The Dynamic Costs of Default

Transcription

1 The Dynamic Costs of Default Dean Corbae University of Wisconsin-Madison and NBER November 4, 2016 Midwest Macro Plenary

2 Why do people in debt pay back rather than file for bankruptcy? Benefits of default: There are relatively small Ch. 7 filing costs, which protect the filer from creditors garnishing their income. Because credit is forward looking, new creditors are not obliged to punish at all. Despite these benefits, filing triggers: Significantly lower credit scores. Consumers with low credit scores face higher interest rates and restricted access to credit in the future.

3 Roadmap 1 Limited Commitment Models with exogenous dynamic punishment 2 Limited Commitment with unobservable type and endogenous dynamic punishment 3 Directions for Future Research

4 Limited Commitment Models

5 What do equilibrium models with limited commitment have to say about the costs of default? Two equilibrium concepts to approach this question: Debt Constrained Asset Markets (DCAM): Example: Kehoe and Levine (1993) Design debt limits on long term contingent contracts so that individual rationality implies consumers choose not to default (punished with permanent autarky). Incomplete Markets with Default (IMD): Example: Eaton and Gersovitz (1981) Design debt contingent pricing schedules on short term contracts that take into account likelihood of default (punished with permanent autarky).

6 What do quantitative equilibrium models with limited commitment say about the costs of default? Equilibrium Concept Application Debt Constrained Mkts Incomplete Mkts Consumer Debt Krueger-Perri (2006) Chatterjee, et. al. (2007) Livshits, et. al. (2007) Sovereign Debt Kehoe-Perri (2002) Arellano (2008) Corporate Debt Albuquerque- Hopenhayn (2004) Corbae-D Erasmo (2015) Note: This, and future tables, provide a short, incomplete list. I welcome suggestions to add to tables.

7 Debt Constrained Asset Markets Individual rationality constraint given by V (y, a) = max {a (y,y), y } u y + a(y) q(y, y)a (y, y) y +βe y y V (y, a (y, y)) u(y) + βe y y V Aut (y ), y. Multiplier on constraint gives shadow cost of default. Risk sharing measure given by 1 var(c)/var(y). Complete markets (autarky) measure is 1 (0). Krueger-Perri (2010, Figure 4) find 0.7 using DCAM model. US data is /1.22 = 0.2

8 Debt Constrained Asset Markets Individual rationality constraint given by V (y, a) = max {a (y,y), y } u y + a(y) q(y, y)a (y, y) y +βe y y V (y, a (y, y)) u(y) + βe y y V Aut (y ), y. Multiplier on constraint gives shadow cost of default. Risk sharing measure given by 1 var(c)/var(y). Complete markets (autarky) measure is 1 (0). Krueger-Perri (2010, Figure 4) find 0.7 using DCAM model. US data is /1.22 = 0.2 DCAM models predict no equilibrium default and constraint binds in high income states.

9 Consumer Default Facts (from Diaz-Gimenez, et. al. (2011)) SCF: the fraction of bankrupt households has shrunk dramatically from 1.76% in 1998 to 0.93% of the sample in Bankrupt HH Solvent HH SCF Income $40,792 $46,009 $83,983 $71,581 Debt $79,201 $63,357 $97,237 $59,742 Debt/Income Bankrupt HHs have lower income and higher debt to income ratios than Solvent HHs, which is inconsistent with predictions from DCAM.

10 A Quantitative Incomplete Markets Model to Understand Unsecured Consumer Default (from CCNR (2007)) An Aiyagari (1994) model with Ch.7 option which is consistent with equilibrium default by high debt to income HHs.

11 A Quantitative Incomplete Markets Model to Understand Unsecured Consumer Default (from CCNR (2007)) An Aiyagari (1994) model with Ch.7 option which is consistent with equilibrium default by high debt to income HHs. Environment: Persistent idiosyncratic earnings shocks y.

12 A Quantitative Incomplete Markets Model to Understand Unsecured Consumer Default (from CCNR (2007)) An Aiyagari (1994) model with Ch.7 option which is consistent with equilibrium default by high debt to income HHs. Environment: Persistent idiosyncratic earnings shocks y. If solvent, HHs choose a at price q(a, y).

