12.10 A Model of Debt Crises

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1 632 Chapter 12 BUDGET DEFICITS AND FISCAL POLICY and intermediaries are not bankrupted by the crisis, the worsening of their financial positions magnifies the effects of financial-market imperfections. One effect of these financial-market disruptions is that investment is lower. This effect, however, can be offset by expansionary (or less contractionary) monetary policy. But another effect is that for a given amount of investment, the average quality of projects is lower, since the financial system now allocates capital less effectively. Similarly, output is lower for a given level of employment, since many firms with profitable production opporrunities are unable to produce because of bankruptcy or an inability to obtain loans to pay their wages and purchase inputs. Bernanke (1983b) argues that such financial-market disruptions played a large role in the Great Depres sian. And they appear to have been important in more recent fiscal crises as well. In Indonesia in 1998, for example, a large majority of firms were at least technically bankrupt, although many continued to function in some form. At the microeconomic level, crises can cause large redistributions with severe consequences. For example, suppose a government that is borrowing to pay for pensions and medical care for the elderly faces a sudden default that makes it unable to do any further borrowing. One result is likely to be a sudden drop in the standards of living of the elderly, along with those whose wealth holdings were concentrated in government debt. Fiscal crises can have other costs as well. Since fiscal crises are unexpected, trying to follow an unsustainable policy increases uncertainty. Default and other failures to repay its debts can reduce a government's ability to borrow in the future. 33 Finally, a crisis can lead to harmful policies, such as broad trade restrictions, hyperinflation, and very high tax rates on capital. One way to summarize the macroeconomic effects of a fiscai crisis is to note that it typically leads to a sharp fall in output followed by only a gradual recovery. This summary, however, overstates the costs of embarking on unsustainable fiscal policy, for two reasons. First, unsustainable fiscal policy is usually not the only source of a crisis; thus it is not appropriate to attribute the crisis's full costs to fiscal policy. Second, there may be benefits to the policy before the crisis. For example, it may lead to real appreciation, with benefits that are the converse of the costs of real depreciation, and to a period of high output. Nonetheless, the costs of an attempt to pursue unsustainable fiscal policy that ends in a crisis are almost surely substantial A Model of Debt Crises We now turn to a simple model of a government attempting to issue debt. We focus on the questions of what can cause investors to be unwilling to 33 Because there is no authority analogous to domestic courts to force borrowers to repay, there are some important issues specifically related to international borrowing. See Obstfeld and Rogoff (1996, Chapter 6) for an introduction.

2 12.10 A Model of Debt Crise. 633 buy the debt at any interest rate, and of whether such a crisis is likely to occur unexpectedly. 34 Assumptions Consider a government that has quantity D of debt coming due. It has no funds immediately available, and so wants to roll the debt over (that is, to issue D of new debt to payoff the debt corning due). It will be obtaining tax revenues the following period, and so wants investors to hold the debt for one period. The government offers an interest factor of R; that is, it offers a real interest rate of R - 1. Let T denote tax revenues the following period. T is random, and its cumulative distribution function, F(-), is continuous. If T exceeds the amount due on the debt in that period, RD, the government pays the debtholders. If T is less than RD, the government defaults. Default corresponds to a debt crisis. Two simplifying assumptions make the model tractable. First, default is all or nothing: if the government cannot pay RD, it repudiates the debt entirely. Second, investors are risk neutral, and the risk free interest factor, R, is independent of R and D. These assumptions do not appear critical to the model's main messages. Analyzing the Model Equilibrium is described by two equations in the probability of default, de noted TT, and the interest factor on government debt, R. Since investors are risk neutral, the expected payoff from holding government debt must equal the risk free payoff, R. Government debt pays R with probability 1 -IT and o with probability IT. Thus equilibrium requires (1 - IT)R= R. (12.41) For comparison with the second equilibrium condition, it is useful to reo arrange this condition as an expression for IT as a function of R. This yields R- R IT= (12.42) R The locus of points satisfying (12.42) is plotted in (R,IT) space in Figure When the government is certain to repay (that is, when IT = 0), R equals R. As the probability of default rises, the interest factor the government must 34 See Calvo (1988) and Cole and Kehoe (2000) for examples of richer models of debt crises.

