Debt default and the insurance of labor income risk.

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

Recent research (e.g., Chatterjee et al. 2007, Livshits, MacGee, and Tertilt 2007) has found that allowing for debt default, such as through the relatively lenient U.S. bankruptcy code, is likely to improve ex ante welfare relative to more strict forms of debt forgiveness. The welfare gains come from improved consumption insurance provided by the option to not repay debt in some circumstances. Thus far, however, all instances where quantitative work finds a beneficial role for default have been ones with large and transitory shocks directly to household consumption expenditures. It is clear therefore that these "expense shocks" that lead to involuntary reductions in net worth are sufficient, given the specification of non-expense-related income risk in current models, to justify debt relief in forms resembling U.S. personal bankruptcy provisions.

The availability of bankruptcy, and more generally, default, will be reflected in the pricing on consumer debt, and so will affect households' ability to smooth consumption across dates and states of nature. It is therefore important to note that a significant amount of the risk to lifetime household resources may come from persistent shocks to labor income (Huggett, Ventura, and Yaron 2010). As a result, to the extent that one might be able to locate other, more targeted, ways of insuring expense shocks, it is useful to better understand how effective debt forgiveness is for managing income risk in isolation.

In this article, we evaluate in detail the role of debt forgiveness in altering the impact of income risk in the absence of expense shocks. The experiments we present can be thought of as asking: "If we insure the out-of-pocket expenses that constitute expenditure shocks, is there still a role of debt relief as a form of insurance against 'pure labor income risk'?" We address this question by studying a range of specifications for households' attitudes toward the intra-and intertemporal properties of income, when expense shocks are not present. Our main finding is that, absent expenditure shocks, the ability to default very generally hinders the ability of households to protect themselves against labor income risk.

Despite the nature of our results, we stress that our work is not to be taken as a strong statement about the overall desirability of U.S. personal bankruptcy law, for two reasons. First, to the extent the expense shocks are a feature of reality, our model is missing a feature known to be capable of justifying bankruptcy protection. Second, informal default or "delinquency" whereby a borrower simply ceases making payments (and leaves themselves open to legally protected collections efforts) may simply increase if formal bankruptcy is made stricter or disallowed altogether. Indeed, in ongoing work (Athreya et al. 2013), we find that this channel is quantitatively relevant. These related, and coexisting, options to avoid debt repayment are not modeled here. Instead, our results apply more narrowly: They suggest that labor income risk alone may not provide a strong rationale for allowing households to default. In other words, our findings suggest that the scope of shocks that debt forgiveness is providing insurance against is limited, perhaps limited principally to relatively catastrophic outcomes.

It is interesting to note that similar results are now being located in the literature on sovereign debt. Namely, it has proved very difficult to find plausible circumstances in which the benefits to being able to repudiate debts (or perhaps more accurately, the costs of being unable to commit to repayment of sovereign debt) are positive. The reasons for the similarity of the results are natural. Most importantly, the models themselves are largely isomorphic in the optimization problems they lead to, and do not differ substantially enough in their quantitative specification of either preferences or risk. Moreover, even though sovereign debt models differ somewhat in the interpretation of the debt itself (i.e., that is public debt, not private), the standard assumption in that literature is that government is benevolent and seeks to borrow on behalf of households who themselves wish to smooth consumption. This blurs the distinction between the path of public debt the government chose and that which households would have chosen.

Our results come from comparing allocations arising from two underlying trading environments. First, we study allocations arising from what we will refer to as the textbook, or "standard model" (SM), of consumption and saving in which households face uninsurable earnings risks with persistent and transitory components. In this model, households can only borrow using nondefaultable debt and also face liquidity constraints. Canonical examples of SM include those laid out in Deaton (1992, chapter 7) and Carroll (1997). To be consistent with the view that borrowing limits should be endogenously determined by repayment incentives, under SM, we investigate primarily the so-called "natural borrowing limit" case. (1)

The second trading arrangement we consider is one where, as before, households face life-cycle consumption/savings problems in which they encounter identical risks as in SM, but can issue defaultable debt. We will refer to this as the "default model" (DM). Benchmarks in this literature are Chatterjee et al. (2007) and Livshits, MacGee, and Tertilt (2007). Following these articles, default in the DM will be represented as a procedure whereby those with negative net worth can stop paying obligations, subject to any costs that may be present. The two trading arrangements we consider are thus clearly different. Nonetheless, they are related in a simple way: SM is the limiting case of DM as default becomes prohibitively costly.

To focus directly on the role of default in insuring labor income risk relative to the SM, we take two steps. First, as already noted, we deliberately set aside expenditure shocks. The presence of such shocks rules out the comparison of models with default against the standard model as budget sets would be empty for some dates and states were it not for the possibility of default. Second, we will examine a wider array of household preferences than has been done in the literature thus far. Specifically, we (i) separate risk aversion from the intertemporal willingness of households to substitute consumption, and (ii) evaluate the role of ambiguity aversion (or uncertainty aversion) when households are unsure of the stochastic environment they populate.

