L + 2L2+ 3L3+ ::::Observe that (1  L)1 + L + 2L2+ 3L3+ :::= 1:Fact #2. Let yt= (1  L)xt; where jj > 1: Thenxt= 1L1(1  1L1)1yt;where(1  1L1)1= 1 + 1L1+ 2L2+ 3L3+ ::::1To establish this last formula, note that(1  1L1)1 + 1L1+ 2L2+ 3L3+ ::::= 1:It is not correct to write xt= (1   L)1ytin this case because jj > 1,which means that (1  L)1does not exist. However, if jj > 1, then1< 1, and yt= (1  L)xt= L1L1 1= L1  1L1xt.Because 1  1L1is invertible when1< 1; we can therefore writext= 1L1(1  1L1)1yt.Now, the model of durables. An individual maximizes1Xt=0t[u(ct) + v(st)]where ctis nondurable consumption and stis the stock (measured at the startof period t) of a consumer durable yielding a ‡ow of services proportionalto st. Let ztbe purchases of durables (which may be resold on a secondarymarket): if durables do not depreciate, thenst+1= st+ zt:Let atbe the value (in terms of consumption c, at the start of period t) ofthe individual’s …nancial assets, which have a constant real gross per periodyield of 1 + r. If (for simplicity) we assume that the durable good’s price interms of c is constant at 1, then (make sure you see why)at+1= (1 + r)at+ yt ct zt;where ytis an exogenous ‡ow of income.(a) Using any metho d you wish, derive and interpret the following …rst-order conditions for the consumer’s problem:u0(ct) = (1 + r)u0(ct+1);u0(ct) = u0(ct+1) + v0(st+1):(b) Write the second f.o.c. above as v0(st) = (1  1L)u0(ct) andshow that u0(ct) = v0(st+1) + 2v0(st+2) + :::; using the preceding lag-leadformalism. Interpret this condition.2(c) Show that the equationv0(st) = (1 + r)v0(st+1)holds at the individual optimum. Thus, when (1 + r) = 1; the consumerwill smooth the marginal utility of durable services.(d) What does this …nding imply about the smoothness of total spendingc + z? To think about that question, let (1 + r) = 1 and assume thatu(c) =  log(c); v(s) = (1  ) log(s). Then solve explicitly for the paths ofc; z; and s.(e) How would the problem change if we allowed explicitly for a rentalmarket in durables?2. The Lucas “tree”model from Econometrica, 1978. Consider a world witha single representative agent, in which a random and exogenous amount ofperishable output ytfalls from a fruit tree each period t. (There is no otheroutput.) Output follows the stochastic processlog yt= log yt1+ "t; Et1"t= 0; (1)where the i.i.d. shock "tis drawn from a N(0; 2) normal distribution. Thereis no way to grow more fruit trees — the supply is …xed.The agent’s lifetime utility function isEt(1Xs=te(st)u(cs));where  > 0 is the rate of time preference. Assume that there is a competitivestock market in which people can trade shares in the fruit tree, whose priceon date t is pt. This is the ex dividend price: if you buy a share on date t,you get your …rst dividend on date t + 1.(a) Show that an individual will choose contingent consumption planssuch that on each date.ptu0(ct) = eEtf(yt+1+ pt+1)u0(ct+1)g : (2)(You can use the individual …nance constraint that ct+ ptxt+1 (yt+ pt) xt;where xtis the share of the fruit tree the individual holds at the end of periodt  1.)3(b) Show that in equilibrium, the “fundamentals”price of the tree isp t= Et(1Xs=t+1e(st)u0(ys) ysu0(yt)):Can you interpret this price in terms of expected dividends and risk factors?What is the sign of these risk factors on the trees?(c) Let u(c) = c1=(1) for  > 0. Show that the normality assumptionin (1) implies (for s > t):Ety1s= y1te2(1)22(st):(Use the lognormal distribution: if "  N(; 2), then e"has a lognormaldistribution with E fe"g = e+122:)(d) Deduce from part (c) that if  > 2(1  )2=2, then p t= yt, where =1fe[2(1)2=2] 1g:(e) Now return to a general strictly concave utility function u(c). Let btbe the random variable Ayt=u0(yt); where  =p2=2and A is an arbitraryconstant. Show that p t+ btwill satisfy the individual’s Euler equation (2)in equilibrium, so that btis a bubble.(f) Show that pt= p t+ btviolates the (equilibrium) transversality condi-tion:limT !1e(T t)Etfu0(yt+T)pt+Tg = 0: (3)(g) Together with the equilibrium Euler equation [equation (2) with ysubstituted for c],ptu0(yt) = eEtf(yt+1+ pt+1)u0(yt+1)g ; (4)condition (3) is su¢ cient for a stochastic price path fptg to be an equilibriumof the Lucas model. In this part of the homework we will show that thecondition is also necessary.Iterate (4) forward to deriveptu0(yt) = Et(1Xs=t+1e(st)u0(ys) ys)+ limT !1e(T t)Etfu0(yt+T)pt+Tg :4Argue that free disposal of output ensures that the limit in condition (3)must be nonnegative. Argue that if the limit is strictly positive, we cannotbe looking at an equilibrium because individuals can raise expected lifetimeutility through the following strategy: sell a tiny amount of the fruit treetoday and consume the proceeds now, never repurchasing the portion ofthe fruit tree just sold (that is, reduce xt, which equals 1 in the hypothesizedequilibrium, permanently). Why is the Euler equation (4) alone not su¢ cientto rule