Optimal Bankruptcy Code: A Fresh Start for Some∗ Grey Gordon† April 18, 2016 Abstract What is the optimal consumer bankruptcy law? I answer this question using an incomplete markets life-cycle model with a planner who can choose state-contingent bankruptcy costs. I develop two key theoretical characterizations. First, the optimal policy is bang-bang: Households are either given a fresh start or prevented from filing. Second, bankruptcy is always allowed for highly-indebted households. Quantitatively, the optimal policy induces more debt, more default, and lower interest rates than the U.S. with an ex-ante welfare gain of 12.4%. A simple rule captures key features of the optimal policy and produces a welfare gain of 11.6%. JEL Codes: D14, D52, D91, E21, K35 Keywords: Bankruptcy, Life-cycle Models, Incomplete Markets

∗

I thank Kartik Athreya, Satyajit Chatterjee, Daphne Chen, Bulent Guler, Aaron Hedlund, Juan Carlos Hatchondo, Aubhik Khan, Dirk Krueger, Amanda Michaud, Victor R´ıosRull, Mich`ele Tertilt, Julia Thomas, and Eric Young. I also thank conference participants from the European Meeting of the Econometric Society 2014, Computing in Economics and Finance 2014, Midwest Macro 2013 and 2014, and a UIUC mini-conference in 2014, as well as seminar participants at Florida State University, the Ohio State University, and Purdue University. † Indiana University, Dept. of Economics, 100 S. Woodlawn Ave., Bloomington, IN 47405. Email: [email protected].

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1

Introduction

Bankruptcy policy varies greatly by time and location. In many European countries, there is little to no debt forgiveness. Bankruptcy laws in the United States, on the other hand, are widely considered pro-debtor. Moreover, views on the proper amount of debt forgiveness have changed dramatically over the last two hundred years. In the U.S., debtors’ prisons have been replaced with a relatively swift bankruptcy process, which, until recently, offered a nearcomplete discharge to almost everyone. In 2005, the Bankruptcy Abuse Prevention and Consumer Protection Act restricted this near-complete discharge to only those with below-median income, forcing above-median income households to pay all their disposable income (in the specific sense defined by the law) for five years.1 Which of the many possible bankruptcy laws—ranging from complete discharge for all to no discharge at all—is best? To answer this question, I use an incomplete markets life-cycle model of bankruptcy and allow a planner to choose state-contingent bankruptcy penalties. The planner specifies whether a household may file for bankruptcy, any pecuniary costs associated with filing, and how long a bankruptcy remains on a household’s credit record. In choosing these penalties, the planner faces a trade-off between improving credit and adding state-contingency to debt repayment. I analytically characterize the planner’s optimal policy for a broad class of utility functions, social welfare functions, and labor efficiency processes. I find two main results. First, the optimal policy has a bang-bang property: The planner either gives a household a fresh start, i.e., a costless bankruptcy process, or forbids it from filing. The fresh start in the optimal policy deviates substantially from current U.S. policy in that there are no filing costs and a bankruptcy filing never impacts a household’s credit record. Second, I 1

Robe, Steiger, and Michel (2006) document both the historical and geographical variation in bankruptcy laws, and Coleman (1974) documents the variation in U.S. bankruptcy laws over time. As Wang and White (2000) summarize, “the United States is extremely unusual in having very prodebtor bankruptcy laws” (p. 255). For details on the Bankruptcy Abuse Prevention and Consumer Protection Act, see White (2007).

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find that bankruptcy should always be allowed in some states. In particular, if a household cannot repay their debt or would otherwise prefer to informally default via an outside option, bankruptcy should be allowed. As a corollary of this result a natural borrowing limit economy—an economy where households have maximal commitment to repay their debt—is suboptimal. I show theoretically that the outside option can be made irrelevant, as if not available to households. Quantitatively, I find the optimal policy produces a consumption equivalent welfare gain of 12.4% relative to an economy calibrated to match the U.S. It does so by allowing young households to work more in midlife, when labor efficiency is highest, and move those extra earnings forward in time to increase consumption when young. In the optimal policy, households typically accumulate debt equal to half of average earnings by their mid-30s and only begin accumulating retirement savings starting in their mid-40s. The optimal policy results in a small increase in the bankruptcy filing rate, a fifteen-fold increase in debt, and 50% lower interest rates and charge-off rates. The optimal policy is mostly intuitive. Specifically, it allows a fresh start to those with bad persistent shocks, high debt levels, and also those whose are older than 40 (when borrowing for life-cycle reasons transitions to saving for life-cycle reasons). Perhaps surprisingly, the optimal policy is much less sensitive to transitory shocks than persistent ones. The reason is the former can be easily self-insured through credit while the latter cannot. I also find that a simple rule, specifically a linear function of the shocks, logged debt, and an age dummy, generates a welfare gain of 11.6%, only one percentage point less than the 12.4% from the optimal rule. In addition to being almost completely inelastic to transitory shocks, the simple rule is more cautious in allowing bankruptcy than the optimal rule. This caution lets the planner provide bankruptcy where it is almost certainly useful while improving credit and self-insurance elsewhere.

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1.1

Related Literature

This paper is most closely related to Grochulski (2010) and Chen and Corbae (2011). In the former, a social planner faces the Mirrleesian problem of delivering consumption subject to an incentive compatibility constraint (derived from truthful reporting of household income). The optimal policy is solved for and then implemented using bankruptcy. The focus in this paper is on full information, which is natural since current U.S. bankruptcy law requires substantial information disclosure, and what the planner can legislate. Grochulski (2010) also requires a two-state income process and a smooth utility function whereas the present paper allows any discrete Markov efficiency process and any continuous, increasing utility function that is unbounded below. Chen and Corbae (2011) use a calibrated model to investigate the optimal length of time for a bankruptcy to remain on a household’s credit record. Interestingly, they find that welfare monotonically decreases in the duration. This result is analogous to the present paper’s theoretical finding that a duration of zero is optimal. The life-cycle bankruptcy model I employ is essentially Livshits, MacGee, and Tertilt (2007) augmented to have elastic labor supply (with inelastic as a special case) but without expenditure shocks, i.e., direct shocks to a households net worth.2 Livshits et al. (2007), like Grochulski (2010) and Chen and Corbae (2011), are part of a large literature investigating the effects of various consumer bankruptcy policies. A non-exhaustive list of these papers is Li and Sarte (2006); Chatterjee, Corbae, Nakajima, and R´ıos-Rull (2007); Athreya, Tam, and Young (2009a,b); Chatterjee and Gordon (2012); and Gordon (2015). The overriding theme is that harsher bankruptcy laws almost always improve 2

Labor efficiency shocks, with their implied reductions in lifetime income, can capture some of the risk inherent in expenditure shocks. However, for very large (and hence rare) expenditure shocks, the risk may be too great to be captured via earnings risk alone. If households hit by these large shocks were allowed a fresh start and allowed to keep their assets, then the risk from these shocks would be small. Additionally, interest rates would be only slightly higher by virtue of the shocks being rare. Consequently, the results would seemingly differ little if expenditure shocks were included but (1) this type of policy applied to households hit by large shocks and (2) this paper’s optimal policy applied to all other households.

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welfare, even to the point that eliminating bankruptcy greatly improves welfare in the absence of shocks to household expenditures. My findings agree with these results in that eliminating bankruptcy produces large welfare gains relative to the U.S. However, I show theoretically and quantitatively that it is always better to allow bankruptcy in some states. Beginning with Li and Sarte (2006), a number of papers (Chatterjee and Gordon, 2012; Athreya, Sanchez, Tam, and Young, 2012a, 2014; Chen, 2013; Li and White, 2009; Li, White, and Zhu, 2011; Mitman, 2011, among them) have examined the effects of having a default option in addition to Chapter 7 bankruptcy. They have found that the inclusion of alternate default options substantially alters equilibrium behavior. The present paper’s inclusion of an outside option is meant to represent, in a reduced form way, a household’s best informal default option, i.e., the best non-bankruptcy default option.3 By allowing the outside option value to be very high or low, and by not placing any regularity conditions on it, it can capture many types of default options. The theory assumes creditors receive nothing in the case of default, which does limit its generality. However, recovery rates on defaulted debt tend to be quite low: Chatterjee and Gordon (2012) report the average net recovery rate on defaulted debt is only 12-14%.

2

Model

The model is a version of Livshits et al. (2007) having state-contingent default costs and endogenous labor supply.

2.1

Model Setup

The economy is populated with a continuum of households who live for at most T periods. The household labor efficiency e is strictly positive and follows a 3

While one could potentially think of the outside option as representing Chapter 13 bankruptcy, it is better to think of Chapter 7 and 13 as being replaced by a unified bankruptcy policy.

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finite-state Markov chain πee0 |t that is age-dependent. Newborn households draw e from a distribution πe . Markets are incomplete with households only having access to a bond a ∈ A with a < 0 being unsecured debt (negative net worth) and a ≥ 0 being savings. A is a finite set containing negative and positive elements, as well as zero. For simplicity, I assume households may not default on savings a ≥ 0. Newborn households have a = 0. Households have a bankruptcy flag h indicating whether a bankruptcy is on their credit record, h = 1, or not, h = 0. All households begin life with h = 0. When a household defaults or has a bankruptcy record, it is assumed that they may not borrow. This restriction is for tractability, and it means that only one bankruptcy law is needed (since households will not default on a ≥ 0) rather than two. Preferences are additively time-separable over consumption c and labor n with a discount factor β > 0. Labor is chosen from a finite set N ⊂ R++ , and I assume the period utility function u(c, n) is well-defined for any c > 0 and any n ∈ N . The period utility u is continuous and strictly increasing in c—but not necessarily concave or differentiable—and decreasing in n. Additionally, it is unbounded below with limc↓0 u(c, n) = −∞ for all n. Having utility unbounded below is important for the theoretical results. However, it does mean u(0, n) is not defined, and so I require households choose c > 0 (the choice set is still compact because a0 is restricted to lie in a finite set). Note that the commonly used constant relative risk aversion (CRRA) preferences are also not defined at zero when CRRA is greater than one. Additionally, I allow for psychic costs of default, κ(e) ≥ 0, that depend on household efficiency. These are assumed invariant to planner policy. A household’s state is (a, e, t, h). The price of a discount bond with face value a0 is qt (a0 , e). For savings, a0 ≥ 0, default is not allowed, and so the price for such a bond is simply equal to the risk-free rate of transferring resources across time, q¯. For debt, a0 < 0 (which implies h = 0), creditors expect a

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repayment rate pt (a0 , e). Consequently, a no-arbitrage condition has qt (a0 , e) = q¯pt (a0 , e).

(1)

When the repayment rate pt is low, so is the discount qt , which implies a high interest rate 1/qt − 1. Equilibrium requires the repayment rate pt (a0 , e) is consistent with household default decisions. For convenience, I define pt (a0 ≥ 0, e) = 1 so that qt (a0 , e) = q¯pt (a0 , e) for all a0 , e. Bankruptcy policy is defined by the instruments available to the planner. Specifically, the planner can do all of the following: 1. Specify whether a household is allowed to file for bankruptcy, Dt (a, e, n) = {0, 1}, or not, Dt (a, e, n) = {0}. 2. Confiscate any fraction χt (a, e, n) ∈ [0, χ] of a household’s earnings in the period of filing with χ < 1. 3. Charge a lump sum bankruptcy filing cost ζt (a, e, n) ≥ 0. 4. Retain a bad credit record with probability λt ∈ [0, 1]. Note that (1) the planner is allowed to condition on n and e and that (2) n and e could be inferred from earnings and hours worked. All the policies condition only on the household’s current state rather than the household’s entire history, and this has two major consequences. First, implementing the bankruptcy policy does not require “full” information, but only perfect information at the time of a household’s bankruptcy filing. The latter assumption is reasonable since, under current law, households disclose their financial records at the time of filing.4 Second, potentially interesting policies, such as those conditioned on the number of times a household has filed, are ruled out by assumption. The 4

Additionally, creditors need enough information to price risk appropriately. While asymmetric information has important implications for consumer bankruptcy, as seen in Drozd and Nosal (2008), S´ anchez (2010), Narajabad (2012), Athreya, Tam, and Young (2012b), and Livshits, MacGee, and Tertilt (2014), it is challenging theoretically and computationally. A worthwhile topic for future research is determining how asymmetric information changes the optimal policy. It should be noted that the Livshits et al. (2007) model adopted here, along with its assumption of full information, is very commonly used in the literature.

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policies available to the planner cover, as special cases, many of the types of penalties that the literature has used to model bankruptcy. I assume that the risk-free rate is invariant to planner policy. This could be potentially important as bankruptcy policy, in this paper, causes substantial changes in aggregate savings. However, this paper does not try to match the earnings and wealth distributions. In a similar model that does, Gordon (2015) finds that bankruptcy policy has very limited general equilibrium effects because wealth is concentrated among households who are unlikely to ever borrow or default. Another potentially important assumption is that the planner cannot partially forgive debt. That is, he must either forgive all of a household’s debt or not forgive any of it. However, this is not as restrictive as it seems because the optimal policy can effectively incorporate partial forgiveness via a lottery.5 Households always have access to an outside option delivering VtO (a, e) in lifetime discounted utility terms. I make three key assumptions regarding the outside option. First, if a household exercises its outside option, the creditor gets nothing. While this is restrictive, recovery rates on defaulted debt can be as low as 12-14% net of collection costs (Chatterjee and Gordon, 2012). Second, VtO (a, e) is less than or equal to an “autarky” value VtA (a, e). I deviate from the usual definition of autarky to allow for savings at the risk-free rate and apply a psychic cost of default in the first period of autarky. That is, VtO (a, e) ≤ VtA (a, e) := Xt (0, e) − κ(e) where Xt (a, e) =

max u(c, n) + β 0

a ∈A,n∈N

X

πee0 |t Xt+1 (a0 , e0 ) (2)

e0

0

0

s.t. c + q¯a = en + a, c > 0, a ≥ 0 5

For instance, suppose the efficiency process is a discretized version of e = exp(ε) with

iid

ε ∼ U [0, σ]. If σ is close to zero, then the risk due to fluctuations in the endowment is negligible. By then choosing a cutoff θ ∈ [0, σ] such that Dt (a, e, n) = {0} if e ≥ exp(θ) and Dt (a, e, n) = {0, 1} if e < exp(θ), the planner can forgive a fraction θ/σ of the debt −a. Assuming households file for bankruptcy when allowed to (and ignoring the outside option), the price of credit would then be q¯(1 − θ/σ), exactly the same as under partial repayment. At the cost of additional notation, this argument generalizes to any efficiency process that incorporates a transitory iid shock.

