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336 CHAPTER 11. BALANCE OF PAYMENTS CRISESObstfeld’s Multiple Devaluation Threshold ModelAll variables are in logarithms. Let p t be the domestic price level ands t be the nominal exchange rate. Set the (log) of the exogenous foreignprice level to zero and assume PPP, p t = s t . Output is given by aquasi-labor demand schedule which varies inversely with the real wageiidw t − s t , and with a shock u t ∼ N(0, σu)2y t = −α(w t − s t ) − u t . (11.23)Firms and workers agree to a rule whereby today’s wage was negotiatedand set one-period in advance so as to keep the ex ante real wageconstantw t =E t−1 (s t ). (11.24)Optimal Exchange Rate ManagementWe Þrst study the model where the government actively manages, butdoes not actually Þx the exchange rate. The authorities are assumedto have direct control over the current-period exchange rate.The policy maker seeks to minimize costs arising from two sources.The Þrst cost is incurred when an output target is missed. Notice that(11.23) says that the natural output level is E t−1 (y t )=0. Weassumethat there exists an entrenched but unspeciÞed labor market distortionthat prevents the natural level of output from reaching the sociallyefficient level. These distortions create an incentive for the governmentto try to raise output towards the efficient level. The government setsa target level of output ȳ>0. When it misses the output target, itbears a cost of (ȳ − y t ) 2 /2 > 0.The second cost is incurred when there is inßation. Under PPPwith the foreign price level Þxed, the domestic inßation rate is thedepreciation rate of the home currency, δ t ≡ s t − s t−1 . Together, policyerrors generate current costs for the policy maker `t, accordingtothequadratic loss function`t = θ 2 (δ t) 2 + 1 2 [ȳ − y t] 2 . (11.25)Presumably, it is the public’ desire to minimize (11.25) which it achievesby electing officials to fulÞll its wishes.