4. (Kaminsky and Peruga [82]). Suppose that the data generating process for observations on consumption growth, inflation, and exchange rates is given by the lognormal distribution, and that the utility function is u

(c) = c1−γ. Let lower case letters denote variables in logarithms. We have ∆ct+1 = ln(Ct+1/Ct) be the rate of consumption growth, ∆st+1 = ln(St+1/St) be the depreciation rate, ∆pt+1 = ln(Pt+1/Pt) be the inflation rate, and ft = ln(Ft) be the log one-period forward rate. If ln(Y ) ∼ N(µ, σ2), then Y is said to be log-normally distributed and E h eln(y) i = E(Y ) = e h µ+σ2 2 i . (6.74) Let Jt consist of lagged values of ct, st, pt and ft be the date t information set available to the econometrician. Conditional on Jt, let yt+1 = (∆st+1, ∆ct+1, ∆pt+1)0 be normally distributed with conditional mean E(yt+1|Jt)=(µst, µct, µpt)0 and conditional covariance matrix Σt =    σsst σsct σspt σcst σcct σcpt σpst σpct σppt    . Let at+1 = ∆st+1 − ∆pt+1 and bt+1 = ft − st − ∆pt+1. Show that µst − ft = γσcst + σspt − σsst 2 . (6.75)