Self-Fulfilling Debt Crises

Optimal policy in an environment where crises remain possible in the future, not just today. “The distinguishing feature of this paper is that it examines optimal policy within an environment in which not only can crises occur in the first period, but crises can occur in the future with positive probability. This is important since we show that previously proposed policies that seek to avert a crisis by reacting contemporaneously, such as pegging the interest rate on the government debt or lengthening the maturity of the debt being sold once a crisis has started, are ineffectual in our model. Instead, it is only preemptive policies, which seek to remove the conditions that make future crises possible, that can be effective.” The antecedents are named precisely: Calvo (1988) is “the first work that makes this point within the context of debt repudiation,” showing “in a simple two-period model of sovereign lending that for certain parameter values, two Pareto-ranked equilibria exist”; Alesina, Prati and Tabellini (1990) is “the most closely related paper,” showing that “even if the government faces a discrete cost of defaulting, its need to roll over its debt can lead to two equilibria” and that lengthening initial maturity from one to two periods “can in some cases remove the possibility of the crisis equilibrium.”

Four stated advantages, plus robustness to within-period timing. “While the use of a fully specified dynamic, stochastic general equilibrium model comes at some technical cost, it has at least four advantages: First, it leads to a model that is easy to compare with models used in other areas of modern macroeconomics, such as business cycle theory. Second, it leads to a model whose features are easily matched with the data. Third, the use of a stochastic model allows to analyze the optimal policy response to the threat of a future crisis. Fourth, the use of a fully specified general equilibrium model forces us to be explicit about the assumptions that go into constructing the model and their role in deriving the results.” The concern being addressed is specific: “As the discussions in Krugman (1996) and Kehoe (1996) show, dynamic models that rely on reduced form loss functions can produce results that depend, more or less arbitrarily, on the specification of within-period timing. The results of the model in this paper are, at least at a qualitative level, robust to this sort of detail.” Because it is the government’s inability to commit ex ante to a repayment policy that makes confidence crises possible, the authors “follow Lucas and Stokey (1983) and Chari and Kehoe (1990) in examining the optimal time-consistent policy of the government.”

Consumers, international bankers and a benevolent government, with capital, a proportional income tax, and a one-period defaultable bond. Consumers are a unit continuum of identical infinitely lived agents with utility E Σ β^t (c_t + v(g_t)) – linear in private consumption, with v continuously differentiable, strictly concave, monotonically increasing and satisfying v(0) = −∞ – facing c_t + k_{t+1} ≤ (1−θ)a_t f(k_t), where θ is a constant proportional tax on domestic income and a_t is a productivity factor “that depends on whether or not the government has ever defaulted.” International bankers are a unit continuum of risk-neutral agents endowed with x̄ each period who buy government bonds at price q_t that pay b_{t+1} if the government repays and zero otherwise, subject to a no-Ponzi bound. The government is benevolent – it maximises consumer utility, and “the welfare of the bankers does not enter the government’s objective” – and each period chooses new borrowing B_{t+1}, the default decision z_t ∈ {0,1}, and spending g_t subject to g_t + z_t B_t ≤ a_t θ f(k_t) + q_t B_{t+1}. Default has two consequences: “productivity drops from a_t = 1 to a_t = α < 1 from period t onward,” and “the government loses access to international borrowing and lending after default.” No borrowing constraint on the government is needed, “because, if the government tries to sell too much new debt B_{t+1}, its price q_t falls to zero.”

Because it is what creates a rollover need, and the rollover need is what makes a self-fulfilling crisis possible. The order within each period is: the sunspot is realised and the aggregate state s_t = (B_t, K_t, a_{t−1}, ζ_t) is determined; the government, taking the price schedule as given, chooses B_{t+1}; bankers, taking the price as given, choose b_t; the government chooses whether to default and how much to consume; and consumers choose consumption and next period’s capital. “An important feature of our model is the timing of the government’s decisions within a period. This timing enables the government to issue new debt before retiring the old debt, while having a maturity of one period on the debt. The need to roll over old debt into new is what allows us to have a self-fulfilling crisis. We could also obtain this feature by having longer maturity debt that is only partially rolled over in any period, but this would complicate the analysis.” The authors are explicit about the limits of this robustness: alternative timing assumptions for consumers’ choices or the imposition of the default penalty “would change the quantitative results of the model, they would not change its qualitative features.”

