Debt default and the insurance of labor income risk.

Athreya, Kartik B. ; Tam, Xuan S. ; Young, Eric R. 等

All of these results are obtained without recalibrating the model. To ensure that our findings are not particularly sensitive to this strategy, we also recalibrate the model for different values of p and [sigma], to the extent that this recalibration is possible; Table 1 contains the new parameter values that best fit the targets under alternative settings. By doing so, we attempt to shut off the extensive margin, although we are not completely successful. When we recalibrate, we find that with high EIS all welfare gains from eliminating default are substantially reduced, with both noncollege types now barely benefiting at all (see Table 4), while for high risk aversion the welfare gains increase slightly. As noted above, this welfare change is entirely due to the shifts in the pricing function that higher EIS and/or higher risk aversion engender. Thus, for no parameter combination that we consider do we observe welfare gains from retaining the default option.

Table 4 Welfare Gains (with Recalibration)

[sigma] =2& EIS = 0.5      College  High School  Non-High School
DM [right arrow] SM          1.21%        0.54%            0.52%

[sigma] = 2 & EIS = 0.67   College  High School  Non-High School
DM [right arrow] SM          0.28%        0.05%            0.04%

[sigma] = 5 & EIS = 0.5    College  High School  Non-High School
DM [right arrow] SM          1.28%        0.57%            0.56%

A summary of findings thus far is that default significantly worsens allocations for income risk and preference parameters that are empirically plausible for U.S. data, as well as for more extreme values of preference parameters within the class of Epstein-Zin non-expected utility preferences. We turn now to the question of whether such policies continue to remain desirable under two additional (and more substantial) departures from the settings studied so far.

Is the Standard Model Ever Worse?

We begin this section by allowing for the underlying volatility of income to be driven by relatively more and less persistent income shocks. For this experiment, we hold the unconditional variance of labor income fixed and vary the relative contributions of the persistent component e and the transitory component v. We then ask whether a relaxation in the household's understanding of the probabilistic structure of earnings risk can open the door for welfare-improving default. For this experiment, we allow for households to display ambiguity aversion in the sense of Klibanoff, Marinacci, and Mukerji (2009). (20)

The Roles of Persistent and Transitory Income Risk

It has long been known that self-insurance, and therefore also the benefit of insurance markets, hinges critically on the persistence of the risks facing households. As a general rule, the more persistent are shocks, the more difficult they are to deal with via the accumulation of assets in good times and decumulation and borrowing in bad times. In contrast, purely transitory income shocks can typically be smoothed effectively. In a pure life-cycle model, however, there are additional impediments to self-insurance: Young households are born with no wealth and often face incentives to borrow arising from purely intertemporal considerations. In particular, those with relatively high levels of human capital, especially the college-educated, can expect age-earnings profiles with a significant upward slope into late middle age. As a result, such households would like to borrow even in the absence of any shocks to income, often substantially, against their growing expected future income. In contrast, those households with low human capital face a far less income-rich future, and as a result borrow primarily to deal with transitory income risk.

In order to understand the role that the persistence of income risk plays in the welfare gains or losses arising from U.S.-style bankruptcy and delinquency, we now evaluate the effects of changes in the persistent component of household income risk for all three classes of households. However, in order to avoid conflating persistence and overall income volatility, we adjust the variance of transitory income volatility such that the overall variance of log labor income remains constant. (21) Figure 10 and Tables 5 and 6 present the welfare and consumption smoothing implications of the standard model under varying income shock persistence. The first column of each table documents the fraction of total variance contributed by the persistent component.

[FIGURE 10 OMITTED]

Table 5 Consumption Smoothing (DM)
                Intra                   Inter                   Total

       Coll        HS     NHS    Coll      HS     NHS    Coll      HS

1.0%   0.0306  0.0462  0.0575  0.0359  0.0364  0.0386  0.0665  0.0826

10.0%  0.0377  0.0561  0.0872  0.0343  0.0367  0.0357  0.0720  0.0938

20.0%  0.0459  0.0807  0.1092  0.0336  0.0347  0.0325  0.0795  0.1154

30.0%  0.0538  0.0884  0.1367  0.0327  0.0327  0.0297  0.0865  0.1211

40.0%  0.0619  0.1013  0.1472  0.0316  0.0301  0.0280  0.0925  0.1314

50.0%  0.0700  0.1146  0.1613  0.0305  0.0284  0.0263  0.1005  0.1430

60.0%  0.0779  0.1280  0.1797  0.0294  0.0264  0.0241  0.1065  0.1544

70.0%  0.0859  0.1413  0.1992  0.0283  0.0247  0.0224  0.1141  0.1660

80.0%  0.0946  0.1543  0.2182  0.0272  0.0231  0.0211  0.1218  0.1774

90.0%  0.1053  0.1681  0.2368  0.0258  0.0212  0.0199  0.1311  0.1893

99.0%  0.1248  0.1863  0.2566  0.0235  0.0187  0.0180  0.1483  0.2050

       Coll       NHS

1.0%   0.0306  0.0961

10.0%  0.0377   01229

20.0%  0.0459  0.1417

30.0%  0.0538  0.1664

40.0%  0.0619  0.1752

50.0%  0.0700  0.1876

60.0%  0.0779  0.2038

70.0%  0.0859  0.2216

80.0%  0.0946  0.2393

90.0%  0.1053  0.2567

99.0%  0.1248  0.2680

Table 6 Consumption Smoothing (SM)

