Loan guarantees for consumer credit markets.

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

Two specific subsets of the U.S. population--the young and those with temporarily low income (and wealth)--have long been identified as pervasively facing liquidity constraints. Empirical work has long measured the fraction of constrained households at close to 20 percent of the U.S. population (see Zeldes 1989; Jappelli 1990; and Hubbard, Skinner, and Zeldes 1994). More recent work again places importance on the inability to cheaply access unsecured credit when needed (see, e.g., Gross and Souleles 2002 and Telyukova 2013). While the preceding work takes substantial care to arrive at estimates, even the simplest summary measures in U.S. data suggest a lack of access to consumer credit for the groups mentioned above. For example, among families with heads of household who have income between $25,000 to $50,000, 34 percent have no credit card at all. Moreover, 60 percent roll over debt and pay interest rates of approximately 15 percent per year, a clear indication of their inability to access cheaper alternatives. Among younger families, those with heads of household of age 35-44, similar patterns emerge: 32 percent have no credit card, while 32 percent roll over debt and also pay the same (high) interest rates. Lastly, poorer households fare badly: 55 percent of those with income between $10,000 and $25,000 have no credit card, while 55 percent of renters don't have a credit card. (1) The lack of wealth among both young and low-income households also precludes the use of far less expensive secured credit, such as home equity loans or lines of credit. Thus, despite the apparent ubiquity of consumer credit, the young and the poor both appear to face tight restrictions on access to the principal source of credit available to them.

The populations most routinely identified as credit constrained are also precisely those groups who are generally most lacking in wealth that could be pledged as collateral. For example, the young and poor do not possess sufficient collateral. In the student loan market, the private sector's inability to attach human capital in the event of default has been viewed as a basis for credit policy since at least Becker (1967). More recently, and very specifically to our inquiry, quantitative work suggests that the market for unsecured consumer credit is significantly hindered by the availability of low-cost personal bankruptcy (see, e.g., Chatterjee et al. 2007; Livshits, MacGee, and Tertilt 2007; or Athreya 2008) and by the presence of private information about borrower default risk (e.g., Sanchez 2009; Athreya, Tam, and Young 2012b). The absence of collateral is important: Full collateralization will, by definition, make both limited-commitment and private-information frictions irrelevant to lending decisions. In other words, it is the unsecured credit market whose functioning is likely to be most important for the populations identified above.

Given that the unsecured credit market is the one most central to the consumption-smoothing objectives of a significant share of U.S. households, the question is then: What, if anything, can be done in this market to improve outcomes? A first answer might be to make bankruptcy law tougher: If limited commitment is the problem, why not directly address the issue by making debt harder to repudiate? The problem is that while formal bankruptcy is a currently important source of credit losses, informal default remains, in practice, a clear option. Recent work of Athreya, Tam, and Young (2012b) suggests that this option seriously limits the power of bankruptcy policy to diminish incentives for debt repudiation, and, hence, to mitigate limited commitment as an impediment to unsecured consumer lending. That is, once calibrated to match the salient facts on consumption and borrowing, informal default remains a viable option that borrowers choose in the face of even modest increases in the cost of formal bankruptcy. This is seen in the data in terms of the high rate of bankruptcy and delinquency in unsecured credit in U.S. data. However, even if incentives to default are difficult to alter, an alternative already employed in a wide array of settings--but not yet for unsecured consumer credit--seems promising: public loan guarantees.

Such guarantees work by using public funds to defray private losses from default. In the United States, the most obvious loan guarantee programs for households are those that accompany home loans. For example, the Federal Housing Administration (FHA) and the Veterans Administration both offer loan guarantees to private lenders, and, in both 2009 and 2010, the FHA alone issued roughly two million guarantees and, as of 2010, insured nearly one-tenth of the stock of outstanding U.S. mortgage debt. Similarly, the U.S. Student Loan Administration (Sallie Mae) is active in arranging guaranteed loans, with recent flows on the order of $100 billion annually and a stock of approximately $500 billion. Loan guarantees also play a sizeable role in credit to households attempting self-employment, with the U.S. Small Business Administration's (SBA) 7a loan program guaranteeing roughly $100 billion in credit per decade since 1990. (2) However, despite their similarity to the programs we study in this article, the closest analogy might be instead to flood insurance. The reason for this will be made explicit below but stems from the fact that in our model, loan guarantees act in a manner that lowers the cost of moving consumption across both time and states-of-nature, in much the same manner as a subsidized form of insurance might.

The goal of this article is not to analyze a specific extant policy but rather to take a first step, within a specific model class, toward understanding the potential gains from extending loan guarantees to unsecured consumer credit markets. As such, and especially because they are currently not in use, some motivation for why one might study loan guarantees, as opposed to any number of other interventions in consumer credit markets, deserves discussion.

One important reason to view loan guarantees as potentially valuable in improving credit access is that under competitive conditions, loan guarantees decouple loan pricing from credit risk. This is relevant for two reasons. First, a growing body of work shows that in the absence of complete insurance markets, risk-averse households can benefit from the state contingency introduced by the option to default in bad states of the world (see, e.g., Zame 1993; Dubey, Geanakoplos, and Shubik 2005; Chatterjee et al. 2007; and Livshits, MacGee, and Tertilt 2007). What this means, intuitively, is that while nondefaultable debt requires the borrowers to always repay debt as promised, once default is allowed, matters are not so stark. Why? Because a borrower in dire straits can now invoke the option to not repay debt if doing so in the current period would expose them to severe hardship. This is what is meant by "state contingency." In a world where such an option is present, given the absence of other more explicit forms of financial contracts to help deal with risk, most notably insurance contracts against income loss, defaultable debt can be beneficial to borrowers.

