International macroe.. - Free

11.2. A SECOND GENERATION MODEL 337The static problem is the only feasible problem. Inanidealworld,the government would like to choose current and future values of theexchange rate to minimize the expected present value of future costs⇐(225)E t ∞ Xj=0β j`t+j ,where β < 1 is a discount factor. The problem is that this opportunityis not available to the government because there is no way that theauthorities can credibly commit themselves to pre-announced futureactions. Future values of s t are therefore not part of the government’scurrent choice set. The problem that is within the government’s abilityto solve is to choose s t each period to minimize (11.25), subject to(11.24) and (11.23). This boils down to a sequence of static problemsso we omit the time subscript from this point on.Let s 0 be yesterday’s exchange rate and E 0 (s) be the public’s expectationof today’s exchange rate formed yesterday. The government Þrstobserves today’s wage w =E 0 (s), and today’s shock u, then choosestoday’s exchange rate s to minimize ` in (11.25). The optimal exchangeratemanagement rule is obtained by substituting y from (11.23) into(11.25), differentiating with respect to s and setting the result to zero.Upon rearrangement, you get the government’s reaction functions = s 0 + α θ[α(w − s)+ȳ + u] . (11.26)Notice that the government’s choice of s depends on yesterday’s predictionof s by the public since w =E 0 (s). Since the public knows thatthe government follows (11.26), they also know that their forecasts ofthe future exchange rate partly determine the future exchange rate. Tosolve for the equilibrium wage rate, w =E 0 (s), take expectations of(11.26) to getw = s 0 + αȳθ . (11.27)Tocutdownonthenotation,letλ =α2θ + α 2 .