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with XT (a, e) := maxn∈N,en+a>0 u(en + a, n). Third, VtO (a, e) is invariant to policy. While this might seem restrictive, it need not be. For instance, if choosing the outside option results in being permanently barred from credit markets, then the outside option would be independent of planner policy as it operates only through credit and default. Consequently, this assumption is valid to the extent that exclusion from credit due to informal default is long-lasting. While the inclusion of an outside option is not completely standard, it is necessary to have household behavior well-defined everywhere. Without it, the planner could deny a household access to bankruptcy despite debt repayment being infeasible. Were that the case, the household problem would not be well-defined. One potential way to resolve that is to assume households have full commitment to only enter states where a feasible policy exists. However, that assumption seems unnatural given this paper’s focus on lack of commitment. Additionally, a number of recent papers (Chatterjee and Gordon, 2012; Athreya et al., 2012a, 2014) have highlighted the importance of informal default. At any rate, Proposition 2 shows the outside option can be set low enough so as to never be chosen along the equilibrium path (for any planner policy).

2.2

Household Problem

Given the policy instruments and prices, the value function of a household with a, e, t who has no record of a bankruptcy is6 Vt (a, e, h = 0) = max oVtO (a, e) + dVtD (a, e) + (1 − o)(1 − d)VtR (a, e) o,d [ s.t. o ∈ {0, 1}, d ∈ Dt (a, e, n), od = 0

(3)

n∈N 6 To cutdown on notation, the definition below treats bankruptcy and repayment both as feasible options. If bankruptcy is not feasible, then the value function should be reformulated without reference to V D and similarly for V R . Additionally, default options are only available if a < 0. To handle the case of t = T , take VT +1 = 0 and qT = 0.

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where the value of repaying debt is VtR (a, e) =

max u(c, n) + β 0

X

a ∈A,n∈N

πee0 |t V (a0 , e0 , 0)

e0 0

(4)

0

s.t. c + qt (a , e)a = en + a, c > 0 and the value of defaulting via bankruptcy is VtD (a, e) =

max u(c, n) − κ(e) + β 0

X

a ∈A,n∈N

πee0 |t ((1 − λt )Vt (a0 , e0 , 0) + λt Vt (a0 , e0 , 1))

e0

s.t. c + q¯a0 + ζt (a, e, n) = en(1 − χt (a, e, n)), c > 0, a0 ≥ 0. Dt (a, e, n) = {0, 1} (5) Note that since bankruptcy policy conditions on labor, to obtain a discharge the household must choose labor such that Dt (a, e, n) = {0, 1}. If a household’s budget constraint would be empty upon choosing to repay, they must choose either the outside option or, if the planner lets them, bankruptcy. The value of having a bankruptcy record, h = 1, is Vt (a, e, h = 1) =

max u(c, n) + β 0

a ∈A,n∈N

X

πee0 |t ((1 − λt )Vt (a0 , e0 , 0) + λt Vt (a0 , e0 , 1))

e0

c + q¯a0 = en + a, c > 0, a0 ≥ 0. (6) Note that I do not allow households in bad standing access to an outside option. This is without loss of generality (wlog) because VtO (a, e) is less than the value of autarky and Vt (a, e, h = 1) is greater than it.

2.3

Equilibrium

Steady state equilibrium for a given set of policy instruments, Dt , λt , χt , ζt and a risk-free price q¯ is a set of policies, ct , nt , a0t , dt , ot , value functions Vt , prices qt , and a distribution of households such that 1. Households optimize taking prices as given. 2. The price schedule qt is given by no arbitrage: qt (a0 , e) = q¯pt (a0 , e). 10

3. Repayment rates pt are consistent: pt (a0 , e) =

X

πee0 |t (1 − max{dt+1 (a0 , e0 ), ot+1 (a0 , e0 )}).

(7)

e0

4. The distribution of households is invariant.

2.4

Planner’s Problem

The planner solves max

χ,ζ,λ,D

X

αt (a, e)Vt (a, e, h = 0)

a,e,t

s.t. Dt (a, e, n) ∈ {{0, 1}, {0}}, λt ∈ [0, 1], χt (a, e, n) ∈ [0, χ], ¯ ζt (a, e, n) ≥ 0 ∀ a, e, n, t (8) where αt (a, e) ≥ 0 is the weight placed on type (a, e, t, h = 0). Note that I restrict the weights to be zero for h = 1. This is not to say the planner does not care about utility in the h = 1 states: He implicitly does if they are visited by a household having a state (a, e, t, h = 0) with αt (a, e) > 0. However, I show theoretically that for any policy with λt > 0, there is another policy with λt = 0 generating the same planner utility. Since the case of λt = 0 results in h = 1 being off equilibrium, restricting αt to only place weight on h = 0 is convenient.

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Theoretical Results

This section characterizes the optimal policy theoretically. To simplify the proofs, households are assumed to repay when indifferent between bankruptcy and repayment. If indifferent between bankruptcy and the outside option, they are assumed to file for bankruptcy if it is an option. All proofs are relegated to Appendix A. Before characterizing the optimal policy, it is essential to establish that equilibrium exists given planner instruments. The only way this would not be the case is if a solution to the household problem did not exist. However, since 11

households only have a finite number of choices, and they always have at least one feasible choice (specifically, never borrowing or saving), this is guaranteed. Proposition 1. For any policy choice of the planner, an equilibrium exists. As discussed above, the inclusion of an outside option is not completely standard in the default literature. However, Proposition 2 shows that the outside option can be made irrelevant: If V O is sufficiently negative, the outside option will never be chosen in equilibrium. One consequence of this result is that the planner can completely eliminate equilibrium default when V O is sufficiently low: By specifying Dt (a, e, n) = {0} for all a, e, n, t, households always choose to repay their debt and the model is equivalent to a standard Aiyagari (1994) model. The key to the proof is that a lower bound on lifetime utility would be violated if the outside option were chosen in equilibrium and was sufficiently unattractive. Proposition 2. There exists a δ such that VtO (a, e) < δ for all a, e, t implies the outside option is never chosen in equilibrium. As in the models of Chatterjee et al. (2007) and Livshits et al. (2007), bankruptcy entails a proportional earnings cost. However, with the flexible specification for the lump-sum filing cost, the planner need not use it: Households only care about the after-cost earnings schedule en(1 − χt (a, e, n)) − ζt (a, e, n), not the exact composition of it. Hence, χt (a, e, n) can be set to zero with ζt (a, e, n) increased to generate the exact same schedule. This is formally established by Lemma 1, and attention from now on is restricted to policies with a zero proportional cost for earnings. Lemma 1. Wlog, the planner policy specifies χt (a, e, n) = 0 for all a, e, n, t. A principal difficulty in characterizing bankruptcy policy is that improveR D ments in Vt also improve Vt−1 and Vt−1 through continuation utility. Depending R D on how Vt−1 − Vt−1 changes, default may become more attractive, worsening credit and potentially welfare for age t − 2 households. However, Lemma 2 D problem can be undone shows the increase in continuation utility in the Vt−1 12

through a commensurate increase in the ζt−1 filing cost. This causes the spread R D Vt−1 − Vt−1 to never decrease, which effectively allows Vt increases to trickle down to younger households. Lemma 2. Fix some (a, e, t) with t < T and suppose VtD (a, e) is well-defined in that it has at least one feasible choice. If the continuation utility of the VtD (a, e) problem increases due to some policy change, VtD (a, e) can be held constant by increasing ζt (a, e, n) for all n. Almost as a corollary of this result is that λt > 0, i.e., retaining a bankruptcy flag, is completely unnecessary. Specifically, lowering λt to zero increases continuation utility in the VtD problem, but that effect can be undone via higher filing costs. Proposition 3 formally establishes that λt = 0 is weakly optimal. This result stands in sharp contrast to the U.S. system where a bankruptcy filing affects a household’s credit report for ten years. Proposition 3. For any policy with λt > 0, there exists another policy with λt = 0 generating the same planner utility. There are two caveats to this result. First, it relies on full information. Because of it, there are no bad borrowers: Borrowers prone to default are known and the risk is priced perfectly. If some borrowers were more prone to default, this propensity was private information, and the propensity persisted over time (i.e., was not iid), there would be some incentive to exclude these bad borrowers. Second, it relies on partial equilibrium. That is because excluding households would induce precautionary savings and thereby increase the aggregate capital stock. In general equilibrium, that could be valuable. However, given how little low-income households contribute to the capital stock, this effect would likely be small. From this point on, I will wlog consider only policies that have λt = 0 for all t. A convenient property of this is that there is no longer a need to carry around a bankruptcy flag (as households are born with h = 0 and never transition to h = 1). So, I write Vt (a, e) in place of Vt (a, e, h = 0) and similarly for the policy functions, never referring to Vt (a, e, h = 1) or its policy functions. 13

Another convenient property is that the continuation utilities of the repayment and bankruptcy filing problems are now the same. Lemma 3 establishes a key method for the planner to improve on an existing policy. Specifically, suppose the planner can improve Vt while lowering max{dt , ot }. In this case, two indirect benefits accrue to age t − 1 households. First, they have improved continuation utility. Second, they have improved D R . By in, and Vt−1 debt pricing. These effects else equal increase Vt−1 , Vt−1 D fixed (Lemma 2), Vt−1 increases and creasing t − 1 filing costs to hold Vt−1 max{dt−1 , ot−1 } decreases. That is, improved welfare and credit for age t households can be translated into improved welfare and credit for age t − 1 households. Using backwards induction, these improvements can be applied to all younger households. ˜ D) ˜ be a policy that for Lemma 3. Consider an arbitrary policy (ζ, D). Let (ζ, some t has Vt˜(a, e) higher—relative to (ζ, D)—for all a, e, t˜ ≥ t and max{dt (a, e), ot (a, e)} lower for all a, e. Then there exists a policy (ζ ∗ , D∗ ) satisfying ˜ t˜ for all t˜ ≥ t 1. ζt˜∗ = ζ˜t˜ and Dt˜∗ = D 2. Dt˜∗ = Dt˜ for all t˜ < t. that has higher Vt˜(a, e)—relative to (ζ, D)—for all a, e, t˜. Moreover, for all ˜ D) ˜ induce the same Vt˜(a, e). a, e, t˜ ≥ t the policies (ζ ∗ , D∗ ) and (ζ, Proposition 4 shows that the planner should not tie debt forgiveness to labor supply. The intuitive reason is that only allowing bankruptcy for some amounts of labor (or, equivalently, earnings) distorts the household’s decision and inflicts a dead-weight cost. Specifically, if the planner wants a household to file in some state a, e, t, then choosing Dt (a, e, n) = {0, 1} for only some values of n constrains the household’s labor choice and reduces VtD relative to a policy with Dt (a, e, n) = {0, 1} for all n. Hence, using instead Dt (a, e, n) = {0, 1} for all n improves Vt (a, e) without changing max{dt (a, e), ot (a, e)} = 1. On the other hand, if the planner does not want a household to file, he may simply set Dt (a, e, n) = {0} for all n without affecting welfare or credit. Consequently, there is no reason to tie debt forgiveness to hours worked. 14

Proposition 4. For any policy (ζ, D) specifying Dt (a, e, n) varying in n for some a, e, t, there is a policy (ζ ∗ , D∗ ) with Dt˜∗ (˜ a, e˜, n) invariant to n and Vt˜(˜ a, e˜) higher for all a ˜, e˜, t˜. Without loss of generality, I now restrict the planner to choosing policies with Dt (a, e, n) invariant to n for all a, e, t. Note that in Lemma 3, if D and ˜ are invariant to n, then the D∗ policy is as well. D Another dead-weight loss is induced by having strictly positive filing costs. Specifically, if the planner does not want a household to file, he may simply not let them by setting Dt (a, e) = {0}. On the other hand, if he does want them to file, then the damage to creditors, max{dt (a, e), ot (a, e)}, is not mitigated by having ζt (a, e, n) > 0. Hence, in either case it is optimal to set ζt (a, e, n) = 0. Proposition 5 establishes this important result. Proposition 5. Consider a policy (ζ, D) and define t = max{0}∪{t | ∃ a, e, n with ζt (a, e, n) > 0}. If t > 0, then there is a policy (ζ ∗ , D∗ )—identical to (ζ, D) for t > t—with ζ ∗ equal to zero that has higher Vt (a, e) for all a, e, t. If under the original policy ζt (a, e, nt (a, e)) > 0 and dt (a, e) = 1, then Vt (a, e) is strictly higher under the new policy. Propositions 3, 4, and 5 provide the first main result of the paper: The planner should either allow a household to file for bankruptcy, making them as well of as possible, or completely prevent them from filing. The optimal policy thus differs from U.S. bankruptcy policy in two key ways. First, bankruptcy in the U.S. has both direct costs in the form of filing costs and indirect costs in the form of exclusion. The optimal policy has neither and allows—in the truest sense—a fresh start. Second, U.S. law with few exceptions allows every household to file. In the optimal policy, access to bankruptcy is restricted. Because the planner’s problem reduces to choosing Dt (a, e, n) ∈ {{0}, {0, 1}} for all a, e, t (and an arbitrary n), the planner’s choice set is finite. Moreover, every choice is feasible as Proposition 1 shows an equilibrium exists for any planner policy. Consequently, an optimal policy exists, and this is established formally in Proposition 6.