Both the separation of risk aversion from intertemporal elasticities and the possibility of ambiguity have been previously identified with a beneficial role for debt default. However, neither has been studied formally. The logic for suspecting that they may be important in delivering a welfare-enhancing role for default is as follows. First, the tradeoff between intertemporal and intratemporal smoothing was first suggested in Livshits, MacGee, and Tertilt (2007) in a life-cycle model of personal default. Assessing the relative importance of these motives therefore requires allowing for preferences in which the two attitudes can be distinct, irrespective of the uncertainty surrounding income. However, prior work has employed constant relative risk aversion (CRRA) preferences that conflate the two aspects of household preferences. In contrast, we employ Epstein-Zin recursive utility (Epstein and Zin 1989), which we select because of its tractability and demonstrated ability to improve the performance of asset pricing models, of which defaultable debt is a special case.

Second, with respect to the role of ambiguity in determining the value of an option to default, the legal and political history of bankruptcy law suggests that allowing for the release of debtors subject only to modest penalties is a policy that improves welfare if households are not perfectly sure of the probabilistic structure of income risk (see Jackson (2001) for one example). (2) This view is not confined to legal experts. As noted as early as Friedman (1957), agents will typically be unsure about the process that generates their labor income shocks, instead accepting that a family of potential distributions that may be difficult to distinguish are possible. Within this class of preferences, an agent who displays ambiguity aversion (Epstein and Schneider 2003) will solve a max-min problem--the agent will choose the member of the class that makes utility lowest and then choose consumption and savings in order to deliver the highest utility in this worst case.3 It is precisely this feature of the problem that will allow for a more nuanced understanding of how penalties can be "excessive" and thereby welfare-reducing: Eliminating default through harsh penalties may leave the agent unwilling to borrow at all. As a result, such a policy could perversely inhibit both intertemporal and intratemporal consumption smoothing, despite "mechanically" alleviating the limited commitment problem that the young and poor face. U.S. bankruptcy law, for instance, appears directly predicated on the idea that penalties can indeed be excessive, in the sense that they may leave would-be borrowers unwilling to do so (see Jackson (2001)).

The potential role for ambiguity in altering the welfare implications of having defaultable debt is also suggested by the observation that, in all extant work on consumer default, the relative gains seen in the SM relative to DM strongly depend on the "worst case" for household income. In particular, the large welfare losses in the DM relative to SM stem from the ability of young agents to borrow out to the natural debt limit. The natural debt limit is, however, extremely sensitive to small changes in the value of the worst-possible labor income realization, particularly for (i) young agents for whom the annuity value of future labor income is particularly high, and (ii) all agents when the risk-free borrowing rate is low.4 This lower bound is difficult to estimate accurately (see Deaton [1992] or Pemberton [1998]) and the worst-case outcomes are the primary focus of ambiguity-averse agents; thus, it seems important to understand whether the superiority of SM hinges entirely on the lowest value of income.

Our main finding along these dimensions is that even in the presence of very high levels of uninsurable labor income risk, high risk aversion, an unwillingness to substitute intertemporally, and the presence of ambiguity, the ability of households to default on debt leads to allocations that all households prefer less than the outcome that arises when they retain full commitment to repay. The intuition for our welfare results involves the relationship between the current economic situation of the borrower and the price of debt. When short-term debt is used in a setting with household labor income risk that is persistent, limited commitment to debt repayment will make credit expensive anytime the household experiences a negative shock; pricing "moves against" the unlucky borrower. (In Athreya, Tam, and Young [2009], we argue that unsecured credit markets are not insurance markets for precisely this reason.) As a result, agents who most "need" debt to smooth consumption are exactly those that find themselves unable to obtain it, because they also pose the highest risk of default. Tam (2009) extends this result to longer-term arrangements; specifically, he finds that competitively priced longer-period debt (in which the pricing function is held fixed over a number of periods) is welfare-dominated by one-period debt.

In contrast, the possibility of welfare gains from lowering penalties by enough to yield default in equilibrium was first suggested by Dubey, Geanakoplos, and Shubik (2005). Theirs was a setting where borrowers of differential default risk were pooled together and thereby did not pay the individually actuarially fair price for their debt issuance. As a result of the stylized nature of their two-period model, it is not suitable for determining whether defaultable debt is welfare-improving in a more quantitatively oriented model economy. In some quantitative settings where pooling is imposed exogenously, Athreya (2002) and Mateos-Planas and Seccia (2006) find that welfare is higher in SM than DM. More recently, in a setting where private information allows for equilibrium pooling, the findings of Athreya, Tam, and Young (2009) suggest again that, as a quantitative matter, short-term defaultable debt is unlikely to be able to function as a form of insurance. Viewing these findings as a whole, they support the notion that the benefits of slacker borrowing constraints outweigh the costs of having no default option.

Lastly, with respect to political support for a policy allowing debt default, in addition to the welfare gains from having defaultable debt available in the presence of expense shocks, it seems possible that such provisions would enjoy support even in their absence. One obvious possibility is that the current regime may simply reflect objectives other than the maximization of the welfare of newborn agents. We therefore ask if ex post welfare can account for the evident political support enjoyed by proponents of relatively lax rules on default. Specifically, we ask whether model agents would choose to allow the option to default on debt in an economy where it was not already present (taking into account all changes resulting from the policy change). We find some support for such a change, but it falls well short of a majority. Support for the default option comes from relatively unlucky middle-aged college graduates: These are agents who borrowed a lot when young, in (rational) anticipation of higher income in middle age. When realized income did not materialize as expected, such households have significant debt as they approach retirement, and so will benefit from having debt obligations removed. Young agents, by contrast, are almost uniformly opposed to allowing defaultable debt, and even less-educated workers do not generally support it.