13 A Quantitative Incomplete Markets Model to Understand Unsecured Consumer Default (from CCNR (2007)) An Aiyagari (1994) model with Ch.7 option which is consistent with equilibrium default by high debt to income HHs. Environment: Persistent idiosyncratic earnings shocks y. If solvent, HHs choose a at price q(a, y). If default, HHs punished (cost) with one period autarky and default flag h = 1.

14 A Quantitative Incomplete Markets Model to Understand Unsecured Consumer Default (from CCNR (2007)) An Aiyagari (1994) model with Ch.7 option which is consistent with equilibrium default by high debt to income HHs. Environment: Persistent idiosyncratic earnings shocks y. If solvent, HHs choose a at price q(a, y). If default, HHs punished (cost) with one period autarky and default flag h = 1. If h = 1, HHs punished (cost) with zero borrowing constraint.

15 A Quantitative Incomplete Markets Model to Understand Unsecured Consumer Default (from CCNR (2007)) An Aiyagari (1994) model with Ch.7 option which is consistent with equilibrium default by high debt to income HHs. Environment: Persistent idiosyncratic earnings shocks y. If solvent, HHs choose a at price q(a, y). If default, HHs punished (cost) with one period autarky and default flag h = 1. If h = 1, HHs punished (cost) with zero borrowing constraint. Flag removed from credit history with prob (1 ρ) calibrated to match FCRA 10 years on average.

16 A Quantitative Incomplete Markets Model to Understand Unsecured Consumer Default (from CCNR (2007)) An Aiyagari (1994) model with Ch.7 option which is consistent with equilibrium default by high debt to income HHs. Environment: Persistent idiosyncratic earnings shocks y. If solvent, HHs choose a at price q(a, y). If default, HHs punished (cost) with one period autarky and default flag h = 1. If h = 1, HHs punished (cost) with zero borrowing constraint. Flag removed from credit history with prob (1 ρ) calibrated to match FCRA 10 years on average. If a < 0, then q(a, y) = (1 E y y [d(a, y )])/(1 + r) where d(a, y ) is the HH default decision rule which solves their DP problem.

17 CCNR (2007) HH DP problem From state (y, a, h): If good credit history (h = 0), where and V (y, a, 0) = max d {V d=0 (y, a, 0), V d=1 (y, a, 0)}. (1) V d=1 (y, a, 0) = u(y) + βe y y V (y, 0, 1) (2) V d=0 (y, a, 0) = max a u(y + a q(a, y)a ) + βe y y V (y, a, 0) (3)

18 CCNR (2007) HH DP problem From state (y, a, h): If good credit history (h = 0), where and V (y, a, 0) = max d {V d=0 (y, a, 0), V d=1 (y, a, 0)}. (1) V d=1 (y, a, 0) = u(y) + βe y y V (y, 0, 1) (2) V d=0 (y, a, 0) = max a u(y + a q(a, y)a ) + βe y y V (y, a, 0) If bad credit history (h = 1), V (y, a, 1) = max a 0 u ( y + a a /(1 + r) ) (4) (3) +βe y y [ ρ V (y, a, 1) + (1 ρ) V (y, a, 0) ].

19 CCNR (2007) Results We find (Table 8) risk sharing measure 1 var(c)/var(y)= 0.42

20 CCNR (2007) Results We find (Table 8) risk sharing measure 1 var(c)/var(y)= 0.42 We used the structural model to evaluate the 2005 BAPCPA which limits above-median-income HHs from filing under Ch. 7.

21 CCNR (2007) Results We find (Table 8) risk sharing measure 1 var(c)/var(y)= 0.42 We used the structural model to evaluate the 2005 BAPCPA which limits above-median-income HHs from filing under Ch. 7. With harsher punishment, we found small changes to aggregate default rates but sizeable drops in interest rates and increases in the fraction of HHs in debt.

22 CCNR (2007) Results We find (Table 8) risk sharing measure 1 var(c)/var(y)= 0.42 We used the structural model to evaluate the 2005 BAPCPA which limits above-median-income HHs from filing under Ch. 7. With harsher punishment, we found small changes to aggregate default rates but sizeable drops in interest rates and increases in the fraction of HHs in debt. Why? Selection effects. While harsher punishment leads the entire schedule of interest rates to drop, HHs select more debt (which induces more default).