15

Proposition 6. An optimal policy with χt (a, e, n) = ζt (a, e, n) = λt = 0 for all a, e, n, t and with Dt (a, e, n) invariant to n for all a, e, t exists. In the absence of filing costs and labor distortions, bankruptcy is always better than autarky, which itself is better than the outside option by assumption. Hence, formal default via bankruptcy is always preferred in welfare terms to informal default via the outside option: V D ≥ V O . Additionally, creditors receive nothing when a household defaults irrespective of whether the default is formal (bankruptcy) or informal (the outside option). Consequently, the planner should always allow bankruptcy whenever the outside option would otherwise be preferable. Proposition 7 formally establishes this second main result. Proposition 7. Suppose a policy (ζ, D) has ζ equal to zero. If Dt (a, e, n) = {0} for some a, e, t where VtO (a, e) > VtR (a, e) or VtR (a, e) is undefined, then there is another policy (ζ ∗ , D∗ ) with ζ ∗ equal to zero that (1) specifies Dt˜∗ (˜ a, e˜, n ˜) = {0, 1} whenever Vt˜O (˜ a, e˜) > Vt˜R (˜ a, e˜) or Vt˜R (˜ a, e˜) is undefined and (2) has higher Vt˜(˜ a, e˜) for all a ˜, e˜, n ˜ , t˜. A trivial consequence of Proposition 7 is that the planner should not set D = {0} for all states: Because V R is undefined if debt is large enough, D = {0, 1} is always optimal for highly indebted states. One reason that is interesting is a natural borrowing limit economy, formally stated below, will never be optimal. Definition. A natural borrowing limit economy is one in which, for all a, e, n, t, Dt (a, e, n) = {0} and VtO (a, e) is sufficiently low, in the sense of Proposition 2, as to make the outside option irrelevant. This definition precisely captures the notion of a natural borrowing limit economy in Aiyagari (1994). Specifically, (1) households always choose to repay their debt; (2) they can borrow at a risk-free rate up to the net present value of the worst possible earnings stream; and (3) they never borrow more than this amount. (While labor is elastically supplied here, one can have inelastic labor supply, like in Aiyagari, 1994, by making N a singleton.) Because a natural 16

borrowing limit requires D = {0} for all states, even for states where debt is extremely large and hence repayment is not feasible, a natural borrowing limit economy is suboptimal. This is formally stated in Corollary 1. Corollary 1. A natural borrowing limit economy is weakly inferior to one that allows bankruptcy whenever VtO (a, e) > VtR (a, e) or VtR (a, e) is undefined. This result settles an important question of whether the natural borrowing limit is optimal. This question has been investigated quantitatively by numerous authors who find that implementing a natural borrowing limit greatly improves welfare relative to economies calibrated to match U.S. statistics. However, this result shows that for a wide class of utility functions and labor efficiency processes—even efficiency processes where the natural borrowing limit is very large—it is always better to allow default in some states. Another trivial consequence of Proposition 7 is stated in Corollary 2: Optimally, the policy results in the outside option never being chosen in equilibrium. This stands in contrast to the U.S. where informal default is a regular occurrence. The basic intuition is that formal default can replicate or improve on whatever informal default accomplishes, both for households and creditors. In the model, informal default, like bankruptcy, results in a total loss for creditors. However, in settings with only partial loss this intuition, that bankruptcy can always do at least as well as informal default, would seem to carry over as long as administrative costs of bankruptcy were not too large. Corollary 2. Without loss of generality, the optimal policy has the outside option never chosen in equilibrium. While the outside option is never chosen, it still plays a critical role. Specifically, it pins down in which states the planner “must” allow a household to default. The planner allows bankruptcy whenever VtR (a, e) < VtO (a, e). If V O is very negative, this only occurs for large levels of debt. If V O equals the value of autarky, this occurs for small amounts of debt. In fact, if the outside option value is equal to autarky and there are no psychic costs, Proposition 8 shows that the planner can do no better than 17

a zero borrowing limit economy. The result obtains because bankruptcy is optimally allowed whenever the outside option is better than repayment. For t = T , there is no borrowing so the value of repaying debt is strictly less than the value of autarky. Consequently, repayment is worse than the outside option and so the planner allows bankruptcy for all indebted households. This then causes qT −1 = 0, resulting in all households aged T − 1 having value of repaying less than the value of autarky and the outside option. This process repeats resulting in a full collapse of the credit market. The result is analogous to that in Bulow and Rogoff (1987) where reputation alone is not enough to sustain debt. Proposition 8. Suppose there are no psychic costs of bankruptcy, κ(e) = 0. If VtO (a, e) = VtA (a, e) for all a, e, t, then an optimum features qt (a0 , e) = 0 for all a0 < 0 and all e, t. That is, an optimum features a zero borrowing limit. Moreover, if αt (a, e) > 0 for all a, e, t, then every policy produces Vt (a, e) identical to that of a zero borrowing limit economy for all a, e, t. If default outside the bankruptcy system is too attractive, nothing can be gained from bankruptcy. So a lesson from this result is that to maximize welfare gains from the bankruptcy system, informal default should be made as unattractive as possible. While the planner cannot affect V O by assumption, if he could, the optimal policy would have V O be irrelevant. In line with this thinking, the quantitative work in the next section will take V O to be irrelevant in the benchmark.

4

Quantitative Results

The theoretical results show the planner’s problem reduces to choosing who may or may not file for bankruptcy. Proposition 7, establishes that the planner specifies Dt (a, e, n) = {0, 1} for all n whenever VtR (a, e) is less than VtO (a, e) or is undefined. Hence, the optimal policy is pinned down for such states. For some other states, specifically those with VtR (a, e) greater than VtD (a, e), households choose dt (a, e) = 0 irrespective of Dt (a, e, n) since repayment is always 18

a feasible option. Hence in those states, planner policy is irrelevant. However, for states where VtR (a, e) is greater than VtO (a, e) but less than VtD (a, e), the theory is incomplete. In such states, households will default if and only if the planner lets them, i.e., they choose dt (a, e) = max Dt (a, e, n). Since the theory does not say what the planner should do in those cases, the model is now brought to the data to better characterize the optimal policy. A model period is three years (as in Livshits et al., 2007). Households age 1 (real age 24) to R − 1 (real age 63) have one labor efficiency process, representing working age risk, and households aged R (real age 66) to T (real age 84) have another, representing guaranteed retirement income. The working age process is estimated from the Panel Study of Income Dynamics (PSID) data. As is typical in the literature, the retirement process has no risk, and it is calibrated to match an average replacement rate.

4.1

Estimation

The PSID data I use is from Heathcote, Perri, and Violante (2010) and ranges from 1967 to 2002. It has been cleaned and processed in a number of ways, such as extrapolation of top-coded values assuming a Pareto distribution and dropping observations with implausible levels of consumption. In addition to these assumptions, I restrict the sample to individuals with ages between 24 and 63 and—for the efficiency process estimation—require the head and, if present, the “wife” (in the PSID sense) work a combined amount of at least 260 hours. I assume the process has the form log ei,t = ν0 + ν1 h + ν2 h2 + ν3 h3 + ui,t ui,t = zi,t + εi,t

(9)

zi,t = ρzi,t−1 + ηi,t 2 ), ηi,t ∼ N (0, ση2 ), εit ∼ N (0, σε2 ) zi,1 ∼ N (0, ση,1

where h is age over 10, t is age minus 23, and all shocks are independently and identically distributed. Allowing for a separate variance of the persistent shock early in life allows, to some extent, for features that are missing from the model 19

such as college attainment, race, and gender. I measure ei,t using equivalized total labor earnings (the head and wife’s joint labor earnings) divided by total hours (head and wife’s joint hours worked). Conceptually, ei,t captures the extra earnings accruing to an individual if hours worked are scaled in proportion to the current division of labor in a household. If the household has size one, it is just the usual definition of the wage. The coefficients are identified following a procedure similar to Heathcote et al. (2010). The age profile coefficients ν1 , ν2 , ν3 are obtained from an OLS regression that controls for time effects (empirically, ν0 is also obtained from OLS, but it is used as a normalization constant in the model). The shock 2 , ση2 , σ2 are identified using the variances Et (u2i,t ) and process parameters ρ, ση,1 the second-order autocovariances Et (ui,t ui,t+2 ) for each age, year, and cohort. As in Heathcote et al. (2010), second-order autocovariances are used because observations only occur every two-years after 1995. In total, 292 moments identify the 4 shock process parameters, and I use the optimal GMM estimator. First-stage regression ν1 ν2 ν3 N R2

-0.959 0.237 -0.017 142595 0.030

(0.080) (0.020) (0.002)

GMM ρ ση2 2 ση,1 σε2

0.969 (0.003) 0.027 (0.002) 0.233 (0.004) 0.065 (0.002)

Table 1: PSID Estimation Table 1 displays the parameter estimates. The first-stage regression coefficients imply mean labor efficiency declines 3% from age 24 to 30 (recall equivalized wages are used) and then grows steadily from age 30 to 64 for a total increase of 36%. The estimated shock process is highly persistent with parameters remarkably similar to those in Storesletten, Telmer, and Yaron (2004). This is despite a large number of differences, foremost being a different dependent variable. As can be seen in Figure 1, the estimates accurately capture the data’s age profile of residual wage variance.

20

0.5

Data Estimates

Residual variance

0.45

0.4

0.35

0.3 30

40

50

60

Age

Figure 1: Life-cycle Residual Wage Variance, Data and Model

4.2

Calibration

To use the estimated efficiency process parameters, I specify the the model efficiency process analogously to (9) and convert the annual estimates to threeyear estimates. Specifically, I take ρ3 as the model persistence, (1 + ρ2 + ρ4 )ση2 2 as the model variance as the model persistent shock innovation variance, ση,1 2 for z1 , and σε as the model variance for the iid shock. This conversion gives that the three-year ahead forecast means and variances are the same in the model and data and that the initial variances are the same. Except for ν0 which is used as a normalization constant, the age-profile parameters carry over directly. For retired households, log ei,t = ν0 + ν1 hR−1 + ν2 h2R−1 + ν3 h3R−1 + zi,R−1 + log(.5) where hR−1 = 63/10. The log(.5) term generates an exogenous 50% decline in labor efficiency and is meant to capture reduced non-asset income in retire-

21

ment. Average earnings in retirement are not sensitive to bankruptcy policy and are roughly 40% of earnings in working life as can be seen in Figure 2. Consistent with the theory, the persistent and transitory shocks are both discretized.7 The transitory shock is discretized using 3 points at 0 and ± 1 standard deviations. The persistent shock is discretized using 7 linearly-spaced points on an age-dependent grid that always covers 3 standard deviations (for retirees, the grid is the same as for R − 1). The transition probabilities are computed using Tauchen (1986)’s method. The labor grid is taken to be arbitrarily fine between .001 and .999 (one can think of the steps from one element in the grid to the next as being set to machine precision). The asset grid cannot be as fine since increasing the grid also increases the search space for the planner’s optimal policy. I use 50 strictly negative points and 250 points overall. The utility function is chosen to be (cθ (1−n)1−θ )1−σ /(1−σ). The parameter σ is set to 1 + 1/θ so that constant relative risk aversion, 1 − θ(1 − σ), equals 2 for any calibrated value of θ. The risk-free price q¯ is set to give a 2% annual real interest rate. For welfare comparisons, I specify an “ex-ante” welfare function where α1 (a = 0, e) = πe for all e and αt (a, e) = 0 for all other a, e, t. This has been the most common choice in the literature for life-cycle models featuring default.8 This welfare measure accounts for the welfare of older households because V1 (a = 0, e) implicitly values utility at all future states reached with positive probability. As a benchmark, I take V O , the value of the outside option, low enough in the sense of Proposition 2 as to make it irrelevant. This is done for two reasons. First, it shows what the planner could achieve if default outside the bankruptcy system could be eliminated. Since the goal is to characterize optimal policy, 7

The grids used in the computation are such that (ε, z) can always be recovered from e. That is, e = f (ε, z) is invertible, which is required to be perfectly consistent with the theory. Although one can construct examples where f is not invertible, Lemma 9 in Appendix A proves the existence of an arbitrarily small perturbation of f that is invertible. 8 For instance, Livshits et al. (2007), Athreya et al. (2009a), and Gordon (2015) all use this. While testing other welfare functions is trivial, this has not been done for brevity.