Two named stories, each of which can be written into the model as a reinterpretation rather than an addition. The first is trade disruption: “a disruption of the country’s ability to engage in international trade reduces the value of output,” which can be formalised by supposing a foreign-produced intermediate good “whose importation is impeded because the foreign creditors can intercept payments made to the producer of this intermediate good,” so that f(k) is output with the intermediate input optimally chosen and αf(k) is output without it. “Among the reasons commonly given for the disruption of trade are confiscation of goods in transit, seizure of trade-related assets, and loss of access to short-term trade credit from intermediaries.” The second is a reputation spillover: “the government’s reputation is damaged by its decision to default, which hurts the government’s ability to operate, and, consequently, output falls. Defaulting on its debts to the international bankers might, for example, lead the government to lose not only its reputation for repaying its debts, but also for paying the wages of government employees. Hence, the government is no longer able to hire workers, and the level of a valuable government input into private production is reduced.” The paper also flags two simplifications it makes for exposition: total loss of market access after default (in fact “the government would not want to borrow again after a default; it would want to lend to smooth post-default government spending,” which is trivial to allow), and a constant tax rate, “because we think that the period over which crises can occur is short compared with that over which tax policy can be changed.”

A recursive (Markov) equilibrium without commitment, deliberately narrower than the sustainable-equilibrium concept. “The government cannot commit itself either to honoring its debt obligations or to following a fixed borrowing and spending path. We therefore define a recursive equilibrium in which there is no commitment and the agents choose their actions sequentially.” Bankers are passive: “they purchase the amount of bonds offered by the government as long as the price of these bonds satisfies q(s,B′) = βEz(…),” i.e. “as long as the expected gross return on these bonds is 1/β,” and the authors note “we could simply reinterpret our model as a small open economy in which the pricing function satisfies this relationship, and we could drop consideration of the bankers completely.” The equilibrium concept “is similar to Chari and Kehoe’s (1990, 1993) definition of a sustainable equilibrium and Stokey’s (1991) definition of a credible equilibrium,” differing in two ways: “First, we have restricted ourselves to a recursive – that is, Markov – equilibrium, and hence the agents’ future conditional plans can be derived from their policy functions. Second, we have allowed the private agents to condition their actions within the period on the government’s new borrowing level, and hence we do not have to assume that the government sets the prices and the private agents set the quantities.” The authors flag the cost of that restriction with a memorable example: “If, for example, we were to include the time period in the definition of the state, we could construct equilibria in which crises are only possible in periods whose dates are prime numbers. If we were to allow for nonrecursive equilibria, even stranger possibilities might arise.”

One in which agents ignore sunspots; debt is constant, capital is at its no-default level, and the only constraint is participation. Consumers’ risk neutrality makes their capital choice a simple rule: if they expect no default, they set k^n satisfying β(1−θ)f′(k^n) = 1; if they expect default, k^d satisfying β(1−θ)αf′(k^d) = 1, and strict concavity of f implies k^n > k^d. The government’s participation constraint (9) requires that the value of not defaulting weakly exceed the value of defaulting, given that it can sell its new debt at price β. When that constraint does not bind, “it is optimal for the government to maintain a constant level of spending and, hence, of its debt,” with g^n(B_0) = θf(k^n) − (1−β)B_0. Proposition 1 then characterises the outcomes: there is a continuous increasing function B̄(K) and a positive debt level B^s with B̄(k^n) > B^s such that, with K_0 = k^n and B_0 ≤ B^s the economy sits in the stationary no-default equilibrium with debt constant at its initial level; with B_0 ≤ B̄(K_0) it converges to that stationary continuation equilibrium “after at most two periods”; and with B_0 > B̄(K_0) the outcome is default. The authors note that “although it is sometimes possible to construct other equilibria, this equilibrium is always an equilibrium.”