23 Some Other IMD Applications and Methodological Contributions

24 Unsecured Consumer Debt with Aggregate Fluctuations Gordon (2015), Nakajima and Rios-Rull (2014) introduce aggregate uncertainty (along the lines of Krusell and Smith (1998)) into CCNR (2007) (N-R with labor supply choice).

25 Unsecured Consumer Debt with Aggregate Fluctuations Gordon (2015), Nakajima and Rios-Rull (2014) introduce aggregate uncertainty (along the lines of Krusell and Smith (1998)) into CCNR (2007) (N-R with labor supply choice). N-R find that the model economy shares the main cyclical properties of borrowing and default in the US: borrowing is procyclical and bankruptcy filings are countercyclical. They find that consumption is much more volatile in the IM model with default than IM models with strict borrowing constraints. Risk sharing measure (Table 4) is 0.6.

26 Multiple Options: Fresh Start vs. Garnishment vs. Delinquency Athreya, et. al. (2016), Bejamin and Mateos-Planas (2014), Chatterjee and Gordon (2012).

27 Multiple Options: Fresh Start vs. Garnishment vs. Delinquency Athreya, et. al. (2016), Bejamin and Mateos-Planas (2014), Chatterjee and Gordon (2012). A-S-T-Y document that while the bankruptcy rate is between 1 2% of the population, the 60+ day delinquency rate is between 6 10%. Delinquency plays a valuable role by giving borrowers the consumption-smoothing benefits of default with a different cost structure than bankruptcy. Interest rates in delinquency are set to maximize ex-post profitability rather than ex-ante profitability. The model predicts HHs start in delinquency and often transit to bankruptcy, as in the data.

28 Secured Consumer Debt: Understanding Foreclosures Several papers use the IMD framework with long term contracts to understand foreclosures (ex. Chatterjee and Eyugungor (2015), Corbae and Quintin (2015), Hatchondo, et. al. (2015)).

29 Secured Consumer Debt: Understanding Foreclosures Several papers use the IMD framework with long term contracts to understand foreclosures (ex. Chatterjee and Eyugungor (2015), Corbae and Quintin (2015), Hatchondo, et. al. (2015)). C-Q: How much of the foreclosure crisis can be explained by the large number of high leverage mortgages originated during the housing boom? Relaxed approval standards (measured by payment to income ratios) during the housing boom lead to selection effects; the pool of borrowers taking high leverage, low downpayment loans gets riskier. In a counterfactual experiment, where approval standards are not relaxed, we find that foreclosure rates would have been 60% lower in the housing price bust.

30 Sovereign Debt Several papers use the IMD framework with long term contracts to understand sovereign spreads (ex. Chatterjee and Eyugungor (2012), Hatchondo and Martinez (2009)).

31 Sovereign Debt Several papers use the IMD framework with long term contracts to understand sovereign spreads (ex. Chatterjee and Eyugungor (2012), Hatchondo and Martinez (2009)). The average duration of sovereign debt is about 5 years. Rather than add finite n period bonds which expands the state space, these papers introduce a perpetual bond and use a recursive formula to price debt. A unit of debt matures probabilistically (λ can be chosen to match average duration). If it doesn t mature (with prob 1 λ), it pays coupon z. Recursive formula: [ (1 d(a, y ) λ + (1 λ) (z + q(a (a, y ] ))) q(a, y) = E y y 1 + r

32 Corporate Debt Several papers use the IMD framework to understand corporate debt dynamics (ex. Corbae and D Erasmo (2015), Hennessy and Whited (2007), Kahn, et. al. (2015)).

33 Corporate Debt Several papers use the IMD framework to understand corporate debt dynamics (ex. Corbae and D Erasmo (2015), Hennessy and Whited (2007), Kahn, et. al. (2015)). Most small firms choose Chapter 7 liquidation (and exit) while large firms choose Chapter 13 reorganization (and continuation). C-D introduce a bargaining problem (renegotiation of the repayment) into a GE model of firm dynamics with bankruptcy choices. In a counterfactual we study the positive and normative effects of a proposed change by the American Bankruptcy Institute to corporate bankruptcy law and find significant effects on the size distribution of firms and allocative efficiency.

