Barro-Gordon model

2. Suppose that the economy is described by the following aggregate demand and aggre- gate supply equations

[AD] : y_t=1/(mt - pt)+dt;

[AS] : y_t=2 (pt - wt)+st;

where wages are given by

[Wages]: wt =E_t-1[pt];

and dt and st are uncorrelated i.i.d. random variables with mean 0 and variance 1.

Further, central bank has loss function

L _t = ^2_t + 1/2 ( y_t - y )² ;

and monetary policy is given by

[Policy]: _t = + dt + st;

where , and are policy parameters which the central bank chooses to minimize

(the expected value of) its loss function.

(a) Derive the (agents') expected value of inflation.

(b) Derive Phillips curve from aggregate supply equation (Hint: small letters correspond to logarithms of the variables.)

(c) Derive the policy rule which minimizes loss function assuming that the central bank is able to credibly commit to a level of inflation. Derive also the corresponding output and expected loss levels.

(d) Derive the policy rule which minimizes loss function assuming that the central bank is able to cheat the agents so that the agents perceive E_t-1 [_t] = 0. Derive also the corresponding output and expected loss levels. Does the central have an incentive to cheat the agents in this setup?

(e) Assume now that the central bank cannot commit to a rule and derive the policy which is optimal at period t when expectation of inflation is taken as given by the central bank (i.e., discretionary monetary policy). Derive also the corresponding output and expected loss levels.