The curve is most understandable at both extremes of income taxation—zero percent and one-hundred percent—where the government collects no revenue. At one extreme, a 0% tax rate means the government's revenue is, of course, zero. At the other extreme, where there is a 100% tax rate, the government collects zero revenue because (in a "rational" economic model) taxpayers presumably change their behavior in response to the tax rate: either they have no incentive to work or they avoid paying taxes, so the government collects 100% of nothing.
It however does not mean that a 50 % tax rate maximises the tax revenue.
The Phillips curve is a historical inverse relation between the rate of unemployment and the rate of inflation in an economy. Stated simply, the lower the unemployment in an economy, the higher the rate of change in wages paid to labor in that economy.
Alban William Phillips, a New Zealand-born economist, wrote a paper in 1958 titled The relationship between unemployment and the rate of change of money wages in the United Kingdom 1861-1957, which was published in the quarterly journal Economica. In the paper Phillips describes how he observed an inverse relationship between money wage changes and unemployment in the British economy over the period examined. Similar patterns were found in other countries and in 1960 Paul Samuelson and Robert Solow took Phillips' work and made explicit the link between inflation and unemployment: when inflation was high, unemployment was low, and vice-versa.
In the 1920s an American economist Irving Fisher noted this kind of Phillips curve relationship. However, Phillips' original curve described the behavior of money wages. So some believe that the Phillips curve should be called the "Fisher curve."
In the years following Phillips' 1958 paper, many economists in the advanced industrial countries believed that his results showed that there was a permanently stable relationship between inflation and unemployment. One implication of this for government policy was that governments could control unemployment and inflation within a Keynesian policy. They could tolerate a reasonably high rate of inflation as this would lead to lower unemployment – there would be a trade-off between inflation and unemployment. For example, monetary policy and/or fiscal policy (i.e., deficit spending) could be used to stimulate the economy, raising gross domestic product and lowering the unemployment rate. Moving along the Phillips curve, this would lead to a higher inflation rate, the cost of enjoying lower unemployment rates.
StagflationIn the 1970s, many countries experienced high levels of both inflation and unemployment also known as stagflation. Theories based on the Phillips curve suggested that this could not happen, and the curve came under concerted attack from a group of economists headed by Milton Friedman—arguing that the demonstrable failure of the relationship demanded a return to non-interventionist, free market policies. The idea that there was a simple, predictable, and persistent relationship between inflation and unemployment was abandoned by most if not all macroeconomists.
 NAIRU and rational expectationsrational expectations and the NAIRU (non-accelerating inflation rate of unemployment) arose to explain how stagflation could occur. The latter theory, also known as the "natural rate of unemployment", distinguished between the "short-term" Phillips curve and the "long-term" one. The short-term Phillips Curve looked like a normal Phillips Curve, but shifted in the long run as expectations changed. In the long run, only a single rate of unemployment (the NAIRU or "natural" rate) was consistent with a stable inflation rate. The long-run Phillips Curve was thus vertical, so there was no trade-off between inflation and unemployment. Edmund Phelps won the Nobel Prize in Economics in 2006 for this.
In the diagram, the long-run Phillips curve is the vertical red line. The NAIRU theory says that when unemployment is at the rate defined by this line, inflation will be stable. However, in the short-run policymakers will face an inflation-unemployment rate tradeoff marked by the "Initial Short-Run Phillips Curve" in the graph. Policymakers can therefore reduce the unemployment rate temporarily, moving from point A to point B through expansionary policy. However, according to the NAIRU, exploiting this short-run tradeoff will raise inflation expectations, shifting the short-run curve rightward to the "New Short-Run Phillips Curve" and moving the point of equilibrium from B to C. Thus the reduction in unemployment below the "Natural Rate" will be temporary, and lead only to higher inflation in the long run.
Since the short-run curve shifts outward due to the attempt to reduce unemployment, the expansionary policy ultimately worsens the exploitable tradeoff between unemployment and inflation. That is, it results in more inflation at each short-run unemployment rate. The name "NAIRU" arises because with actual unemployment below it, inflation accelerates, while with unemployment above it, inflation decelerates. With the actual rate equal to it, inflation is stable, neither accelerating nor decelerating. One practical use of this model was to provide an explanation for Stagflation, which confounded the traditional Phillips curve.
The rational expectations theory said that expectations of inflation were equal to what actually happened, with some minor and temporary errors. This in turn suggested that the short-run period was so short that it was non-existent: any effort to reduce unemployment below the NAIRU, for example, would immediately cause inflationary expectations to rise and thus imply that the policy would fail. Unemployment would never deviate from the NAIRU except due to random and transitory mistakes in developing expectations about future inflation rates. In this perspective, any deviation of the actual unemployment rate from the NAIRU was an illusion.
However, in the 1990s in the U.S., it became increasingly clear that the NAIRU did not have a unique equilibrium and could change in unpredictable ways. In the late 1990s, the actual unemployment rate fell below 4 % of the labor force, much lower than almost all estimates of the NAIRU. But inflation stayed very moderate rather than accelerating. So, just as the Phillips curve had become a subject of debate, so did the NAIRU.
Further, the concept of rational expectations had become subject to much doubt when it became clear that the main assumption of models based on it was that there exists a single (unique) equilibrium in the economy that is set ahead of time, determined independent of demand conditions. The experience of the 1990s suggests that this assumption cannot be sustained.
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