                Intra                   Inter                   Total
         Coll      HS     NHS    Coll      HS     NHS    Coll      HS

1.0%   0.0196  0.0307  0.0474  0.0318  0.0314  0.0120  0.0514  0.0621

10.0%  0.0271  0.0397  0.0577  0.0315  0.0298  0.0124  0.0586  0.0695

20.0%  0.0360  0.0541  0.0771  0.0311  0.0290  0.0131  0.0671  0.0831

30.0%  0.0444  0.0683  0.0971  0.0306  0.0284  0.0137  0.0750  0.0967

40.0%  0.0524  0.0820  0.1173  0.0300  0.0277  0.0144  0.0824  0.1097

50.0%  0.0600  0.0951  0.1364  0.0295  0.0271  0.0151  0.0895  0.1222

60.0%  0.0673  0.1076  0.1550  0.0291  0.0267  0.0158  0.0964  0.1343

70.0%  0.0743  0.1197  0.1729  0.0288  0.0262  0.0164  0.1031  0.1495

80.0%  0.0811  0.1314  0.1903  0.0285  0.0258  0.0171  0.1096  0.1627

90.0%  0.0878  0.1428  0.2072  0.0282  0.0255  0.0178  0.1160  0.1638

99.0%  0.0935  0.1528  0.2218  0.0280  0.0253  0.0182  0.1215  0.1781

         Coll     NHS

1.0%   0.0196  0.0594

10.0%  0.0271  0.0801

20.0%  0.0360  0.0902

30.0%  0.0444  0.1108

40.0%  0.0524  0.1317

50.0%  0.0600  0.1515

60.0%  0.0673  0.1708

70.0%  0.0743  0.1893

80.0%  0.0811  0.2075

90.0%  0.0878  0.2250

99.0%  0.0935  0.2400

Normatively, three findings are noteworthy. First, and perhaps most importantly, the standard model displays higher welfare irrespective of the nature of shocks accounting for observed income volatility. This result strengthens our findings thus far, and it further suggests that defaultable debt is simply unlikely to be useful to households. It is also a particularly important form of robustness, given both the general importance of persistence for the efficacy of self-insurance and borrowing and because estimates of income shock persistence vary dramatically--see Guvenen (2007), Hryshko (2008), or Guvenen and Smith (2009) for discussions of the debate between so-called "restricted income profiles" (RIP), in which all households draw earnings from a single stochastic process, and "heterogeneous income profiles" (HIP), in which households vary in the processes from which they derive earnings. This debate has implications for models like ours because these two models differ, sometimes strongly, in the persistence of earnings shocks their structure implies. Most recent work now suggests that income-process parameters vary over the life cycle as well (Karahan and Ozkan 2009).

Second, the effect of the contribution of persistent shocks to income volatility depends on the education level of households. In particular, when volatility is driven primarily by persistent shocks, the relatively well-educated benefit from the elimination of default substantially more than their less-educated counterparts. Conversely, when most income variability is driven by large but transitory shocks, it is the relatively less-educated who benefit most from the elimination of the default option. The intuition for this result comes from the nature of borrowing: College types borrow primarily to use future expected income today while less-educated types borrow to smooth shocks

Third, within each educational class, the welfare losses from default decline monotonically as the relative contribution of the persistence of the shock grows; default on debt is least (most) useful when income volatility is driven primarily by shocks that are transitory (persistent). What is surprising, but in keeping with the main theme of our results, is that in no case is it true that U.S.-style default is ex ante more desirable than allocations obtaining under the standard model. Moreover, even in the case where essentially all income risk is delivered in the form of persistent shocks where credit markets are least useful in dealing with income risk, outcomes that allow for default are worse for agents than those arising in the standard model. The welfare in the standard model is non-trivially higher, at up to 1.24 percent of consumption for college-educated households (as seen in Figure 10).

In Figures 11 and 12 we display the measure of borrowers at each age and the conditional mean of debt among those who borrow for two levels of the importance of persistent income risk.22 The fact that the losses from allowing default rise for all agent types with the importance of transitory shocks is a consequence of the increased usefulness of credit in dealing with transitory income risk. Conversely, when shocks are primarily persistent, a negative realization requires more frequent borrowing and leads, all else equal, to more debt in middle age; the combination is ultimately unable to stern the transfer of income risk to consumption volatility. In Tables 5 and 6, we see that, irrespective of default policy, persistence translates into higher consumption volatility, and that the presence of lax default policy seen in Table 6 does little to stem the flow of income risk into consumption risk (echoing our previous result in Athreya, Tam, and Young [2009]).

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

We turn next to the relationship between shock persistence and equilibrium default rates, displayed in Figure 13. Default is "U-shaped," with high default rates at both ends. To understand this shape, consider first the case where the labor income shocks are nearly all transitory (the left side of the graph). Here, agents can generally manage their risk effectively via saving and dissaving, but they choose to augment the self-insurance mechanism with default at higher rates than they do in the benchmark setting. The reason they do so is that risk-based pricing is not effective here, because there is no useful information contained in the current labor income of the borrower that would identify future bad risks. In contrast, in the case where labor income is driven entirely by the persistent component (the right side of the graph), high default is the result of agents being generally unable to smooth consumption; persistent shocks are hard to smooth using assets alone (and if permanent are in fact impossible). As a result, despite the pricing effects, borrowers will use default relatively often (and pay the costs to do so). The middle parts of the graph, where default is lowest, balance these two effects.

