International macroe.. - Free

11.1. A FIRST-GENERATION MODEL 333let the economic environment be given bym t = γd t +(1− γ)r t , (11.12)m t − p t = −αi t , (11.13)p t = s t , (11.14)i t = E t (∆s t+1 ). (11.15)Let domestic credit be governed by the random walkd t =(µ − 1 λ )+d t−1 + v t , (11.16)where v t is drawn from the exponential distribution. 2 . Also, assumethat the domestic credit process has an upward drift µ > 1/λ. Attime t, agents attack the central bank if ˜s t ≥ ¯s, where˜s is the shadowexchange rate.Let the publicly available information set be I t and let p t be theprobability of an attack at t + 1 conditional on I t . Then,p t = Pr[˜s t+1 > ¯s|I t ]= Pr[αγµ + m t+1 − ¯s >0|I t ]= Pr[αγµ + γd t+1 − ¯s >0|I t ]· µ ·= Pr αγµ + γ d t + µ − 1 ¸ + v t+1 − ¯s >0|I t¸λ= Pr"v t+1 > 1 γ ¯s − (1 + α)µ − d t + 1 λ |I t#= Pr(v t+1 > θ t |I t )=Z ∞θ tλe −λu du =(e−λθ tθ t ≥ 01 θ t < 0(11.17)where θ t ≡ (1/γ)¯s − (1 − α)µ − d t +(1/λ). The rational exchange rateforecast error isE t s t+1 − ¯s = p t [E t (˜s t+1 ) − ¯s], (11.18)and is systematic if p t > 0.2 A random variable X has the exponential distribution if for x ≥ 0, f(x) =λe −λx . The mean of the distribution is E(X) =1/λ.