The simultaneous satisfaction of the participation constraint and the no-lending condition. The no-lending condition (11) states that “if q = 0, then the government strictly prefers to default,” which is what makes a run self-confirming: “This crisis is then self-fulfilling if the inability to sell its bonds at a positive price induces the government to default.” Its boundary b̄(K) is “the largest value of B for which the government weakly prefers to repay its debt, even if it cannot sell new bonds at a positive price,” and it is increasing in capital over the relevant range. “We refer to the range of debt value for which both the participation constraint and the no-lending condition are satisfied as the crisis zone.” Combining the two conditions under a stationary debt policy yields the compact inequality W_p(B) ≥ W̄ > W_nl(B), where W_p(B) is “the net social benefit in terms of government spending of not defaulting if the government can roll over its debt,” W̄ is “the net social cost in terms of consumption of not defaulting,” and W_nl(B) is “the net social benefit in terms of government spending of not defaulting if the government cannot roll over its debt.” The zone is non-empty when the social cost of default in terms of private consumption is large enough.

Patient ones, and ones that weight private consumption heavily – which the authors themselves call surprising. Parameterising v(g) = γw(g), where γ is the preference for government spending, Proposition 2 states that “for positive γ sufficiently close to 0 or for β < 1 but sufficiently close to 1, there is a nonempty interval of levels of government debt B… for which zero-probability crises are possible.” The intuition is that the two bounds of the zone respond differently: “If the utility function puts sufficient weight on private consumption, then the high cost of default on consumption makes it possible for the government to satisfy the participation constraint with a relatively high level of debt. Such a high level of debt makes a crisis possible if the government cannot roll over its debt. One interpretation of this result is that governments that care more about private consumption than public consumption have a wider crisis zone.” The introduction states the result more bluntly: “We are surprised to find that a government that cares sufficiently more about private than government consumption or is sufficiently farsighted is guaranteed to have a crisis zone. While these governments have a larger lower bound on debt in their crisis zone, they also have an even larger upper bound.”

Because in this model causality runs from expectations to prices, so a peg just produces a buyers’ strike. “Since causality runs from expectations to interest rates in our model, Calvo’s (1988) prescription that the government peg the interest rate simply results in a refusal of the private agents to buy government debt. For example, even if the government pegs the price of its debt at β, the optimal response of the bankers is to buy zero of it if they believe that z′ = 0. This result is consistent with the Mexican government’s inability to issue new debt at prices above its auctions’ reservation price.”

Because it moves both boundaries of the crisis zone, and a country already inside can end up in a worse one. The exercise is to consider “a country that has raised the credibility of its commitment to repay its debts by increasing the cost of defaulting on its debt through, say, increasing its dependence on foreign trade,” which in the model means lowering α. “Raising the cost of defaulting does indeed increase the level of debt that can be sustained if a crisis does not occur, B̄(k^n). It also raises the level of debt at which a crisis is possible, b̄(k^n). It does not remove the possibility of a crisis occurring however, since the gap between b̄(k^n) and B̄(k^n) need not be removed. Hence, if this country’s debt places it within the new crisis zone, the consequences of the acquisition of some additional credibility can be to make the effects of a crisis much worse.” Proposition 3 sharpens this: “For K = k^n and positive α sufficiently close to 0, there is a nonempty interval of levels of government debt B… for which zero-probability crises are possible” – that is, a sufficiently large default cost guarantees a crisis zone exists. The general statement from the introduction: “any policy that increases the costs of a crisis without removing the possibility of one occurring need have no effect on the likelihood of a crisis.”

Lengthening the maturity of prevailing debt can eliminate crises; lengthening the maturity of new debt cannot help today. Converting a stationary stock B of one-period bonds into equal quantities of bonds of maturities 1 through N gives per-maturity holdings B_N = [(1−β)/(1−β^N)]B, and Proposition 4 states that for any debt level in the zero-probability crisis interval “there exists a maturity of the debt, N′, sufficiently long so that no crises are possible for any maturity N ≥ N′.” The intuition: “As the maturity of the debt gets longer, the amount that the government needs to borrow every period get smaller. This decreases the government’s incentive to default, whether or not it can roll over its debt. Furthermore, that international bankers refuse to make new loans to the government has less and less impact on the government payoff, and consequently the payoff from not defaulting if the government can roll over the debt converges to that if it cannot.” The sharp corollary, stated against Alesina-Prati-Tabellini: “it is the maturity structure of the prevailing debt, and not that of the debt being issued, that determines whether or not a crisis is possible. Hence, if there is enough debt coming due to allow for a crisis, then contrary to the suggestion of Alesina, Prati, and Tabellini (1990), the maturity structure of new debt being issued is irrelevant to the possibility of a crisis today, although it may remove the possibility of a crisis in the future.” The authors immediately flag what their model omits: “it is easy for the government to increase the maturity of its debt in this model because there is no cost to doing so. In a more general framework, there may be significant costs. One well-known example of such a cost arises from a worsening of the time-consistent problem associated with setting the inflation tax… Notice, however, that in the case of Mexico, where tesobonos were dollar-indexed, this time-consistency problem was not present.”

