International macroe.. - Free

258 CHAPTER 8. THE MUNDELL-FLEMING MODELSubstitution of (8.65) and (8.66) into (8.62) yieldsà !α(A − θI 2 ) =0. (8.67)β(146)⇒In order for (8.67) to have a solution other than the trivial one (α, β) =(0, 0),requires that0 = |A − θI 2 | (8.68)= θ 2 − Tr(A)θ + |A|, (8.69)where Tr(A) =−π(δ + σ/λ) and|A| = −πδ/λ otherwise, (A − θI 2 ) −1 existswhich means that the unique solution is the trivial one, which isn’t veryinteresting. Imposing the restriction that (8.69) is true, we Þnd that itsroots areThe general solution isθ 1 = 1 2 [Tr(A) − qTr 2 (A) − 4|A|] < 0, (8.70)θ 2 = 1 2 [Tr(A)+ qTr 2 (A) − 4|A|] > 0. (8.71)s(t) = ¯s + α 1 e θ1t + α 2 e θ2t , (8.72)p(t) = ¯p + β 1 e θ1t + β 2 e θ2t . (8.73)This solution is explosive, however, because of the eventual dominance ofthe positive root. We can view an explosive solution as a bubble, in whichtheexchangerateandthepriceleveldivergesfromvaluesoftheeconomicfundamentals. While there are no restrictions within the model to rule outexplosive solutions, we will simply assume that the economy follows thestable solution by setting α 2 = β 2 = 0, and study the solution with thestable rootθ ≡ −θ 1 (8.74)= 1 2 [π(δ + σ/λ)+ qπ 2 (δ + σ/λ) 2 +4πδ/λ]. (8.75)Now, to Þndthestablesolution,wesolve(8.67)withthestablerootà !α0 = (A − θ 1 I 2 )βà !à !−θ1 1/λα=.πδ −θ 1 − π(δ + σ/λ) β(8.76)