To derive the equation for the Phillips curve, we assume that the expected inflation rate (πte) is equal to **Previous period's inflation rate**. This assumption is based on the idea that people form their expectations of future inflation based on past inflation rates since past inflation is often a relevant indicator of what future inflation may be.
The Phillips curve represents the short-run trade-off between inflation and unemployment. It suggests that there is an inverse relationship between these two variables in the short term. The original Phillips curve was based on empirical observations in the mid-20th century and showed that when unemployment is low, inflation tends to be high, and vice versa.
The equation for the Phillips curve is typically represented as:
πt = πte - β(u - un)
Where:
πt = Current inflation rate
πte = Expected inflation rate
β = Coefficient representing the sensitivity of inflation to the unemployment gap (the gap between the actual unemployment rate u and the natural rate of unemployment un)
u = Current unemployment rate
un = Natural rate of unemployment (the rate of unemployment consistent with stable inflation, also known as the non-accelerating inflation rate of unemployment or NAIRU)
The assumption that expected inflation (πte) is equal to the previous period's inflation rate is known as adaptive expectations. It implies that individuals and firms form their expectations of future inflation based on what inflation was in the recent past, without considering other macroeconomic factors like money growth rates or GDP growth rates. While this assumption is simple and has been used in early versions of the Phillips curve, more sophisticated models and expectations formations have been developed over time to better capture the complexity of inflation expectations in modern economies.