34 Static vs. Dynamic Costs of Default Albanesi and Nosal (2015) find that (static) filing costs rose after the BAPCA from roughly 1% to 2% of median income, but this can have a significant effect on bankruptcy (and delinquency) decision of the poorest HHs. What are the other costs? Stigma (non-pecuniary costs; we tie our hands by not introducing them). Lost reputation and signalling value in a dynamic adverse selection environment For other adverse selection environments with bankruptcy see Athreya, et. al. (2012) and Kovrijnykh and Livshits (2013).

35 Informational Spillovers to Other Markets

36 Does default signal something about your unobservable type? Part 1. Auto insurers charge higher premiums to people with low credit scores. Chatterjee et.al. (2008) provide a theory of unsecured consumer debt that does not rely on stigma or on enforcement mechanisms that arise in repeated-interaction settings. Our theory is based on private info about a person s type and on incentives to signal one s type to non-creditors. Timing: credit market decisions precede insurance decisions. Low risk debtors are less likely to default thereby generating a low cost signal of their type to insurers in another market so as to receive better insurance terms.

37 Does default signal something about your unobservable type? Part 2. A 2012 survey by DEMOS found that 25% of low-to-medium income HHs reported having their credit checked for a job app and 10% claimed to be denied a job due to bad credit. Cortes et. al. (2016) document that county-level unemployment rose faster in states that restricted employer credit checks than adjacent counties which did not, suggesting there might be some signalling benefit in the labor market. Chen et. al. (2014) build a GE model where debt and default decisions have signalling content about the HHs unobservable labor productivity, thereby affecting labor demand.

38 A Quantitative Theory of Credit Scoring

39 Chatterjee, et. al. (2016, CCDR): A reputation based theory of default costs People differ in privately observed characteristics that make some of them more prone to future default. Borrowing too much and filing for bankruptcy are signals of a bad type. This deters them from borrowing too much, which can sustain credit. Our theory replicates key patterns in U.S. unsecured credit market data for bankruptcy laws resembling those in the U.S......without imposing exogenous punishments following default.

40 How we make the theory quantitative A Technical Innovation Our paper introduces unobserved shocks as in the discrete choice (logit) literature (e.g. McFadden (1973), Rust (1987)) to an adverse selection competitive equilibrium model with bankruptcy: We do not have to deal with off-path beliefs in our dynamic Bayesian posteriors since all feasible actions are taken with some probability. Actions only partially reveal information about type (semi-separating equilibrium). Competitive lenders offer loans of different sizes at different prices based on credit scores which account for type dependent household likelihood of default. Literature

41 Taking the model to data 1 An Aiyagari model where households have unobservable persistent differences in discount factors which make some more prone to borrow and default. 2 Intermediaries use observable asset and default choices to try to infer borrower type in order to price loans. 3 We measure type heterogeneity using U.S. unsecured credit market data. 4 Using our estimates, we can quantify the value of reputation the value of information

42 Individuals (HH) Unit mass of HH, i, with preferences E 0 [ t=0 βt it u(c it)]: persistent: discount rate β it {β H, β L }, β Q β (β β) transitory: additive, action-specific shocks ɛ it G(ɛ it ), i.i.d (β, ɛ) unobservable, low β more likely to borrow and default. Earnings, y = e + z, comprised of 2 observable components: persistent: e it E = {e 1,..., e E }, drawn from Q e (e e) transitory: z it Z = {z 1,..., z Z }, i.i.d. from H(z) Each period, choose (d, a ): a A = {a 1,..., 0,..., a A }: asset position for next period d {0, 1} (if a < 0): if d = 1, face temporary exclusion [a = 0] and income loss from default [c = (1 η)(e + z)]

43 Intermediaries risk neutral, perfectly competitive (free entry) borrow at r observe ω = (e, z, a, s) and asset choices σ = (d, a )

44 Intermediaries risk neutral, perfectly competitive (free entry) borrow at r observe ω = (e, z, a, s) and asset choices σ = (d, a ) Inference problem: cannot observe β or ɛ (d,a ) when pricing loans β persistent = actions can signal type ɛ transitory = no information, but clouds inference

