### The Phillips Curve

The Phillips curve is an attempt to describe the macroeconomic tradeoff between unemployment and inflation. In the late 1950s, economists such as A.W. Phillips started noticing that, historically, stretches of low unemployment were correlated with periods of high inflation, and vice versa. This finding suggested that there was a stable inverse relationship between the unemployment rate and the level of inflation, as shown in the example above.

The logic behind the Phillips curve is based on the traditional macroeconomic model of aggregate demand and aggregate supply. Since it is often the case that inflation is the result of increased aggregate demand for goods and services, it makes sense that higher levels of inflation would be linked to higher levels of output and therefore lower unemployment.

### The Simple Phillips Curve Equation

This simple Phillips curve is generally written with inflation as a function of the unemployment rate and the hypothetical unemployment rate that would exist if inflation were equal to zero. Typically, the inflation rate is represented by pi and the unemployment rate is represented by u. The h in the equation is a positive constant that guarantees that the Phillips curve slopes downwards, and the u_{n} is the "natural" rate of unemployment that would result if inflation were equal to zero. (This is not to be confused with the NAIRU, which is the unemployment rate that results with non-accelerating, or constant, inflation.)

Inflation and unemployment can be written either as numbers or as percents, so it's important to determine from context which is appropriate. For example, an unemployment rate of 5 percent could either be written as 5% or 0.05.

### The Phillips Curve Incorporates Both Inflation and Deflation

The Phillips curve describes the effect on unemployment for both positive and negative inflation rates. (Negative inflation is referred to as deflation.) As shown in the graph above, unemployment is lower than the natural rate when inflation is positive, and unemployment is higher than the natural rate when inflation is negative.

Theoretically, the Phillips curve presents a menu of options for policymakers- if higher inflation actually causes lower levels of unemployment, then the government could control unemployment via monetary policy as long as it was willing to accept changes in the level of inflation. Unfortunately, economists soon learned that the relationship between inflation and unemployment was not as simple as they had previously thought.

### The Long-Run Phillips Curve

What economists initially failed to realize in constructing the Phillips curve was that people and firms take the expected level of inflation into account when deciding how much to produce and how much to consume. Therefore, a given level of inflation will eventually be incorporated into the decision-making process and not affect the level of unemployment in the long run. The long-run Phillips curve is vertical, since moving from one constant rate of inflation to another doesn't affect unemployment in the long run.

This concept is illustrated in the figure above. In the long run, unemployment returns to the natural rate regardless of what constant rate of inflation is present in the economy.

### The Expectations-Augmented Phillips Curve

In the short-run, changes in the rate of inflation can affect unemployment, but they can only do so if they aren't incorporated into production and consumption decisions. Because of this, the "expectations-augmented" Phillips curve is viewed as a more realistic model of the short-run relationship between inflation and unemployment than the simple Phillips curve. The expectations-augmented Phillips curve shows unemployment as a function of the difference between actual and expected inflation- in other words, surprise inflation.

In the equation above, the pi on the left-hand side of the equation is actual inflation and the pi on the right-hand side of the equation is expected inflation. u is the unemployment rate, and, in this equation, u_{n} is the unemployment rate that would result if actual inflation was equal to expected inflation.

### Accelerating Inflation and Unemployment

Since people tend to form expectations based on past behavior, the expectations-augmented Phillips curve suggests that a (short-run) decrease in unemployment can be achieved via accelerating inflation. This is shown by the equation above, where inflation in time period t-1 replaces expected inflation. When inflation is equal to last period's inflation, unemployment is equal to u_{NAIRU}, where NAIRU stands for "Non-Accelerating Inflation Rate of Unemployment." In order to reduce unemployment below the NAIRU, inflation must be higher in the present than it was in the past.

Accelerating inflation is a risky proposition, however, for two reasons. First, accelerating inflation imposes various costs on the economy that potentially outweigh the benefits of lower unemployment. Second, if a central bank exhibits a pattern of accelerating inflation, it's entirely likely that people will start expecting the accelerating inflation, which would negate the effect of the changes in inflation on unemployment.