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117 (c) Show that if r and are constant over time, the resuh in part (b) impUes that the log of consumption follows a random walk with drift: In Q+i = a + lnCt + Ut+i, where is white noise. (d) How do changes in each of r and o affect expected consumption growth, E,[lnCt+i  InCf]? Interpret the effect of on expected consumption growth in light of the discussion of precautionary saving in Section 7.6. 7.4. A framework for investigating excess smoothness. Suppose that Q equals [r/(l + r)][Af + I,"Lo Et[Y,+s]/a + rYl and that Af+i = (1 + r)(A, + Yt  Q). (a) Show that these assumptions imply that EtiQ+i] = Ci {and thus that consumption follows a random walk) and that = 1 +*]/(! + = A,  X7oEt\Yt+s\/a + rr. (b) Suppose that ) = 1 + , where is white noise. Suppose that , exceeds Eti[Yt] by one unit (that is, suppose u, = 1). By how much does consumption increase? (c) For the case of > 0, which has a larger variance, the innovation in income, Ut, or the innovation in consumption, Q ~ EtilQ]? Do consumers use saving and borrowing to smooth the path of consumption relative to income in this model? Explain. 7.5. Consider the twoperiod setup analyzed in Section 7.4. Suppose that the go\ ernment initially raises revenue only by taxing interest income. Thus the individuals budget constraint is Ci  C2 /[1  (1  ) ] < Fi  /[1  (1  ) , where is the tax rate. The governments revenue is zero in period 1 and ( 1  Cit in period 2, where C° is the individuals choice of Q given this tax rate. Now suppose the government eliminates the taxation of interest income and instead institutes lumpsum taxes of amounts Ti and T2 in the two periods; thus the individuals budget constraint is now CjHCz/d + r) < ( 1 1) ( 2 T,)/{l + r}. Assume that 2, and r are exogenous. (a) What condition must the new taxes satisfy so that the change does not affect the present value of government revenues? (b) If the new taxes satisfy the condition in part id), is the old consumption bundle, (C", C), not affordable, just affordable, or affordable with room to spare? (c) If the new taxes satisfy the condition in part (a), does flrstperiod consumption rise, fall, or stay the same? 7.6. Consiunption of durable goods. (Mankiw, 1982.) Suppose that, as in Section 7.2, the instantaneous utility function is quadratic and the interest rate and the discount rate are zero. Suppose, however, that goods are durable; speciflcalh, Ct = (1  S)C(i + Et, where E, is purchases in period f and 0 < S < 1. (a) Consider a marginal reduction in purchases in period t of dEr. Find values of dEt+\ and dEt+? such that the combined changes in Et, Et+\, and E1+2 leave the present value of spending unchanged (so dEt + d£t+i + dEt+2 = 0) and leave Ct+2 unchanged (so (1 8)dEt + {1  S)dEt+i + dE,+2 = 0). (b) What is the effect of the change in part (a) on C, and Ct+i? What is the effect on expected utility?