22

this seems like the nature case to consider. Second, most of the literature has only one default option, bankruptcy. Hence, it is useful to know what the optimal policy looks like in this context. Section 4.6 considers the opposite extreme where V O = V A . For comparison with the optimal policy and determining the remaining parameter values, I attempt to capture the current U.S. bankruptcy system. In the U.S., any household in good standing can file for bankruptcy, so I set Dt (a, e, n) = {0, 1} for all a, e, n, t. Since a bankruptcy stays on credit reports for 10 years, I set λt = .7 (for all t) to match this duration. Since there are no formal proportional earnings costs in Chapter 7 bankruptcy, I set χt (a, e, n) = 0. Filing costs in the U.S. are comprised of two main components, official filing fees and attorney costs. Official Chapter 7 bankruptcy filing fees in 2016 were $335 dollars but can be waved for debtors earnings less than $150 below the poverty level (U.S. Courts, 2016). Attorney’s fees, which presumably are increasing in labor efficiency, are substantial. White (2007) states that a “typical” debtor’s cost of filing is between $1,800 and $2,800 (p. 192). So, I ¯ The slope coefficient ζ¯ parameterize ζt (a, e, n) to allow for progressive costs ζe. is then chosen to give average filing costs equal to 1% of average earnings. This is based on an $1,800 filing cost as a fraction of 3 years of earnings at $60,000. The psychic costs κ(e) are parameterized as max{0, κ0 + κ1 zi,t + κ2 εi,t }. The 7 remaining parameters, β, θ, ν0 , κ0 , κ1 , κ2 , ζ are used to match 7 moments from the data. The discount factor β is primarily used to match a wealth-income ratio of 1 based on an annual ratio of 3. The Cobb-Douglas utility weight on consumption is used to match the fraction of hours spent working. In the PSID sample with age restrictions but no restriction on hours worked, the fraction of potential work time spent working is 27.6%. This estimate assumes 16 hours of work time a day with 365 work days available per adult. The earnings process parameter ν0 sets average earnings to 1, a normalization. The psychic cost parameters κ0 , κ1 , κ2 are used to match a debt-income ratio of .028, a filing rate of 2.52%, and an annualized interest rate of 12.7%. The first two numbers are triennial conversions of the annual numbers in Livshits et al. (2007) and the last is from Gordon (2015). The pro23

portional filing cost parameter ζ is used to match a filing cost-earnings ratio of 0.01. The calibrated parameters and moments are given in Table 2. The model delivers every targeted moment with mixed performance for the untargeted moments. The population in debt is in the range of 6.7% (Chatterjee et al., 2007) and 17.6% (Wolff, 2010), the former measuring strictly negative debt positions and the latter weakly negative. The annualized charge-off rate is about twice what it should be, but the implied model rate is closely linked to the interest rate because there are no transaction costs.9 As is typically the case for a normally distributed efficiency process, the model under-predicts earnings and wealth inequality. The model’s implied psychic costs are increasing in both persistent and transitory earnings with a mean value of 0.97. This mean value is very large, around 16% in terms of consumption equivalent variation for a newborn, and consequently households with above-median earnings default very infrequently. However, the psychic costs quickly go to zero for households with negative earnings shocks. E.g., for a transitory shock one-standard deviation below its mean, εit = −0.25, the psychic cost is zero if zi,t is zero. Figure 2 plots life-cycle profiles for consumption, earnings, asset holdings, and default rates. Except for a spike at age 27, the model’s profile of bankruptcies is similar to that observed in the data where bankruptcies are most frequent among 30-47 year olds (Livshits et al., 2007, Figure 1, p. 404).

4.3

Baseline cases

Before analyzing the full optimal policy, it is useful to consider some baseline cases. The first baseline is the current U.S. system (referred to as US). The second is a zero borrowing limit (ZBL) economy where qt (a, e) = 0 for all a < 0, e, t, which provides a lower bound on utility. This is the endogenous outcome when there are no psychic costs and Dt (a, e, n) = {0, 1} for all a, e, n, t, but since κ(e) > 0 for some e, I impose it exogenously. A third baseline case is a natural borrowing limit (NBL) economy, which has Dt (a, e, n) = {0} 9

The charge-off rate is computed using the average charge-off rate on credit card loans from 1985Q1 to 2015Q3 (Board, 2015).

24

Statistic Independently determined Annual risk-free real interest rate Bad credit record duration (years) Relative risk aversion Jointly determined Percent of households filing Debt-earnings ratio × 100 Annualized interest rate Wealth-income ratio Filing cost-earnings ratio Labor supply Earnings (normalization) Untargeted moments Population in debt Annualized charge-off rate Debt-income of filers Filers with below-median income Earnings gini Wealth gini Mean-median earnings Mean-median wealth Mean efficiency (normalization)

Target

Model

2% 10 2

2% 10 2

2.52 2.80 0.127 1.00 0.010 0.28 1.00

2.48 2.85 0.129 1.00 0.010 0.28 1.00

6.7%-17.6% 0.05 0.54 0.69 0.61 0.80 1.57 4.03 -

12.8% 0.10 0.93 0.93 0.50 0.73 1.52 2.56 3.27

Param.

Value

q¯ 0.942 λ 0.7 σ 1 + 1/θ κ0 κ1 κ2 β 1/3 ζ¯ θ ν

0.97 1.22 5.63 0.943 0.007 0.32 2.14

Note: Model income is measured as en + (1/¯ q − 1)a and “in debt” is a < 0. Table 2: Calibration

25

for all states. The last baseline must, by Corollary 1, improve on the NBL economy. This policy specifies Dt (a, e, n) = {0} for all states having VtR (a, e) < VtO (a, e) or VtR (a, e) undefined. I refer to this as the D{} economy reflecting that bankruptcy is only allowed when repayment would result in an empty budget set. The results for ZBL, NBL, US, and D{} are summarized in Table 3. The reported welfare gain is the consumption equivalent welfare measure relative to US. ZBL is worse than US with a welfare loss of 1.3%. NBL on the other hand does much better with a welfare gain of 6.0%. Comparing the amounts of debt in US and NBL, debt is roughly 5 times larger in NBL. These findings agree with a large literature that has found the NBL does much better than model economies calibrated to match U.S. data moments because improved credit allows households to self-insure well. Figure 2 shows NBL increases consumption of young households relative to US, reducing savings early in life. Front-loading consumption improves welfare since β/¯ q is significantly less than 1. Statistic

ZBL

US

Welfare gain relative to US -1.29 0.00 Total filings (%) 0.00 2.48 Total debt 0.00 0.03 Pop. in debt (%) 0.00 12.8 Annualized interest rate (%) 12.9 Annualized charge-off rate (%) 10.4 Total assets* 1.15 1.07 Total consumption 1.06 1.06 Total earnings 1.00 1.00 R *Measured as a(1 − d)dµ (capital in general

NBL

D{}

D∗

Simple

5.95 0.00 0.13 31.8 2.00 0.00 0.89 1.06 1.01

6.92 1.02 0.26 30.9 4.86 7.27 0.80 1.06 1.01

12.4 2.83 0.44 42.2 5.85 3.96 0.47 1.05 1.02

11.6 2.19 0.64 41.5 4.57 3.49 0.25 1.05 1.03

equilibrium).

Table 3: Welfare and Allocations from Differing Default Policies D{} generates a 6.9% welfare gain relative to US, 1 percentage point larger than NBL’s gain. While Corollary 1 shows D{} must weakly outperform NBL, quantitatively there is a sizable difference. This is despite D{} and NBL differing only in states where households cannot repay, in which case D{} provides a 26

fresh start and NBL does not. While implementing NBL would be very costly (probably involving a return to debtors’ prisons), implementing D{} would be much less costly: Courts would determine whether a household could repay and, if not, give them a fresh start. Consumption

Earnings 2

1.6

D* 1.4

NBL

1.5

US

1.2 1 1 0.5

0.8 0.6 20

30

40

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60

70

80

0 20

90

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2

0.06

1

0.04

0

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Default Rate

3

-1 20

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0 20

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40

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Figure 2: Life-cycle Profiles for D∗ , NBL, and US

4.4

The optimal bankruptcy rule

The full optimal policy, labeled as D∗ , is now considered. It is computed using a genetic algorithm and multigrid on 26 nodes of a super computer, and interested readers may consult Appendix B for more details. As can be seen in Table 3, D∗ generates a 12.4% welfare gain relative to US, more than doubling NBL’s 6% gain. To see why the welfare gain is so large, first note that relative to US, D∗ generates 15 times more debt and, despite this, similar filing rates with lower interest rates. This debt is evidence of improved consumption smoothing that is also evident in the life-cycle profiles in Figure 2. Under D∗ , an average household accumulates debt exceeding half 27

of mean earnings by age 40. It is not until the mid-40s where deleveraging occurs and mean assets grow in preparation for retirement. In contrast, US and NBL have positive average asset positions throughout the life-cycle. D∗ ’s larger debt amounts in mid-life increase labor through a negative wealth effect on leisure, which results in households working more when their labor efficiency is highest. As a result, average earnings are 2% larger in D∗ than in US (as can be seen in Table 3). The optimal policy improves consumption smoothing by allowing default only for those who benefit the most from it. This can be seen in Figure 3, which plots the optimal policy for select ages. A black circle means that the state is visited in equilibrium with probability greater than 10−5 . A blue plus sign means that the household files for bankruptcy if the planner lets them, i.e., dt (a, e) = max Dt (a, e, n). A red circle means Dt (a, e, n) = {0, 1}, and so the household defaults. These parts of the policy are the most meaningful because changing the policy from {0, 1} to {0} will have a direct effect on default decisions and the planner’s objective function (by virtue of using exante utility and the probability being greater than 10−5 ). The horizontal axis is debt (recall average earnings are roughly 1), and the vertical axis gives the standard deviations from the persistent shock mean. The top and bottom panels present the policy for the median and worst transitory shock values, respectively. The optimal policy tends to allow bankruptcy when, conditional on a level of debt, households receive the worst persistent shock occurring with some nonnegligible probability. In this sense, the planner only allows default for the most unlucky households. Restricting default to only a small fraction of households is important because default rates effectively act as a borrowing tax: Borrowing interest rates relative to risk-free rates are 1/(1 − Ee0 |e,t dt+1 (a0 , e0 )) or roughly Ee0 |e,t dt+1 (a0 , e0 ) for small default rates; hence, a x% default rate translates (roughly) into a borrowing tax of x%. The optimal policy mitigates this distortion by restricting default to those who benefit the most from it. Interestingly, the optimal policy seems to change little in response to a negative transitory shock. This can be seen in comparing the median transitory 28

Stdev from persistent mean Stdev from persistent mean

Median transitory shock, Age 30

Median transitory shock, Age 51

3

3

Pr>.00001

2

2

& d=maxD

1

1

& d=1

0

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-1

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-2

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-3 0

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5

Debt

Figure 3: The Optimal Policy by Age state (the top panels) with the worst transitory state (the bottom panels). While it is true that more people wish to file in response to the negative transitory shock, the planner generally does not let them (i.e., a negative transitory shock creates more blue plus signs but not many more red circles). The optimal policy responds more to persistent shocks than transitory shocks because the latter are easily insured using credit while the former are not. Because transitory shocks have no impact on future earnings, a negative shock hitting an age t household has no impact on qt (a0 , e). Hence, households can readily borrow to smooth out this small reduction in lifetime income. In contrast, persistent shocks are much harder to self-insure via credit. Not only do they reduce lifetime income by a substantial amount, they also reduce qt (a0 , e), which makes borrowing costly precisely when households want to borrow. So, the planner is more likely to offer bankruptcy in response to a negative persistent shock than a negative transitory one. To show this formally and elucidate other properties of the optimal policy,

29

Table 4 presents logit estimates of the optimal policy. The logit specification is P r(max Di,t = 1) = 1/(1 + exp(−δi,t )) where δi,t = b0 + b1 1[agei,t ≥ 40] + b2 log(−ai,t ) + b3 z˜i,t + b4 ε˜i,t + ei,t

(10)

and z˜i,t = zi,t /Stdev(z|t), ε˜i,t = εi,t /Stdev(ε) are the number of standard deviations from the mean conditional on age. Including only an age 40+ bin was done because adding two bins for 40-59 and 60+ had similar coefficients. The observations come from, and are weighted, by the invariant distribution conditioned on debt. When not conditioning on additional information, this specification does a poor job of capturing the optimal policy. For instance, in column (1), the pseudo-R2 measure is only 0.056. Additionally, the coefficient on the persistent shock standard deviation is not statistically significant. (1) max D

(2) max D | ∈ ∆

(3) max D | ∈ ∆, Simple

σ(Persistent) σ(Transitory) Log debt Age 40+ Constant

-0.11 0.43∗ -0.35∗ -0.52∗ -0.99∗

-2.26∗ -0.38∗ 0.86∗ 1.05∗ -5.72∗

-1162.60 -1.06 2085.41 -610.13 -2748.09

Observations Pseudo R2

9774 0.056

2650 0.263

2650 1.000

Note: Estimates are weighted by the invariant distribution for D∗ ; Coefficients in (3) are not uniquely determined; * denotes significance at the 1% level Table 4: Logit Estimation of Optimal Policy However, planner policy does not always have a direct effect on outcomes. E.g., for small amounts of debt, many households will repay their debt whether the planner allows them to file for bankruptcy or not. The states where planner policy does have a direct effect on outcomes are given by ∆ = {(a, e, t)|VtD (a, e) > VtR (a, e) or VtR (a, e) infeasible}. For any state in ∆, households choose dt (a, e) = max Dt (a, e, n) and so changing Dt (a, e, n) from {0, 1} to {0} or vice versa 30

leads to a direct effect on default decisions and planner welfare. When only including observations corresponding to states in ∆, the planner policy is captured fairly well by the logit specification. This is seen in column (2) labeled “max D | ∈ ∆” where the pseudo-R2 in this case is now 0.263. Conditioning on ∆ states not only increases explanatory power but also makes the coefficients more intuitive. Specifically, a negative persistent or transitory shock now significantly increases the likelihood of the planner allowing bankruptcy. Additionally, as debt increases, filing becomes more likely. The age dummy coefficient indicates the planner lets more bankruptcy occur for those age 40 and above, else equal. The intuition for this last part goes back to Figure 2 where individuals begin saving for retirement in earnest at around age 45. Since households naturally wish to borrow less at this age, increasing filing rates and the effective tax on borrowing is less costly in welfare terms. 0.1

0.05

0

-0.05

-0.1

-0.15

-0.2 σ(Persistent)

max D | ∈ ∆ max D | ∈ ∆, Simple

Log debt

σ(Transitory)

Age 40+

Figure 4: Average Marginal Effect on P r(max D|∆) Because the logit is non-linear, the marginal effect of a change in one regressor depends on the values of the other regressors. Figure 4 plots the marginal effect for each regressor averaged across differing values for other regressors 31

(the confidence bands indicate how much the marginal effect depends on the values of the other regressors). The persistent shock standard deviation has the largest marginal effect. In particular, being three standard deviations below would else equal increase the probability max D = 1 by between 36 and 48%. In contrast, the transitory shock standard deviation is much smaller. These results agree with Figure 3 that indicated planner policy responds more to persistent shocks than transitory ones. Figure 4 also reveals that the elasticity of P r(max D) with respect to debt is around 5%. Hence, a doubling of debt would increase the probability of max D = 1 by 5 percentage points, a seemingly small amount. Being 40 years or older also increases the likelihood by about 5%, but the effect is more varied (as seen in the confidence bands being wider).