1. MODEL

Households in the model economy live for a maximum of J < [infinity] periods. We assume that the economy is small and open, so that the risk-free interest rate is exogenous, while the wage rate is still determined by a factor price condition. (5) As a result, our welfare calculations will be biased toward finding a positive role for bankruptcy, since any lost resources arising from the implementation of default procedures like bankruptcy courts and legal costs will be ignored.

Households

Each household of age j has a probability [[psi].sub.j] < 1 of surviving to age j + 1 and has a pure time discount factor [beta] < 1. Households value consumption per household member [c.sub.i/[n.sub.j]] and attach a negative value [[lambda].sub.j, y] (in terms of a percentage of consumption) to all nonpecuniary costs of defaulting, which depend on type y to be defined below. Their (preferences are represented by a recursive utility function U ([{[c.sub.i/[n.sub.j]]}.sub.j = 1.sup.J]) that we detail below. Households retire exogenously at age j* < J.

We follow Chatterjee et al. (2007) in allowing for household-level costs from default that are primarily nonpecuniary in nature. The existence of nonpecuniary costs of default are also suggested by the calculations and evidence in Fay, Hurst, and White (1998) and Gross and Souleles (2002), respectively. The former article shows that a large measure of households would have "financially benefited" from debt default via personal bankruptcy but did not file for protection, while both articles document significant unexplained variability in the probability of default across households even after controlling for a large number of observables. These results suggest the presence of implicit unobserved collateral that is heterogeneous across households, including (but not limited to) any "stigma" associated with default along with any other costs that are not explicitly pecuniary in nature (as in Athreya [2004]). We will therefore sometimes refer to [[lambda].sub.j, y] as stigma in what follows, although we intend it to be more encompassing.

The household budget constraint during working age is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where q is an individual-specific bond price that depends on bond issuance [b.sub.j] and a vector of individual characteristics I. Net worth after the current-period default decision is denoted [a.sub.j], and therefore satisfies [a.sub.j] = [b.sub.j - 1] if the household does not default and [a.sub.j] = 0 otherwise; [DELTA] is the pecuniary cost of filing for default. The last term is after-tax current labor income ([tau] is the tax rate). Log labor income is the sum of five terms: the aggregate wage index W, a permanent shock y realized prior to entry into the labor market, a deterministic age term [w.sub.j, y], a persistent shock e that evolves as an AR(1):

log (e') = [gamma]log (e) + [euro]', (2)

and a purely transitory shock log (v). Both e and log (v) are independent mean zero normal random variables with variances that are y-dependent. (6) The budget constraint during retirement is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where, for simplicity, we assume that pension benefits are composed of a fraction V [member of] (0, 1) of income in the last period of working life plus a fraction [GAMMA] of average income (which is normalized to 1).

The survival probabilities [[psi].sub.j, y] and the deterministic age-income terms [[omega].sub.j, y] differ according to the realization of the permanent shock. We interpret y as differentiating between non-high school, high school, and college education levels, as in Hubbard, Skinner, and Zeldes (1994), and the differences in these life-cycle parameters will generate different incentives to borrow across types. In particular, college workers will have higher survival rates and a steeper hump in earnings; the second is critically important as it generates a strong desire to borrow early in the life cycle. Less importantly, they also face slightly smaller shocks than the other two education groups. The life-cycle aspect of our model is key--in the data, defaults are skewed toward young households (who borrow at least in part for purely intertemporal reasons), particularly those who do not report medical expenses as a main contributor to their default. (7)

Nonpecuniary costs, [lambda], follow a two-state Markov chain with realizations {[[lambda].sub.L, y], [[lambda].sub.H, y]} that are independent across households, but serially dependent with transition matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Due to data limitations, we assume that the transition probability matrix is symmetric and type-invariant, so the only difference across types in terms of stigma costs are their realizations. Our parametrization is more flexible than we used in previous work (Athreya, Tam, and Young 2009, 2012) so that we can match the default rates across education groups. As we show in a subsequent section, the process is still not flexible enough to match all the targets of interest, although it does a reasonable job. Households cannot borrow or save during the period in which they declare default; however, they face no restriction in any subsequent period. (8)

Loan Pricing

We focus throughout on competitive domestic lending. There exists a competitive market of intermediaries who offer one-period debt contracts and utilize available information to offer individualized credit pricing. Let I denote the information set for a lender and [^.[pi]]: b x I [right arrow] [0, 1] denote the function that assigns a probability of default to a loan of size b given information I; [^.[pi]] (b, I) is identically zero for positive levels of net worth and is equal to 1 for some sufficiently large debt level. The break-even pricing function q(*) satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

given [^.[pi]] (b, I).

