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104 ( ) Substitute your result in part (b) into the expression for C, and show that C, = y,/ , where P UjojPJ " djVK id) Use the resuhs in part (b) and part (c) to show that C,j = Zj{Pj / )"( , / ). (e) Compare your results with (6.7) and (6.9) in the text. 6.3. Observational equivalence. (Sargent, 1976.) Suppose that the money supply is determined by = cz,] + e,, where and z are vectors and e, is an i.i.d. disturbance uncorrelated with z, i.e, is unpredictable and unobservable. Thus the expected component of is cZf i, and the unexpected component is e,. In setting the money supply, the Federal Reserve responds only to variables that matter for real activity; that is, the variables in z directly affect y. Now consider the following two models: (i) Only unexpected money matters, so Yt = azti + be, + v,; (ii) all money matters, so = azti + pm, + v,. In each specification, the disturbance is i.i.d. and uncorrelated with Zti and (a) Is it possible to distinguish between these two theories? That is, given a candidate set of parameter values under, say. Model (z), are there parameter values under Model (zz) that have the same predictions? Explain. (b) Suppose that the Federal Reserve also responds to some variables that do not directly affect output; that is, suppose zrz, = czti + \ + and that Models (z) and (zz) are as before (with their distubances now uncorrelated with Wf 1 as weU as with z, ] and e,) In this case, is it possible to distinguish between the two theories? Explain. 6.4. Suppose the economy is described by the model of Section 6.6. Assume, however, that P is the price index described in part (c) of Problem 6.2 (with all the Zjs equal to 1 for simplicity). In addition, assume that moneymarket equilibrium requires that total spending in the economy equal M. With these changes, is it still the case that in equilibrium, output of each good is given by (6.46) and that the price of each good is given by (6.47)? 6.5. Indexation. (See Gray, 1976, 1978, and Fischer, 1977b. This problem follows Ball, 1988.) Suppose production at firm z is given by , = SL", where 5 is a supply shock and 0 < a < 1. Thus in logs, y, = s + a£, Prices are flexible; thus (setting the constant term to zero for simplicity), p, = w, +{1  a)£,  s. Aggregating the output and price equations yields y = s + aL and p = wn (1  a)£ s Wages are partially indexed to prices: w = , where 0 < e < 1. Finally, aggregate demand is given by = m  p. s and m are independent, meanzero random variables with variances and Vm. (a) What are p,y,£, and w as functions of m and s and the parameters a and e? How does indexation affect the response of employment to monetary shocks? How does it affect the response to supply shocks? (Jb) What value of minimizes the variance of employment? (c) Suppose the demand for a single firms output is y, = yviPi p) Suppose all firms other than firm i index their wages to the price level by w = as before, but that firm z indexes its wage to the price level by w, = e, p. Firm z continues to set its price as p, = w, +(1 a)£,  s. The production function and the pricing equation then imply that y, = y (w,  w), where = arj/[a + (l  a)r]]
(z) What is employment at firm /, ,, as a funcnon of m, s, a, rj, , and ft ? (ll) What value of ft minimizes the variance of f, ? ( ) Find the Nash equilibrium value of That is, hnd the value of such that if aggregate indexation is given by , the representative ftrm mm imizes the variance of C, by setting , ~ Compare this value with the value found in part (Jb) 6.6. Synchronized price setting. Consider the Taylor model Suppose, however, that every other period all of the individuals set their prices for that period and the next That is, m period t prices are set for t and f + 1, in f + 1, no prices aie set, m t + 2, prices are set for t + 2 and t + 3, and so on As m the Taylor model, prices are both predetermined and hxed, and individuals set their prices accordmg to (6 60) Finally, assume that m follows a random walk (a) What is the representative individuals price in period f, Xt, as a function of mt, £,mf+b p,, and EiPt+i (Jb) Use the fact that synchronization implies that p, and pt+i are both equal to Xt to sohe for ai terms of m, and E,m,+] (c) What are yt and ,+1? Does the cenlrdi result of the layloi model that nominal disturbances conunue to have real effects after all prices have been changedstill hold Explain intuitively 6.7. The Fischer model with unbalanced price setting. Suppose the economy is as described