4.5

A good simple rule

While the optimal policy is intuitive, implementing the optimal rule in practice would be challenging. It requires that households and creditors have intricate knowledge about not just efficiency processes, but about every aspect of bankruptcy law. This section looks for a parsimonious parameterization of Dt (a, e, n) that comes close to D∗ in terms of welfare. Since the pseudo-R2 in Table 4 indicates that 26% of the variation in Dt (a, e, n) can be accounted for by a simple logit model (conditioning on ∆), a natural way to parameterize Dt (a, e, n) is (

{0, 1} if b0 + b1 1[age(t) ≥ 40] + b2 log(−a) + b3 z˜ + b4 ε˜ > .5 . {0} otherwise (11) This parameterization effectively rounds the probability implied by the logit specification in (10) to the nearest of 0 and 1. To ensure the policy is parsimonious, the theoretical result that bankruptcy be allowed when repayment is infeasible is not used, i.e., (11) applies in all states. The optimal parameters, found via a global maximization algorithm, are (b0 , b1 , b2 , b3 , b4 ) = (−8.45, −1.95, 6.48, −3.55, 0.01). Dt (a, e, n) =

32

As can be seen in Table 3, this simple rule generates a large welfare gain of 11.6%, only one percentage point less than the 12.4% under D∗ . Relative to D∗ , the simple rule produces smaller filing, interest, and charge-off rates and larger amounts of debt. These statistics indicate the simple rule is biased away from intratemporal smoothing via default and towards intertemporal smoothing via credit. To gain more insight into how the simple policy differs from the optimal one, it is useful to re-estimate the logit specification in (10) and recompute the average marginal effects. The logit estimation results are presented in column (3) of Table 4. The logit perfectly captures the specification in (11), which also means the coefficients are not uniquely determined: A small enough change in the coefficients will generically have no effect on (11) and hence the results will still be perfectly predictive. However, the signs and magnitudes reveal some information. Specifically, transitory shocks have an extremely limited effect on outcomes. As discussed above, transitory shocks are easily insured through borrowing. Without fine control of bankruptcy policy, it is optimal to effectively ignore these states. Additionally, the age 40+ dummy now has a negative coefficient. This suggests that the positive effect under the optimal policy hinged on allowing it only for some 40+ year olds, not all of them. The average marginal effects, displayed in Figure 4 alongside those from D∗ , confirm and extend these findings.10 For easier comparison, it is conditioned on the ∆ states coming from D∗ . Along most dimensions, there are noticeable differences from D∗ . Transitory shocks have virtually no effect under the simple rule. The dummy indicator for age 40+ is slightly negative under the simple rule while it is positive under D∗ . Lastly, the simple rule responds less to permanent shocks: A negative three standard deviation persistent shock increases the probability of filing by only around 12 percentage points compared to 45 10

Under the true data generating process, the average marginal effect would be zero since a marginal change in a regressor will not change (generically) the value of Dt (a, e, n) in (11). However, Figure 4 presents the average marginal effect from the logit, which assumes a continuous latent variable. The fact that this is not very close to zero shows the simple rule is targeting the states in ∆. Not conditioning on ∆, the average marginal effects are virtually zero for the simple rule.

33

points under D∗ . The sensitivity to debt is somewhat higher than under D∗ , but overall the best simple rule seems to be less responsive to a household’s state. With the simple rule, the planner is balancing two types of error. First, some households who would optimally be allowed to default will not be. This error results in too much intertemporal smoothing and too little intratemporal. Second, some who would optimally be prevented from defaulting are permitted. This results in the opposite case, too little intertemporal smoothing and too much intratemporal. The results suggest the second type of error is worse: The planner should bias policy towards having too little default and too much credit. One benefit of doing this is that households can increase their own current consumption in a marginal fashion through debt issuance. In contrast, for the planner to increase a household’s consumption by allowing for bankruptcy, the household’s consumption must increase in a discrete fashion. By erring on the side of credit, the planner is effectively delegating consumption smoothing, reducing the cost of credit to help households better self-insure.

4.6

The impact of a larger outside option

Until now, V O has been set low enough to be irrelevant leaving households with, essentially, only one default option, bankruptcy. This is the natural case to consider for two reasons. First, V O places constraints on the planner. The planner can always offer individuals at least as much utility as V O through bankruptcy. Hence, if the planner could lower V O , it would certainly be optimal do so. Consequently, D∗ corresponds to the optimal policy if default outside the bankruptcy system could be completely eliminated. Second, most of the literature has evaluated bankruptcy policies when there is only a formal default option. However, I now consider the implications of a higher outside option by setting it to the maximum allowed by the theory, namely, V O = V A . D∗ in this context provides a lower bound on what optimal bankruptcy policy can accomplish, whereas D∗ with V O irrelevant provides an upper bound. Because the psychic costs of default are not zero, debt can be supported in equilibrium.

34

Table 5 reports the welfare under ZBL, US, NBL, D{} , D∗ , and Simple when V O = V A , as well as for the benchmark of V O irrelevant.11 US now slightly outperforms “NBL,” i.e., Dt (a, e, n) = {0} for all a, e, n, t. NBL has some households choosing the outside option, which induces deadweight loss that makes it slightly worse than US (which allows all households in good standing to file for bankruptcy). That is, US is better than NBL because it replaces V O with V D . However, V D and V O differ very little when V O = V A . Statistic

ZBL

US

NBL

D{}

D∗

Simple

Largest outside option Welfare gain relative to US

-1.32

0.00

-0.08

0.81

0.81

0.06

Irrelevant outside option Welfare gain relative to US

-1.29

0.00

5.95

6.92

12.4

11.6

Table 5: Welfare from Differing Default Policies by Outside Option Value D{} , which allows bankruptcy whenever repayment is infeasible or the outside option is preferred to bankruptcy, generates a 0.8% welfare gain relative to US. Because it is much more targeted than US, which allows bankruptcy in all states, it generates more credit than US. This widens the gap between V D and V O (by improving the value of regaining access to credit), and so D{} noticeably improves on US. Interestingly, D∗ does not noticeably improve on D{} (although it does do slightly better without rounding). While this is surprising, it highlights that for a large outside option value, the gain from improving credit dominates that of increasing access to bankruptcy. Part of the reason for this is that increasing VtO worsens credit for all households age t − 1 and younger households. Specifically, larger VtO makes (informal) default more attractive. The planner then R optimally allows for more bankruptcy, which worsens qt−1 and makes Vt−1 fall. O O This in turn makes Vt−1 relatively more attractive, even if Vt−1 is unchanged, 11

Allocations are not reported because allocations for households choosing the outside option are unspecified. For a well-defined problem, only the lifetime utility derived from the outside option must be specified, not the precise consumption and other allocations associated with it.

35

and causes a type of credit crunch for all younger households. As Proposition 8 shows, this effect is so extreme that—without psychic costs—no debt can be sustained in equilibrium when V O = V A . The calibration’s psychic costs do imply a non-negligible amount of borrowing can be sustained. However, they are not large enough for D∗ to accomplish much. One way of quantifying this problem is by measuring how much debt the planner can sustain in states where κ(e) = 0. A necessary condition for sustaining debt is that VtR (a, e)−VtO (a, e) > 0. Relative to repayment, the benefit of choosing the outside option is −a. The cost is a loss of access to credit. This cost can be quantified as the net present value of the welfare gain of D∗ relative to ZBL. Since this flow gain is 2.16%, the net present value of it (for a P newborn household) is .0216 Tt=1 .942t−1 or 0.27. Hence, the benefit would exceed the cost, and hence default would occur, whenever −a > 0.27. When V O was irrelevant, the average debt position under D∗ was 0.44. In fact, a similar calculation for irrelevant V O suggests debt as large as 1.71 could be sustained, which is more than 6 times larger. Hence, an attractive outside option severely restricts how much debt the planner can sustain. With the same parsimonious parameterization of bankruptcy policy as before, the best simple rule is (b0 , b1 , b2 , b3 , b4 , b5 ) = (2.45, 1.78, 4.08, −5.14, −2.78). This generates a welfare gain of less than 0.1% relative to US. It improves on US because bankruptcy in the simple rule (1) has no filing cost and (2) allows individuals to borrow in the next period. However, neither of these are large improvements. The first is not a significant improvement because the ¯ is already close to zero for housefiling cost ζt (a, e, n), parameterized as ζe, holds considering bankruptcy. The second is not a large improvement as only small amounts of debt can be sustained. For a large outside option, the best “simple” rule is not a policy parameterized by (b0 , b1 , b2 , b3 , b4 , b5 ). Rather, it is D{} : Households should be allowed to file if and only if they cannot repay their debt or would otherwise prefer to default informally.

36

5

Conclusion

Theoretically, optimal bankruptcy law offers a fresh start for a general class of preferences and labor efficiency processes. However, this fresh start is not offered to all households. While it is difficult to say theoretically who should or should not be allowed to file, bankruptcy should always be permitted when a household cannot repay their debt or would otherwise prefer informal default. Consequently, a natural borrowing limit economy is suboptimal. A high outside option value restricts what the planner can accomplish and, without psychic costs of default, prevents debt from being sustained in equilibrium. Quantitatively, the optimal policy generates large welfare gains by restricting default to a small portion of households who benefit the most from it. These households typically have large, negative persistent shocks and large amounts of debt. A simple rule can capture these important features of the optimal policy and come close to the optimal policy’s welfare gain. When the outside option is very attractive, planner policy can accomplish much less. Well-designed bankruptcy policy is capable of producing large welfare gains as long as default outside the bankruptcy system is sufficiently costly.

References S. R. Aiyagari. Uninsured idiosyncratic risk and aggregate savings. Quarterly Journal of Economics, 109(3):659–684, 1994. K. B. Athreya, X. S. Tam, and E. R. Young. Unsecured credit markets are not insurance markets. Journal of Monetary Economics, 56(1):83–103, 2009a. K. B. Athreya, X. S. Tam, and E. R. Young. Are harsh penalties for default really better? Working Paper 09-11, Federal Reserve Bank of Richmond, 2009b. K. B. Athreya, J. M. Sanchez, X. S. Tam, and E. R. Young. Bankruptcy and delinquency in a model of unsecured debt. Working Paper 2012-042C, Federal Reserve Bank of Saint Louis, 2012a. 37

K. B. Athreya, X. S. Tam, and E. R. Young. A quantitative theory of information and unsecured credit. American Economic Journal: Macroeconomics, 4(3):153–183, 2012b. K. B. Athreya, J. M. Sanchez, X. S. Tam, and E. R. Young. Labor market upheaval, default regulations, and consumer debt. Working Paper 2014002A, Federal Reserve Bank of Saint Louis, 2014. Board, 2015. Charge-Off and Delinquency Rates on Loans and Leases at Commercial Banks. Board of Governors of the Federal Reserve System, http://www.federalreserve.gov/releases/chargeoff/chgallsa. htm, 2015. Accessed: 2015-11-24. J. Bulow and K. Rogoff. Sovereign debt: Is to forgive to forget? American Economic Review, 79(1):43–50, 1987. S. Chatterjee and G. Gordon. Dealing with consumer default: Bankruptcy vs garnishment. Journal of Monetary Economics, 59(S):S1–S16, 2012. S. Chatterjee, D. Corbae, M. Nakajima, and J.-V. R´ıos-Rull. A quantitative theory of unsecured consumer credit with risk of default. Econometrica, 75 (6):1525–1589, 2007. D. Chen. The impact of personal bankruptcy on labor supply decisions. Mimeo, 2013. D. Chen and D. Corbae. On welfare implications of restricted bankruptcy information. Journal of Macroeconomics, 33(1):4–13, 2011. P. J. Coleman. Debtors and Creditors in America: Insolvency, Imprisonment for Debt, and Bankruptcy, 1607-1900. The State Historical Society of Wisconsin, Madison, Wisconsin, 1974. L. A. Drozd and J. B. Nosal. Competing for customers: A search model of the market for unsecured credit. Mimeo, 2008.

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G. Gordon. Evaluating default policy: The business cycle matters. Quantitative Economics, Forthcoming, 2015. B. Grochulski. Optimal personal bankruptcy design under moral hazard. Review of Economic Dynamics, 13(2):350–378, 2010. J. Heathcote, F. Perri, and G. L. Violante. Unequal We Stand: An Empirical Analysis of Economic Inequality in the United States: 1967-2006. Review of Economic Dynamics, 13(1):15–51, 2010. W. Li and P. Sarte. U.S. consumer bankruptcy choice: The importance of general equilibrium effects. Journal of Monetary Economics, 53(3):613–631, 2006. W. Li and M. White. Mortgage default, foreclosure and bankruptcy. Working Paper 15472, NBER, 2009. W. Li, M. J. White, and N. Zhu. Did bankruptcy reform cause mortgage defaults to rise? American Economic Journal: Economic Policy, 3(4):123– 147, 2011. I. Livshits, J. MacGee, and M. Tertilt. Consumer bankruptcy: A fresh start. American Economic Review, 97(1):402–418, 2007. I. Livshits, J. MacGee, and M. Tertilt. The democratization of credit and the rise in consumer bankruptcies. Working Paper 14-07, University of Mannheim, 2014. K. Mitman. Macroeconomic effects of bankruptcy and foreclosure policies. PIER Working Paper 11-015, University of Pennsylvania, 2011. B. N. Narajabad. Information technology and the rise of household bankruptcy. Review of Economic Dynamics, 15(4):526–550, 2012. M. A. Robe, E.-M. Steiger, and P.-A. Michel. Penalties and optimality in financial contracts: Taking stock. Working Paper 2006-013, HumboldtUniversit¨at zu Berlin, 2006. 39

J. M. S´anchez. The IT revolution and the unsecured credit market. Working Paper 2010-022B, Federal Reserve Bank of St. Louis, 2010. K. Storesletten, C. Telmer, and A. Yaron. Cyclical dynamics in idiosyncratic labor market risk. Journal of Political Economy, 112(3):695–717, 2004. G. Tauchen. Finite state Markov-chain approximations to univariate and vector autoregressions. Economics Letters, 20(2):177–181, 1986. U.S. Courts, 2016. Chapter 7 - Bankruptcy Basics. United States Courts, http://www.uscourts.gov/services-forms/bankruptcy/ bankruptcy-basics/chapter-7-bankruptcy-basics, 2016. Accessed: 2016-03-11. H.-J. Wang and M. J. White. An optimal personal bankruptcy procedure and proposed reforms. Journal of Legal Studies, 29(1):255–286, 2000. M. J. White. Bankruptcy reform and credit cards. Journal of Economic Perspectives, 21(4):179–199, 2007. E. N. Wolff. Recent trends in household wealth in the united states: Rising debt and the middle-class squeeze—an update to 2007. Working Paper 589, Levy Economics Institute of Bard College, 2010.