45 Intermediaries risk neutral, perfectly competitive (free entry) borrow at r observe ω = (e, z, a, s) and asset choices σ = (d, a ) Inference problem: cannot observe β or ɛ (d,a ) when pricing loans β persistent = actions can signal type ɛ transitory = no information, but clouds inference Reputation: creditor s prior of HH type denoted s = Pr(β = β H ) Bayesian posterior uses observables to revise type score s = ψ (d,a ) (ω)

46 Intermediaries risk neutral, perfectly competitive (free entry) borrow at r observe ω = (e, z, a, s) and asset choices σ = (d, a ) Inference problem: cannot observe β or ɛ (d,a ) when pricing loans β persistent = actions can signal type ɛ transitory = no information, but clouds inference Reputation: creditor s prior of HH type denoted s = Pr(β = β H ) Bayesian posterior uses observables to revise type score s = ψ (d,a ) (ω) Repayment probability p( ) uses type score ψ and decision rules σ: p (0,a ) (ω) = s Pr(s ω) (1 Pr(default on a ω, s )) (5)

47 Intermediaries risk neutral, perfectly competitive (free entry) borrow at r observe ω = (e, z, a, s) and asset choices σ = (d, a ) Inference problem: cannot observe β or ɛ (d,a ) when pricing loans β persistent = actions can signal type ɛ transitory = no information, but clouds inference Reputation: creditor s prior of HH type denoted s = Pr(β = β H ) Bayesian posterior uses observables to revise type score s = ψ (d,a ) (ω) Repayment probability p( ) uses type score ψ and decision rules σ: p (0,a ) (ω) = s Pr(s ω) (1 Pr(default on a ω, s )) (5) Offer discount loans at prices q (d,a ) (ω) = p (0,a ) (ω)/(1 + r).

48 Equilibrium Definition A stationary recursive competitive equilibrium is a vector-valued pricing function q, a vector-valued type scoring function ψ, a vector-valued quantal response function σ, and a cross-sectional distribution µ such that: σ (d,a ) (β, ω f ) satisfies HH optimization, HHprob q (0,a ) (ω) implies lenders break even with objective likelihood of repayment p (0,a ) (ω f ), INTprob ψ (d,a ) β (ω) satisfies Bayes rule, and µ (β, ω f ) is a fixed point of the law of motion for the cross-sectional distribution. Details Existence Theorem There exists a stationary recursive competitive equilibrium. Proof

49 Credit Scores around Default in the Data (Jagtiani and Li (2015, ABLJ))

50 Dynamics of Debt and Default After calibrating the model to match standard credit market moments (default rate, interest rates, net worth to income, fraction in debt, debt to income), we conduct the same event analysis using model generated panel data. percentile in pop. assets, a 25th / 75th percentile credit score, ξ periods after default avg. interest rate, 1/q periods after default HH debt = credit score = higher rates CS (IR) tanks (spikes) following default Figure Panel Construction Calibration

51 How much does Info Asymmetry Matter? Full information environment: β observable = no inference problem ɛ still unobservable and transitory obviates type scoring = no ψ( ), no s Key insights: high (low) β type with full info case face more (less) favorable price schedules than high (low) s type in benchmark Prices high (low) β take on more (less) debt to income and default more (less) than in benchmark, important selection effects. Moments on average, HH are slightly better off in full info, but low β types in debt prefer benchmark Welfare Analysis

52 How Much is Reputation Worth? Question: How much must a HH be compensated to accept being assigned the lowest possible type score? Answer: Define for each state (β, ω) a number τ such that W (β, e, z, a, s) = W (β, e, z, a + τ(β, e, z, a, s), s min ) Aggregating, we find: τ (%) agg. a < 0 s = s max, a < 0 s = s max, a = a min agg β H β L small numbers in aggregate reflect small fraction in debt Across states

53 CCDR Conclusion Developed model of unsecured consumer credit in which agents have option to default, and do so in equilibrium unobservable preference shocks impose an inference problem on intermediaries who price debt credit scoring helps solve this problem Calibrated the model to key credit market moments to show default behavior by credit score closely matches data asymmetric info expands the fraction of economy in debt (selection effects matter), but reduces welfare relative to full info. reputation matters in that many borrowers would require significant compensation to be labeled as bad

54 Short Run Directions for Future Research Enriching the Model: Add another state variable for household balance sheet (i.e. debt and assets rather than net assets). Add credit limits into the theory.

