In this appendix, we examine more formally the time-inconsistency argument for rules rather than discretion. This analysis is relegated to an appendix because it requires some calculus.8

Suppose that the Phillips curve describes the relationship between inflation and unemployment. Letting *u* denote the unemployment rate, *u ^{n}* the natural rate of unemployment,

*u* = *u ^{n}* −

Unemployment is low when inflation exceeds expected inflation and high when inflation falls below expected inflation. The parameter *α* determines how much unemployment responds to surprise inflation.

For simplicity, suppose also that the Fed chooses the rate of inflation. In reality, the Fed controls inflation only imperfectly through its control of the money supply. But for purposes of illustration, it is useful to assume that the Fed can control inflation perfectly.

The Fed likes low unemployment and low inflation. Suppose that the cost of unemployment and inflation, as perceived by the Fed, can be represented as

*L*(*u*, *π*) = *u* + *γπ*^{2},

where the parameter *γ* represents how much the Fed dislikes inflation relative to unemployment. *L*(*u*, *π*) is called the *loss function*. The Fed’s objective is to make the loss as small as possible.

Having specified how the economy works and the Fed’s objective, let’s compare monetary policy made under a fixed rule and under discretion.

We begin by considering policy under a fixed rule. A rule commits the Fed to a particular level of inflation. As long as private agents understand that the Fed is committed to this rule, the expected level of inflation will be the level the Fed is committed to produce. Because expected inflation equals actual inflation (*Eπ* = *π*), unemployment will be at its natural rate (*u* = *u ^{n}*).

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What is the optimal rule? Because unemployment is at its natural rate regardless of the level of inflation legislated by the rule, there is no benefit to having any inflation at all. Therefore, the optimal fixed rule requires that the Fed produce zero inflation.

Now let’s consider discretionary monetary policy. Under discretion, the economy works as follows:

Private agents form their expectations of inflation

*Eπ*.The Fed chooses the actual level of inflation

*π*.Based on expected and actual inflation, unemployment is determined.

Under this arrangement, the Fed minimizes its loss *L*(*u*, *π*) subject to the constraint that the Phillips curve imposes. When making its decision about the rate of inflation, the Fed takes expected inflation as already determined.

To find what outcome we would obtain under discretionary policy, we must examine what level of inflation the Fed would choose. By substituting the Phillips curve into the Fed’s loss function, we obtain

*L*(*u*, *π*) = *u ^{n}* −

Notice that the Fed’s loss is negatively related to unexpected inflation (the second term in the equation) and positively related to actual inflation (the third term). To find the level of inflation that minimizes this loss, differentiate with respect to *π* to obtain

*dL*/*dπ* = −*α* + 2*γπ*.

The loss is minimized when this derivative equals zero.9 Solving for *π*, we get

*π* = *α*/(2*γ*).

Whatever level of inflation private agents expected, this is the “optimal” level of inflation for the Fed to choose. Of course, rational private agents understand the objective of the Fed and the constraint that the Phillips curve imposes. They therefore expect that the Fed will choose this level of inflation. Expected inflation equals actual inflation [*Eπ* = *π* = *α*/(2*γ*)], and unemployment equals its natural rate (*u* = *u ^{n}*).

Now compare the outcome under optimal discretion to the outcome under the optimal rule. In both cases, unemployment is at its natural rate. Yet discretionary policy produces more inflation than does policy under the rule. *Thus, optimal discretion is worse than the optimal rule*. This is true even though the Fed under discretion was attempting to minimize its loss, *L*(*u*, *π*).

At first it may seem strange that the Fed can achieve a better outcome by being committed to a fixed rule. Why can’t the Fed with discretion mimic the Fed committed to a zero-inflation rule? The answer is that the Fed is playing a game against private decisionmakers who have rational expectations. Unless it is committed to a fixed rule of zero inflation, the Fed cannot get private agents to expect zero inflation.

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Suppose, for example, that the Fed simply announces that it will follow a zero-inflation policy. Such an announcement by itself cannot be credible. After private agents have formed their expectations of inflation, the Fed has the incentive to renege on its announcement in order to decrease unemployment. [As we have just seen, once expectations are determined, the Fed’s optimal policy is to set inflation at *π* = *α*/(2*γ*), regardless of *Eπ*.] Private agents understand the incentive to renege and therefore do not believe the announcement in the first place.

This theory of monetary policy has an important corollary. Under one circumstance, the Fed with discretion achieves the same outcome as the Fed committed to a fixed rule of zero inflation. If the Fed dislikes inflation much more than it dislikes unemployment (so that *γ* is very large), inflation under discretion is near zero, because the Fed has little incentive to inflate. This finding provides some guidance to those who have the job of appointing central bankers. An alternative to imposing a fixed rule is to appoint an individual with a fervent distaste for inflation. Perhaps this is why even liberal politicians (Jimmy Carter, Bill Clinton) who are more concerned about unemployment than inflation sometimes appoint conservative central bankers (Paul Volcker, Alan Greenspan) who are more concerned about inflation.10