by the model of Section 6 7, except that instead of half of the individuals setting their prices each period, fraction f set their prices in odd periods and fraction 1 f sel their prices in e\ en periods Thus the price level is /Pf + (1 ~ t)pf if f is even and (1 Ppj + fpf if t is odd Derive expressions analogous to (6 S7) and (6 58) for Pi and y, for even and odd periods 6.8. The instability of staggered pricesetting. (See Fethke and Policano, 1986, Ball and Cecchetu, 1988, and Ball and D Romer, 1989 ) Suppose the economy is described as in Problem 6 7, and assume for simplicity that is a random walk (so mt  mt i + Ut, where is white noise and has a constant variance) Assume that the amount of prohts an individual loses over two periods rel ative to always having p, = p* is proportional to (p,f  p,*) + ( ,, p*,) If f < and < 1, is the expected value of this loss larger for the mdivid uals who set their pnces in odd periods or for the individuals who set their pnces in even penods? In hght of this, would you expect to see staggered pnce setting if < 1 6.9. Consider the 1 aylor model with the money stock white noise rather than a random walk, that is, mt = Et, where Et is serially uncorrelated Solve the model using the method of tmdetermmed coefftcients (Hint m the equation analogous to (6 63), is it still reasonable to impose A + i  1) 6.10 Repeat Problem 6 9 using lag operators 6.11. (This follows Ball, 1994a) Consider a continuous time version of the Taylor model, so that p{t) = (1/ ) j/ oX(f  ) 1 , where T is the interval between each individuals price changes and x(t  ) is the price set by individuals
who set their prices at lime f  . Assume that 4> = I, so that p*{t) = mit); thus xit) = (1/T) jXc£tf«(f + T)dT. (a) Suppose that Initially mit) = gt ig > 0), and that Etmit + ) is therefore (f + ) . What are ), p(t), and y(t) = ( )  p(f)? (b) Suppose that at time zero the government announces that it is steadily reducing money growth to zero over the next interval T of time. Thus mit) = f[l it/2T)]g for 0 < f < T, and mit) = /2 for t > T. The change is unexpected, so that prices set before = 0 are as in part ia). ii) Show that if :( ) = /2 for all t > 0, then p(f) = mit) for all t > 0, and thus that output is the same as it would be without the change in policy. (il) For 0 < t < , are the prices that Arms set more than, less than, or equal XogT/27 What about for < f < 2T? Given this, how does output during the period (0,2T) compare with what it would be without the change in policy? 6.12. Statedependent pricing with both positive and negative inflation. (This follows Caplin and Leahy, 1991.) Consider an economy like that of the CaplinSpulber model. Suppose, however, that m can cither rise or fall, and that Arms therefore follow a twosided Ss policy: if p,  ,*( ) reaches either S or ~S, firm i changes p, so that p,  ,*( ) equals zero. As in the CaplinSpulber model, changes in m are continuous. Assume for simplicity that p,*(t) mit). In addition, assume that p, p,*(r) is initially distributed uniformly over some intenal of width 5; that is. Pi  P*it) is distributed uniformly on IX,X + S] for some X between S and 0. This is shown in Figure 6.1U: the distribution of p,  p*(f) is an "elevator" of height S in a "shaft" of height 2S. ia) Explain why, given these assumptions, p,  p*(t) continues to be distributed uniformly over some interval of width S. (In terms of the diagram, this means that although the elevator may move in the shaft, it remains of height S.) ib) Are there any positions of the elevator (that is, any values of X) where an infinitesimal increase in m of dm raises average prices by less than dm 7 by more than dm 7 by exactly dm 7 Thus, what docs this model imply about the real effects of monetary shocks? 6.13. Consider an economy consisting of some firms with flexible prices and some with rigid prices. Let p denote the price set by a representative flexibleprice firm and p the price set by a representative rigidprice firm. Flexibleprice firms set their prices after m is known; rigidprice firms set their prices before m is known. Thus flexibleprice firms set p = p* = (1  ) + , and rigidprice firms set p = Ep* = (1  ) + , where E denotes the expectation of a variable as of when the rigidprice firms set their prices. Assume that fraction q of firms have rigid prices, so that p = qp" i aq)p. ia) Find pf in terms of p*, m, and the parameters of the model ( and q). ib) Find p in terms of Em and the parameters of the model.
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