In terms of loan pricing, some remarks are in order. In earlier work, Athreya (2002) specified an exogenous credit limit and then limited the sensitivity of loan pricing by forcing all loans to be priced identically. This approach has the benefit of plausibly capturing the "optionality" of the typical unsecured debt contract, whereby households can count on being able to borrow at a predetermined interest rate up to a predetermined credit limit, i.e., a credit "line." A second benefit from this approach is that it might allow a shortcut to analyzing pooling outcomes that arise from private information on borrower characteristics. However, there are clear drawbacks to this approach as well. First, for the counterfactuals we are interested in, we desire a setting in which both the supply side of the credit market and prices jointly respond to changes in borrowing and repayment incentives. By contrast, in Athreya (2002), only prices responded. For large changes in default incentives, such as what we will examine, this is not a desirable limitation. More recently, Mateos-Planas and Seccia (2006) extended the approach of Athreya (2002) to allow for changes in credit limits, but both it and Athreya (2002) in the end employ a framework substantially different enough to make the comparison to the existing models described at the outset difficult. Second, from even a purely empirical perspective, there are reasons to avoid the use of pooling contracts. As documented in Livshits et al. (2012), and Athreya, Tam, and Young (2012), among others, the variation in unsecured credit terms is now large and appears sensitive to household-level conditions. Lastly, while not directly observable, it is plausible that while individual credit contracts are best characterized by a single interest rate and credit limit, the proper interpretation of credit in the model is the sum of all credit available to the household. In this case, then, the question is the extent to which the household would have to pay more, sooner or later, to acquire additional credit. Our chosen approach features pricing that responds to default in a manner that yields supply-side effects and makes the marginal cost of credit an increasing function.

Returning to the model, r is the exogenous risk-free saving rate and 0 is a transaction cost for lending, so that r + [empty set] is the risk-free borrowing rate; the pricing function takes into account the automatic default by those households that die at the end of the period. (9) We assume I contains the entire state vector for the household: I = (a, y, e, v, [lambda], j). Zero profit for the intermediary requires that the probability of default used to price debt must be consistent with that observed in the stationary equilibrium, implying that

[^.[pi]] (b, I) = [[SIGMA].sub.e', v', [lambda']][[pi].sub.e] (e'|e) [[pi].sub.v](v') [[pi].sub.[lambda]] ([lambda]'|[lambda])d(b (a, y, e, v, [lambda], j), e', v', [lambda']). (5)

Since d (b, e', v', [lambda']) is the probability that the agent will default in state (e', v', [lambda']) tomorrow at debt level b, integrating over all such events tomorrow produces the relevant default risk. This expression also makes clear that knowledge of the persistent component e is critical for predicting default probabilities; the more persistent e is, the more useful it becomes in assessing default risk.

Government

The only purpose of government in this model is to fund pension payments to retirees. The government budget constraint is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The left-hand side is the total revenue obtained by levy of a flat tax rate [tau] on all working agents, where the distribution of working households (those for whom j < j*) over productivity levels and age is given by [GAMMA] (*). The right-hand side is the total expenditure on retirees (those for whom j [greater than or equal to] j*). Recall that to provide a tractable representation of social security and retirement benefits, we assume that retirement income is composed of a fraction v [member of] (0, 1) of income in the last period of working life plus a fraction [gamma] of average income (which is normalized to 1).

Price Determination

We assume that the risk-free rate r is exogenous and determined by the world market for credit. Given r, profit maximization by domestic production firms implies that

W (1 - [alpha]) [([gamma]/[alpha]).sup.[alpha]/[alpha]-1]

where [alpha] is capital's share of income in a Cobb-Douglas aggregate production technology. Our assumption that the risk-free rate is exogenous deserves discussion. It is certainly reasonable to assume that the U.S. capital market is open, so empirically it is not implausible. Furthermore, if we close the economy we confront the high concentration of wealth puzzle directly--the median-wealth agent in the United States has little or no wealth and thus cares about default policy, since they may borrow in the future if unlucky, while the mean agent holds substantial wealth and is unlikely to be concerned with the default policy in place. (10) There is a caveat, however. Li and Sarte (2006) is an early article that establishes a role for general equilibrium feedback effects that overturn partial equilibrium implications. Though we suspect our findings are robust to the determination of the risk-free rate via general equilibrium restrictions, it is not known for sure whether this is the case.

Preferences

Here we present the recursive representations of the preferences we study.

Constant Relative Risk Aversion

The agent's problem is standard under CRRA preferences, with the Bellman equation for a household of age j given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

v (b, y, e', v', [lambda]', j + 1) = (1 - d (e', v', [lambda]')) v (b, y, e', v', [lambda]', j + 1) +

d(e', v', [lambda]') [v.sup.D] (O, y, e' v', [lambda]' j + 1), (6)

subject to the budget constraint given in (1) and (3), depending on their age.

The value function for a household that defaulted in the current period is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

1 - [rho] [greater than or equal to] 0 is the coefficient of relative risk aversion and also the inverse of the elasticity of intertemporal substitution. Given our assumptions, the budget constraints remain the same as for all other agent types, aside from current net worth being zero as a result of the default.

