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3.3. THE MONETARY MODEL UNDER FLEXIBLE EXCHANGE RATES87the fundamentals, just as stock prices are much more volatile thandividends. Before exploring further the relation between the exchangerate and the fundamentals, consider what happens if the transversalitycondition is violated.Rational bubbles. If the transversality condition does not hold, it ispossible for the exchange rate to be governed in part by an explosivebubble {b t } that will eventually dominate its behavior. To see why, letthe bubble evolve according tob t =(1/ψ)b t−1 + η t , (3.13)where η tiid∼ N(0, σ 2 η). The coefficient (1/ψ) exceeds 1 so the bubbleprocess is explosive. Now add the bubble to the fundamental solution(3.12) and call the resultŝ t = s t + b t . (3.14)You can see that ŝ t violates the transversality condition by substituting(3.14) into (3.11) to getψ t+k E t ŝ t+k = ψ t+k E t s t+k +ψ t+k E t b t+k = b t .| {z }0However, ŝ t is a solution to the model, because it solves (3.9). You cancheck this out by substituting (3.14) into (3.9) to gets t + b t =(ψ/λ)f t + ψ[E t S t+1 +(1/ψ)b t ].The b t terms on either side of the equality cancel out so ŝ t is indeed isanother solution to (3.9) but the bubble will eventually dominate andwill drive the exchange rate arbitrarily far away from the fundamentalsf t . The bubble arises in a model where people have rational expectationsso it is referred to as a rational bubble. What does a rationalbubble look like? Figure 3.3 displays a realization of a ŝ t for 200 timeperiods where ψ =0.99 and the fundamentals follow a driftless randomwalk with innovation variance 0.035 2 . Early on, the exchange rateseems to return to the fundamentals but the exchange rate diverges astime goes on.