55 Long Run Directions for Future Research Design the costs of Default: How does non-exclusivity of contracts affect punishment and market structure? Parlour and Rajan (2001) show that non-exclusivity can create imperfect competition in the credit market. Implementation of mechanism design problems with limited commitment (Kehoe and Levine (2006)) or moral hazard (Grochulski (2010)). Designing how we unwind banks can affect risk taking prior to default.

56 Long Run Directions for Future Research Endogenize the costs of Default: Credit scoring is being used more and more in other markets so there are potentially important dynamic costs. Persistent net-debt and delinquency will magnify those costs (Athreya, et. al. (2016) find that HHs in financial distress today are 10% more likely to be in financial distress 6 years from now). Adverse selection and signalling are also important in other applications like sovereign default (e.g. D Erasmo (2011)) and corporate finance which use credit ratings (as in Jovanovic (1982)).

57 Appendix APPENDIX

58 Appendix Back Borrower Characteristics and Credit Terms

59 Appendix Borrower Characteristics and Credit Terms

60 Appendix Unobserved Heterogeneity From HKL (Table 3 and p.23 ): Notably, despite the extensive set of explanatory variables and flexible specifications, our models explain only a relatively small portion of the overall variation in contract terms. For instance, the largest R-squared we obtain is for the interest rate spread, where we can explain only 35 of the variation using observable borrower and geographic characteristics (and time fixed effects). Our low R-squareds are remarkable because the amount of information we use is similar to what a lender would have at its disposal in screening a consumer without a prior business relationship. Back

61 Appendix More Detailed Literature Review Equilibrium Models of Bankruptcy Full info, Exogenous Punishment: Chatterjee et al. (2007), Livshits et al. (2007) Asymmetric info, Static Signaling, Exogenous Punishment: Athreya et al. (2009, 2012), Livshits et al. (2015) Asymmetric info, Dynamic Signaling, Endogenous Punishment (Reputation): Chatterjee et al. (2008). Important Issue with Asym Info: Off-Path Beliefs Discrete Choice Models McFadden (1973), Rust (1987). Discrete choice models specify the probability that an individual chooses an action among a set of feasible alternatives. = no off-path beliefs to specify for feasible actions, imperfect separation

62 Appendix Budget Feasibility and Actions Set of all possible default and asset choices: Y = { (d, a ) : (d, a ) {0} A or (d, a ) = (1, 0) } Given observable state ω and a set of equilibrium functions f the set of feasible actions is F(ω f ) Y that contains all actions (d, a ) Y such that c (d,a ) > 0 Consumption is pinned down by the budget constraint: e + z + a q (0,a ) (ω) a for d = 0, a < 0 c (d,a ) = e + z + a a /(1 + r) for d = 0, a 0 (1 η)(e + z) for d = 1, a = 0 Back to HH problem Back to HH policies

63 Appendix Existence of a Solution to HH Problem Theorem Given f, there exists a unique solution W (f ) to the individual s decision problem in (9) to (11) and W (f ) is continuous in f. Sketch of proof: Apply Contraction Mapping Theorem defining the operator (T f )(W ) : R B+ Ω R B+ Ω. To prove continuity of W (f ), show that the operator T f is continuous in f. Follows given continuity of u with respect to c, c (d,a ) with respect to q for (d, a ) F(ω f ) and Q s with respect to ψ. Since R M+K is a Banach space, then apply Theorem in Hutson and Pym (1980). Back to HH policies Back to theorems

64 Appendix Extreme Value Shocks 101 Extreme Value Distribution with location parameter = 0 and scale parameter 1 α where higher α implies lower variance. Can show that σ(d,a ) (β,ω) α takes the sign of [ v (d,a ) (β, ω) v ( d,ã ) ] (β, ω) ( d,ã ) F(ω) { ( exp α v (d,a ) (β, ω) + v (ˆd,â ) )} (β, ω) so more likely to take optimal action the lower is variance. From the formula for σ( ) we have arg max (d,a ) F(ω) σ(d,a ) (β, ω) = arg max v (d,a ) (β, ω), (d,a ) F(ω) so the optimal action without extreme value shocks is the modal action in our paper. Back to HH policies Figure