[FIGURE 13 OMITTED]

Intuitively, in the standard model, borrowers realize that debt must be repaid, and under high persistence, heavy borrowing in response to a negative shock makes low future consumption relatively likely. Nonetheless, credit markets are willing to lend to such households at the risk-free rate (adjusted for any transactions costs of intermediation), making total debt rise. When default is available, borrowing today to deal with persistent income risk does not expose the borrower to severe consumption risk in the long term as default offers an "escape valve," but it does expose lenders to severe credit risk in the near term. Creditors then price debt accordingly; as seen in Figure 14, when shocks are primarily persistent, as the current shock deteriorates so do the terms at which borrowers can access credit. Moreover, under a bad current realization of income, households facing persistent risk see a disproportionate decline in the price of any debt they may issue, while the reverse occurs in the event of a good current realization of income; the pricing functions essentially "switch places."

[FIGURE 14 OMITTED]

Yet, despite the increased sensitivity of loan pricing to the borrower's current income state under relatively high persistence, the welfare gains under the SM, though still positive, fall. This result obtains because of the reduction in the ability of self-insurance, inclusive of borrowing, to prevent income fluctuations from affecting consumption. To sum up, income risk is quantitatively relevant in governing the losses conferred by default, but irrelevant for altering the qualitative welfare property that, in the absence of expense shocks, the default option lowers welfare.

Ambiguity Aversion

We turn next to the question of whether default can improve outcomes when households are not perfectly certain about the probabilistic structure of income risk. Households that face ambiguity are uncertain about the probability process for their incomes; if ambiguity-averse, these households behave pessimistically and therefore adopt views about their income that would, for example, imply that it would mean-revert more slowly from low realizations. In such a situation, borrowing to smooth away temporary falls may not be optimal, since asset decumulation is not effective against permanent shocks, and therefore in the absence of a default option households may be unwilling to do so. In contrast, if default is an option, the household may be willing to borrow since, even if their pessimism is validated, consumption can be protected via discharge. We formalize this idea, as in Klibanoff, Marinacci, and Mukerji (2009), by assuming agents are averse to ambiguity. In this formulation, a household of age j solves the dynamic programming problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

EU = [[SIGMA].sub.e', v', [lambda]'] [[pi].sub.e](e'|e) [[pi].sub.v] (v') [[pi].sub.[lambda]] ([lambda]' | [lambda]) x V (b, y, e', v', [lambda]', j + 1) (10)

subject to budget constraints, (1) and (3), where ([empty set])(*) is given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

determines preferences over ambiguity. [eta] [greater than or equal to] 0 controls the attitude toward ambiguity; as [eta] increases, the household becomes more averse to ambiguous stochastic processes. The restrictions on the choices of p (e', v' | e, v) are that they must sum to 1 for each (e, v) and every element must lie in some set P [subset] [0, 1]; we nest the standard model by setting the P to be an arbitrarily small interval around the objective probabilities. (23) We use to denote objective probabilities and p to denote subjective ones; note that households are assumed to be uncertain only about the distribution of income shocks, not the process for [lambda].

Because we are interested in these preferences only to the extent that they may provide an environment in which relatively low-cost default and debt discharge are welfare-enhancing, we will deliberately take the most extreme case of [eta] = [infinity], yielding the max-min specification from Epstein and Schneider (2003):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

EU = [[SIGMA].sub.e', v', [lambda]'] p (e', v' | e, v) x [[pi].sub.[lambda]] V (b, y, e', v', [lambda]', j + 1)

V (b, y, e', v', A', j + i) = (i - d (e', v', [lambda]')) v (b, y, e', v', [lambda]', j + i) + d(e', v', [lambda]') [v.sup.D] (0, y, e', v', [lambda]', j + 1), (11)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

EU = [[SIGMA].sub.e', v', [lambda]'] p (e', v' | e, v) x [[pi].sub.[lambda]] ([lambda]' | [lambda])v (0, y, e', v', [lambda]', j + 1) (12)

is the value of default.

The min operator that appears in front of the summation reflects the agent's aversion to uncertainty; as shown by Epstein and Schneider (2003), a household who is infinitely uncertainty-averse chooses the subjective distribution of future events that is least favorable and then makes their decisions based on that subjective distribution. The size of the set of possible processes P measures the amount of ambiguity agents face; a typical [p.sub.ij] element lies in the interval [[p.sub.1.sup.ij], [p.sub.2.sup.ij]] [subset] [0, 4] (24)

Standard ambiguity aversion models imply that households will learn over time and reject stochastic processes that are inconsistent with observed data (for example, a household who initially entertains the possibility of permanently receiving the worst possible income level forever will dismiss this process as soon as one non-worst realization occurs). For simplicity, we will focus our attention on a special case of extreme ambiguity aversion in which this learning does not occur; if default is not useful in this environment, it is likely of less use to households than when they face less uncertainty over time. The intuition is that the income process we buffet agents with is a non-unit process. To the extent that households would realize by a certain age that the data they've received makes unit-root earnings unlikely, they would be able to rule out such a persistent process and thereby smooth more effectively, and as a result, may not value default as much as someone viewing shocks as permanent.