Moreover, in existing work, consumers have been shown to benefit despite the presence of loan pricing that moves "against" the riskiest borrowers. However, these gains are not necessarily accessible in all a priori plausible environments. In recent quantitative work on the value of defaultable consumer debt, a variety of authors (such as Athreya, Tam, and Young 2009) have found that in many cases the ability of lenders to reprice loans at the same frequency as the arrival of new information on income risk undoes insurance benefits altogether. In other words, every time a consumer is hit by a persistent (but not permanent) bad shock, she will find her ability to commit to loan repayment eroded, and any borrowing she might attempt will become expensive. From the perspective of borrowers, if competitive lenders are made partially whole, they cannot "risk adjust" interest rates as much and so such loans will better assist households in consumption smoothing. Indeed, in the context of boosting aggregate consumption, researchers have recently started considering ways to direct unsecured credit to households at "favorable interest rates," with the public sector bearing default risk, exactly as would occur under the loan guarantees to consumers we study. (3)

Second, if information on borrowers has improved secularly over time, as has been suggested by many as having occurred in recent decades in U.S. consumer credit markets (see, e.g., Sanchez 2009; Athreya, Tam, and Young 2012b; and Narajabad 2012), then loans are now priced more accurately. This is likely to make relatively risky borrowers' access either worse or improve it by less than that of safer borrowers. Indeed, Sanchez (2009) suggests that this will be the case. Moreover, improvements in information will certainly bring the risk sensitivity of loan pricing closer to what we study below in the "full information" (FI) case. In these cases, as noted above, standard models suggest that unsecured credit will not work well as a smoothing device. Thus, policies that allow for default but break the link between credit risk and credit pricing are promising candidates to improve allocations--at least to borrowers. (4,5)

Despite their likely benefits, loan guarantees will create costs, particularly in two places. First, default rates are likely to rise, generating more deadweight loss (whether pecuniary or nonpecuniary in nature). The rise in default rates occurs for the very reason that loan guarantees "work": They lead to the systematic underpricing of loans by lenders, given their risk. Relatively larger loans will now attract relatively high-risk borrowers. As a result, the more effective any loan guarantee scheme is in spurring borrowing and consumption, the more prevalent that default and deadweight losses will be on the equilibrium path. In the context of loan guarantees for entrepreneurial ventures, the work of Lelarge, Sraer, and Thesmar (2010) documents precisely this type of response in a near-natural French experiment. As they note, "it [loan guarantee] significantly increases their probability of default, suggesting that risk-shifting may be a serious drawback for such loan guarantee programs." This inevitable tradeoff means that the real questions are: "By how much?" and "does risk-shifting happen, and if so, is it welfare-enhancing?" (6)

Additionally, tax revenue must be raised to finance transfers to lenders ex post. Under incomplete markets, the taxes used to finance these transfers have two opposing effects on welfare. First, if, as was the case in the in study of Lelarge, Sraer, and Thesmar (2010), a relatively large fraction of households faces a tax that a relatively small proportion benefit most significantly from, the introduction of a publicly funded loan guarantee program will reduce the mean level of income for many households. In particular, if it is a relatively small measure of households who run up substantial debts that, absent the guarantee, would demand high interest rates, they then receive a transfer from all other households. Second, nonregressive taxes reduce the variance of after-tax income, especially when one's expected lifetime income (as captured by ex ante uncertainty over one's eventual educational attainment) is uncertain.

While ours is not a policy evaluation article, the model class we study contains features that we believe will be essential to include in any empirically relevant policy related to consumer credit access. Specifically, our model contains a well-defined life cycle for household income that motivates credit use for intertemporal smoothing and uninsurable risk that motivates the use of credit to smooth across states. Importantly, our model features credit constraints that are endogenously derived in response to a limited commitment friction and, in other cases, to asymmetric information as well. Before proceeding, we also note that there is a distinction between what we study here and a more complicated alternative that in some ways may be more natural: We are interested in the implications of the replacement of the current nonguaranteed system with one in which guarantees necessarily apply to loans below a certain size threshold. Future work will, ideally, allow for the addition of guaranteed consumer lending as an option, with households self-selecting into programs with and without loan guarantees.

Our results can be summarized as follows. First, we find in our model that loan guarantees are powerful in influencing allocations: Even modest limits on qualifying loan size invite very large borrowing--as perhaps intended by proponents. However, these same limits also bring very large increases in default rates relative to a world without guarantees and, as a result, transfer resources in significant amounts from the ex post lucky to the ex post unlucky, in addition to transferring wealth across education types. Indeed, this is the key tradeoff that differentiates our work from existing research on unsecured credit, such as Chatterjee et al. (2007) or Livshits, MacGee, and Tertilt (2007). These articles find some gains from lax bankruptcy despite the fact that such rules make borrowing more expensive and, hence, tighten credit constraints, as such rules provide valuable insurance. By contrast, our article shows that loan guarantees can improve welfare despite greater ex-post deadweight loss due to bankruptcies, as they make borrowing cheaper and, hence, provide insurance as they relax borrowing constraints.

