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Both the short run and the long run Phillips curves are meant to show the relationship between unemployment and inflation. They argue that there is a great difference between the relationship of those two things in the short run and in the long run. This is very similar to how there is a difference between the short run aggregate supply (AS) curve and the long run AS curve.
In the short run, there is an inverse relationship between unemployment and inflation. The more unemployment there is, the less inflation. This is because unemployment is often caused by a lack of aggregate demand (AD). When AD is low, inflation will be low as well.
However, Milton Friedman and other economists have argued that this relationship does not hold in the long run. In the long run, they say, the Phillips curve is actually vertical. There is the same amount of inflation at every price level. This is very similar to how the short and long run AS curves differ. In the short run, the AS curve is upward sloping. However, in the long run, the AS curve is vertical. These two facts mean the same thing: in the long run, there is no point in having the government try to change AD. Actual output (and unemployment) will stay the same regardless of the level of AD. All that will change if the government manipulates AD is the price level.
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