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8.4. VAR ANALYSIS OF MUNDELL—FLEMING 257Appendix: Solving the Dornbusch ModelFrom (8.9) and (8.11), we see that the behavior of i(t) is completely determinedby that of s(t). This means that we need only determine the differentialequations governing the exchange rate and the price level to obtain acomplete characterization of the system’s dynamics.Substitute (8.9) and (8.11) into (8.6). Make use of (8.13) and rearrangeto obtainús(t) = 1 [p(t) − ¯p]. (8.58)λTo obtain the differential equation for the price level, begin by substituting(8.58) into (8.9), and then substituting the result into (8.8) to get ⇐(144)úp(t) =π[δ(s(t) − p(t)) + (γ − 1)y − σi ∗ − σ (p(t) − ¯p)+g]. (8.59)λHowever, in the long run0=π[δ(¯s − ¯p)+(γ − 1)y − σr ∗ + g], (8.60)the price dynamics are more conveniently characterized byúp(t) =π·δ(s(t) − ¯s) − (δ + σ )(p(t) − ¯p)¸, (8.61)λ⇐(145)which is obtained by subtracting (8.60) from (8.59).Now write (8.58) and (8.61) as the systemà ! à !ús(t) s(t) − ¯s= A, (8.62)úp(t) p(t) − ¯pwhereà !0 1/λA =.πδ −π(δ + σ/λ)(8.62) is a system of two linear homogeneous differential equations. We knowthat the solutions to these systems take the forms(t) = ¯s + αe θt , (8.63)p(t) = ¯p + βe θt . (8.64)We will next substitute (8.63) and (8.64) into (8.62) and solve for theunknown coefficients, α, β, andθ. First, taking time derivatives of (8.63)and (8.64) yieldsús = θαe θt , (8.65)úp = θβe θt . (8.66)