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The Phillips curve represents the relationship between inflation and unemployment in an economy. It was first proposed by economist A.W. Phillips in 1958. In the short run, the trade-off between inflation and unemployment implies that a decrease in unemployment results in increased inflation and vice versa. However, in the long run, this relationship changes, leading to the concept of the long-run Phillips curve.

The short-run Phillips curve is commonly known as the "Keynesian" Phillips curve, which is downward-sloping, suggesting a trade-off between inflation and unemployment. On the other hand, the long-run Phillips curve is vertical at the natural rate of unemployment, which represents the idea that inflation does not affect the long-term unemployment rate. The long-run version is also known as the "neoclassical" Phillips curve.

The neoclassical long-run Phillips curve is a vertical line. This vertical shape indicates that there is no trade-off between inflation and unemployment in the long run. The location of the curve depends on the natural rate of unemployment, i.e., the lowest level of unemployment that can be sustained without causing inflation to accelerate.

The following economic assumptions form the basis for the neoclassical long-run Phillips curve shape: 1. Rational expectations: People have access to all relevant information and use it to adjust their expectations about future inflation, resulting in no systematic errors in predicting inflation. 2. Price and wage flexibility: Prices and wages adjust to clear markets, meaning that they quickly respond to changes in demand and supply conditions. 3. The natural rate of unemployment: The economy has a unique "natural rate" of unemployment which is determined by structural factors such as labor market institutions, demographics, and technology. This natural rate is unaffected by the inflation rate. 4. Monetary policy neutrality: In the long run, monetary policy cannot affect real variables such as output or unemployment; it can only influence nominal variables, such as the price level or the inflation rate. In conclusion, the neoclassical long-run Phillips curve is vertical at the natural rate of unemployment, which signifies that there is no long-term trade-off between inflation and unemployment. This shape is based on assumptions of rational expectations, price and wage flexibility, the existence of a natural rate of unemployment, and monetary policy neutrality in the long run.