Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks

The authors argue that merely pointing to the self-fulfilling nature of speculative beliefs leaves open, first, why an attack occurs precisely when it does – by most accounts both the European ERM and the Mexican peso were “ripe” for attack well before the crises that actually brought them down (citing Eichengreen and Wyplosz 1993 and Dornbusch and Werner 1994 for ERM/Mexico timing), yet conventional theory must appeal to forces “outside the theoretical model” to explain the timing of the shift in expectations – and second, it leaves the policy debate over curbing attacks (e.g., proposals to “throw sand” into currency markets via transaction costs, citing Eichengreen et al. 1995) without theoretical purchase, since an account that cannot say which equilibrium obtains cannot evaluate whether a given policy moves the system toward or away from crisis (Introduction).

The economy is characterized by a fundamental state theta, uniformly distributed on [0,1]; the floating exchange rate absent intervention is f(theta), strictly increasing in theta. The government defends a peg e* and derives value v > 0 from doing so, but incurs a defense cost c(alpha, theta) that is increasing in the proportion alpha of speculators who attack and decreasing in theta (Sec. I). Each of a continuum of speculators may short-sell the currency at transaction cost t > 0; if the peg is abandoned, a speculator who attacked earns e* - f(theta) - t, while an unsuccessful attack costs -t and not attacking always pays zero. Three regularity assumptions (c(0,0) > v; c(1,1) > v; e* - f(1) < t) guarantee the problem is economically interesting, defining a “theta-bar” below which the government abandons the peg regardless of speculator behavior (defense cost exceeds value even absent attack) and a “theta-double-bar” above which speculators can never profit from attacking even if they succeed.

In the “ripe for attack” interval (theta-bar, theta-double-bar), the government defends the peg if few speculators attack and abandons it if enough do, and a speculator who believes the peg will be abandoned finds attacking individually rational – so both “all speculators attack, peg is abandoned” and “no speculators attack, peg survives” are self-fulfilling, mutually consistent equilibria for the identical fundamental state theta (Sec. I.A). Given this multiplicity, “no definitive prediction can be made as to whether the currency will come under attack or not” for any theta in this range – this is the baseline indeterminacy the paper’s main theorem removes.

Each speculator instead observes a private signal x, drawn uniformly from [theta - epsilon, theta + epsilon] for small epsilon > 0, independently and identically across speculators conditional on theta (Sec. I.B). Theorem 1 states: there is a unique theta such that, in any equilibrium of the game with imperfect information, the government abandons the peg if and only if theta (cid:18) theta.** The proof proceeds through three lemmas: Lemma 1 shows attacking decisions are strategic complements (a speculator’s expected payoff to attacking is weakly increasing in how many other speculators attack, for any fixed message x); Lemma 2 shows that under the simple “attack if signal below k” strategy profile, the expected payoff to the marginal speculator u(k, I_k) is continuous and strictly decreasing in k (stronger fundamentals make the marginal attacker’s expected payoff lower); Lemma 3 combines these via a supremum/infimum argument on the equilibrium strategy to show any equilibrium must itself be the simple threshold (step-function) strategy at the unique x* solving u(x*, I_{x*}) = 0, from which the unique abandonment threshold theta* follows by matching the aggregate short-sales schedule against the government’s abandonment condition.

Theorem 2 states that in the limit as epsilon -> 0, theta is given by the unique solution to f(theta) = e* - 2t.** The intuition offered: the marginal speculator who observes signal x = theta* knows, for small epsilon, that the true theta is close to theta*; since the government abandons the peg exactly when theta is below theta*, this speculator attaches equal probability to the peg being defended and abandoned, so his expected payoff to attacking is (1/2)(e* - f(theta*)), which he sets equal to the certain cost t – yielding the stated equation.

The authors argue that even with noise arbitrarily small, it is never common knowledge that theta is not in the “unstable” region [0, theta-bar]: observing a signal above theta-bar + epsilon lets you personally infer theta (cid:18) theta-bar, but knowing that everyone else also observed such a signal requires a signal above theta-bar + 3*epsilon, and so on – for any fixed epsilon and signal x there is some order n of iterated knowledge that fails, so full common knowledge that fundamentals are sound never obtains (Sec. II, following the main proofs). The paper’s interpretation is that what triggers a crisis is not necessarily new information about fundamentals per se, but “informational events” – news that different speculators interpret slightly differently – that remove common knowledge that the peg is sustainable; the authors cite reported rumors of political trouble in Chiapas before the 1994 Mexican crisis, and uncertainty around Maastricht/German unification/Bundesbank pronouncements before the 1992 ERM crises, as possible instances, while being explicit this is an interpretive suggestion rather than a tested empirical claim.

Totally differentiating f(theta) = e - 2t in the small-epsilon limit gives d(theta*)/dt = -2/f’(theta*), so increasing transaction costs always reduces the range of fundamentals over which an attack occurs** (Sec. III.B). The size of this effect depends on f’(theta): when f’ is small (a weak link between fundamentals and the floating exchange rate), a given increase in transaction costs produces a large reduction in the crisis threshold, precisely when the consequences of a speculative attack, were it to occur, would be mild* – conversely, when attacks are predicted to produce drastic exchange-rate effects (f steep), a small transaction-cost tax does little. The paper offers this as a qualification to proposals for small, uniform “sand in the wheels” taxes.

An increase in aggregate speculator wealth lowers the critical proportion of speculators alpha(theta) needed to force abandonment (since the government’s decision depends on the absolute level of short sales), which unambiguously shifts the crisis threshold theta to the right – enlarging the set of states in which attacks occur* (Sec. III.C). Crucially, the magnitude of this effect depends on the size of the noise epsilon: in the limiting case as epsilon -> 0, a shift in the alpha(.) function has no effect on the switching point theta at all, since the equilibrium short-sales schedule becomes a pure step function* – so changes in aggregate wealth matter most when fundamentals are relatively opaque (large epsilon) and barely matter when fundamentals are highly transparent (small epsilon). The authors draw the policy implication that direct capital controls aimed at limiting speculative firepower “work best” precisely when transparency is low, and are “far less effective” when fundamentals are relatively transparent to all market participants, since in that regime strategic considerations swamp uncertainty about fundamentals.

The paper argues its analysis “suggests an important role for public announcements by the monetary authorities, and more generally, the transparency of the conduct of monetary policy and its dissemination to the public” – the policy instruments that stabilize the market are those that restore common knowledge of the fundamentals, with the canonical device being a “town hall meeting” announcement that is commonly known to convey the same information to all participants simultaneously, as opposed to fragmented communication in small groups where common knowledge is extremely difficult to achieve (Sec. III.A, citing Rubinstein’s 1989 e-mail game and Chwe 1996 on communication structures). The paper explicitly offers the Clinton administration’s 1995 announcement of a $40 billion rescue package for the Mexican peso as an illustration, arguing its effectiveness “derived more from its very public nature, rather than the actual sum of money involved” – a specific interpretive claim the paper advances but does not independently test against the theoretical model’s mechanism.