Second, we find that loan guarantees yield significant benefits as long as they are not too generous (whereby only small loans qualify). At low levels of the guarantee, this welfare gain is disproportionately experienced by low-skilled households who face flat paths for their average income over the life cycle and the risk of relatively large shocks. As loan guarantees are made more generous, however, higher-skilled types rapidly begin to experience welfare losses. This occurs because loan guarantees induce a transfer from skilled to unskilled households, which can be substantial, while the gains to skilled households from improved loan pricing as a result of guarantees are relatively small.

Third, we show that restricted guarantees clearly dominate unrestricted programs in terms of welfare. As noted, households in our model face risk, including that of large negative income shocks. It is plausible, therefore, that a more conditional loan guarantee program, available only to households hit by such shocks, might allow policymakers to provide the benefits from guarantees while limiting their cost. The model thus suggests that this intuition is likely to be valid.

In the case of asymmetric information, the size of this friction will be endogenous as well and will depend on how heterogeneous borrowers are, not only in terms of both exogenous shocks, but also in terms of endogenously determined and unobservable net asset positions. One of the earliest studies of loan guarantees is that of Smith and Stutzer (1989). These authors show in a stylized model with two types of borrowers (high risk and low risk) that the reduction in the sensitivity of loan interest rates to default that accompanies loan guarantees also reduces the high-risk types' incentives to reduce their borrowing inefficiently simply by mimicking the low-risk types. This incentive effect contributes positively to the welfare impact of loan guarantees. And while private information may not currently be a crucial problem in U.S. consumer credit markets, given extensive recordkeeping and information sharing via credit bureaus, it was likely present both in the United States in earlier periods (see Sanchez 2009 and Athreya, Tam, and Young 2012b), and plausibly remains an impediment in developing countries currently. A goal of this article is to measure the effects of loan guarantees under these more difficult circumstances. We find that under private information, the gains from guarantees are meaningfully larger than in the absence of private information, quantitatively consistent with the prediction of Smith and Stutzer (1989).

It is important to recognize that loan guarantees can only matter for allocations and welfare when debt is imperfectly collateralized. Even when a loan guarantee program is nominally targeted at a secured form of lending, such as mortgage loan guarantees, they can only alter allocations because there is a positive probability of the loan becoming at least partially unsecured ex post. This leads us to focus on the effects of introducing guarantees for uncollateralized consumer lending, which is the most prominent form of unsecured credit. (7) Nonetheless, our setting will clearly be informative for the effects of loan guarantees in any market in which there exists states of nature where repayment is less attractive than paying the costs of default. (8)

Our article is linked to three strands of research in public interventions in credit markets. First, our focus on consumer credit with default risk connects this article closely to recent work of Chatterjee et al. (2007); Livshits, MacGee, and Tertilt (2007); Athreya, Tam, and Young (2012b); and Narajabad (2012). In this line of work, however, guarantees are not studied, but both voluntary default and asymmetric information have been shown to matter for the allocation of consumer credit in the absence of guarantees (see, e.g., Sanchez 2009 and Athreya, Tam, and Young 2012b). As noted earlier, our research is novel in studying a distinct mechanism from this strand of work, whereby welfare can be improved by relaxing constraints, rather than tightening them through the promotion of debt forgiveness.

Second, our work is clearly connected to more recent research on quantitative analysis of the allocational consequences of loan guarantees. This work began, to our knowledge, with Gale (1990) and was followed by the rich, fully dynamic, and relatively tractable incomplete-market models developed in Li (1998) and Jeske, Krueger, and Mitman (2010). The last article is the first to focus centrally on credit markets in a consumption smoothing context. However, with respect to modeling default, in all the preceding work, default is involuntary. (9) Our article is also related to recent work of Jia (2013), who in turn builds on Li (1998) to study loan guarantees for firms in a setting where the government's relative (and absolute) financing advantage in recessions can be put to use by providing loan guarantees for small businesses. Our focus, by contrast, is on households, and a main goal is to provide a quantitative analysis that is rich and aims to evaluate consumption smoothing, not investment in small business as these other articles do.

Third, our work relates to an earlier, relatively stylized class of articles that focus on the role of interventions, including loan guarantees, on outcomes for a general problem of risky-investment in static or near-static settings under asymmetric information. Key landmarks in this category are Chaney and Thakor (1985), Smith and Stutzer (1989), Gale (1990), Innes (1990), and Williamson (1994). (10) Most of this work abstracts from the financing of guarantee programs as well. By contrast, these costs will feature prominently in our analysis. (11)

1. ILLUSTRATIVE MODEL

The friction faced by households stems, ultimately, from their inability to explicitly insure income shocks and their inability to credibly promise to always repay loans. Given default risk, competitive lenders will be forced to price loans in a way that allows them to break even. As a result, in general, households with differing levels of default risk will face different prices for credit. However, the fact that loans in our model will be priced to reflect default risk also means that some borrowers will find themselves facing expensive credit terms precisely when they most need to borrow. It is these groups who will find guarantees most helpful.

Before turning to the quantitative setting in the next section, it is useful to describe a simple two-period variant of our model to more clearly identify the types of individuals who are affected by risk-based pricing that, by definition, makes borrowing expensive when future income levels might remain low, and who may therefore gain from loan guarantees. Let [c.sub.i] denote consumption in period i = 1, 2, [e.sub.i] denote the endowment of the consumption good received by the agent in period i = 1, 2, and [d.sub.2] [member of] {0,1} denote the default decision in period 2. Defaulting implies that the consumer incurs a nonpecuniary cost 1/[lambda]. To remain consistent with the quantitative model on which the final results are based, and for mnemonic ease, a high value of [lambda] implies a high risk of default, all else equal, because it implies a low value for the term 1/[lambda], which is what gets subtracted from utility in the second period in the event of default. This cost is a stand-in for the variety (and entirety) of costs associated with defaulting and is meant to tractably encompass not only the explicit costs (e.g., bankruptcy filing costs, legal costs, etc.) but also the nonpecuniary costs (e.g., difficulty renting durable goods, obtaining employment, emotional distress, etc.). The dependence of default risk on loan size leads loan prices to depend on loan size.