Epstein-Zin

Under Epstein-Zin preferences, a household of age j solves the dynamic programming problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

subject to the usual budget constraints, and where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

is the value of default. [sigma] [greater than or equal to] 0 governs the household's aversion to fluctuations in utility across states of nature while [rho] [less than or equal to] 1 controls the substitutability between current and future utility; specifically, [sigma] is the coefficient of relative risk aversion with respect to gambles over future consumption and 1/1-[rho]is the elasticity of intertemporal substitution in consumption. When [rho] 1 - [sigma], these preferences generate the same ordering over stochastic streams of consumption as expected utility does.

2. RESULTS

The results are organized into two subsections. First, we study the roles played by aversion to fluctuations in consumption over time and across states-of-nature. We begin with expected utility preferences. We then relax this by employing Epstein-Zin preferences. Throughout this subsection, we consider parameter values that lie near the values implied by the benchmark calibration; these values ensure that model outcomes remain in congruence with cross-sectional facts on consumption and income inequality. We show that welfare under the default option is lower, at least ex ante. Second, based on this result, we ask the "inverse" question: Are there economies in which welfare in the standard model is worse? In this subsection, we no longer restrict ourselves to parameters dictated by U.S. data; rather, our goal is to understand whether any parameterizations within the parametric classes we study are capable of generating lax default as a welfare-improving policy. Specifically, we consider shocks with counterfactually large persistent and transitory components and preferences that display ambiguity aversion.

As noted at the outset, our approach throughout will shut down expense shocks in an otherwise standard consumption smoothing problem. A caveat is in order. While we have argued that this is informative about a case in which insurance is introduced where it was previously missing, it should be recognized that this is not necessarily identical to that case. In particular, the most direct route to addressing the question of whether default would remain useful if society located a way to insure what are presently uninsurable expenses is to explicitly model such an option. We opt for a simpler approach here in part because the form of such insurance, were it to become available, is not obvious a priori. This is primarily because it is unlikely to be provided privately, given that it has not emerged to this point. As a result, the form it takes will likely be as part of a tax-transfer scheme. Our model lacks the detail needed to address the associated incentive-related costs. Our approach is therefore similar to the thought-experiment of Lucas (1987) in which the costless removal of risks was employed as a benchmark for the gains from business cycle stabilization. Still, the reader should keep in mind the indirect nature of our approach and the limits it places on the interpretation of our results. In particular, our approach leads us to calibrate more than once, sometimes with only partial success, depending on the case under study, as opposed to calibrating once at the outset. We acknowledge this limitation and leave the alternate approach for future work.

Does Default Help Insure Labor Income Risk?

In this subsection, we evaluate the implications of default relative to the standard model for a variety of empirically plausible values for agent attitudes toward intra- and intertemporal consumption smoothing. Before evaluating these alternatives, we present our argument for why default regimes must be a matter of policy rather than an endogenous outcome of decentralized trading arrangements. The most prevalent form of explicitly unsecured credit is that arising from the open-ended revolving debt plan offered by credit card lenders. Credit card lending, in turn, has been (certainly since the mid-1990s) extremely competitive. (11) The relevance of the competitiveness of the U.S. unsecured lending industry is that the credit market cannot be punitive in its treatment of those who default. That is, no single firm would be willing to treat an individual borrower any worse than the current assessment of their state would justify. As a result, a household contemplating default in such a setting can safely rule out being "punished" for it. In the case where default conveys no additional information to a lender than what it was able to observe ex ante, there is literally no change in terms that are "caused" by the act of reneging on a payment obligation. Conversely, when default does reveal information, the change in terms is again not "punitive" in nature, but instead reflects an updated assessment of default risk. As a result, "high" ex post interest rates following default are implausibly ascribed to deadweight loss-inducing penalties. In the symmetric-information and competitive setting we study, punishments that are ex post inefficient will not be sustainable. Even if any single lender could withhold credit after default, the presence of other lenders would undermine the possibility of anything purely punitive. As a result, default costs capable of sustaining unsecured credit markets are likely to require intervention by policymaking authorities. (12) Thus, in the market for unsecured consumer debt, it is likely that any costs of default filing that are in any way punitive have to be policies. (13)

At the outset, we noted that for plausible parameterizations of preferences that admit an expected utility representation, the standard model typically maximizes welfare. Our first step is to understand whether this argument against default obtains only because of the restriction to expected utility or is a more fundamental property of models of life-cycle consumption smoothing. To collapse the model to the standard model, the specific quantitative experiment we consider is the imposition of a cost of default [DELTA] that is large enough to eliminate all default on the equilibrium path. (14) Before proceeding, we note the following property of our model.

Proposition 1 For each (a, y, e, v, j) there exists [DELTA] large enough that [^.[pi]](b) = 0.

This result relies on the nonnegativity condition for consumption--if [DELTA] exceeds the labor income of the household in the current period, default cannot occur since consumption would have to be negative. Given that total labor income is bounded (by assumption) and borrowing is proscribed in the period of default, we can always impose a cost of filing sufficient to generate zero default along the equilibrium path. We then compute the change in lifetime utility for each individual given a [DELTA] that exceeds the maximum required; in the absence of general equilibrium effects, we can compute these changes for each individual, rather than simply for newborns, without the need to track transitional dynamics. We will focus in general on ex ante welfare of newborns.

