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86 We can use (6.22) to find E[p]. Ex post, after m is determined, the two sides of (6.22) are equal. Thus it must be that ex ante, before m is determined, the expectations of the two sides are equal. Taking the expectations of both sides of (6.22), we obtain E[p] = E[m]. (6.25) Using (6.25) and the fact that m = E[m] + (m  E[m\), we can rewrite (6.22) and (6.23) as p = E[m] + Y(m  E[m]), (6.26) = Yim  Elm]). (6.27) Equations (6.26) and (6.27) show the key impUcations of the model: the component of aggregate demand that is observed, E[m ], affects only prices, but the component that is not observed, m  E[m\, has real effects. Consider, for concreteness, an unobserved increase in m that is, a higher realization of m given its distribution. This increase in the money supply raises aggregate demand, and thus produces an outward shift in the demand curve for each good. Since the increase is not observed, each suppliers best guess IS that some portion of the rise in the demand for his or her product reflects a relative price shock. Thus producers increase their output. The effects of an observed increase in m are very different. Specifically, consider the effects of an upward shift in the entire distribution of m, with the realization of m  E[m] held fixed. In this case, each supplier attributes the rise in the demand for his or her product to money, and thus does not change his or her output. Of course, the taste shocks cause variations in relative prices and in output across goods (just as they do in the case of an unobserved shock), but on average real output does not rise. Thus observed changes in aggregate demand affect only prices. To complete the model, we must express b in terms of underlying parameters rather than in terms of the variances of p and r,. RecaU that b = llKyrnVr/iVr + Vp)] (see [6.20]). Equation(6.26)implies Vp = Vm/a + b). The demand curve, (6.7), and the supply curve, (6.20), can be used to find Vr, the variance of p,  p. Specifically, we can substitute = b{p E[p\) into
(l+b)2" (6.28) Equation (6.28) implicitly defines b in terms of Vz.Vm, and y, and thus completes the model. It is straightforward to show that b is increasing in and decreasing in Vm  In the special case of r] = 1, we can obtain a closedform expression for b: 1 V4 7 1 Vz  V; Finally, note that the results that p = E[m] + [1/(1 + b)](m  E[m]) and Ki = Zi /(r] + b) imply that p and r, are linear functions of m and z,. Since m and z, are independent, p and r, are independent; and since linear functions of normal variables are normal, p and r, are normal. This confirms the assumptions made above about these variables. 6.4 Implications and Limitations The Phillips Curve and the Lucas Critique Lucass model implies that unexpectedly high realizations of aggregate demand lead to both higher output and higherthanexpected prices. As a result, for reasonable specifications of the behavior of aggregate demand, the model implies a positive association between output and inflation. Suppose, for example, that m is a random walk with drift: ntt = mti + + Ut, (6.30) where is white noise. Thus the expectation of mt is ntti + c, and the unobserved component of mt is . Thus, from (6.26) and (6.27), Pt = mt i + c + Yut, (6.31) Since the model also implies that Pfi = mtz + + [Ufi/(l + b)], the rate of inflation (measured as the change in the log price level) is (6.7) to obtain q, = b(p  E[p]) + Zi  r](pi ~ p), and we can rewrite (6.20) as £i = Hp, p) + Hp  E[p]). Solving these two equations for p,  p then yields Pi  p = z, /{t] + b). Thus VV = VzKv + bf. Substituting the expressions for Vp and Vy into the definition of b (see [6.20]) yields
(6.33) Note that appears in both (6.32) and (6.33) with a positive sign, and that Ut and Uf i are uncorrelated. These facts imply that output and inflation are positively correlated. Intuitively, high unexpected money growth leads, through the Lucas supply curve, to increases in both prices and output. The model therefore implies a positive relationship between output and inflationa Phillips curve. But although there is a statistical outputinflation relationship, there is no exploitable tradeoff between high output and low inflation. Suppose that policymakers decide to raise average money growth (for example, by raising in equation [6.30]). If the change is not pubUcly known, there is an interval when unobserved money growth is typically positive and output is therefore usually above normal. Once individuals determine that the change has occurred, however, unobserved money growth is again on average zero, and so average real output is unchanged. And if the increase in average money growth is known, expected money growth jumps immediately and there is not even a brief interval of high output. The idea that the statistical relationship between output and inflation may change if policymakers attempt to take advantage of it is not just a theoretical curiosity: as we saw in Chapter 5, when average inflation rose in the late 1960s and early 1970s, the traditional outputinflation relationship collapsed. The central idea underlying this analysis is of wider relevance. Expectations are likely to be important to many relationships among aggregate variables, and changes in policy are likely to affect those expectations. As a result, shifts in policy can change aggregate relationships. In short, if policymakers attempt to take advantage of statistical relationships, effects operating through expectations may cause the relationships to break down. This is the famous Lucas critique (Lucas, 1976). The Phillips curve is the most famous application of the Lucas critique. Another example is temporary changes in taxes. There is a close relationship between disposable income and consumption spending. Yet to some extent this relationship arises not because current disposable income determines current spending, but because current income is strongly correlated with permanent income (see Chapter 7)that is, it is highly correlated with households expectations of their disposable incomes in the future. If policymakers attempt to reduce consumption through a tax increase that is known to be temporary, the relationship between current income and expected future income, and hence the relationship between current income and spending, will change. Agam this is notjust a theoretical possibiUty. The United States enacted a temporary tax surcharge in 1968, and the impact
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