Unemployment and its relationship with Inflation Can anybody please explain me "Inflation-Unemployment Trade-off under Adaptive Expectation and under Rational Expectation"?
When agents in economy have adaptive expectations they predict future in the basis of what happened in the past. If we consider an inflation adjustment scenario, the expected inflation by the agents is directly dependent on the inflation that prevailed in the immediately previous period and the output gap.

$$Y = Y^0 + b(P - P^e)$$

where $P^e$ is expected price level.

When agents have adaptive expectations there is an inflation inertia.
To reduce unemployment rate the government will try to stimulate the economy and if the economy was at its natural level of unemployment the will be inflationary pressure an inflation will increase. If the agents do not correctly predict the inflation they are likely to produce more. Say for example agents have an expectation that inflation will be 4% based on past experience. And with the government expansion output increases and inflation actually increases to say, 8%. Their nominal wages will adjust according to 4% but the real wage will actually decrease. As a rule of thumb you can remember that if the expected inflation is lower than the actual inflation then output increases and producers hire more workers thereby reducing unemployment. So, there is a trade off here between unemployment rate and inflation level as suggested by the Okun's law and Philips curve.
However this is not the end of story. As soon as the workers realize that their real wages have reduced, they bargain for higher wages and eventually the whole change is reverted as firms' costs increase and they reduce output and fire workers to match the cost. This is also the reason why the long run supply curve is vertical.

With rational expectations ,however, it is a different story altogether. Rational expectations mean that the agents in economy apply all levels of reasoning to the available information to figure out the future scenario. They know the policy of central bank and government, they can correctly predict their actions and hence they know that inflation will be higher and simply ask for more wages. So no effect on output or employment and hence no tradeoff but only an increase in inflation. The policy will not be effective here. This is a concept in neo classical economics and is sometimes called as policy ineffectiveness proposition.