"The Phillips Curve in a Liquidity Trap
PC: Infl(t) = a*E(infl(t+1)) + b*OutputGap; a > 0, b < href="http://canucksanonymous.blogspot.com/2009/05/consumption-euler-equations-explained.html">consumption Euler equation, (see here as well), the other is the Fisher equation:
FE: 1+i = (1+r)*(1+E(infl)); i is the nominal interest rate, r is the real interest rate.
It is not correct to base an analysis on the Phillips curve alone, the time path of consumption, inflation and interest rates (both real and nominal) must satisfy all three equations simultaneously.
The Effect of an Increase in Expected Inflation
The first thing to notice is that the equation PC, on its own, implies that increasing expected inflation does not change output at all, it simply raises current inflation. The effect on output of an increase in expected inflation depends on the reaction of the nominal interest rate.
In particular, if the nominal interest rate does not change (perhaps because the central bank is targeting the nominal rate) then the Fisher equation implies that the real rate has fallen and this feeds into the consumption Euler equation to increase aggregate demand.
On the other hand, if the central bank increases the nominal rate one for one with expected inflation then the real rate and AD are unchanged. Finally, if the central bank increases nominal rates more than one for one then the real rate is actually raised and thus AD falls.
Two Important Conclusions for Policy in a Liquidity Trap
The Phillips curve and its relation to the other two equations makes clear two important points.
1) Higher inflation should be expected to precede a real recovery, it is unlikely to be the case that inflation only picks up after the output gap closes.
2) The real recovery will only happens if the central bank keeps nominal rates low for a time after inflation begins to rise. The central bank must accommodate some inflation for a period of time in order to increase output.
The three equations taken together imply that in a liquidity trap there is simply no way for monetary policy to generate a real recovery without accommodating a sufficient amount of inflation.