Given the qualifications and considerations discussed above, we now evaluate outcomes in the standard model in the case where P = [0, 1], the most extreme case possible (households behave as if the minimum income draw will be realized with probability 1 next period). The intuition is that such a case offers the possibility, discussed at the outset, that lax penalties for default might actually encourage the use of credit for consumption in a setting where the agent's aversion to ambiguity would otherwise preclude becoming indebted. And in fact, we do find that this case delivers default as welfare-improving for some agents (see Table 7). However, this finding is very limited: Benchmark default costs improve welfare for only the college type and the welfare gain is tiny (under 0.2 percent of consumption). As a result, unconditional ex ante welfare is negative since college types are not a large enough group to overcome the losses to the remainder of the population. It is interesting to see, however, that the welfare changes from allowing default are now reversed--the largest gains are experienced by the most educated, while the least educated suffer more. Part of the intuition for this result is that it is the best educated who face the steepest mean age-earnings profiles. Therefore, these agents would have the strongest purely intertemporal motives to borrow, absent any ambiguity. Low default costs mitigate the effect of ambiguity and allow for states in which a temporarily unlucky college-educated agent would find borrowing desirable.

Table 7 Welfare Effects Under Ambiguity Aversion

P = [0, 1]              Non-High School  High School   College
DM [right arrow] SM              0.215%       0.189%   -0.185%

P = min(1, [pi] + 0.5)  Non-High School  High School   College
DM [right arrow] SM              0.296%       0.219%    0.044%

Pricing is presented in Figures 15 and 16. Notice that for the low realization of e, the pricing function under ambiguity aversion is everywhere below the baseline expected utility case, but for the higher realization they switch places; ambiguity-averse agents with high income actually pose less of a default risk. The difference in pricing stems only from a difference in the households' willingness to default next period for a given b. Since default has a fixed cost component ([DELTA]), households want to time their usage of default; in particular, households must balance the gains from defaulting tomorrow from those arising from waiting until additional shocks have been realized. This fact places the expectations of income in periods after tomorrow at the heart of the timing of default decisions, and here households who face ambiguity about the income process act quite differently from those in the benchmark economy. (25)

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

Take first the household with low e. For a "rational expectations" household, income in the distant future is expected to be better than whatever is realized tomorrow, as e is persistent but mean-reverting; for the household facing ambiguity, however, income is actually expected to be no better, or even worse, than tomorrow's realization. Since ambiguity-averse households do not think the future will be better, they may as well default next period if the realization of income is bad; lenders must therefore offer them higher interest rates to break even. In contrast, the ambiguity-averse household with higher e views a realization near the mean for next period as unexpectedly good, but does not expect better times in the more-distant future. Default in the next period is therefore not as valuable as waiting for a future period when those bad states are expected to occur. In contrast, without ambiguity a bad realization will induce the household to substantially revise their future expectations downward, making default today more attractive (the decline in future income makes the fixed cost of default worth paying). (26) The result is that ambiguity-averse households with high current income obtain better terms.

Is such extreme ambiguity aversion "reasonable?" It seems highly unlikely that households entertain a stochastic process in which they receive the worst possible outcome forever with probability one as reasonable, at least not for long--after all, they need only observe the fact that their income is occasionally higher than the lower bound to discard this process empirically. As we noted above, we could introduce this learning into the model--since the households are simply learning about an exogenous process, it can be done "offline"--but it is computationally quite burdensome to condition the set of permissive stochastic processes on the history of observations. (27) It is also the case that this extreme ambiguity leads to a discrepancy between model and data in terms of borrowing patterns; there is far too little debt, which lessens our interest in making this economy "more realistic." If we consider smaller limits for P, such as 10 percent above or below the objective value, we find that default is welfare-reducing for all education levels. Thus, while ambiguity aversion provides a theoretical foundation for default options, it does not appear to provide an empirically tenable one.

3. CONCLUDING REMARKS

We have studied the efficacy of default in helping households better insure labor income risk in a large range of settings in which risk aversion, intertemporal smoothing motives, income risk, and uncertainty--and attitudes to uncertainty--over income risk itself were all varied. Our findings here suggest that within the broad class of models used thus far to develop quantitative theory for unsecured consumer credit and default, relatively generous U.S.-style default does not appear to be capable of providing protection against labor income risk.

Despite the fact that we find that labor income risk is not well hedged from the ex ante perspective, we also show that there are ex post beneficiaries from allowing default as it currently is; specifically, we show that the standard model generates a positive measure of agents ex post who would vote to introduce default. Our calibrated model predicts that these agents do not constitute a majority, though, since they are primarily college-educated middle-aged households who have been unlucky enough to still have significant debt. This result warrants further investigation since it may help explain why default penalties are becoming less stringent over time (with the exception of some aspects of the most recent reform).

Our results also suggest that "expense" shocks or catastrophic movements in net worth are likely to be essential to justify the view of default as a welfare-improving social institution. To the extent that uninsured, catastrophically large, and "involuntary" expenditures are indeed a feature of the data, a natural question is whether consumer default is the best way to deal with such events. Given the nature of resource transfers created by default and the constraints that it imposes on the young, who disproportionately account for both the income-poor and uninsured, this statement seems unlikely.

With respect to future work, it is worth stressing that since expense shocks and their absence seem so important to the implications of the class of models considered here, the value of purely empirical work better documenting the nature of expense shocks, and their (a priori plausible) connection to income shocks (for example, job loss leading to insurance loss, which in turn exposes households to out of pocket expenditures), is high. Relatedly, the pivotal role played by borrowing costs "moving against" unlucky borrowers seems important to independently substantiate. In the absence of such work, it remains a possibility that the welfare findings of this article (and essentially all others) hinges too much on an institutional arrangement for borrowing that is inaccurate. Use of detailed household level credit card pricing and income information seems productive.

