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136 Assumptions Kydland and Prescott consider an economy where aggregate demand disturbances have real effects and expectations concerning inflation affect aggregate supply. We can capture both of these effects by assuming that aggregate supply is given by the Lucas supply curve (see equations [5.38] and [6.21]): y = y + b(iT 77% b>0, (9.8) where is the log of output and is the log of its flexibleprice level. Kydland and Prescott assume that the flexibleprice level of output is less than the socially optimal level. This could arise from positive marginal tax rates (so that individuals do not capture the full benefits of additional labor supply), or from imperfect competition (so that firms do not capture the fuU benefits of additional output). In addition, they assume that inflation above some level is costly, and that the marginal cost of inflation increases as inflation rises. A simple way to capture these assumptions is to make social welfare quadratic in both output and inflation. Thus the policymaker minimizes: L={yy*) + a{7r7T*f, y*>y, a>0. (9.9) The assumption tliat only unexpected inflation matters is not essential. For example, a model along the lines of equation (5.39), in Section 5.5, where core inflation is given by a weighted average of past inflation and expected inflation, has similar implications. of a sfiortrun tradeoff is irrelevant to the determination of average inflation. Consider, for example, two monetary policies that differ only because money growth is lower by a constant amount in every situation under one policy than the other. If the public is aware of the difference, there is no reason for output to behave differently under the lowinflation policy than under the highinflation one. In a famous paper, however, Kydland and Prescott (1977) show that the inability of policymakers to commit themselves to such a lowinflation policy can give rise to excessive inflation despite the absence of a longrun tradeoff (see also Barro and Gordon, 1983a). Kydland and Prescotts basic observation is that if expected inflation is low, so that the marginal cost of additional inflation is low, policymakers will pursue expansionary policies to push output temporarily above its normal level. But the publics knowledge that policymakers have this incentive means that they will not in fact expect low inflation. The end result is that policymakers ability to pursue discretionary policy results in inflation without any increase in output. This section presents a simple model that formalizes this idea.
Equation (9.9) is intended to reflect not just the policymakers preferences, but also the representative mdividuals. The reason that the decentralized equiUbrium with flexible prices does not achieve the ftrstbest level of output is that (because of the taxes or imperfect competition) there are positive externalities from higher output. That is, neglecting inflation for the moment, we can think of the representative individuals welfare as depending on his or her own output (or labor supply), y,, and average economywide output, y: (7, = V(y,,y). The assumption underlying (9.9) is that is the Nash equilibrium (so V, (y, y) = 0 and Vt 1 (y, y) < 0, where subscripts denote partial derivatives), but is less than the social optimum (so .( , ) > 0). The parameter a reflects the relative importance of output and inflation in social welfare. Finally, the policymaker controls money growth, which determines the behavior of aggregate demand. Since there is no uncertainty, we can think of the policymaker as choosing inflation directly, subject to the constraint thai inflation and output are related by the aggregate supply curve, (9.8). Analyzing the Model To see the models imphcations, consider two ways that monetary policy and expected inflation could be determined. In the first, the policymaker makes a binding commitment about what inflation will be before expected inflation is determined. Since the commitment is binding, expected inflatton equals actual inflation, and so (by [9.8]) output equals its natural rate. Thus the policymakers problem is to choose to minimize (y  y*)/2 + (  *)/2. The solution is simply = *. In the second situation, the policymaker chooses inflation taking expectations of inflation as given. This could occur either if expected inflation is determined before money growth is, or if and are determined simultaneously. Substituting (9.8) into (9.9) implies that the policymakers problem is mml;ly + b(7T7T)y*f + la(7T7T*f. (9.10) The firstorder condition is [y + b(7T ~ )  y*]b + a{7T  TT*) = 0. (9.11) Solving (9.11) for yields " + * + b{y*  ) (9.12)
= TT* + (y*  y) (9.13) If expected inflation exceeds this level, actual inflation is less than individuals expect, and thus the economy is not in equilibnum. Similarly, if * is less than *, exceeds ". Thus the only equilibrium is for tt and * to equal *, and for to therefore equal y. Intuitively, expected inflation rises to the point where the policymaker, taking * as given, chooses to set tt equal to 77"=. In short. FIGURE 9.4 The determination of inflation in the absence of commitment Figure 9.4 plots the policymakers choice of tt as a function of . The relationship is upward sloping with a slope less than 1. The hgure and equation (9.12) show the policymakers incentive to pursue expansionary pohcy. If the public expects the policymaker to choose the optimal rate of inflation, TT*, the marginal cost of slightly higher inflation is zero, and the marginal benefit of the resulting higher output is positive. Thus in this situation the policymaker chooses an inflation rate greater than *. Since there is no uncertainty, equilibrium requires that expected and actual inflation are equal. As Figure 9.4 shows, there is a unique inflation rate for which this is true. If we impose = in (9.12) and then solve for this inflation rate, we obtain
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