3 634 Chapter 12 BUDGET DEFICITS AND FISCAL POLICY 1 TT O~--~R~ ~R FIGURE 12.3 The condition for investors to be willing to hold government debt offer rises; thus the locus is upward-sloping. Finally, R approaches infinity as the probability of default approaches l. The other equilibrium condition comes from the fact that whether the government defaults is determined by its available revenues relative to the amount due bondholders. Specifically, the government defaults if and only if T is less than RD. Thus the probability of default is the probability that T is less than RD. Since T's distribution function is F(o), we can write this condition as 1T= F(RD). (12.43) The set of points satisfying (12.43) is plotted in Figure If there are minimum and maximum possible values of T, I and f, the probability of default is 0 for R < lid and 1 for R > f / D. And if the density function of T is bell-shaped, the distribution function has an S shape like that shown in the figure. Equilibrium occurs at a point where both (12.42) and (12.43) are satisfied. At such a point, the interest factor on government debt makes investors wiliing to purchase the debt given the probability of default, and the probability of default is the probability that tax revenues are insufficient to pay off the debt given the interest factor. In addition to any equilibria satisfying these rwo conditions, however, there is always an equilibrium where investors are certain the government will not payoff the debt the following period and are therefore unwiliing to purchase the debt at any interest factor. If investors refuse to purchase the debt at any interest factor, the probability of default is 1; and if the probability of default is 1, investors refuse to purchase the debt at any interest factor. Loosely speaking, this equilibrium corresponds to the point R = 00, 1T = 1 in the diagram It is straightforward to extend the analysis to the case where default is not all-ornothing. For example, suppose that when revenue is less than RD, the government pays all of it to debtholders. To analyze the model in this case, define rr as the expected fraction of

4 12.10 A Model of Debt Crises TT ol-+-~~ ~~ o IID TID R FIGURE 12.4 The probability of default as a function of the Interest factor Implications The model has at least four interesting implications. The first is that there is a simple force tending to create multiple equilibria in the probability of default. The higher the probability of default, the higher the interest factor investors demand; but the higher the interest factor investors demand, the higher the probability of default. In terms of the diagram, the fact that the curves showing the equilibrium conditions are both upward sloping means that they can have multiple intersections. Figure 12.5 shows one possibility. In this case, tbere are three equilibria. At Point A, the probability of default is low and the interest factor on government debt is only slightly above the safe interest factor. At Point B, there is a substantial chance of default and the interest factor on the debt is well above the safe factor. Finally, there is the equilibrium where default is certain and investors refuse to purchase the government's debt at any interest factor.'6 the amount due to investors, RD, that they do not receive. With this defmition, the condition for investors to be wuling to hold government debt, (1 - rr)r = R. is the same as before, and so equation (12.42) holds as before. The expression for the expected fraction of the amount due to investors that they do not receive as a function of the interest factor the government offers is now more complicated than (12.43). It still has the same basic shape in (R, TT) space, however: it is 0 for R suffidently small, upward-sloping, and approaches 1 as R approaches infinity. Because this change in assumptions does not change one curve at all and does not cbange the other's main features, the model's main messages are unaffected. 36 One natural Question is whether the government can avoid the multiplicity by issuing its debt at the lowest equilibrium interest rate. The answer depends on how investors form their expectations of the probability of default. One possibility is that they tentatively as sume that the government can successfully issue debt at the interest factor it is offering: they then purchase the debt if the expected return given this assumption at least equals the rtsk~tum. In this case, the government can issue debt at the lowest interest factor where the two curves intersect. But this is not the only possibility. For example, suppose each investor believes that others believe the government will default for sure, and that others are therefore unwilling to purchase the debt at any interest factor. Then no investor JJ:wchases the debt, and so the beliefs prove correct.