Calibration

We consider a benchmark case of expected utility, where [rho] = 1 - [sigma] = -1. We choose ([beta], [[lambda].sub.L, y], [[lambda].sub.H, y], [pi]) to match the default rates of each type y, the measure of negative net worth as a fraction of gross domestic product for each type y, the fraction of borrowers, and the discharge ratio (mean debt removed via default divided by mean income at time of filing). Table 1 contains the constellation of parameters that fits best (when viewed as exactly identified generalized method of moments with an identity weighting matrix). Other parameters are identical to those in Athreya, Tam, and Young (2009)--these include the resource cost of default [DELTA], the income processes faced by each type, the measure of each type, and the parameters of the retirement system ([theta], [THETA]). (15)

Table 1 Calibration

Case                             [rho] = -1,        [rho] = -0.5,
                                 [sigma] = 2        [sigma] = 2

Parameter, Target            Parameter  Outcome  Parameter  Outcome

[[lambda].sub.nhs.sup.h],       0.8972    0.31%     0.8668    1.24%
[[pi].sub.nhs] = 1.03%

[[lambda].sub.nhs.sup.l], E     0.7624   0.2071     0.6929   0.2104
[(b/y | b < 0).sub.nhs] =
0.1552

[[lambda].sub.nhs.sup.h],       0.8832    0.97%     0.8064    1.29%
[[pi].sub.hs] = 1.11%

[[lambda].sub.hs.sup.l], E      0.7135   0.1835     0.6933   0.1825
[(b/y | b < 0).sub.hs] =
0.5801

[[lambda].sub.coll.sup.h],      0.7067    0.63%     0.7136    0.79%
[[pi].sub.coll] = 0.57%

[[lambda].sub.coll.sup.l],      0.5698   0.1504     0.6352   0.1506
E [(b/y | b < 0).sub.coll]
= 0.7251

[beta], Pr(b < 0) = 12.5%       0.9765    17.5%     0.9895    13.3%

[[rho].sub.[lambda]], [E (b     0.8597   0.3986     0.6655   0.4073
| d = 1)]/[E (y | d = 1)]
= 0.56

                                [rho] = -1
Case                           [sigma] = 5

Parameter, Target            Parameter  Outcome

[[lambda].sub.nhs.sup.h],       0.9376    0.51%
[[pi].sub.nhs] = 1.03%

[[lambda].sub.nhs.sup.l], E     0.7538   0.1561
[(b/y | b < 0).sub.nhs] =
0.1552

[[lambda].sub.nhs.sup.h],       0.8872    1.31%
[[pi].sub.hs] = 1.11%

[[lambda].sub.hs.sup.l], E      0.6236   0.2553
[(b/y | b < 0).sub.hs] =
0.5801

[[lambda].sub.coll.sup.h],      0.7055    0.76%
[[pi].sub.coll] = 0.57%

[[lambda].sub.coll.sup.l],      0.4205   0.2194
E [(b/y | b < 0).sub.coll]
= 0.7251

[beta], Pr(b < 0) = 12.5%       0.9532    12.5%

[[rho].sub.[lambda]], [E (b     0.7658   0.4630
| d = 1)]/[E (y | d = 1)]
= 0.56

Our model is not capable of exactly matching the entire set of moments--for example, we underpredict default rates and discharge, generally underpredict debt-to-income ratios, and overpredict the measure of borrowers. This inability arises because the model actually places very tight links between some variables, restricting the minimization routine's ability to independently vary them. (16) In the end, one either accepts that expense shocks do indeed play a very dominant role in default data, or one is left with a puzzle relative to standard consumption-savings models. Still, we note that the qualitative findings from our analysis do not appear to depend on our specification of the stochastic process for [lambda]. (17)

Expected Utility and Ex Ante Welfare

We consider two environments--one with the calibrated value for [DELTA] and one with a cost [DELTA] sufficient to eliminate default on the equilibrium path. Table 2 contains the welfare gain from the standard model in which it is infeasible for any household to declare default. Consistent with our previous work, we find that welfare is higher in the standard model ex ante for every newborn (independent of type). College types benefit the most from the change, and their welfare gain is substantial (1.2 percent of lifetime consumption). To aid the discussion in subsequent sections where we alter preference parameters, we quickly summarize the reasons for the welfare gains here.

Table 2 Welfare Gains (without Recalibration)

[sigma] = 2 & EIS = 0.5     Coll      HS      NHS
DM [right arrow] SM        1.21%   0.54%    0.52%

[sigma] = 2 & EIS = 0.67    Coll      HS      NHS
DM [right arrow] SM        0.58%   0.21%    0.13%

a = 5 & EIS =0.5            Coll      HS      NHS
DM [right arrow] SM        0.47%   0.16%    0.13%

In the standard model, the loss of resources generated by the filing cost is not present. Since we do not impose an economy-wide resource constraint, these lost resources are not important. Instead, the welfare gain is driven by an improved allocation of consumption. By the law of total variance, the variance of consumption over the life cycle can be decomposed into two components:

Var (log (c)) = Var (E [log (c) [age]) + E [Var (log (c) [age]].