Households are modeled as borrowing through the issuance of debt with a face value b < 0. "Face value" refers to the amount that the household is obligated to repay and is the value that it would deliver if it did not default. However, the household may, in period 2, elect to exercise its default option. As a result, the face value of debt, by virtue of being risky, will be discounted by lenders. The term q (b) [member of] [0,1] is the discount factor applied to a debt issuance of face value b and is determined by competitive markets. To see how this discount is determined, consider a lender wishing to price a loan with face value b. Let the default probability for this loan be given by [pi](b). Thus, the expected value of the loan is (1 - [pi](b))b. In facing this, the lender must decide what discount q(b) to apply. Competition among lenders implies that the discount allow the bank to, at best, break even on average. This implies that q(b)b, the real expected value of resources transferred to the borrower, must have the same cost for the lender to obtain as the expected value of the loan. Let the cost of funds for the lender be given by (1 + r). Thus, the cost of making a loan with face value b with discount q(b) is (1 + r)q(b)b. Equating this with the expected value of the loan, (1 - [pi](b))b, and simplifying gives

q(b) = 1 - [pi](b)/1 + r.

This is intuitive. As [pi](b) rises, q(b) falls and hits zero when [pi](b) reaches unity (certain default). For loans where [pi](b) = 0, the discount is simply 1/1+r, the competitive price of a risk-free loan.

Given this credit market and endowment structure, households choose consumption, borrowing, and saving to maximize standard expected utility preferences:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Default risk arises in the model from the fact that endowments are probabilistic and structured as follows: All households receive [e.sub.1] in the first period and this value is a known constant. Households face uncertainty with respect to income only in the second period--[e.sub.2] is drawn from a two-point distribution: [e.sub.2] [member of] {[e.sub.L], [e.sub.H]} with probabilities [p.sub.L] and 1 - [p.sub.L].

Households may save or borrow in the first period, denoted b, with [b.sub.1] > 0 corresponding to saving and [b.sub.1] < 0 being borrowing. In period 2, they first draw income [e.sub.2], and then elect to default ([d.sub.2]([e.sub.2]) = 1) or not ([d.sub.2]([e.sub.2]) = 0). Note clearly that if they default, they repay nothing (default is total). This is a useful simplification, but can be relaxed to allow partial default. If they do not default, households must repay the face value they issued in the first period, b. As a result, the households' choices are restricted by the following pair of budget constraints. In period 1,

[c.sub.1] + q (b) b [less than or equal to] [e.sub.1].

In period 2, they face

[c.sub.2] ([e.sub.2]) [less than or equal to] b (1 - [d.sub.2] ([e.sub.2])) + [e.sub.2].

Consider a case with two types of households. Let one type be those whose second-period endowments [e.sub.2] have a high mean relative to their period 1 value and (relatively) small variance. In other words, income in the future is expected to be higher than today and relatively safe as well; this group roughly corresponds in the data to relatively highly educated borrowers. This group values access to credit because it helps them bring their high, safe, future income into the present. Thus, loan prices for such households will be at the risk-free rate for a relatively wide range of borrowing levels, as the household will elect to repay irrespective of the realization of [e.sub.2], and then fall abruptly when the loan size reaches a threshold where households would first begin to consider default. For households facing only small uncertainty about [e.sub.2], this means that once default in one state becomes attractive, it generally becomes attractive in the other state (since they are, by construction, similar). This is easiest to see in the limit where [e.sub.2] is known in period 1 with certainty: For a given loan size b, default in period 2 either occurs none of the time or all of the time.

[FIGURE 1 OMITTED]

The second type of household we are interested in has small mean and large variance of [e.sub.2]; one can think roughly of this group as being relatively less educated and facing greater risk of unemployment in period 2. For these households, borrowing is not particularly useful, but if undertaken, can yield more variation in terms because the default decisions will differ more substantially across realizations of second period income [e.sub.2].

Figure 1 shows a typical situation faced by either type of household. (12) The indifference curves are monotone (over the range of interest at least), and reflect the fact that a household can, in principle, receive additional consumption today in two ways: hold b fixed, as long as it faces a higher q, or increase borrowing b, but accept that the discount q will fall as default risk rises (though not so rapidly that q(b)b, which is what the household receives, falls). (13) At the optimum the household is constrained, in the sense that additional borrowing is desired but not feasible due to the increase in the probability of default; this situation will be typical in the quantitative model as well. Thus, local to that optimal b there are welfare improvements available to households if q can be held fixed while b is increased. This is an obvious point, perhaps, but it is useful to keep in mind as it is the source of the ability of loan guarantees to assist households in smoothing consumption.