It depresses investment and the bond price in proportion to π, and it does so before any crisis occurs. With the sunspot uniform on [0,1], “π is both the crucial value of ζ, and the probability that ζ ≤ π.” When the crisis probability is π, consumers’ first-order conditions give a stationary capital stock k^π satisfying β(1−π) + παf′(k^π) = 1, with consumption c^π(k) = (1−θ)f(k) − k^π, and bankers “purchase whatever amount of bonds the government offers up to x̄ as long as q = β(1−π).” The authors are explicit about which assumption delivers this tractability: “It is here that the risk neutrality of utility in consumption plays its role: if consumers are risk averse, then optimal investment and consumption are not stationary.” The real-side implication stated in the introduction is that “within the crisis zone, the country’s economic activity is depressed in proportion to the probability that a crisis will take place.”

Choose how many periods T to take in running debt down to the crisis-zone floor – possibly one, possibly never. In period 0 the government faces the choice to “default now; plan to run the debt down to b̄(k^n) or less in T periods if no crisis occurs, T = 1, 2, …; or never run the debt down,” and picks the maximum expected payoff among these. Proposition 6 establishes such a maximising T exists and that “as B_0 increases, T(B_0) passes through critical points where it increases by one period,” with the full range T = 1, 2, …, ∞ arising for π close to zero. The payoff of exiting has a distinctive shape: “lowering the debt to the crucial value at which crises cannot occur leads to an investment boom in period T−1. In period T, consumption and government spending increase, and the equilibrium becomes the stationary no-crisis equilibrium.” The introduction states the comparative static: “if the probability that the agents in equilibrium assign to a crisis is high, then the government will want to adjust its debt so as to exit the crisis zone immediately, but that as the level of the government’s debt increases and the probability that the agents assign to a confidence crisis occurring decreases, the number of periods that this transition out of the crisis zone takes can become arbitrarily large.”

Not because of the lower price it receives on debt, but because of the benefits of exiting – and the authors go out of their way to establish this. The first observation is that near the floor, selling less debt can raise more cash: “for some initial levels of the debt B larger than b̄(k^n), the government actually receives more when it sells new debt if it tries to sell less debt,” since β̂B < β b̄(k^n) when B is just above the floor; and because b̄(k^n)/(1−π) → ∞ as π → 1, “the region for which the government receives less and, hence, prefers to run its debt down to b̄(k^n) in one step will come to include all of the region of the crisis zone… as the probability of a crisis gets large.” The second observation isolates the motive: in the case T(B_0) = ∞ “the government never exits the crisis zone, and its optimal debt policy is to keep the debt constant and thereby smooth government consumption. This outcome arises despite the fact that government discount rate is β and the market is paying β̂ per unit of debt because the government only assigns probability (1−π) to the possibility that it has to repay its debt. The government therefore discounts future repayment at rate β̂ as well. Hence, the government’s motivation to run down its debt and thereby deviate from the sort of simple government consumption smoothing policy comes solely from the benefits of exiting the crisis zone.” One further texture: “the government’s optimal consumption level is not smooth, since it rises when the debt level hits the threshold b̄(k^n). This level is, however, smooth both during the transition and after the boom, but not equal across these intervals.”

One in which investors’ fear of a future default cuts investment, which lowers the crisis threshold and makes the fear come true. Adding a second sunspot realised after the government sells its debt but before consumers act, “consider an outcome in which consumers’ fears lead them to lower their investment level to k^{π_ζ}. If the government’s new borrowing satisfies B′ > b̄(k^{π_ζ}), then a self-fulfilling crisis next period would be possible given that K′ = k^{π_ζ}, since b̄′ > 0 and, hence, b̄(k^{π_ζ}) < b̄(k^n). Therefore, consumers’ fears that they could be in the crisis zone next period could be self-fulfilling, even if K = k^n and they were out of the crisis zone today.” The probability of these fears “can be quite arbitrary,” any value between zero and one. Two consequences follow: “the equilibrium outcome could potentially exhibit wide swings in investment before a financial crisis that forces the government into default finally arises”; and the safe debt level tightens, since as the probability approaches one, capital approaches k^d and “only debt levels at or below b̄(k^d) are sufficiently low so as to ensure that a self-fulfilling crisis is not possible.” The authors connect this to the Mexican episode: “as is noted in the International Monetary Fund (1995) report, domestic investment fell before the debt crisis occurred.”