(c) What condhion must Q and Et[C,+i] satisfy for the change in part (a) not to affect expected utility? Does follow a random walk? (d) Does E follow a random walk? (Hint: write E,  Et~i in terms of Q  Q i and Cfi  Cf 2) Explain intuitively. If ¸ = 0, what is the behavior of E? 7.7. Consider a stock that pays dividends of D, in period f and whose price in period f is Pf. Assume that consumers are riskneutral and have a discount rate of r; thus they maximize ElZTo Cr/d + r)]. (a) Show that equilibrium requires Ft = EdiDt+i + P(+i)/(l + r)] (assume that if the stock is sold, this happens after that periods dividends have been paid). (b) Assume that limsoo £r[Pt+s /(1 i Y] = 0 (this is a nobubbles condition; see the next problem). Iterate the expression in part (a) forward to derive an expression for Pt in terms of expectations of future dividends. 7.8. Bubbles. Consider the setup of the previous problem without the assumption that lim, c» £f[P(+s/(l + ] = 0. (a) Deterministic bubbles. Suppose that Pf equals the expression derived in part (b) of Problem 7.7 plus {1\r)b, b > 0. (z) Is consumers firstorder condition derived in part (a) of Problem 7.7 still satisfied? (zz) Can b be negative? (Hint: consider the strategy of never selling the stock.) (b) Bursting bubbles. (Blanchard, 1979.) Suppose that P, equals the expression derived in part (b) of Problem 7.7 plus q,, where q, equals (1 + r)q,i/a with probability a and equals zero with probability I  a. ii) Is consumers firstorder condition derived in part (a) of Problem 7.7 still satisfied? (zz) If there is a bubble at time t (that is, if q, > 0), what is the probability that the bubble has burst by time r i s (that is, that qt+s = 0)? What is the limit of this probability as s approaches infinity? (c) Intrinsic bubbles. (Froot and Obtsfeld, 1991.) Suppose that dividends follow a random walk: D, = Dti + e,, where e is white noise. (z) [n the absence of bubbles, what is the price of the stock in period f? (zz) Suppose that P, equals the expression derived in (z) plus bt, where bt = {l r)b,j + cet, > 0. Is consumers firstorder condition derived in part (a) of Problem 7.7 still satisfied? In what sense do stock prices overreact to changes in dividends? 7.9. The Lucas assetpricing modeL (Lucas, 1978.) Suppose the only assets in the economy are inftnitelylived trees. Output equals the fruit of the trees, which is exogenous and cannot be stored; thus C, = Y,, where Y, is the exogenously determined output per person and C, is consumption per person. Assume that initially each consumer owns the same number of trees. Since all consumers are assumed to be the same, this means that, in equilibrium, the behavior of the price of trees must be such that, each period, the representative
consumer does not want to either increase or decrease liis or her holdings of trees. Let P, denote the price of a tree in period f (assume that if the tree is sold, the sale occurs after the existing owner receives that periods output). Finally, assume that the representative consumer maximizes E[Xtlo 1" Q/(1 + )  (a) Suppose the representative consumer reduces his or her consumption in period f by an infinitesimal amount, uses the resultmg saving to increase his or her holdings of trees, and then sells these additional holdings in period f + 1. Find the condition that Q and expectations involving Yt+i, Pf+i, and Cti must satisfy for this change not to affect expected uHlity. Solve this condition for Pt in terms of Y, and expectations involving Yt+j, Pt+i, and Ct+i. (b) Assume that limsc E,[{PtJ Yt+s)/{I + p)4 = 0. Given this assumption, iterate your answer to part (a) forward to solve for Pf. (Hint: use the fact that Cf+s = Tf+s for all s.) (c) Explain intuitively why an increase in expectations of future dividends does not affect the price of the asset. id) Does consumption follow a random walk in this model? 7.10. The equity premium and the concentration of aggregate shocks. (Mankiw, 1986b.) Consider an economy with two possible states, each of which occurs with probabiUty . In the good state, each individuals consumption is 1. In the bad state, fraction A of the population consumes 1  ( /A) and the remainder consumes 1, where 0 < < 1 and < \ < 1. measures the reduction in average consumption in the bad state, and A measures how broadly that reduction is shared. Consider two assets, one that pays off 1 unit in the good state and one that pays off 1 unit in the bad state. Let p denote the relative price of the badstate asset to the goodstate asset. (a) Consider an individual whose initial holdings of the two assets are zero, and consider the experiment of the individual marginally reducing (that is, selling .short) his or her holdings of the goodstate asset and using the proceeds to purchase more of the badstate asset. Derive the condition for this change not to affect the individuals expected utility. (b) Since consumption in the two states is exogenous and individuals are ex ante identical, p must adjust to the point where it is an equiUbrium for individuals holdings of both assets to be zero. Solve the condition derived in part (a) for this equilibrium value of p in terms of . A, [/(1), and Lf(l  ( / )). (c) Find clp/cl\. id) Show that if utility is quadratic, / \ = 0. (e) Show that if !/"(•) is everywhere positive, / \ < 0.
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