Editing Rational expectations (section)

== Mathematical derivation ==
If rational expectations are applied to the Phillips curve analysis, the distinction between long and short term will be completely negated, that is, there is no Phillips curve, and there is no substitute relationship between inflation rate and unemployment rate that can be utilized.

The mathematical derivation is as follows:

Rational expectation is consistent with objective mathematical expectation: 

<math>E\dot{P}_t=\dot{P}_t+\varepsilon_t</math>

'''Mathematical derivation (1)'''

We denote unemployment rate by <math>u_t</math>. Assuming that the actual process is known, the rate of inflation (<math>\dot P_t</math>) depends on previous monetary changes (<math>\dot M_{t-1}</math>) and changes in short-term variables such as X (for example, oil prices):

(1) <math>\dot{P}=q\dot M_{t-1}+z\dot{X}_{t-1}+\varepsilon_t</math>

Taking expected values,

(2) <math>E\dot{P}_t=q\dot M_{t-1}+z\dot X_{t-1}</math>

On the other hand, inflation rate is related to unemployment by the Phillips curve:

(3) <math>\dot{P}_t=\alpha-\beta u_t+\gamma E_{t-1}(\dot{P}_t)</math> , <math>\gamma=1</math>

Equating (1) and (3):

(4) <math>\alpha-\beta u_t+q\dot M_{t-1}+z\dot X_{t-1}=q\dot M_{t-1}+z\dot{X}_{t-1}+\varepsilon_t</math>

Cancelling terms and rearrangement gives

(5) <math>u_t=\frac{\alpha-\varepsilon_t}{\beta}</math>

Thus, even in the short run, there is no substitute relationship between inflation and unemployment. Random shocks, which are completely unpredictable, are the only reason why the unemployment rate deviates from the natural rate.

'''Mathematical derivation (2)'''

Even if the actual rate of inflation is dependent on current monetary changes, the public can make rational expectations as long as they know how monetary policy is being decided:

(1) <math>\dot{P}_t=q\dot{M}_t+z\dot{X}_{t-1}+\varepsilon_t</math>

Denote the change due to monetary policy by <math>\mu_t</math>.

(2) <math>\dot{M}_t=g\dot{M}_{t-1}+\mu_t</math>

We then substitute (2) into (1):

(3) <math>\dot{P}_t=qg\dot{M}_{t-1}+z\dot{X}_{t-1}+q\mu_t+\varepsilon_{t}</math>

Taking expected value at time <math>t-1</math>,

(4) <math>E_{t-1}\dot{P}=qg\dot{M}_{t-1}+z\dot{X}_{t-1}</math>

Using the Phillips curve relation, cancelling terms on both sides and rearrangement gives

(5) <math>u_t=\frac{\alpha-q\mu_t-\varepsilon_t}{\beta}</math>

The conclusion is essentially the same: random shocks that are completely unpredictable are the only thing that can cause the unemployment rate to deviate from the natural rate.