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A

Appendix: Proofs and Omitted Results [Not for Publication]

Proof of Proposition 1. Consider t = T and any a, e, h. Continuation utilities are zero and households solve static problems taking qT := 0 as given. If h = 0, then the outside option is a feasible choice. If h = 1, then a0 = 0 is a feasible choice for any a ≥ 0 (recall Vt (a, e, h = 1) is only defined for a ≥ 0). Since there are only a finite number of choices a0 , n, d, and o and the choice set is non-empty, a maximum exists. So, VT (a, e, h) is well-defined for all a, e, h. Moreover, the maximization problem delivers decisions dT (a, e) and oT (a, e). Since a and e were arbitrary, qT −1 (a0 , e) is well-defined for all a0 , e. Now, consider t < T . For induction, assume Vt+1 and qt are well-defined. Consider an arbitrary a, e, h. If h = 0, taking the outside option is a feasible choice. If h = 1, choosing a0 = 0 is a feasible option. Since there are only a finite number of choices, a maximum exists. A solution to the maximization problem gives decisions dt (a, e) and ot (a, e). Since a and e were arbitrary, qt−1 (a0 , e) is well-defined for all a0 , e. This completes the induction argument. Lemmas 4 and 5 establish some basic properties of the household value function. Lemma 4 shows higher prices, i.e., lower interest rates, for debt improve household welfare. Lemma 5 shows households are weakly better off not having a bankruptcy record. The proof is straightforward since households have an option value of borrowing if they do not have a bankruptcy record. Lemma 4. Whenever well-defined, VtR (a, e) is weakly increasing in qt (a0 , e) for any a0 ≤ 0, e and all a. Proof of Lemma 4. A larger qt (a0 , e) for a0 ≤ 0 increases the budget constraint of a household choosing to repay their debt. Hence, this must weakly increase VtR (a, e) (for all a). Lemma 5. Vt (a, e, h = 0) ≥ Vt (a, e, h = 1) for all a ≥ 0. Proof of Lemma 5. Any feasible policy for a household of type (a, e, h = 41

1, t) can also be chosen by a household with type (a, e, t, h = 0) by restricting themselves to a0 ≥ 0. Lemma 6 shows the value function is bounded above and below when households have positive assets. It also shows these bounds are independent of planner policy and the outside option. This lemma is essential to the proof of Proposition 2. Lemma 6. For a ≥ 0, Vt (a, e, h) is uniformly bounded above and below. Moreover, defining the maximum and minimum values possible for e as e and e and likewise for n, these bounds depend only on e, e, n, n, min A, q¯, β, and u. Proof of Lemma 6. Define a = min A, n = min N , n = max N , y := e · n, and y := e · n. For h = 0 or h = 1, one feasible policy when a ≥ 0 is to choose a0 = 0 in all subsequent periods and never default. Since earnings are bounded below by y := e · n, this policy results in lifetime utility greater than P or equal to Tτ=t β τ −t u(y, n). So, Vt (a, e, h) is bounded below. Additionally, the most consumption possible is bounded above by −¯ q a + y + a. So, Vt (a, e, h) is PT q a + y + a, n). These bounds are functions of bounded above by τ =t β τ −t u(−¯ t, but taking the max and min over all t gives uniform bounds on V . Proof of Proposition 2. For the outside option to be chosen in equilibrium, there must be a state (a, e, t) reached with probability p > 0 from an initial state a1 = 0, e1 , h1 = 0 such that o(a, e, t) = 1. Consequently, as viewed by an (a1 , e1 , t = 1) household, the expected utility derived from this state is pβ t−1 VtO (a, e). Let the uniform bounds on V be denoted V and V . Wlog, take V to be positive. Then expected utility from all other states, as viewed by an (a1 , e1 , t = P 1, h1 ) household, is not more than Tt=1 β t−1 V (a very loose upper bound). P So, Vt=1 (a1 , e1 , h1 ) ≤ Tt=1 β t−1 V + pβ t−1 VtO (a, e). Because a1 = 0, Lemma 6 P has Vt=1 (a1 , e1 , h1 ) ≥ V . Consequently, V ≤ Vt=1 (a1 , e1 , h1 ) ≤ Tt=1 β t−1 V + pβ t−1 VtO (a, e) implying VtO (a, e)

≥

V −

42

PT

t=1 β pβ t−1

t−1

V

.

Since p must be larger than the least likely history of shocks which itself occurs with positive probability, call it p, we have VtO (a, e)

≥

V −

PT

t=1 β pβ t−1

t−1

V

≥

V −

PT

t=1 β pβ t−1

t−1

V

since the numerator is negative.  PT t−1 Consequently, for any δ ≤ V − t=1 β V /(pβ t−1 ), VtO (a, e) < δ implies the outside option is not chosen in equilibrium. Proof of Lemma 1. Fix some policy (D, λ, ζ, χ) that has χt (a, e, n) > 0 for at least one a, e, n, t. Consider the V D problem: VtD (a, e) =

max u(c, n) − κ(e) + β 0

a ∈A,n∈N

X

πee0 |t ((1 − λt )Vt (a0 , e0 , 0) + λt Vt (a0 , e0 , 1))

e0

s.t. c + q¯a0 + ζt (a, e, n) = en(1 − χt (a, e, n)), c > 0, a0 ≥ 0. Dt (a, e, n) = {0, 1} (12) ˜ χ) Define a new policy (D, λ, ζ, ˜ that has χ˜t (a, e, n) = 0 for all a, e, n, t and ˜ ζt (a, e, n) = enχt (a, e, n))+ζt (a, e, n). Then, the budget constraint in bankruptcy is unchanged, and so must be the optimal value and policies (assuming a fixed tie-breaking rule). Hence, the two policies generate the same value functions and prices and consequently the same planner utility. Lemma 7 establishes the continuity of VtD in the minimum of the bankruptcy filing cost across all choices of n. That is, defining ζ t (a, e) = minn∈N ζt (a, e, n) and decomposing ζt (a, e, n) into (ζt (a, e, n) − ζ t (a, e)) + ζ t (a, e), the lemma shows VtD (a, e) moves continuously in ζ t (a, e). While intuitively obvious, the proof is non-trivial because the budget constraint can move discontinuously as a reduction in ζ can make new choices feasible. To ensure that these choices are not optimal, the proof uses the Inada condition on u. Additionally, Lemma 7 establishes that if the minimum filing cost is made large enough, utility from filing for bankruptcy becomes negative infinite.

43

Lemma 7. Wlog, the planner’s filing cost choice can be replaced by a choice of ζ t (a, e) ≥ 0 and ζˆt (a, e, n) ≥ 0 with ζˆt (a, e, n) = 0 for some n with ζt (a, e, n) given implicitly as ζˆt (a, e, n) + ζ t (a, e). VtD (a, e) is continuous in ζ t (a, e) wherever well-defined. Additionally, if dt (a, e) = 1 is feasible in some state a, e, t, there exists a ξt (a, e) > ζ t (a, e) such that the limit of VtD (a, e) as ζ t (a, e) ↑ ξt (a, e) (holding the policy in all other states fixed) is −∞. Proof of Lemma 7. For the decomposition, the goal is to show that the possible budgetary impacts of the filing cost ζ (i.e., the household’s net-offiling-cost income) is exactly equivalent to the possible budgetary impacts of the filing costs ζˆ and ζ. This may be obvious, but a proof is as follows. It is sufficient to show, since ζt (a, e, n) may be any non-negative number for each a, e, n, t, that (1) the conditions on ζ t (a, e) and ζˆt (a, e, n) imply ζ t (a, e) + ζˆt (a, e, n) ≥ 0 and (2) for any a, e, t and z : N → R+ , there exists valid values of ζ t (a, e) and ζˆt (a, e, n) such that z(n) = ζ t (a, e) + ζˆt (a, e, n) for all n. Because ζ t (a, e) ≥ 0 and ζˆt (a, e, n) ≥ 0, (1) is clearly satisfied. Now, to show (2), consider an arbitrary z(n) : N → R+ . Define ζ t (a, e) = minn∈N z(n) and ζˆt (a, e, n) = z(n) − ζ t (a, e). Then clearly they are both positive. Moreover, ζ t (a, e) + ζˆt (a, e, n) = z(n). Lastly, ζˆt (a, e, n) = 0 for some n. So, the planner’s choice can be replaced with a choice of ζ t (a, e) and ζˆt (a, e, n). Fix some a, e, t and suppress dependence on it (e.g., let Dt (a, e, n) be denoted as D(n)). If t = T , then the optimal a0 choice is always zero, and so V D = maxn∈N u(en− ζ, n) s.t. D(n) = {0, 1}. Since u is continuous in c, V D is continuous in ζ (where defined). Now suppose t < T . For all a0 ∈ A, a0 ≥ 0, n such that D(n) = {0, 1}, and ˆ with en − q¯a0 − ζ − ζ(n) > 0, define ˆ V D (a0 , n, ζ) = u(en − q¯a0 − ζ − ζ(n), n) + W (a0 ) where W is the continuation utility associated with a choice of a0 less the psychic costs κ. So V D (a0 , n, ζ) is the lifetime utility associated with a choice 44

of a0 , n given the minimum filing costs ζ. Note two things. By the continuity of u(c, n) in c, V D (a0 , n, ζ) is continuous ˆ in ζ wherever defined. Moreover, as ζ ↑ en − q¯a0 − ζ(n), V D (a0 , n, ζ) goes to −∞ by the Inada condition on u. Now, define feasible choices of a0 , n for a given minimum filing cost as ˆ B(ζ) = {(a0 , n)|a0 ∈ A, a0 ≥ 0, n ∈ N, D(n) = {0, 1}, ζ < en − q¯a0 − ζ(n)} (*) Making the dependence of VtD (a, e) on ζ explicit by writing V D (ζ), one has V D (ζ) =

max

(a0 ,n)∈B(ζ)

V D (a0 , n, ζ)

whenever B(ζ) 6= ∅ (i.e., wherever well-defined). Because V D (a0 , n, ζ) is continuous in ζ wherever defined and because the choice set is finite, V D (ζ) is continuous at all ζ such that B(ζ) is invariant to small changes in ζ (since the upper envelope of a finite number of continuous functions is continuous). What then remains to establish is that V D (ζ) is also continuous at values of ζ such that B(ζ) is not locally invariant. First, I establish the left-hand continuity. So consider a ζ such that for all  > 0, B(ζ − ) 6= B(ζ) (otherwise for small enough , B(ζ − ) = B(ζ) and the left-hand continuity obtains). For any choice (a0 , n) ∈ A × N satisfying a0 ≥ 0 and D(n) = {0, 1}, the choice is ˆ in B(ζ − ) if and only if − < en − q¯a0 − ζ − ζ(n). Similarly, the choice is in ˆ B(ζ) if and only if 0 ≥ en − q¯a0 − ζ − ζ(n). Putting these together, ˆ q a0 −ζ−ζ(n) ∈ (−, 0]} B(ζ−)\B(ζ) = {(a0 , n)|a0 ∈ A, a0 ≥ 0, n ∈ N, D(n) = {0, 1}, en−¯ Since we are considering the case that B(ζ − ) 6= B(ζ) for all  > 0 and B is decreasing, the infinite intersection B(ζ − ) \ B(ζ) over all  > 0 is nonempty. Moreover, because the choice set is always finite, there exists a δ > 0 such that  < δ has \ ˜ B(ζ) := B(ζ − ) \ B(ζ) = B(ζ − ) \ B(ζ). >0

45

Wlog, consider  < δ. Then V D can be written as ( V D (ζ − ) = max

) max

(a0 ,n)∈B(ζ)

V D (a0 , n, ζ − ),

max

˜ (a0 ,n)∈B(ζ)

V D (a0 , n, ζ − )

˜ All elements in Consider the implied consumption for any element of B(ζ). 0 ˆ ˆ ˜ B(ζ) have en− q¯a −ζ − ζ(n) ∈ (−, 0], which implies c = en− q¯a0 −(ζ −)− ζ(n) ˜ has c ∈ (0, ]. As  goes to zero, so does consumption for every element in B. ˜ as  ↓ 0. At the Consequently, V D (a0 , n, ζ − ) goes to −∞ for each (a0 , n) ∈ B same time, all (a0 , n) ∈ B(ζ) have continuation utility that is bounded (Lemma 6) and consumption bounded away from zero since to be in B(ζ) requires ˆ > 0. So max(a0 ,n)∈B(ζ) V D (a0 , n, ζ −) consumption at ζ satisfy en− q¯a0 −ζ − ζ(n) is bounded below as  goes to zero. Therefore, V D (ζ − ) =

max

(a0 ,n)∈B(ζ)