Loan Guarantees

We now introduce a publicly funded (via taxes that, in this section, will remain unmodeled) loan guarantee into this model economy. Loan guarantees will be defined by two parameters: (i) a "replacement rate" [theta] that determines the fraction of defaulted obligations b that the lender receives as a transfer from the government, and (ii) a "coverage limit" v that determines the largest (riskiest) loan that the government will insure. Only loans smaller than, or equal to, v in size qualify for any compensation; lenders making loans larger than the ceiling receive nothing in the event of default. (14) Households may borrow more than v, but if they do, the discount on these loans will jump discretely downward as expected rewards to lenders fall discretely due to the "non-conforming" nature of loans exceeding the program limit.

Given that the loan guarantee covers [theta] percent of the repayments lost to default for the portion of any loan less than v, competitive pricing of loans for a conforming loan--one with face value b [less than or equal to] v--must obey

q (b) = 1 - [pi](b)/1 + r + [pi](b)/1 + r [theta].

Non-conforming loans are priced exactly as if there was no loan guarantee program in place. In the absence of taxes needed to compensate lenders for default under any guarantee program, it is clear that both types of households would gain from the introduction of loan guarantees. Assuming default probabilities don't change, the guarantee increases the bond price for the first group from 0 to [theta]/[1 + r] and for the second group from 1-[p.sub.L]/1+r to 1-[p.sub.L]+[p.sub.L][theta]/1+r. This increase expands the set of feasible consumption paths and raises welfare; default probabilities will not increase if [theta] is small enough due to the discreteness of the income process. To illustrate how a loan guarantee works, Figure 1 shows how the pricing function shifts weakly upward, which clearly raises utility because the household is currently constrained and the deadweight loss from default is the same.

Asymmetric Information

Our analysis has so far focused on limited commitment alone as an impediment to credit access. We now allow for asymmetric information to further hinder lending. To adapt the model to deal with asymmetric information, suppose now that default risk varies according to some characteristic that is not observable to the lender; for concreteness, let there be two such groups, and keep in mind that within each, there are two groups of borrowers with respect to their current cost of default [lambda]. Private information forces (barring a rich menu of screening contracts) the lender to offer a uniform pricing function to both types of households based on the invariant measure of each type (let [delta] [member of] (0,1) be the measure of the first type); the function is contingent on the costly signal b sent by the household. (15) The pricing function without the guarantee would be

q(b) = [delta] [??](b|1)/1 + r + (1 - [delta]) [??](b|2)/1 + r,

where the hatted variable [??] reflects the fact that under asymmetric information default, probabilities are no longer necessarily known with certainty since the agent who asked for the loan is not necessarily known, in equilibrium, with certainty. Instead, default risk is an imperfect estimate that reflects the uncertainty over which agent-type attempted to take a given loan. For a risk-neutral lender, what is then relevant is the conditional probability of a given loan request having come from either type of borrower (and reflecting the fact that each type will not default with the same probability at any given level of debt). In a more elaborate model (such as the one used later for quantitative analysis), the debt level b would lead lenders to update the estimate of default risk since not all types would find it optimal to issue b; in such a case one can think of lenders computing conditional probabilities of the borrower being of a particular type given their requested loan size b (that is, updating [delta]) and using those probabilities to compute default risk. Equilibrium then requires that updated beliefs are consistent with the population of borrowers of a given type issuing b.

The "bad" type of borrower--that is, the borrower with the high value of [??]--will want to reduce b in order to look more like the good borrower, all things being equal. As discussed more completely in Athreya, Tam, and Young (2012b), pooling is potentially an equilibrium if the pricing function is relatively flat just to the right of the equilibrium choice; in that case, the indifference curves of both types lie above the break-even curve for the lenders so deviations to lower debt levels do not occur. Separating equilibria occur when pricing functions are steep (relative to indifference curves), because then the good type would be better off reducing b while the bad type would not. Loan guarantees reduce the desire of bad types to pool with good types because they break the link between pricing and type; this disincentive is welfare-improving because it improves the allocation of consumption, and so under asymmetric information loan guarantees will have even better welfare properties. But as before, we must consider whether the costs outweigh the gains, and under asymmetric information the costs will increase more than under symmetric information because default is initially lower. Whether the costs or benefits are larger is the main focus of our quantitative model, which we describe in the next section.

The Irrelevance of Actuarially Fair Loan Guarantees

In this article, we study fully subsidized guarantees. However, in practice, many loan guarantee schemes ask the borrower to pay the guarantee fee (such as SBA loans and FHA-guaranteed home loans). (16) An important point we now develop is that any private loan guarantee scheme that is also actuarially fair, and therefore will survive competition, will necessarily be irrelevant.

To see this result, consider a competitive economy in which, notionally, the borrower is obligated to pay the loan guarantee fee, as observed in practice. Let [tau](b) be the insurance premium on a loan with face value (i.e., what is paid outside of default) b. Let [q.sub.f] be the reciprocal of the risk-free interest rate, i.e., [q.sub.f] = 1/(1 + r), where r is the risk-free rate of interest on savings. As before, let [pi](b) be the probability of default on a loan of size b.

A borrower who issues b units of face value then gets, after the insurance payment of [tau](b), [q.sub.f] b - [tau](b) units of resources in period t, and is free to default or not in period t + 1. So what does the guarantee fee have to be? If it is set to break even across all borrowers of the given type of borrower who issued b units of debt, then the premium must be [tau](b) = [pi](b)[bq.sub.f] (the last term appears since the lender will only get paid next period and so must discount), which equals the expected loss on the loan. Therefore, the net resources an agent gets for issuing b units of debt, after paying the loan guarantee premium, is [q.sub.f] b - [pi](b)[bq.sub.f], or [q.sub.f] b (1 - [pi](b)). But this is exactly the pricing function that would arise in a competitive setting without guarantees.