In addition to the preceding, in light of the findings of this article and the larger quantitative theory of consumer default, two directions seem particularly useful. First, a more "normative" approach that asks if observed default procedures can arise an optimal arrangement under plausible frictions, may yield different conclusions. One interesting example of the latter approach is the theoretical work of Grochulski (2010), where default is shown to be one method for decentralizing a constrained Pareto optimum in the presence of private information. Quantifying models with default and endogenously derived asset market structures may lead to better understanding of policy choices in this area (such as why Europe has chosen to make default available under very strict conditions, and social insurance generous, while the United States has chosen the opposite).

Second, with respect to the experiments we studied, we were led to allow for two specific preference extensions beyond CRRA expected utility in order to accurately assess the particular tradeoffs created by default. While we emphatically did not attempt to turn the article into a survey of any larger variety of non-expected utility preferences, some further extensions seem potentially important: disappointment aversion (Gul [1991] or Routledge and Zin [2010]), deviations from geometric discounting (Laibson 1997), habit formation (Constantinides 1990), and loss aversion (Barberis, Huang, and Santos 2001). Why these preferences specifically? In each case, the more general preference structure breaks the link between risk aversion and intertemporal substitution (and generally makes risk aversion state-dependent), and some (such as nongeometric discounting and loss aversion) provide arguments for government intervention; there is also extensive empirical work supporting many of them. A recent contribution to this literature is Nakajima (2012), who investigates whether the temptation preferences of Gul and Pesendorfer (2001) alter the consequences of default reform. (28) We suspect other work will follow.

APPENDIX: COMPUTATIONAL CONSIDERATIONS

We make some brief points here regarding the computation of the model. The model is burdensome to calibrate, and all programs are implemented using Fortran95 with OpenMP messaging.

In all the models we study, the objective function (the right-hand side of the Bellman equation) is not globally concave, since the discrete nature of the bankruptcy decision introduces convex segments around the point where the default option is exercised (we find that, as in Chatterjee et al. [2007], the default decision encompasses an interval and in our case it extends to b = - [infinity] as [delta] is smaller than even the worst income realization). The nonconcavity poses a problem for local optimization routines, so we approach it using a global strategy. We use linear splines to extend the value function to the real line and a golden section search to find the optimum, with some adjustments to guarantee that we bracket the global solution rather than the local one. It is straightforward to detect whether we have converged to the local maximum at any point in the state space, as the resulting price function will typically have an upward jump.

For the ambiguity aversion case we have a saddlepoint problem to solve. By the saddlepoint theorem we can do the maximization and minimization in any order; the minimization (conditional on b and d) is a linear program that we solve using a standard simplex method conditional on some b (as in Routledge and Zin [2009]). We then nest this minimization within our golden section search, again with adjustments to deal with the presence of the local maximum. For our model, this linear program turns out to be extremely simple to solve--the household puts as much weight as allowed on the worst possible outcome, then as much weight as allowed on the next worst, and so on.

To impose boundedness on the realizations of income, we approximate both e and v by Markov chains using the approach in Floden (2008). Having income be bounded above is convenient since it implies that there always exists a cost of default [delta] such that bankruptcy is completely eliminated because it becomes infeasible. Quite naturally, bankruptcy is also likely not to occur when [delta] is high enough even if filing is feasible for some types; in general, households with high income are not interested in the default option in our mode1. (29)

Figure 17 shows a typical objective function for a household in our benchmark case (expected utility with [sigma] = [p.sup.-1] = 2). The objective function has three distinct segments. The first segment is at the far right, where the values for both the low- and high-cost types coincide. In this region, default is suboptimal because borrowing either does not or barely exceeds [delta]. The second segment is at the other end, where q (b) = 0; although impossible to see in the picture, the low-cost default experiences slightly more utility in this region since default is less painful. The action is all in the middle segment. For this particular individual, the high-cost type ([[lambda].sub.L]) borrows significantly more than the low-cost type; this extra borrowing reflects primarily the pricing function (as seen in the lower panel) and not any particular desire to borrow. High-cost types have more implicit collateral and are less likely to default at any given debt level, so they face lower interest rates. As a result, high-type borrowers today who become low-type borrowers tomorrow are a main source of default in our model--they both have debts and are not particularly averse to disposing of those debts through the legal system. Since type is persistent, low-type borrowers today will not generally make the same choice--the supply side of their credit market will contract.

[FIGURE 17 OMITTED]

Athreya is an economist at the Richmond Fed; Tam is affiliated with the University of Cambridge; Young is an economist at the University of Virginia. This article previously circulated under the title "Are Harsh Punishments for Default Really Better?" We would like to thank seminar and conference participants at UT-Austin, the Board of Governors, the Cleveland Fed, Georgetown University, the Philadelphia Fed, and Queen's University for comments on the earlier versions. We thank the EQ committee, especially Huberto Ennis, for detailed comments. Tam thanks the John Olin Foundation for financial support. The opinions expressed here do not reflect those of the Federal Reserve System or the Federal Reserve Bank of Richmond. All errors are the responsibility of the authors. E-mail: [email protected].

REFERENCES

Athreya, Kartik. 2002. "Welfare Implications of the Bankruptcy Reform Act of 1999." Journal of Monetary Economics 49 (November): 1,567-95.