We label the first term the "intertemporal" component of consumption smoothing; it represents how expected consumption differs across time periods. The second term is the "intratemporal" component; it measures how much consumption varies across agents of a given age. Roughly speaking, how costly the first component is in terms of welfare depends on the elasticity of intertemporal substitution, because it measures the deterministic variance of consumption over time, whereas the welfare cost of the second part is governed by static risk aversion. In Figure 1 we see that the standard model, or "no-default" case (SM), improves intertemporal smoothing (the curve gets flatter) because all lending becomes risk-free. Thus, as we noted in the introduction, the only debt limit that is relevant is the natural debt limit, which is very large in our model for newborn agents. Turning to the intratemporal component, in Figure 2 we see that the SM improves this as well, restating the analysis in Athreya, Tam, and Young (2009) that unsecured credit markets do not provide insurance. Here, bad shocks trigger tightening of credit constraints, making consumption smoothing across states of nature more difficult. As a result, young agents are unable to respond effectively to bad income realizations when they can default, causing their consumption to be highly volatile. Under the SM, the natural debt limit is sufficient to protect them against adverse shocks; by middle age, default has ceased to be relevant and thus the two cases largely coincide. (18)

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The differences in outcomes across the DM and SM cases are given in Figures 3, 4, and 5, and are driven by changes in the pricing functions agents face. In Figure 3 we show the pricing functions in the low costs of default environment facing a young college agent across realizations of the persistent shock e. The initial flat segment is driven by [lambda] and is increasing in the current realization of the persistent shock e. As debt increases, more realizations of e' would trigger default, causing q to decline until it reaches zero; looking across e values we see that higher e realizations permit more borrowing. Of course, higher e realizations in our model are typically associated with less, not more, borrowing, so these increased debt limits are not particularly valuable; instead, the tightening of credit limits when e is low generates substantial costs for poor agents. In contrast, under SM pricing is flat out to the natural debt limit. Crucially, transitory shocks do not impact pricing; because v' cannot be predicted using v, the current transitory shock has no effect on the default decision tomorrow conditional on b (b is changed by the transitory shock, however).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The potential tradeoff between the two components of smoothing motivated the life-cycle analysis of Livshits, MacGee, and Tertilt (2007) and Athreya (2008), so why doesn't default generate this tradeoff? As discussed in Athreya, Tam, and Young (2009), default can either help or hinder intratemporal smoothing, depending on which agent you ask. An agent facing an income process with low intertemporal variance but high intratemporal variance--that is, tomorrow's expected income is close to current income but tomorrow's income has substantial risk--may benefit from default; the intertemporal distortion is minimal while the potential to truncate the consumption distribution at the low end conveys significant benefits (even once pricing is taken into account). In contrast, an agent facing the opposite process--income that grows over time and is relatively safe--generally does not benefit; default is not used because pricing prevents it and the intertemporal distortion is substantial, leading to significant welfare losses. In our model, a young agent is of the second type, especially a college-educated one, while older households are members of the first type.

Ex Post Welfare--Voting over Default Policy

Because we study a small open-economy model in which the risk-free rate is fixed, but also allow all pricing to be individualized, there are no "pecuniary" externalities. We can therefore compute the welfare consequences of policy changes for any agent at any point in the state space; since the distribution plays no role in pricing (and therefore no role in welfare), we do not need to calculate the transitional dynamics of the model to get the welfare changes. We ask agents of a given age and type whether, conditional on their cm-rent state, they would be in favor of eliminating the option to declare default. Figure 6 displays the measure of each type, conditional on age, that would support retaining default with the calibrated [delta]. A substantial portion of college types oppose elimination, but they are all middle-aged and have experienced histories of bad shocks; the peak in opposition occurs earlier for high-school types and later for non-high-school types, with correspondingly fewer such households opposing overall. For the convenience of the reader, Table 3 presents the aggregate measures of each type that oppose eliminating default (the column labeled "DM Regime"); they are small for each education group. Furthermore, as is clear from the figures, almost no newborns oppose eliminating the option.

[FIGURE 6 OMITTED]

Table 3 Measure of Agents in Favor of Bankruptcy

Education        DM Regime   SM Regime
College              6.45%       4.09%
High School          4.05%       3.26%
Non-High School      0.16%       0.24%
Total                4.05%       2.98%

We now consider the inverse of the preceding experiment: Agents of different ages and types are asked if they would prefer to introduce default (again, with [DELTA] set to its calibrated value) into a setting in which it is currently prohibited. As seen in Figure 7, a nontrivial fraction of agents would like to introduce default. The intuition here is that the no-default case allows significant borrowing at the risk-free rate. As a result, many households, especially the college-educated, borrow when young in anticipation of higher earnings. The relatively unlucky among them then find themselves indebted by middle age and thereby will benefit from the discharge of debts. Moreover, by virtue of being middle-aged, these households place relatively low value on being able to access the cheap unsecured debt later in life. This effect is especially strong for the college-educated, for whom purely intertemporal consumption smoothing motives dictate a strong effort to save for retirement beyond middle age. As a result, a substantial proportion of high-school- and college-educated household groups would support the introduction of default when they reach middle age. In contrast, those who have not completed high school support the introduction of default only late in working life, when the subsequent increase in borrowing costs is not long-lasting. However, as Table 3 shows (the column "SM Regime"), the aggregate number of agents who vote in favor of introduction falls well short of majority status.