This result follows naturally from competition between lenders: If the borrower pays the insurance premium and leaves the lender insured, the loan is then risk free to the lender. As a result, a lender can, under competitive conditions, only charge the risk-free rate for the loan. Thus, if loan guarantee schemes are to matter for allocations, they must carry a subsidy with them, such as the one that comes with public provision of the guarantee. This implication is why we study fully subsidized guarantees (no premium). Our approach ensures that guarantees don't merely lead to a reinterpretation of existing contracts, but rather are capable of changing household budget sets. (17)

In our quantitative model we study programs that insure only a fraction 6 of loan losses, using tax revenue to fund payments to lenders. The argument here is unchanged--the private sector cannot offer meaningful guarantees for the fraction 1 - [theta] of the loan that the government does not cover.

2. QUANTITATIVE ANALYSIS

Given that the implications of guarantees depend on the opposing forces we have isolated above, we now turn to the quantitative analysis of guarantees for consumer credit in order to determine the ultimate effects they may have. The general framework we employ is a standard life-cycle model of consumer debt with default, and aside from the budget-constraint-related complications arising from loan guarantees, the model we use is essentially identical to that of Athreya, Tam, and Young (2012b). (18)

Because the contribution of our article lies in its application of a standard type of model to understand loan guarantees, we relegate the technical description of the model and its parameterization, except for the pricing implications of guarantees, to the Appendix. We will note here only the essential model features, as follows. First, there is a large number (continuum) of households who each live for a finite number of periods. Second, households differ, ex ante, and permanently, in their earnings prospects. This is to reflect differences in the population with respect to educational attainment, and we will therefore allow for three classes of households: those who have not completed high school ("NHS"), those who have completed high school ("HS"), and those who have completed college ("Coll"). Third, households face shocks to their labor productivity throughout life, with shocks having both transitory and persistent components. Fourth, households face the risk of needing to spend suddenly in a manner that is involuntary. These "expenditure shocks" capture the idea that households sometimes face sudden health care expenses or legal obligations that make spending more or less involuntary. Such expenses have been viewed by researchers as relevant in at least a portion of observed default so, for completeness, we include them. Fifth, households have access to credit markets in which they may borrow (and where they will receive guarantees), and may save in a single risk-free asset that earns a constant interest rate.

The features just described lead households in our model to solve a consumption-savings problem over a well-defined life cycle in which their productivity has both deterministic and stochastic components. The risks faced by households include, most importantly, those that alter labor earnings, but also those that govern the marginal value of default. Because of default risk, lenders discount household promises according to their estimate of repayment likelihood.

We assume that the economy is small and open, so that the risk-free rate is exogenous. There is a representative firm that takes prices and wages as given, and demands labor as a function of its relative price (the wage). In equilibrium, the wage rate is part of the fixed point with the property that the representative firm's first-order condition governing the level of desired labor input at that wage is equal to household labor supply at that wage.

We will restrict attention throughout to stationary equilibria of the model--i.e., steady states. Stationary equilibria are, as usual, those outcomes where prices and the distribution of households over the state space remain constant under optimal household and firm decision-making. More intuitively, aggregate outcomes from our model can be viewed as averages (across households of all ages) for a single large cohort whose members each begin life with zero wealth, draw initial shocks from the unconditional distribution of shocks, and then draw shocks according to the stochastic processes we will specify further below.

To generate predictions from the model, we parameterize the environment to match a set of salient features, as displayed in Table 1. These targets are the ones most relevant for our analysis and collectively cover debt use and default-related features. The main message of Table 1 is that our baseline quantitative accurately captures key features of the data, and hence can be seen as reliable in its implications for the counterfactual exercises we will examine for alternative loan guarantee programs. Table 2 displays the specific parameters used in the quantitative model. We turn next to a description of how, in the quantitative model, loan guarantees will affect the terms on which credit is available to households.

Loan Pricing and Loan Guarantees

Loan guarantee regimes are defined by two parameters: the "replacement rate" [theta] and the "coverage limit" v. Only loans smaller than v qualify for any compensation; lenders making loans larger than the ceiling receive nothing in the event of default. (19) Conditional on default occurring, the lender, having made a loan of qualifying size, will receive partial compensation whereby the fraction [theta] will be paid to the lender for each unit of face value. (20)

We focus throughout on competitive lending whereby intermediaries utilize all available information to offer one-period debt contracts with individualized credit pricing that is subject to meeting a zero profit condition. Denote by I the information set of lenders. The information set is, under symmetric information, the entire state vector as understood by the household. In other words, I is the set of items that fully summarizes default risk to a lender for whatever level of borrowing the household requests.

Under asymmetric information, only a subset of these features is known. Specifically, the household's cost of default will not be known to the lender, who must then conjecture it, given the signal embedded in the households requested borrowing amount b. Denote this estimate by [??] (b, I). It is the lenders' best estimate of a household's default risk on a debt issuance with face value b. Of course, [??] (b, I) is identically zero for positive levels of net worth, and is also equal to 1 for some sufficiently large debt level. Denote by r the exogenous risk-free saving rate. In order to capture the costs associated with lending, we will also assume henceforth that lenders face a constant (i.e., proportional) transaction cost when lending. This implies that r + [phi] is the risk-free borrowing rate.