5 636 Chapter 12 BUDGET DEFICITS AND FISCAL POLICY 1 IT ITAt-~~A~~ ~=- IID R TID R FIGURE 12.5 The determination of the interest factor and the probability of default Under plausible dynamics, the equilibrium at B is unstable and the other two are stable. Suppose, for example, investors believe the probability of default is slightly below ITB. Then at the interest factor needed to induce them to buy the debt given this belief, the actual probability of default is less than what they conjecture. It is plausible that their estimate of the probability of default therefore falls, and that this process continues until the equilibrium at Point A is reached. A similar argument suggests that if investors conjecture that the probability of default exceeds ITB, the economy converges to the equilibrium where investors will not hold the debt at any interest factor. Thus there are two stable equilibria. In one, the interest factor and the probability of default are low. In the other, the government cannot get investors to purchase its debt at any interest factor, and so it defaults immediately on its outstanding debt. In short, there can be a self-fulfilling element to default. The second implication is that large differences in fundamentals are not needed for large differences in outcomes. One reason for this is the multiplicity just described: two economies can have the same fundamentals, but one can be in the equilibrium with low R and low IT and the other in the equilibrium where investors refuse to buy the debt at any interest factor. A more interesting source of large differences stems from differences in the set of equilibria. Suppose the two curves have the form shown in Figure 12.5, and suppose an economy is in the equilibrium with low R and low IT at Point A. A rise in R shifts the IT= (R - R)/R curve to the right. Similarly, a rise in D shifts the IT = F(RD) curve to the left. For small enough changes, IT and R change smoothly in response to either of these developments. Figure 12.6, for example, shows the effects of a moderate change in R from Ro to R,. The equilibrium with low R and low IT changes smoothly from A to N. But now suppose R rises further. If R becomes sufficiently large-if it rises to R" for example-the two curves no longer intersect. In this situation, the only equilibrium is the one where investors will not buy the debt. Thus two economies can have similar fundamentals, but in one there is an equilibrium where the government can issue debt at a low interest rate while in

6 12.10 A Model of Debt Crises IT TID FIGURE 12.6 The effects of increases in the safe interest factor R the other the only equilibrium is for the government to be unable to issue debt at any interest rate. Third, the model suggests that default, when it occurs, may always be quite unexpected. That is, it may be that for realistic cases, there is never an equilibrium value of IT that is substantial but strictly less than 1. lf there is little uncertainty about T, the revenue the government can obtain to payoff the debt, the IT = F(RD) locus has sharp bends near IT = 0 and IT = 1 like those in Figure Since the IT = (R - R)!R locus does not bend sharply, in this case the switch to the situation where default is the only equilibrium occurs at a low value of 1T. That is, there may never be a situation where investors believe the probability of default is substantial but strictly less than 1. As a result, defaults are always a surprise. The final implication is the most straightforward. Default depends not only on self-fulfilling beliefs, but also on fundamentals. In particular, an increase m the amount the government wants to borrow, an increase in the safe interest factor, and a downward shift in the distribution of potential revenue au make default more likely. Each of these developments shifts either the IT = (R - R)/ R locus down or the IT = F(RD) locus up. As a result, each development increases IT at any stable equilibrium. In addition, each development can move the economy to a situation where the only equilibrium is the one where there is no interest factor at which investors will hold the debt. Thus one message of the model is that high debt, a high required rate of return, and low future revenues au make default more likely. Multiple Periods A version of the model with multiple periods raises some interesting additional issues. For instance, suppose the government wants to issue debt for two periods. The government inherits a stock of debt in period 0, Do. Let R, denote the interest factor it pays from period 0 to period 1, and R, the interest factor from period 1 to period 2. For Simplicity, the government

