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342 CHAPTER 11. BALANCE OF PAYMENTS CRISES= P[δ = δ 0 ]δ 0 +P[δ = δ 1 ]E[(λ/α)(αδ e +ȳ +E(u|u >ū))]= P[u>ū](λ/α)[αδ e +ȳ +E(u|u >ū)].Solving for δ e as a function of ū yieldsδ e =λP(u >ū)1 − λP(u >ū)1[ȳ +E(u|u >ū)] . (11.40)αTo proceed further, you need to assume a probability law governing theoutput shocks, u.Uniformly distributed output shocks. Let u be uniformly distributed onthe interval [−a, a]. The probability density function of u isf(u) =1/(2a) for −a ū) =1/(a − ū). It follows thatP(u >ū) =Z aūZ a(1/(2a))dx =(a − ū), (11.41)2a(a +ū)E(u|u >ū) = x/(a − ū)dx = . (11.42)ū2Substituting (11.41) and (11.42) into (11.40) givesδ e = f δ (ū) =λ(a − ū)2αa⎛⎝ȳ + a+ū21 − λ(a−ū)2a⎞⎠ . (11.43)Notice that δ e involves the square terms ū 2 . Quadratic equations usuallyhave two solutions. Substituting δ e into (11.37) givesū = −αf δ (ū) − ȳ +s2cdλ , (11.44)(229)⇒where f δ (ū) isdeÞned in (11.43). (11.44) has two solutions for ū, eachof which trigger a devaluation. For parameter values a =0.03, θ =0.15,c =0.0004, α =1,ȳ =0.01 solving (11.44) yields the two solutionsū 1 = −0.0209 and ū 2 =0.0030. (11.44) is displayed in Figure 11.3 forthese parameter values.Using (11.43), the public’s expected depreciation associated with ū 1is 2.7 percent whereas δ e associated with ū 2 is 45 percent. The high