Athreya, Kartik. 2004. "Shame As It Ever Was: Stigma and Personal Bankruptcy." Federal Reserve Bank of Richmond Economic Quarterly 90 (Spring): 1-19.

Athreya, Kartik. 2008. "Default, Insurance, and Debt over the Life-Cycle." Journal of Monetary Economics 55 (May): 752-74.

Athreya, Kartik, Juan M. Sanchez, Xuan Tam, and Eric R. Young. 2013. "Bankruptcy and Delinquency in a Model of Unsecured Debt." Federal Reserve Bank of St. Louis Working Paper 13-2.

Athreya, Kartik, Xuan S. Tam, and Eric R. Young. 2009. "Unsecured Credit Markets Are Not Insurance Markets." Journal of Monetary Economics 56 (January): 83-103.

Athreya, Kartik, Xuan S. Tam, and Eric R. Young. 2012. "A Quantitative Theory of Information and Unsecured Credit." American Economic Journal: Macroeconomics 4 (July): 153-83.

Ausubel, Lawrence M., and Amanda E. Dawsey. 2008. "Penalty Interest Rates, Universal Default, and the Common Pool Problem of Credit Card Debt." Unpublished manuscript, University of Maryland and University of Montana.

Barberis, Nicholas, Ming Huang, and Tano Santos. 2001. "Prospect Theory and Asset Prices." Quarterly Journal of Economics 116 (February): 1-53.

Bewley, Truman F. 2002. "Knightian Decision Theory, Part I." Decisions in Economics and Finance 25 (November): 79-110.

Campanale, Claudio. 2008. "Learning, Ambiguity, and Life-Cycle Portfolio Allocation." Mimeo, Universidad de Alicante.

Carroll, Christopher D. 1997. "Buffer-Stock Saving and the Life Cycle/Permanent Income Hypothesis." The Quarterly Journal of Economics 112 (February): 1-55.

Chatterjee, Satyajit, P. Dean Corbae, Makoto Nakajima, and Jose-Victor Rios-Rull. 2007. "A Quantitative Theory of Unsecured Consumer Credit with Risk of Default." Econometrica 75 (November): 1,525-90.

Constantinides, George M. 1990. "Habit Formation: A Resolution of the Equity Premium Puzzle." Journal of Political Economy 98 (June): 519-43.

Deaton, Angus. 1992. Understanding Consumption. New York: Oxford University Press.

Dubey, Pradeep, John Geanakoplos, and Martin Shubik. 2005. "Default and Punishment in General Equilibrium." Econometrica 73 (January): 1-37.

Epstein, Larry G., and Martin Schneider. 2003. "Recursive Multiple-Priors." Journal of Economic Theory 113 (November): 1-31.

Epstein, Larry G., and Stanley E. Zin. 1989. "Substitution, Risk Aversion, and the Temporal Behavior of Consumption Growth and Asset Returns I: A Theoretical Framework." Econometrica 57 (July): 937-69.

Evans, David S., and Richard L. Schmalensee. 2005. Paying with Plastic: The Digital Revolution in Buying and Borrowing. Cambridge, Mass.: MIT Press.

Fay, Scott A., Erik Hurst, and Michelle J. White. 1998. "The Bankruptcy Decision: Does Stigma Matter?" University of Michigan Working Paper 98-01 (January).

Floden, Martin. 2008. "A Note on the Accuracy of Markov-Chain Approximations to Highly Persistent AR(1) Processes." Economics Letters 99 (June): 516-20.

Friedman, Milton. 1957. A Theory of the Consumption Function. Princeton, N. J.: Princeton University Press.

Furletti, Mark. 2003. "Credit Card Pricing Developments and Their Disclosure." Federal Reserve Bank of Philadelphia Payment Cards Center Discussion Paper 03-02 (January).

Grochulski, Borys. 2010. "Optimal Personal Bankruptcy Design under Moral Hazard." Review of Economic Dynamics 13 (April): 350-78.

Gross, David B., and Nicholas S. Souleles. 2002. "An Empirical Analysis of Personal Bankruptcy and Delinquency." Review of Financial Studies 15 (1): 319-47.

Gul, Faruk. 1991. "A Theory of Disappointment Aversion." Econometrica 59 (May): 667-86.

Gul, Faruk, and Wolfgang Pesendorfer. 2001. "Temptation and Self-Control." Econometrica 69 (November): 1,403-35.

Guvenen, Fatih. 2007. "Learning Your Earning: Are Labor Income Shocks Really Very Persistent?" American Economic Review 97 (June): 687-712.

Guvenen, Fatih, and Anthony A. Smith, Jr. 2009. "Inferring Labor Income Risk from Economic Choices: An Indirect Inference Approach." Mimeo, University of Minnesota and Yale University.

Hansen, Lars Peter, and Thomas J. Sargent. 2007. Robustness. Princeton, N. J.: Princeton University Press.

Hryshko, Dmytro. 2008. "RIP to HIP: The Data Reject Heterogeneous Labor Income Profiles." Mimeo, University of Alberta.

Hubbard, R. Glenn, Jonathan Skinner, and Stephen P. Zeldes. 1994. "The Importance of Precautionary Motives in Explaining Individual and Aggregate Saving." Carnegie-Rochester Conference Series on Public Policy 40 (June): 59-126.

Huggett, Mark, Gustavo Ventura, and Amir Yaron. 2011. "Sources of Lifetime Inequality." American Economic Review 101 (December): 2,923-54.