7 638 Chapter 12 BUDGET DEFICITS AND FISCAL POLlCY receives tax revenue only in period 2. Thus it pays off the debt in period 2 if and only if its available revenues, T, exceed the amount due, R,R,Do. Finally, since the multiperiod version does not provide important additional insights into the possibility of multiple equilibria, assume that the equilibrium with the lowest IT (and hence the lowest R) is selected when there is more than one equilibrium. The most interesting new issues raised by the multiperiod model concern the importance of investors' beliefs, their beliefs about other investors' beliefs, and so on. The question of when investors can have heterogeneous beliefs in equilibrium is difficult and important. For this discussion, however, we simply assume that heterogeneous beliefs are possible. Consider an investor in period O. In the one-period case with the issue of multiple equilibria assumed away, the investor's beliefs about others' beliefs are irrelevant to his or her behavior. The investor holds the debt if the interest factor times his or her estimate of the probability that tax revenues will be suffident to payoff the debt is greater than or equal to the safe interest factor. But in the two-period case, the investor's willingness to bold the debt depends not only on R, and the distribution of T, but also on what R, will be. This in turn depends on what other investors will believe as of period I about the distribution of T. Suppose, for example, that for some R" the investor's own beliefs about F(.) imply that If the government offered an R, only slightly above the safe factor, the probability of default would be low, so that it would be sensible to hold the debt. Suppose, however, he or she believes that others' beliefs will make them unwilling to hold the debt from period 1 to period 2 at any interest factor. Then the investor believes the government will default in period 1. He or she therefore does not purchase the debt in period 0 despite the fact that his or her own beliefs about fundamentals suggest that the government's policy is reasonable. Even a belief that there is a small chance that in period 1 others' beliefs will make them unwilling to hold the debt at any interest rate can matter. Such a belief increases the R, that investors require to buy the debt in period O. This raises the amount of debt the government has to roll over in period 1, which reduces the chances that it will be able to do so, which raises R, further, and so on. The end result is that the government may not be able to sell its debt in period O. With more periods, even more complicated beliefs can matter. For example, if there are three periods rather than two, an investor in period 0 may be unwilling to purchase the debt because he or she believes that in period 1 others may think that in period 2 investors may believe that there is no interest factor that makes it worthwhile for them to hold the debt. This discussion implies that it is rational for investors to be concerned about others' beliefs about governments' solvency, about others' beliefs about others' beliefs, and so on. Those beliefs affect the government's ability to service its debt and thus the expected return from holding debt. An additional implication is that a change in the debt market, or even a crisis,

8 Problems 639 can be caused by information not about fundamentals, but about beliefs about fundamentals, or about beliefs about beliefs about fundamentals. Problems The stability of fiscal policy. (Blinder and Solow, 1973.) By definition, the budget deficit equals the rate of cbange of the amount of debt outstanding: 5(,) = D(t). Define d(,) to be the ratio of debt to output: d(t) = D(t)/ Y(t). Assume that Y(t) grows at a constant rate 9 > O. (a) Suppose that the deficit-to-output ratio is constant: 6(t)/Y(t) = a, where Q > O. (i) Find an expression for d(t) in tenns of a, g, and d(t). (ii) Sketch d(t) as a function of d(,). Is this system stable? (b) Suppose that the ratio of the primary deficit to output is constant and equal to a > O. Thus the total deficit at " 5(t), is given by 5(,) = ay(') + r(t)d(t). where r(t) is the interest rate at t. Assume that r is an increasing function of the debt-to-output ratio: r(,) = r(d(t)), where r '(o) > 0, r "(o) > 0, lim... _~ r(d) < g,lim... ~ r(d) > g. (i) Find an expression for d(t) in terms of a, g, and d(t). (ii) Sketch d(t) as a function of d(t). In the case where a is sufficiently small that d is negative for some values of d, what are the stability properties «?f the system? What about the case where a is sufficiently large that d is positive for all values of d? Precautionary saving. non-lump-sum taxation, and Ricardian equivalence. (Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts Yl and Y2 in the two periods. Yl is known, but Yz is random; assume for simplicity that E[Yzl = Y t. The government taxes income at rate T\ in period 1 and T Z in period 2. The individual can borrow and lend at a fixed interest rate, which for simplidty is assumed to be zero. Thus second period consumption is C2 = (1 - Tl) Yl - C1 + (1 - T2)Y2- The individual chooses C, to maximize expected lifetime utility, U(C,)+ E(U(C,»). (a) Find the first-order condition for C,. (b) Show that E(C,) = c, if Y, is not random or if utility is quadratic. (c) Show that if U III(e) > 0 and Yz is random, E(CzI > C 1. (d) Suppose that the government marginally lowers Tt and raises T2 by the same amount, so that its expected total revenue, Tj Yt + TzE[Yzl, is un changed. Implicitly differentiate the first-order condition in part (a) to find an expression for how C 1 responds to this change. (e) Show that C, is unaffected by this change if Y, is not random or if utility is quadratic.