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332 CHAPTER 11. BALANCE OF PAYMENTS CRISESWhen reserves are exhausted, r(t) =0, and the money supply becomesm(t) =γd(t) =γ[d(0) +Substitute m(t) into(11.8)togetZ t0ú d(u)du] =γ[d(0) + µt].˜s(t) =γ[d(0) + µt]+αγµ. (11.9)Setting ˜s(t A )=¯s = m(0) = γd(0) + (1 − γ)r(0) and solving for thetime of attack givest A =(1 − γ)r(0)γµ− α = t N − α. (11.10)The level of reserves at the point of attack isr(t A )=r(0) − µγ1 − γ t A = µαγ1 − γ> 0. (11.11)Figure 11.1 illustrates the time-path of money and its componentswhen there is an attack. One of the key features of the model is thatepisodes of large asset market volatility, namely the attack, does notcoincide with big news or corresponding large events. The attack comessuddenly but is the rational response of speculators to the accumulatedeffects of domestic credit creation that is inconsistent with the Þxedexchange rate in the long run.One dissatisfying feature of the deterministic model is that the attackis perfectly predictable. Another feature is that there is no transferof wealth. In actual crises, the attacks are largely unpredictable andtypically result in sizable transfers of wealth from the central bank (withcosts ultimately borne by taxpayers) to speculators.AstochasticÞrst-generation model.Let’s now extend the Flood and Garber model to a stochastic environment.We will not be able to solve for the date of attack but wecan model the conditional probability of an attack. In discrete time,