The Hypothesis of Inflation-Unemployment Trade-off:
This hypothesis about formation of expectations is therefore known as the hypothesis of adaptive expectations. The hypothesis implies that if the actual rate of inflation is always greater than the expected rate, then the expected rate would be rising overtime. Therefore, in order to maintain a constant rate of unemployment lower than the natural rate, the actual rate of inflation must be rising overtime. Otherwise, the difference between the expected real wage rate of workers and the actual real wage rate (expected by firms) would be falling in the economy.
Since ω'/ω> 1, the coefficient of Φe(t) in the above first-order difference equation is greater than one and the constant term on the right hand side is positive.
The stable relation between inflation and unemployment suggested by the Phillips curve is therefore illusory. The same rate of unemployment, if lower than the natural rate, would be associated with increasing rates of inflation over time. Similarly, it can be shown that a rate of unemployment greater than the natural rate; must, in the above case be associated with an increasing rate of deflation over time. The only rate of unemployment which can be maintained in the long run with a constant rate of inflation is the natural rate, where the actual rate of inflation is equal to the rate historically expected by workers. Thus, it follows that there exists no policy trade-off between inflation and unemployment in the sense that a permanently lower rate of unemployment can be established through policy at the expense of a permanent but fixed increase in Rational Expectations and the rate of inflation in the economy.