[FIGURE 7 OMITTED]

Separating Risk Aversion from Intertemporal Substitution

As discussed above, the two pieces of the variance decomposition have welfare costs that depend (mainly) on different aspects of preferences. Our benchmark case using CRRA expected utility restricted these two aspects of preferences to be reciprocals of each other. Here, we relax that requirement by using the Epstein-Zin preference structure, and consider two particular deviations. First, we make households more tolerant of intertemporal variance than in the expected utility benchmark by employing a high value for p. Second, the default option may shrink the volatility of intratemporal consumption, at least for some ages. Given this, making intratemporal variance more painful to households may help us explain the presence of low default costs. We therefore select a relatively high value for [sigma]. It is important to note that this particular combination of insensitivity to the timing of consumption and sensitivity to the income state in which it occurs is the arrangement that gives default its best chance of improving ex ante welfare and does not lie within the class of expected utility preferences.

The specific experiments we investigate involve changing p and [sigma] without recalibrating the entire model. This type of change generates two effects--an effect conditional on borrowing (which we call the price effect) and an effect caused by changes in the number of borrowers (the extensive effect). We then compare the results with cases where the model is recalibrated (to the extent that is possible) in an attempt to isolate the two effects.

We first consider changes in p. To understand how this change affects welfare, it is helpful to first consider the extreme case of p = 1, making the household infinitely willing to move consumption deterministically through time. As p[right arrow] 1, the Bellman equation converges to the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, the household will either completely frontload or backload consumption, depending on the relationship between the discount factor and the interest rate. For the parametrization we use, the effective discount factor ([beta] times the survival probability) lies between the risk-free saving and borrowing rates for almost every age, meaning that households don't wish to borrow and, critically, do not value the default option no matter how risk averse they are. For some older households, whose survival probabilities are relatively low, the effective discount factor is sufficiently low that they want to borrow and "frontload" their consumption; the option to default makes borrowing expensive enough to render complete frontloading impossible. This, in turn, reduces the welfare of these households--since they face no uncertainty, default is either probability zero or one and pricing therefore eliminates it. Obviously such extreme consumption behavior is inconsistent with U.S. cross-sectional facts; in particular, the model with p = 1 would miss very badly on the life-cycle pattern of consumption inequality, which in the data is substantially smaller than income inequality.

Returning to less extreme values for p, Figure 8 displays the pricing function across several different values of p and demonstrates the effect on loan prices. As [rho] increases, the pricing function shifts downward because at any given level of debt an agent with a higher p is more willing to default. The intuition for this result is not straightforward. When p increases, the household is more willing to accept variability in consumption across time. If a household enters the current period with some debt and wishes not to lower debt, they have two options: (i) borrow more if possible or (ii) default and void those obligations. Borrowing more is only feasible if there is a reasonable commitment to repay. But since a bad shock would lead to low mean consumption, default becomes attractive, and households lack strong commitment to repay debt. As a result, they cannot borrow easily. For the cases with "intermediate" values for p, the creation of strong default incentives makes intertemporal smoothing more costly, but the latter is relatively unimportant.

[FIGURE 8 OMITTED]

Consider next an experiment where [sigma], the risk aversion with respect to gambles over future utility, is increased. Again, turning first to the polar case, let [sigma] [right arrow] [infinity], so that the household becomes infinitely risk averse. In this case, the limiting household Bellman equation takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to the usual budget constraints seen earlier in equations (1) and (3).

When households are infinitely risk averse, they choose not to borrow for the reasons outlined in Athreya, Tam, and Young (2009)-- unsecured credit markets do not provide insurance and thus agents will be unwilling to pay the transaction cost to borrow. As a result, there is a welfare gain to living in the standard model, as no household has negative net worth. Again, extreme preferences render the model grossly inconsistent with cross-sectional facts; here consumption inequality would be essentially zero over all ages.

Returning again to more intermediate cases, we see that changes in risk aversion generates two effects. The extensive margin effect is similar to increasing p, but for different reasons. When [sigma] is large, households have a strong demand for precautionary savings; for [sigma] = 5, for example, we see a clear decline in the measure of total borrowers, again making default overall less damaging. The pricing effect is also similar; by increasing risk aversion, we make the household less willing to have consumption differ across states of the world tomorrow. Conditional on borrowing, the pricing functions reveal a stronger desire to default--for any given b, the price of debt is decreasing in [sigma] (see Figure 9). As above, there are only two options for a household with debt; since even a moderately bad outcome will cause a highly risk-averse agent to default, commitment is not possible, leaving default as the only option for smoothing consumption across states. (19) Combining these results into one statement, we see that no combination of (p, [sigma]) leads to default being a welfare-improving policy, although for extreme cases it will be nearly innocuous.

[FIGURE 9 OMITTED]

Table 2 shows that welfare is higher (for newborns) in the standard model (SM), but that the gains from (imposing the high [delta]) decline with risk aversion and elasticity of intertemporal substitution (EIS). p> 1 - [sigma], which is satisfied when either parameter increases, implies the household has a preference for early resolution of uncertainty; thus, default appears to be least damaging when households prefer to resolve their risk early rather than late.