65 Appendix Figure: Impact of Extreme Value Shocks mode, argmax (d,a )σ (d,a ) (β,e,z,a) High α = β = 0.97 β = a β = 0.97 β = 0.8 Low α = a mean, E[a (β,e,z,a)] β = 0.97 β = β = 0.97 β = a a Back

66 Appendix Bayesian Type Assessment Updating and Pricing Details Probability that an agent will be of type β tomorrow given by Bayes rule: ψ (d,a ) β (ω) = β Q β (β β) σ (d,a ) (β, ω) s(β) [ ] ˆβ σ (d,a ) ( ˆβ, ω) s( ˆβ) for each type β. Since ψ may not lie on the grid S, for the two nearest grid points s i ψ (d,a ) (ω) s j compute χ(ψ) = s j ψ s j s i = Q s (s ψ) = χ(ψ) if s = s i 1 χ(ψ) if s = s j 0 otherwise Then repayment probabilities given by: p (0,a ) (ω) = Q s (s ψ (d,a ) (ω)) Q e (e e) H(z ) (6) s,e,z [ ] s (1 σ (1,0) (β H, ω )) + (1 s )(1 σ (1,0) (β L, ω )) Back

67 Appendix Cross-sectional Distribution Let µ(β, ω f ) be the measure of individuals in state (β, ω) today for a given set of equilibrium functions f. The distribution evolves according to µ (β, ω f ) = T (β, ω β, ω; f ) µ(β, ω f ). (7) where (β,ω) B Ω T (β, ω β, ω; f ) = σ (d,a ) (β, ω f ) Q s (s ψ (d,a ) (ω)) (8) Q β (β β) Q e (e e) H(z ) An invariant distribution is a fixed point µ( ) of (7). Existence Back to equilibrium definition

68 Appendix Existence of an Invariant Distribution Lemma There exists a unique invariant distribution µ. Sketch of proof: Use Theorem 11.2 in Stokey and Lucas (1989) to establish this result. µ is critical for computing cross-sectional moments map model to data No other equilibrium objects functions f, the value function V ( ) or the decision rule σ ( ) ( ) take µ( ) as argument simplifies computation Back to distribution Back to theorems

69 Appendix Existence of Equilibrium pt. 1 Theorem There exists a stationary recursive competitive equilibrium. Sketch of proof: Let f be the vector composed by stacking q [0, 1] K and ψ [0, 1] M so f [0, 1] K+M and let W = W (f ) : [0, 1] K+M R B+ Ω be the solution established in Theorem 2. Given W, use (10) to construct the vector-valued function v = J 1 (W ) : R B+ Ω R M Given v, use (13) to construct the vector-valued function σ = J 2 (v) : R M (0, 1) M. Pt. 2 Back to equilibrium definition

70 Appendix Existence of Equilibrium - pt. 2 Given σ and ψ, use the mapping in (16) to construct the vector-valued function p = J 3 (σ, ψ) : (0, 1) M [0, 1] M [0, 1] A Ω. Given p and σ, use the mapping in (15) and (14) to construct the K + M vector f new = (q new, ψ new ) = J 4 (p, σ) : [0, 1] A Ω [0, 1] M [0, 1] K+M. Let J(f ) : [0, 1] K+M [0, 1] K+M be the composite mapping J 4 J 3 J 2 J 1 W. By Theorem 2 W (f ) is continuous and the functions J i, i {1, 2, 3, 4} are also continuous. Hence J is a continuous self-map. Since [0, 1] K is a compact and convex subset of R K, the existence of f = J(f ) is guaranteed by Brouwer s FPT. Pt. 1 Back to equilibrium definition

71 Appendix Theorems 1 HH solution: Given f, there exists a unique solution W (f ) to the individual s decision problem in (9) to (11) and W (f ) is continuous in f. Existence of HH solution 2 Stationary distribution: There exists a unique invariant distribution µ. Existence of stationary distribution 3 Equilibrium existence: There exists a stationary recursive competitive equilibrium. Existence of equilibrium