The qualitative results survive, and a richer state space would let the model speak to reputation. “Assume that a falls from 1 to α for only a finite number of periods, after which it returns to 1 until a subsequent default occurs… Then an equilibrium of the model would simply involve restarting after a returns to 1, in which case [the default payoff] would be higher, but the qualitative characteristics of the no-crisis and the π-probability-of-a-crisis equilibrium would be unchanged. If we expand the state space to include the number of past defaults of the government, however, it would be possible to construct sunspot equilibria in which the probability of a crisis depends on the number of past defaults by the government. Such a model would allow one possible interpretation of a reputation for repayment by the government, especially if this probability were increasing in the number of past defaults.”

Four lessons, one overall prescription, and a carefully conditional case for a lender of last resort. The four, stated in the conclusion: “First, standard prescriptions, such as increasing the cost of default to increase credibility, may be a bad idea for the government, since it does not eliminate the possibility of a crisis, but only increases its severity. Second, since causality runs from expectations to interest rates in our model, pegging the interest rate simply results in a refusal by the private agents to buy government debt. Third, if there is enough debt coming due to allow for a crisis, then the maturity structure of new debt being issued is irrelevant to the possibility of a crisis today, although it may remove the possibility of a crisis in the future. Fourth, good governments, in the sense that they are more concerned about private consumption than government consumption or are more patient, are more likely to have a nonempty crisis zone.” The summary prescription: “the only way to avoid debt crises is to avoid the conditions on fundamentals that make them possible: in particular, relatively high levels of debt with a short maturity structure.” On bailouts, the case is made and then qualified twice: “since debt crises are due to a coordination failure among the private lenders, an international agency that stands ready to act as a lender of last resort would be welfare enhancing… It is important to note, however, that for these sorts of bailouts to be successful, the policy makers must be able to distinguish between confidence crises and crises in which the government would have an incentive to default even if it could sell new debt. Furthermore, reducing the likelihood of a crisis actually leading to a default reduces the cost to a government of being in the crisis zone, and that this will in turn reduce a government’s incentive to run its debt down and exit the crisis zone.”

Through maturity, not through debt levels – and the authors note the quantitative work is in a companion paper, not this one. On Mexico: “In the Mexican crisis, the fear of a government default led to the inability of the government to issue new debt, which in turn seemed about to confirm the fears of a default until the United States intervened with a rescue package. The crisis occurred despite the fact that the Mexican government’s fiscal behavior in terms of standard measures, such as its debt/GDP ratio, appears to have been responsible with respect to both its own past behavior and the behavior of many other governments that, at least as yet, have not experienced similar crises. At the time of the crisis, however, the average maturity of Mexico’s debt had become very short. When the crisis occurred, the Mexican government found itself unable to sell either dollar-indexed tesobonos or domestically denominated debt, which seems hard to explain on the basis of currency risk, but easy to explain on the basis of default risk.” The quantitative claim is attributed to the companion paper: in Cole and Kehoe (1996)’s model calibrated to 1994 Mexico, “the magnitude and maturity structure of Mexico’s debt are consistent with Mexico being in the crisis zone. The model predicts that doubling the average maturity of this debt from roughly 200 days to 400 days, however, would have eliminated the possibility of a crisis.” On Asia, the extension is offered as a conjecture rather than a result: Korea, Indonesia, Malaysia and Thailand “appear to have experienced financial problems that are similar to those experienced by Mexico, except that financial crises seem to have originated in the banking systems of these countries. Here, too, the key element allowing these crises to occur is the large amount of short-term liabilities. The threat that this maturity mismatch poses is exacerbated by the decline in the value of the local currency, since the banks’ loans are denominated in local currency, while their debts are denominated in dollars. We think that a version of our model that incorporates a private banking system with explicit or implicit government-provided insurance to the foreign lenders could account for these crises as well.”