Jackson, Thomas H. 2001. The Logic and Limits of Bankruptcy Law. Hopkins, Minn.: Beard Books.

Karahan, Fatih, and Serdar Ozkan. 2009. "On the Persistence of Income Shocks over the Life Cycle: Evidence and Implications." Penn Institute for Economic Research PIER Working Paper 09-05 (December).

Klibanoff, Peter, Massimo Marinacci, and Sujoy Mukerji. 2009. "Recursive Smooth Ambiguity Preferences." Journal of Economic Theory 144 (May): 930-76.

Krueger, Dirk, and Harald Uhlig. 2006. "Competitive Risk Sharing Contracts with One-Sided Commitment." Journal of Monetary Economics 53 (October): 1,661-91.

Laibson, David A. 1997. "Golden Eggs and Hyperbolic Discounting." Quarterly Journal of Economics 112 (May): 443-77.

Li, Wenli, and Pierre-Daniel Sarte. 2006. "U.S. Consumer Bankruptcy Choice: The Importance of General Equilibrium Effects." Journal of Monetary Economics 53 (April): 613-31.

Livshits, Igor, James MacGee, and Michele Tertilt. 2011. "Costly Contracts and Consumer Credit." Working Paper 17448. Cambridge, Mass.: National Bureau of Economic Research (September).

Ljungqvist, Lars, and Thomas J. Sargent. 2004. Recursive Macroeconomic Theory. Cambridge, Mass.: MIT Press.

Livshits, Igor, James MacGee, and Michele Tertilt. 2007. "Consumer Bankruptcy: A Fresh Start." American Economic Review 97 (March): 402-18.

Livshits, Igor, James MacGee, and Michele Tertilt. 2008. "Costly Contracts and Consumer Credit." Mimeo, University of Western Ontario and Stanford University.

Lucas, Robert E., Jr. 1987. Models of Business Cycles. Oxford: Basil-Blackwell Ltd.

Mateos-Planas, Xavier, and Giulio Seccia. 2006. "Welfare Implications of Endogenous Credit Limits with Bankruptcy." Journal of Economic Dynamics and Control 30 (November): 2,081-115.

Miao, Jianjun, and Neng Wang. 2009. "Risk, Uncertainty, and Option Exercise." Manuscript, Boston University and Columbia University.

Nakajima, Makoto. 2012. "Rising Indebtedness and Temptation: A Welfare Analysis." Quantitative Economics 3 (July): 257-88.

Obiols-Homs, Francesc. 2011. "On Borrowing Limits and Welfare." Review of Economic Dynamics 14 (April): 279-94.

Pemberton, James. 1998. "Income Catastrophes and Precautionary Saving." Mimeo, University of Reading.

Routledge, Bryan R., and Stanley E. Zin. 2009. "Model Uncertainty and Liquidity." Review of Economic Dynamics 12 (October): 543-66.

Routledge, Bryan R., and Stanley E. Zin. 2010. "Generalized Disappointment Aversion and Asset Prices." Journal of Finance 65 (August): 1,303-32.

Sullivan, Teresa A., Elizabeth Warren, and Jay Lawrence Westbrook. 2000. The Fragile Middle Class: Americans in Debt. New Haven, Conn.: Yale University Press.

Tam, Xuan S. 2009. "Long-Term Contracts in Unsecured Credit Markets." Mimeo, University of Virginia.

(1.) See, e.g., Ljungqvist and Sargent (2004, p. 577)

(2.) Miao and Wang (2009) study the decision to exercise an option under ambiguity. Due to the presence of fixed costs, bankruptcy has option value. We focus on a related setting but are interested in the quantitative aspects associated with household consumption smoothing.

(3.) These preferences are a special case of the more general ambiguity-averse preferences axiomatized by Klibanoff, Marinacci, and Mukerji (2009).

(4.) Denoting by [y.sub.min] > 0 the lowest realization of potential labor income and r the risk-free interest rate on debt, the natural borrowing limit for an infinitely lived agent is given by [[b.bar].sub.nat] [equivalent to] [y.sub.min]/r, a function that asymptotes to--(infinity) as interest rates go to zero. Assuming a credit card interest rate of 14 percent (the modal interest rate in Survey of Consumer Finances data in 1983 adjusted for a measure of realized inflation), the natural debt limit moves roughly seven times as much as the minimum income level. For good borrowers, for whom interest rate discounts have recently appeared (Furletti 2003; Livshits, MacGee, and Tertilt 2008), the natural debt limit will be even more sensitive.

(5.) In our previous work we introduce a class of "special" agents who hold large amounts of capital for the purpose of endogenously obtaining a low, risk-free rate in the presence of low asset holdings for the median agent. Here we ignore the general equilibrium determination of returns and thus drop the special households from the model because their presence is irrelevant to the question at hand.

(6.) We approximate both e and v with finite-state Markov chains. This approximation has the convenient property that income is bounded.

(7.) See Sullivan, Warren, and Westbrook (2000).

(8.) That is, exclusion from credit markets beyond the initial period is not sustainable as a punishment.

(9.) We assume any savings of households who die is taxed at 100 percent and used to fund wasteful government spending.

(10.) Chatterjee et al. (2007) calibrate their model to match the wealth distribution in the United States in a dynastic setting. As we have argued, life-cycle considerations are important for assessing the welfare effects of bankruptcy.

(11.) The average interest rate on credit card balances is high--currently 14 percent--relative to more secured forms of debt. As Evans and Schmalensee (2005) have pointed out, however, it is straightforward to account for the interest rate after funding costs, transactions costs, and, most crucially, default costs are taken into account, without relying on market power distortions.