Given the preceding and the loan guarantee program parameters ([theta], v), the break-even pricing function on loans (b < 0) will depend on the size of the loan relative to the guarantee limit v as follows. Letting, as before, q(*) denote the price, or discount, applied to a bond issuance by a household, we note first that since only loans smaller than the guarantee ceiling entitle lenders to compensation, qualifying loans (those with b [member of] (-v, 0)) are priced as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

The first term, [[psi].sub.j+1|j], is new and represents the conditional probability of surviving to age j + 1 given survival to age j. Its presence in the pricing of loans reflects the fact that repayment occurs, if at all, only if the borrower survives. Conditional on survival, the payoff to a loan of face value b will be complete in the event of no default, which occurs with probability 1 - [??] (b, I), and partial, according to the guarantee, if default occurs. These payoffs are then discounted according to the cost of funds, inclusive of transactions costs, 1 + r + [phi]. For any loans exceeding the guarantee qualification threshold, lenders will receive nothing in the event of default. As a result, the preceding zero-profit loan price collapses (the second term goes to zero), yielding the simpler expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Lastly, savings are trivial to price, as they carry no transactions costs or default risk. Therefore, for b [greater than or equal to] 0, we have

q (b, I) = 1/[1 + r] if b [greater than or equal to] 0.

As for the fiscal implications of loan guarantees, the budget constraint for the government is straightforward: The flat tax rate on all earnings must be sufficient to make payments to all lenders whose borrowers defaulted on their debts.

Allocations

We will study four types of allocations. First, we examine our benchmark setting, where information is symmetric and there is no loan guarantee program. Second, we introduce various loan guarantee programs and examine how credit market aggregates, default rates, and welfare are altered. Third, we relax the assumption of symmetric information and study allocations without loan guarantees; in this setting we permit lenders to use all observable characteristics to infer as much as they can about borrowers. Finally, we examine the introduction of loan guarantees into this asymmetric information environment. We will refer to these four allocations as full information without loan guarantees (FI), full information with loan guarantees (FI-LG), asymmetric or "partial" information without loan guarantees (PI), and partial information with guarantees (PI-LG).

To preview the results, we find that introducing a small loan guarantee program into a symmetric information economy (comparing FI with FI-LG) can benefit all households, independent of type, but that increasing generosity quickly eliminates the gains for skilled types. To be clear, our measures of welfare, throughout, will be "ex ante": They are the gains of losses that a household entering the economy at the beginning of its life, i.e., as "newborns", as it were, would obtain. In the environments with asymmetric information (comparing PI to PI-LG), welfare gains are larger for any given generosity, but the same pattern emerges. Thus, a general lesson from these experiments is that loan guarantees are welfare-improving, and in fact can be welfare-improving for all newborns, provided they are not too generous.

Symmetric Information

As we noted at the outset, unsecured credit markets are most vital for the consumption smoothing needs of the least wealthy members of any society. This is obvious for any household with liquid wealth, but even those whose wealth is illiquid will, in general, be able to pledge at least a portion of that wealth to obtain credit. Moreover, as we noted, existing work suggests that information asymmetries may not be central, relative to the limited-commitment problem, in explaining current U.S. unsecured credit market activity. We therefore first isolate the role that loan guarantees and limited commitment play in dealing with the effects of such a friction by studying the model under symmetric information. Moreover, since loan guarantee programs require two parameters for their specification, we simplify the results by focusing throughout the analysis--and unless otherwise stated--on the case where the replacement rate is set to cover 50 percent of lender losses, i.e., [theta] = 0.5.

Allocations and Pricing

Our first main result is that loan guarantees are powerful tools in altering the use of unsecured credit. In Table 3, we see that as we move away from the case with no loan guarantees (v = 0), equilibrium borrowing rises for all households and the increase in debt is nonlinear. In particular, for small qualifying loan sizes (e.g., v = 0.1, or $4,000), allocations are fairly similar to a setting with no guarantees. In large part, this similarity reflects the presence of bankruptcy costs that serve as a form of implicit collateral. In particular, the fixed cost component of bankruptcy ([LAMBDA]) will ensure the existence of a region of risk-free debt. Therefore, under a small qualifying loan size, few individuals will see their access to credit substantially altered; in fact, setting V < -[LAMBDA] would have no effect on credit, since those loans are always risk free. Once the qualifying loan size grows large enough to make large loans "cheap" relative to default risk, matters are different. The compensation to lenders for default disproportionately subsidizes large loans and thereby generates the significant additional default seen in Table 3.

The differential distortion to loan pricing is displayed in Figure 2, and our model suggests that this feature helps account for the striking distributional consequences seen in Table 3. In particular, borrowing behavior changes in different ways across the education groups. Relative to income, debt rises by far the most for the least skilled (NHS) households. The differential increase in debt relative to income for the lowest skilled is also reflected in the disproportionate rise in bankruptcy rates within this group. While remaining modest under small qualifying loan ceilings, more generous ceilings create greatly increased default rates. The preceding suggests in part that the pricing of debt is a meaningful barrier to nearly all households, but especially NHS households. An additional force at work is that high-skilled households have less reason to use unsecured credit beyond early life. As a result, any distortion in the pricing of debt will affect them less than their NHS counterparts. In particular, all NHS households who have income below their age-specific mean will find "artificially" cheap credit useful, while the well-educated, many of whom wish to save less for precautionary reasons (i.e., to hedge against possible bad outcome for income in the future) and more for life-cycle (keeping consumption stable as household age) reasons, will be less sensitive to credit conditions. The latter insensitivity arises from the fact that an individual or household with a pronounced hump in their average earnings shown in Figure 3 will wish to save less or borrow when young, and save in the peak earnings years in order to have a comfortable retirement period (which the model captures by making households incapable of working beyond a certain age). Lastly, under high ceilings for qualifying loan guarantees, the high tax rate will also meaningfully compress the intertemporal profile of earnings, and therefore attenuate the incentives of the skilled to borrow for pure life-cycle smoothing. This will make loan guarantees even less valuable.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Welfare