V D (a0 , n, ζ − ),

which implies V D moves continuously for all  (less than δ). Hence, V D is left-continuous at ζ. Second, I establish the right-hand continuity. Right-hand continuity would only possibly fail if B(ζ) 6= B(ζ +) for arbitrarily small , otherwise the choice set would be constant for small enough  making V D locally the upper envelope of a finite number of continuous functions. However, I will show that this is impossible. Since B is decreasing, B(ζ) 6= B(ζ + ) if and only if B(ζ) \ B(ζ + ˆ and ) 6= ∅. For (a0 , n) to be in B(ζ)\B(ζ +), it must satisfy ζ < en− q¯a0 − ζ(n) ˆ ˆ ζ + ≥ en− q¯a0 − ζ(n). Equivalently, it must satisfy  ≥ −ζ +en− q¯a0 − ζ(n) > 0. 0 0 Hence, for every (a , n) there exists a threshold (a , n) = −ζ + en − q¯a0 − ˆ ζ(n) > 0 such that  < (a0 , n) implies (a0 , n) ∈ / B(ζ) \ B(ζ + ). But then 0  < min(a0 ,n)∈A×N (a , n) implies B(ζ) \ B(ζ + ) is empty. So, V D is rightcontinuous. Therefore, V D is both left and right-continuous, and so V D is continuous at ζ whenever B(ζ) 6= ∅, i.e., a feasible choice exists. ˆ ζ) and any a, e, t with dt (a, e) = Now, I will show that for any policy (D, λ, ζ, S 1 a feasible choice (which implies Dt (a, e, n) = {0, 1} and restricts ζˆt (a, e, n)+ 46

ζ t (a, e) to not be too large), there exists an ξ > ζ t (a, e) such that the limit as ζ t (a, e) ↑ ξ has VtD (a, e) converge to −∞. First, by virtue of dt (a, e) = 1 being feasible, there is a feasible choice when filing for bankruptcy. Again fixing an a, e, t and suppressing dependence on it, B(ζ) 6= ∅. Consider any (a0 , n) ∈ B(ζ). It remains feasible if ζ is replaced by ˆ So define ξ(a0 , n) = en − q¯a0 − some z ≥ ζ as long as z < en − q¯a0 − ζ(n). ˆ ζ(n). Then (a0 , n) is feasible for all z ∈ [ζ, ξ(a0 , n)) (note that ξ(a0 , n) > ζ). ˆ − z converges to Additionally, as z ↑ ξ(a0 , n), consumption c = en − q¯a0 − ζ(n) 0, and so V D (a0 , n, z) converges to negative infinity. Now, define ξ = max(a0 ,n)∈B(ζ) ξ(a0 , n). Then it is clear that for any z ∈ [ζ, ξ), V D (z) = max(a0 ,n)∈B(z) V D (a0 , n, z) is well-defined (B(z) is non-empty). It is also clear that as z ↑ ξ, the number of feasible choices dwindle to arg max(a0 ,n)∈B(ζ) ξ(a0 , n) (which may or may not be a singleton). However, for each element in this argmax, utility converges to −∞. Hence, V D (z) converges to negative −∞ as z ↑ ξ.

Proof of Lemma 2. Wlog, use the decomposition in Lemma 7 of ζt (a, e, n) into ζˆt (a, e, n) + ζ t (a, e) with ζ t (a, e) ≥ 0, ζˆt (a, e, n) ≥ 0, and ζˆt (a, e, n) = 0 for some n. Note that for VtD (a, e) to be well-defined, Dt (a, e, n) = {0, 1} for at least one choice of n. Fix an a, e, t and suppress dependence on it. Let the value of V D before the policy change be given by V0D , and similarly for the continuation utility less psychic costs W and for ζ. Then V0D =

max u(c, n) + W0 (a0 )

a0 ∈A,n∈N

ˆ + ζ = en, c > 0, a0 ≥ 0 s.t. c + q¯a0 + ζ(n) 0 Dt (a, e, n) = {0, 1}. Now, let the new, larger continuation utility be W1 (a0 ). Conditional on ζ, the

47

new value V D (ζ) is V D (ζ) =

max u(c, n) + W1 (a0 )

a0 ∈A,n∈N

ˆ + ζ = en, c > 0, a0 ≥ 0 s.t. c + q¯a0 + ζ(n) Dt (a, e, n) = {0, 1}. It is evident then that V D (ζ 0 ) − V0D ≥ 0 since W1 ≥ W0 . Lemma 7 established the existence of a ξ > ζ 0 such that the limit of V D (ζ) as ζ converges up to ξ is negative infinite. So, for ζ sufficiently close to ξ, V D (ζ) − V0D < 0. Since Lemma 7 also established the continuity of V D (ζ), the intermediate value theorem gives the existence of a ζ 1 ∈ [ζ 0 , ξ) such that V D (ζ 1 ) = V0D . Proof of Proposition 3. Beginning with a policy that has λt > 0 for potentially all t, I will construct a new policy having λt = 0 for an arbitrary t, but resulting in the exact same value and policy functions for households with h = 0 (as well as price schedules). Consider a policy with λt > 0. Setting λt = 0 has no effect on future value, policy, or price functions. It also has no effect on VtR (a, e) for any a, e. By Lemma 5, it does weakly increase VtD (a, e) for each a, e. However, Lemma 2 shows this increase can be undone through a commensurate increase in ζ t (a, e). Consequently, VtR , VtD , Vt (·, ·, h = 0), and dt can all be held constant. This also implies all value functions for τ < t are unchanged, and so the planner is completely indifferent. ˜ D), ˜ which increases Vt (a, e) Proof of Lemma 3. For t = 1, (ζ ∗ , D∗ ) = (ζ, for all a, e, t and so trivially increases the social planner’s utility. Now let Wt˜(a0 , e) be the continuation utility conditional on being in a state with e, t˜ and choosing a0 (recall that because λ = 0, the repayment and bankruptcy value functions have the same continuation utility). Consider t ≥ 2 and suppose, for induction, that the hypothesis holds for R D ˜ D) ˜ has Vt−1 t−1. Note that (ζ, higher and Vt−1 higher relative to (ζ, D) because Wt−1 and qt−1 are larger (the hypothesis has Vt larger and max{dt , ot } smaller). R D Because both Vt−1 and Vt−1 increase, dt−1 might increase, which could make 48

t − 2 households worse off. To circumvent defined by  ˜ ˜   (ζt˜, Dt˜) (ζt˜+ , Dt˜+ ) = (z, Dt−1 )   arbitrary

this, consider a policy (ζ + , D+ ) if t˜ ≥ t if t˜ = t − 1 otherwise

D where z(a, e) is such that Vt−1 (a, e) is the same under (ζ + , D+ ) as under (ζ, D) D for all a, e. Such a z(a, e) exists because the effect of Vt−1 due to the increased continuation utility Wt−1 under (ζ + , D+ ) can be offset through a filing cost increase (Lemma 2). D R the same higher and Vt−1 By construction, the policy (ζ + , D+ ) has Vt−1 relative to (ζ, D) because Wt−1 and qt−1 are larger. This must induce less default, i.e., max{dt−1 , ot−1 } is smaller, and greater utility Vt−1 . Additionally, ˜ D) ˜ for all t˜ ≥ t, which is at least as (ζ + , D+ ) generates the same utility as (ζ, large as that under (ζ, D). Hence, from the induction hypothesis, there exists a policy (ζ ∗ , D∗ ) satisfying

1. ζt˜∗ = ζt˜+ and Dt˜∗ = Dt˜+ for all t˜ ≥ t − 1 2. Dt˜∗ = Dt˜ for all t˜ < t − 1. + that has higher Vt˜(a, e)—relative to (ζ, D)—for all a, e, t˜. By definition, Dt−1 = ∗ ∗ ∗ Dt−1 , and so Dt˜ = Dt˜ for all t˜ < t. Lastly, for all t˜ ≥ t, one has (ζt˜ , Dt˜ ) = ˜ t˜). Hence, (ζt˜+ , Dt˜+ ) which by the definition of (ζt˜+ , Dt˜+ ) implies (ζt˜∗ , Dt˜∗ ) = (ζ˜t˜, D the policy (ζ ∗ , D∗ ) satisfies

˜ t˜ for all t˜ ≥ t 1. ζt˜∗ = ζ˜t˜ and Dt˜∗ = D 2. Dt˜∗ = Dt˜ for all t˜ < t. and generates higher utility (relative to (ζ, D)) for all states, which completes the induction hypothesis. The last claim (the hypothesis’ “moreover” statement) is quite obvious, but a proof is as follows. Note that for all a, e, t˜ ≥ t, the policies (ζ ∗ , D∗ ) and ˜ D) ˜ are equivalent. Hence, given constant tie-breaking rules, if prices are (ζ, the same at t, households make the same decisions at t, which then induce the 49

same prices at t − 1. Because the age T decisions do not depend on prices, backwards induction shows none of the decisions or prices can change, and so the two policies must induce the same Vt˜(a, e) for all a, e, t˜ ≥ t. Proof of Proposition 4. Define t(D) = max{0} ∪ {t | ∃ a, e with Dt (a, e, n) varying in n}. Note that t(D) = 0 if and only if D is invariant to n for all a, e. I will show that if for some t ≥ 1 a policy (ζ, D) has t(D) = t, then there is a better policy (ζ ∗ , D∗ ) with t(D∗ ) ≤ t − 1. This then gives the desired result of there existing a better policy (ζ ∗∗ , D∗∗ ) with t(D∗∗ ) = 0. Consider some (ζ, D) that for some t ≥ 1 has t(D) = t. Fix an arbitrary (a, e). If the policy induces dt (a, e) = 0, then the household is unaffected by setting Dt (a, e, n) = {0} for all n. That is, VtD (a, e) and Vt (a, e) are unchanged as well as max{dt (a, e), ot (a, e)}. On the other hand, if under such a policy dt (a, e) = 1, then setting Dt (a, e, n) = {0, 1} for all n weakly increases VtD (a, e) and Vt (a, e). But note that dt (a, e) = 1 and max{dt (a, e), ot (a, e)} = 1 are unchanged since VtD (a, e) increased without VtR (a, e) or VtO (a, e) changing. Let this modification of (ζ, D) applied to all a, e be denoted (ζ + , D+ ). By construction, (ζ + , D+ ) has t(D+ ) ≤ t − 1 and, as argued, improves Vt˜ for all t˜ ≥ t and leaves max{dt , ot } unchanged. Hence, by Lemma 3, there is a policy (ζ ∗ , D∗ ) with weakly higher utility in all states—relative to (ζ, D)—that has Dt˜∗ = Dt˜+ for all t˜ ≥ t and Dt˜∗ = Dt˜ for all t˜ < t. Hence, t(D∗ ) ≤ t − 1. Induction then gives desired result of there existing a better policy (ζ ∗∗ , D∗∗ ) with t(D∗∗ ) = 0. Proof of Proposition 5. The proof has a similar structure to the proof of Proposition 4. Define t(ζ) = max{0} ∪ {t | ∃ a, e, n with ζt (a, e, n) > 0}. Note that t(ζ) = 0 if and only if ζ is identically equal to zero.

50

I will show that if for some t ≥ 1 a policy (ζ, D) has t(ζ) = t, then there is a better policy (ζ ∗ , D∗ ) with t(ζ ∗ ) ≤ t − 1. This then gives the desired result of there existing a better policy (ζ ∗∗ , D∗∗ ) with t(ζ ∗∗ ) = 0. Consider some (ζ, D) that for some t ≥ 1 has t(ζ) = t. Fix an arbitrary (a, e). I will show that replacing ζt (a, e, n) with 0 for all n and setting Dt (a, e, n) either to {0} for all n or {0, 1} for all n improves Vt (a, e) while lowering max{dt (a, e), ot (a, e)}. This will then allow for the application of Lemma 3. I proceed in two cases. First, suppose that (ζ, D) induces dt (a, e) = 0. Then setting ζt (a, e, n) = 0 and Dt (a, e, n) = {0} for all n has no effect on outcomes. Second, suppose that instead dt (a, e) = 1, which implies Dt (a, e, n) = {0, 1} for all n (since we have restricted attention to this class of policies). Then setting ζt (a, e, n) = 0 for all n without changing D increases VtD (a, e) and leaves VtR (a, e) or VtO (a, e) unchanged. This increases Vt (a, e)—strictly if ζt (a, e, nt (a, e)) > 0 under (ζ, D)— without changing max{dt (a, e), ot (a, e)} = 1. This (potential) modification of ζt (a, e, n) and Dt (a, e, n) was for arbitrary (a, e). Let the modification applied to each (a, e) pair define a new policy (ζ + , D+ ). Then (ζ + , D+ ) is identical to (ζ, D) except for time t values, and has weakly higher Vt˜(a, e) and weakly lower max{dt (a, e), ot (a, e)} for all a, e and t˜ ≥ t. Hence, Lemma 3 shows there exists a policy (ζ ∗ , D∗ ) that has higher utility and the same filing costs for t˜ ≥ t. Since t(ζ) = t, and t(ζ + ) ≤ t − 1 by construction, this implies t(ζ ∗ ) ≤ t−1. Moreover, Lemma 3 states that Vt (a, e) is the same under (ζ + , D+ ) and (ζ ∗ , D∗ ) for each a, e. Induction then gives the desired result of there existing a better policy (ζ ∗∗ , D∗∗ ) with t(ζ ∗∗ ) = 0. For the last claim, first note that (ζ ∗∗ , D∗∗ ) is identical to (ζ + , D+ ) for all t˜ ≥ t. Second, (ζ + , D+ ) had a strict increase in Vt (a, e) if dt (a, e) = 1 and ζt (a, e, nt (a, e)) > 0 relative to (ζ, D). Consequently, so does (ζ ∗∗ , D∗∗ ). Therefore, in this case welfare is strictly improved by the new policy as long as αt (a, e) > 0. Proof of Proposition 6. Any policy is weakly inferior to one having χt (a, e, n) = ζt (a, e, n) = λt = 0 for all a, e, n, t. So, if a maximum exists restricting the choice space to χt (a, e, n) = ζt (a, e, n) = λt = 0 for all a, e, n, t, then 51