(12.) Most dynamic contracting models of limited borrower commitment, for example, currently use implicit or explicit appeals to public institutions with commitment to punish, in order to motivate penalties for the value of autarky. In recent work, Krueger and Uhlig (2006) show that the inability of the supply side of the credit market to commit to punishments can have severe implications for the existence of the market itself. In the "normal" case, Krueger and Uhlig (2006) show that competition in fact collapses credit and insurance markets completely even without informational frictions.

(13.) We want to be clear that what we call "penalties" differs from the usage in Ausubel and Dawsey (2008), where rates imposed after late or missed payments are labeled punitive. They attribute the high values of such rates to a common agency problem. Modeling the bilateral contracting problem that would arise in the presence of noncompetitive intermediation is well beyond the goals for this article. We are pursuing the endogenous determination of interest rate hikes for delinquent borrowers in other work.

(14.) Similar results would obtain if the government could impose "shame" on households by choosing values for [lambda], provided it could make [lambda] large enough to guarantee zero default on the equilibrium path. In our model, the Inada condition on consumption implies that such a [lambda] always exists.

(15.) Specifically, we set v = 0.35, Y = 0.2, [empty set] = 0.03, [DELTA] = 0.03, [xi] = 0.95, [[sigma].sub.n, [member of].sup.2] = 0.033, [[sigma].sub.n, v.sup.2] = 0.04, [[sigma].sub.h, [member of].sup.2] = 0.025, [[sigma].sub.h, v.sup.2] = 0.021, [[sigma].sub.c, [member of].sup.2] = 0.016, and [[sigma].sub.c, v.sup.2] = 0.014.

(16.) Consider an attempt to improve the model's prediction for the measure of borrowers by increasing [beta]. Holding all other parameters constant, this reduces default rates and debt-to-income ratios for all types (and these variables are generally already too small). To counteract this effect, one might then move [lambda] for each type and each state. Consider first increasing both [[lambda].sub.i.sup.H] and [[lambda].sub.i.sup.L] for one type i. While this change would increase the default rate--default becomes less costly--it would via a supply-side effect tend to reduce debt levels (see Athreya [2004]). By contrast, suppose we increase [[lambda].sub.i.sup.H] and decrease [[lambda].sub.i.sup.L]; this change has countervailing effects on both default rates and debt levels and default rates could rise because it becomes cheaper for H types, but fall as it becomes more expensive for L types. A similar tension exists for debt-to-income ratios--driving it up for one type tends to drive it down for the other.

(17.) In the real world, "stigma" may also be a function of aggregate default rates (an agent cares less about default if everyone else is defaulting), in which case this invariance may break. To analyze this case would be of interest, but it poses some challenges with respect to calibration. We therefore defer it to future work.

(18.) The figures are drawn for the aggregate, since the results are the same for each type qualitatively. Figures decomposed by type are available from the authors upon request.

(19.) Our model satisfies the conditions noted in Chatterjee et al. (2007) that imply default occurs only if current debt cannot be rolled over: If d ([member of]', v', [lambda]') > 0 for some [member of]', v', [lambda]', then there does not exist b such that a + y - q (b, Y)b > 0 for total income Y.

(20.) There are connections between ambiguity aversion and the concept of Knightian uncertainty from Bewley (2002), although the latter concept does not permit preferences to be represented by a utility function and is therefore hard to analyze quantitatively. There are also connections between ambiguity aversion and robust decisionmaking as defined by Hansen and Sargent (2007).

(21.) Athreya, Tam, and Young (2009) are primarily concerned with the role of income variance in models of default.

(22). From the perspective of a newborn, the measure of borrowers of a given age equals the probability of the newborn borrowing at that age.

(23.) We do not require that the household assume that the probabilities of the independent events are independent in every distribution that is considered. That is, the household may be concerned that the independence property is misspecified and therefore select a worst-case distribution in which the events are correlated.

(24.) Hansen and Sargent (2007) provide an interpretation of P in terms of detection probabilities.

(25.) The exposition is simpler if we refer to the expectations of the households facing ambiguity as coinciding with the choice of p, because the ambiguity-averse agents act as if those probabilities were the objective ones. Of course, if one were to ask ambiguity-averse agents about their forecasts of future income, they would use the true objective probabilities; they just do not use these probabilities for decisions. The proper phrasing of our statement "ambiguity-averse agents expect low future income" would be the more cumbersome "ambiguity-averse agents act as if they expect low future income." We abuse the notion of expectation slightly as a result, and beg for the reader's indulgence on this matter.

(26.) The median e has the pricing functions crossing, so that agents who face ambiguity are more likely to default on small debts but less likely to default on large ones.

(27.) Since this learning is not Bayesian, it can be quite difficult to write recursively, and, in any case, learning about discrete processes generally involves a large number of states. Campanale (2008) investigates non-Bayesian learning in a two-state model where the approach taken introduces only one additional state.

(28.) Nakajima (2009) finds that increasing borrowing constraints in a model with quasi-geometric discounting is not always welfare-improving, similar to Obiols-Homs (2011).

(29.) Households with high income realizations do not want to pay the stigma cost (which is proportionally higher for them) even if they are currently carrying a large amount of debt (which is very rare due to persistence). Thus, our model does not predict any "strategic" default, which can arise in models that rely on exclusion as a punishment for bankruptcy.