Having shown results suggesting that loan guarantees will likely have sizeable and nonlinear effects on credit use and default, we now turn to the issue that motivated us at the outset: Can loan guarantees, by breaking the link between credit risk and loan pricing, improve welfare? And if so, for whom? Our metric for measuring welfare is standard: it is the change to consumption at all dates and states needed to make the household indifferent, in terms of ex ante expected utility, between the benchmark economy and the one with loan guarantees.

A fact that will be important for welfare is that households in our economy who borrow are always constrained. Figure 4 plots indifference curves in (b, q) space along with the zero-profit pricing function; the optimal amount of borrowing and the resulting price lies where the highest indifference curve intersects this zero-profit curve. At this point, the slope of the indifference curve is strictly smaller than the slope of the pricing function (which is infinite). This implies that borrowing more is desirable at the current interest rate, but the increase in the default rate that a marginal increase in b would generate means that lenders must charge a higher rate. As a result, by reducing the slope of the pricing function at the optimal point, loan guarantees can improve utility at the margin. What we are contemplating, however, are not marginal changes; thus, whether a discrete change is welfare-improving is a quantitative question.

[FIGURE 4 OMITTED]

We see first, from Table 3, that more generous loan guarantees come with higher taxes, and that the taxes also naturally reflect the nonlinearity in household borrowing and default behavior. However, not all households pay the same amount in taxes, and, as we noted, proportional taxes--which are used here--will by themselves provide some risk-sharing benefits. Moreover, the loan guarantee may allow for an effective form of insurance for some households, especially the low-skilled. The transfers from loan guarantees come "at the right time" for households, but require households to pay a cost, which, intuitively, is akin to a deductible on an insurance policy. Therefore, while households pay more in taxes under a generous loan guarantee scheme, they also receive transfers in a manner that is effective in providing insurance.

Turning to welfare in Table 4, we see that this is precisely what is at work. In this table a positive value indicates a gain to welfare from moving to loan guarantees, and vice versa. In particular, we see that generous loan guarantee schemes mainly represent transfers to the very unskilled. These are, in turn, the groups with the most to gain from improved credit access. As a result, the most skilled households lose in welfare terms from any qualifying loan sizes in excess of approximately $4,000 (v = 0.1). Conversely, HS households continue to gain, and gain substantially in welfare terms, from loan guarantees of up to $16,000 (v = 0.4). Most strikingly, NHS households gain for very large loan guarantee levels, even to levels exceeding their mean income level. In summary, our results suggest that modest loan guarantee programs can improve welfare for all households, even those households who likely will pay the bulk of the taxes needed to finance them. However, our model also suggests that qualifying loan size is likely to be quite important in determining whether a particular guarantee program serves all households or instead functions as a very significant redistributive mechanism. In the absence of definitive means for detecting the sensitivity of aggregate credit use and default to the size of qualifying loans, instituting a program that is too generous will lead to significant welfare losses for some groups.

Where do the welfare gains come from? Table 5 shows mean consumption and decomposes the variance of consumption into two moments: the variance of mean consumption by age, a measure of intertemporal consumption smoothing, and the mean of consumption variance by age, a measure of intratemporal consumption smoothing. (21) What we mean here is the following. "Intertemporal" smoothing refers to how much variation of consumption or living standards individuals experience. We measure it by calculating how much average consumption varies over the life cycle, and we average consumption as a natural measure of what the individual can expect at any given age. This is an intuitive measure of consumption smoothing through time: If the variance of average consumption over the life cycle were high, this would mean that young and old households were, on average, consuming quite different amounts. As for "intratemporal" smoothing, our measure answers the question of how much variability there is among households of any given age, when averaged across individuals of all ages. In other words, what is the average variability of consumption that one would expect to observe if one drew a sample of households of any given age? The decomposition of total variance into these components is a complete one: Together they account for the total variance of consumption in the model (and this is due to a simple statistical fact known as the "law of total variance").

Loan guarantees reduce average consumption due to the combination of higher taxes, more borrowing, and more frequent default. The gain comes through a better distribution of consumption over the life cycle. We see here that this gain is driven entirely by a reduction in the intertemporal dimension as intratemporal consumption volatility actually increases.

We note here that our welfare results differ significantly from those in Athreya et al. (2012), where the role of the out-of-pocket costs of default, A, in restricting access to bankruptcy is explored. High values of A restrict access to bankruptcy to high income types (who typically do not want to default), and in a wide range of models the optimal value (from an ex ante perspective) is infinite for all types; that is, from the perspective of a newborn household, permitting any bankruptcy in equilibrium is suboptimal. The largest gains are experienced by the college types, because they have the strongest demand to borrow for purely intertemporal reasons (i.e., reasons unrelated to the effects of uncertainty) and this demand is thwarted by risk-based pricing. There are a number of reasons to view that result as impractical from a policy perspective. Loan guarantees, in contrast, are clearly policy-feasible and benefit the least-skilled more than the more-skilled. (22)