a maximum exists. In this case, the planner problem reduces to choosing Dt (a, e, n) ∈ {{0}, {0, 1}} for all a, e, t. Since there is only a finite number of choices (and all of them are feasible), a maximum exists. Lemma 8. Any policy having χt (a, e, n) = ζt (a, e, n) = 0 for all a, e, n, t with Dt (a, e, n) invariant to n has VtD (a, e) ≥ VtO (a, e) (for all a, e, t) whenever VtD (a, e) is feasible (i.e., whenever Dt (a, e, n) = {0, 1} for all n). Proof of Lemma 8. Fix some a, e, t and suppress dependence on it. Recall that V O ≤ V A (the value of autarky) by assumption. In the case of χ and ζ uniformly equal to zero with D(n) = {0, 1} for all n, the optimal policy in autarky is also a feasible plan for a bankrupt household and delivers the same utility. Therefore, V D ≥ V A and V O ≤ V A giving the desired result. Proof of Proposition 7. With ζ = 0 (and χ = λ = 0), V D ≥ V O (Lemma 8). When indifferent between the outside option and bankruptcy, households are assumed to file for bankruptcy. Also, when indifferent between default and repayment, they are assumed to repay. Because of all this, for D invariant to n, VtO (a, e) > VtR (a, e) or VtR (a, e) undefined with Dt (a, e, n) = {0} implies ot (a, e) = 1, and ot (a, e) = 1 implies that VtO (a, e) > VtR (a, e) or VtR (a, e) is undefined and Dt (a, e, n) = {0}. Define t(ζ, D) = max{0} ∪ {t | ∃ a, e with ot (a, e) = 1} (with ζ and D implicitly determining ot (a, e)). Note that t(ζ, D) = 0 if and only if (ζ, D) has ot (a, e) = 0 for all a, e, t. I will show that if for some t ≥ 1 a policy (0, D) (meaning filing costs are zero) has t(0, D) = t, then there is a better policy (0, D∗ ) with t(0, D∗ ) ≤ t−1. This then gives the desired result of there existing a better policy (0, D∗∗ ) with t(0, D∗∗ ) = 0. Consider a policy (0, D) that for some t ≥ 1 has t(0, D) = t. Consider any (a, e) such that ot (a, e) = 1. Then making Dt (a, e, n) = {0, 1} for all n makes Vt (a, e) higher (because of increased option value) and cannot possibly increase max{dt (a, e), ot (a, e)}. Now let the modification applied to each (a, e) 52

pair define a new policy (0, D+ ). Then (0, D+ ) is identical to (0, D) except for time t values, and has weakly higher Vt˜ and weakly lower max{dt (a, e), ot (a, e)} for all a, e and t˜ ≥ t. Hence, Lemma 3 shows there exists a policy (ζ ∗ , D∗ ) that has higher utility and identical to (0, D+ ) for t˜ ≥ t. Note that by construction t(ζ ∗ , D∗ ) ≤ t − 1 as ot (a, e) = 1 has been ruled out and future policies remain the same. A complication is that ζt˜∗ may be greater than zero for t˜ < t. If not, then the proof is done as (ζ ∗ , D∗ ) = (0, D∗ ) and t(0, D∗ ) ≤ t − 1. So suppose that ζt∗2 (a2 , e2 ) > 0 for some (a2 , e2 ) and t2 < t. Then Proposition 5 shows an even better policy (0, D∗∗ ) exists, and moreover it is identical to (ζ ∗ , D∗ ) for t˜ > t2 . Since (0, D∗∗ ) is identical to (ζ ∗ , D∗ ) for t˜ > t2 and because t2 < t, (0, D∗∗ ) is identical to (ζ ∗ , D∗ ) for t˜ ≥ t. By virtue of this last fact, (0, D∗∗ ) and (ζ ∗ , D∗ ) induce the same policies for ages t and above. In particular, they both have ot˜(˜ a, e˜) = 0 for all a ˜, e˜, t˜ ≥ t. Hence, t(0, D∗∗ ) ≤ t − 1 and (0, D∗∗ ) delivers higher utility in all states. Proof of Proposition 8. Recall the definition of the autarky value as VtA (a, e) = Xt (0, e) − κ(e) where Xt defined in (2) can equivalently be defined as Xt (a, e) =

max u(c, n) + β 0

a ∈A,n∈N

X

πee0 |t Xt+1 (max{0, a0 }, e0 ) (*)

e0 0

0

s.t. c + q¯a = en + a, c > 0, a ≥ 0 with XT (a, e) := maxn∈N,en+a>0 u(en + a, n). Consider t = T and an arbitrary a, e. In this case, VT (a, e) = VTR (a, e) = XT (a, e). For a < 0 with no psychic costs, VTO (a, e) = VTA (a, e) = VTD (a, e) = XT (0, e) > XT (a, e) = VTR (a, e). By Proposition 7, it is then weakly optimal to have DT (a, e, n) = {0, 1} for all n. Then, dT (a, e) = 1 because VTA (a, e) = VTD (a, e) in this case and households file for bankruptcy when indifferent between bankruptcy and the outside option. Because dT (a, e) = 1 for all a < 0, e, qT −1 (a0 , e) = 0 for all a0 < 0, e and VT (a, e) = XT (max{0, a}, e) for all a, e. For induction, suppose that for some t one has qt (a0 , e) = 0 for all a0 < 0, e

53

and Vt+1 (a, e) = Xt+1 (max{0, a}, e) for all a, e. Fix some a, e. Conditional on repaying, a choice of a0 < 0 results in the same consumption as a choice of a0 = 0 because both have qt (a0 , e)a0 = 0. Moreover, the continuation utilities are the P P same: e0 πee0 |t Vt+1 (a0 , e0 ) = e0 πee0 |t Xt+1 (0, e0 ) for all a0 ≤ 0. Therefore, the household is indifferent over all a0 ≤ 0. Consequently, the household problem conditional on repayment can be written as VtR (a, e) =

max u(c, n) + β 0

a ∈A,n∈N

X

πee0 |t Vt+1 (a0 , e0 )

e0

0

s.t. c + q¯a = en + a, c > 0, a0 ≥ 0 where the choice set is restricted to a0 ≥ 0. Since Vt+1 (a0 , e0 ) = Xt+1 (max{0, a0 }, e0 ), it is clear from comparison with (*) that VtR (a, e) = Xt (a, e) for all a ≥ 0. Since default is not allowed for a ≥ 0, Vt (a, e) = VtR (a, e) and so Vt (a, e) = Xt (a, e). Likewise, for a < 0, either VtR (a, e) < Xt (0, e) or VtR (a, e) is undefined (since repaying strictly positive amounts of debt is worse than autarky when the continuation utilities are the same). Then because VtO (a, e) = Xt (0, e) for a < 0, the outside option is preferable to repayment. Hence Proposition 7 gives that it is then weakly optimal for the planner to have Dt (a, e, n) = {0, 1} for all n. Then, with Dt (a, e, n) = {0, 1} for all n, the problem conditional on default when there are no psychic costs is VtD (a, e) =

max u(c, n) + β 0

a ∈A,n∈N

X

πee0 |t Vt+1 (a0 , e0 )

e0

s.t. c + q¯a0 = en, c > 0, a0 ≥ 0 Again, it is clear from comparison with (*) that VtD (a, e) = Xt (0, e) for all a < 0 (i.e., wherever V D is defined) since the continuation utilities are the same. Putting these results together, Vt (a, e) = VtD (a, e) = Xt (0, e) for a < 0 and Vt (a, e) = VtR (a, e) = Xt (a, e) for a ≥ 0. So, Vt (a, e) = Xt (max{0, a}, e) for all a, e. Additionally, dt (a, e) = 1 (for all a < 0, e). Moreover, qt−1 (a0 , e) = 0 for all a0 < 0, e since dt is uniformly equal to one. This completes the induction argument. 54

So, an optimal policy has Vt (a, e) = Xt (max{0, a}, e) for all a, e, t. In any policy, the household always has Vt (a, e) ≥ max{VtO (a, e), VtR (a, e)} for a < 0 and Vt (a, e) = VtR (a, e) for a ≥ 0. Since VtO (a, e) = Xt (0, e) for a < 0 and VtR (a, e) ≥ Xt (a, e) for a ≥ 0, it must be that Vt (a, e) ≥ Xt (max{0, a}, e). So, if αt (a, e) > 0 for each state, then there cannot be a policy having Vt (a, e) > Xt (max{0, a}, e) in some state because such a policy would “Pareto dominate”— in the sense of Vt (a, e) being weakly higher for each state and strictly higher for some state—the optimal policy with Vt (a, e) = Xt (max{0, a}, e) for every state. So, every optimal policy has Vt (a, e) = Xt (max{0, a}, e) for every state. But every policy also has Vt (a, e) ≥ Xt (max{0, a}, e) for all every state. So, every policy generates the same welfare, which is the same as welfare in the zero borrowing limit economy. Lemma 9. Let z ∈ Z ⊂ Rd with Z finite. Suppose there is a mapping f from z to e that is not invertible. Then for any  > 0 there is an invertible mapping g such that maxz∈Z |f − g| < . Proof. Let n be the cardinality of Z and let zi denote the ith element of some arbitrary ordering of Z. is invertible If f is identically equal to some constant c, then g(zi ) = c+ i−1 n and maxz∈Z |f − g| < . Now, suppose f is not identically equal to a constant. Define ρ(i, j) = |f (zi ) − f (zj )|. Let N = {(i, j)|ρ(i, j) 6= 0, i 6= j} and Y = {(i, j)|ρ(i, j) = 0, i 6= j}. Then Y is non-empty since f is not invertible and N is non-empty since f is not identically equal to a constant. Define δ = mini,j|(i,j)∈N ρ(i, j). satisfies maxz∈Z |f − g| <  and Then I claim g(zi ) = f (zi ) + min{, δ} i−1 n is invertible. The first claim is obvious, so suppose it is not invertible. Then there exists a j 6= i such that g(zi ) = g(zj ). This implies f (zi )+min{, δ} i−1 = n

55

f (zj ) + min{, δ} j−1 , hence n j − 1 i − 1 − min{, δ} ρ(i, j) = min{, δ} n n = min{, δ}

|i − j| n

∈ (0, δ) Then because ρ(i, j) > 0, ρ(i, j) ∈ N . But by definition, δ = mini,j|(i,j)∈N ρ(i, j). Hence, δ = mini,j|(i,j)∈N ρ(i, j) ≤ ρ(i, j) < δ, a contradiction.

56

B

Appendix: Computation [Not for Publication]

This appendix outlines some of the more important parts of the computation, including the algorithm used to compute the optimal policies.

B.1

Discretization

The asset grid is 50 strictly negative points and 250 total points. The points range from −25 to 15 (recall average earnings are normalized to 1 in the US calibration) and are very concentrated about zero. Specifically, the spacing between points from −25 to 0 shrinks geometrically at rate 0.85, while the spacing from 0 to 15 grows geometrically at rate 1.01. The persistent shock z is discretized with 7 points and the transitory shock ε is discretized with 3 points as described in the main text.

B.2

Algorithm for finding the full optimal policy

The full optimal policy requires the planner choose Dt (a, e, n) ∈ {{0}, {0, 1}} for every a < 0, e, t and an arbitrary value of n. Consequently, the planner’s −− choice space is very large at 2#A ×#E×T and an exhaustive search of the choice space is impossible. However, genetic algorithms are designed for optimization problems such as this. In the language of genetic algorithms, the planner’s policy can be represented as a “gene,” a sequence of 0-1 bits with each bit representing the value of max Dt (a, e, n) for one a, e, t (and all n). The algorithm works as follows. An initial population of genes is chosen. Over time, each gene mutates (i.e., switches from 0 to 1 or vice versa) and reproduces with other genes in the population. The probability that a given gene reproduces is an increasing function of the gene’s fitness. For the planner problem, I define a gene’s fitness P by − a,e,t αt (a, e)Vt (a, e) since the period utility function is negative. There are many variants of genetic algorithms corresponding to different ways of selecting genes for reproduction and how the reproduction is carried 57

out. I use a double crossover variation that generates two children from two parents via cutting the parent at two random positions and exchanging the middle section. Then each child is mutated with the probability of a single bit mutation set to 0.1%. The total population of genes is set to 1600 genes (and is maintained at each iteration). I use elitism, a variant that saves the best gene by dropping the worst gene at each iteration and replacing it with the best one (consequently, the search is hill-climbing). I also employ a multigrid approach to more accurately recover the optimal policy. Specifically, I use 6 different and progressively finer partitions of {1, . . . , #A−− }×{1, . . . , #E}×{1, . . . , T }. Within each partition, max Dt (a, e, n) is forced to be identical. The number of partitions is initially 105, then 245, 735, 3,675, 11,025, and finally 22,050 (equal to #A−− ×#E ×T ) , which means the final grid does not restrict the planner at all. At each multigrid step (i.e., for each partitioning), the genetic algorithm evaluates the planner policy (i.e., solves for household value functions given planner policy) one million times. This multigrid approach allows for a more exhaustive search of the state space than would otherwise be possible. The first few partition refinements noticeably improve on one another, but in going from 3,675 partitions to 22,050 there is only a small welfare gain. While the search space is massive in theory, in practice it is substantially smaller. As argued in the main text, the most relevant parts are the states in ∆ that are visited in equilibrium. Based on the logit estimation sample size in Table 4, only 2650 states are in ∆ and visited with non-zero probability. Of these, only a small fraction have a non-negligible effect on the planner’s ex-ante welfare measure.

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