International macroe.. - Free

10.2. THE CONTINUOUS—TIME MONETARY MODEL 311International asset-market equilibrium is given by uncovered interestparityi(t) − i ∗ (t) = ús(t). (10.12)The model is completed by invoking PPPCombining (10.10)-(10.13) you gets(t)+p ∗ (t) =p(t). (10.13)s(t) =f(t)+α ús(t), (10.14)where f(t) ≡ m(t) − m ∗ (t) − φ[y(t) − y ∗ (t)] are the monetary-model‘fundamentals.’ Rewrite (10.14) as the Þrst-order differential equationús(t) − s(t)αThe solution to (10.15) is 3Z ∞= −f(t)α . (10.15)s(t) = 1 e (t−x)/α f(x)dxα t= 1 Z ∞α et/α e −x/α f(x)dx. (10.16)A stochastic setting. The stochastic continuous-time monetary modelistm(t) − p(t) = φy(t) − αi(t), (10.17)m ∗ (t) − p ∗ (t) = φy ∗ (t) − αi ∗ (t), (10.18)i(t) − i ∗ (t) = E t [ ús(t)], (10.19)s(t)+p ∗ (t) = p(t). (10.20)3 To verify that (10.16) is a solution, take its time derivativeús(t) = 1 α et/α · ddtZ ∞= − 1 α f(t)+ 1 α 2 et/α Z ∞= − 1 α f(t)+ 1 α s(t)Therefore, (10.16) solves (10.15).t¸ ·Z ∞¸e −x/α f(x)dx + e −x/α f(x)dx α −2 e t/αte −x/α f(x)dxt