72 Appendix Computational Algorithm and Estimation Algorithm: tiered-loop grid search 1 create grids for β, e, z, a, s (earnings calibrated outside model) 2 start with initial guesses of f = f i 3 compute feasible set F(e, z, a, s f i ) 4 value function iteration = σ(β, e, z, a, s f i ) 5 σ = f i+1 6 if max{ f i+1 f i } < tol, continue; else, go back to 2 7 compute µ, moments Estimation: 2-stage SMM 1 set W 0 = I 5, embed above algorithm in DFBOLS optimization procedure of Zhang et al. (2010) to get parameter estimates ˆθ 0 2 simulate N T panel from the model under ˆθ 0 to compute efficient weighting matrix W, repeat stage 1 procedure to get final estimates ˆθ and standard errors from W

73 Appendix Parameterization Details Grid Size Range Details β 2 {0.89, 0.97} bivariate type = scalar ψ, Γ e 3 [0.58,1.74] Floden and Lindé (2009) z 3 {-0.182,0,0.182} z = +/ 3/ a 151 [-0.25,7.0] 50 neg pos s 50 [0.04, 0.90] [Q β (β L β H), 1 Q β (β L β H)] Earnings details e Q e (e e) e 1 e 2 e 3 e e e Back

74 Appendix Definitions of Key Model Moments Default rate = β,ω σ(1,0) (β, ω) µ(β, ω) Median net worth to median income - straightforward Fraction of HH in debt = β,e,z,s a<0 µ(β, e, z, a, s) Average debt to income ratio = β,e,z,a<0,s Average chargeoff rate and total debt total debt defaulted, where β,e,z,s a e+z+(1/q( ) 1) a total debt = a a<0 µ(β, e, z, a, s) µ(β,e,z,a,s) ˆβ,ê,ẑ,â<0,ŝ µ( ˆβ,ê,ẑ,â,ŝ) total debt defaulted = a σ (1,0) µ(β, e, z, a, s) (β, e, z, a, s) a<0 β,e,z,s ˆβ,ê,ẑ,ŝ µ(. ˆβ, ê, ẑ, a, ŝ) Back

75 Appendix Targeted Moments Moment [Source] Data Model Default rate (%) Chatterjee et al. (2007) Average interest rate (%) Chatterjee and Eyigungor (2009) Median net worth / median income Chatterjee and Eyigungor (2009) Fraction of households in debt (%) Chatterjee et al. (2007) Back Average debt-to-income ratio (%) Chatterjee et al. (2007)

76 Appendix Credit Score Distribution Model Data (TransUnion) 30 Default rate (%) % of population % Delinquency rates by FICO score delinquency rate (%) % people Mass and default rate by credit score bin % 51% 31% 27% 18% 15% 15% 12% 13% 8% 5% 5% 2% 2% 1% 0 <90.4 [90.4,95.1) [95.1,95.3) [95.3,95.8) [95.8,96.1) [96.1,96.7) [96.7,98.1) >98.1 Credit score (ξ) range 0 up to ! ! ! ! ! ! FICO score range compute credit scores within the model by integrating choice-specific default probabilities over choice probabilities Model credit score details Back

77 Appendix Credit Scores Mapping from model credit scorecard to analog of real world credit score is not trivial can t just use type score: β L types have higher propensity to default = priced out = default less...so who s the bad type? can t just use the repayment probability: p is action-specific, not like FICO or anything... Proper procedure is to integrate over actions, conditional on going into debt define credit score function ξ( ) as ξ(ω) = p (0,a ) β ( ) σ (0,a ) (β, ω) µ(β, ω) ( ) a <0 σ (0,â ) ( ˆβ, ω) µ( ˆβ, ω) Back ˆβ,â <0

78 Appendix Construction of Panel In order to construct the figures that map out the prices and states before and after default, we follow the procedure: 1 draw a set of N = 5000 initial conditionals for (β, ω) from the stationary distribution µ( ) 2 for T = 100 periods, use the decision rule σ( ) and the exogenous transitions to map HH s flows through states 3 isolate all the default events, and the HH s state in t = 5 periods before and t + periods after 4 average over all relevant variables and compute desired confidence intervals Back

79 Appendix Impact of Default on Price Menu price, q Before After asset choice, a default raises entire menu of interest rates. Construction Back

80 Appendix Asymmetric vs. Full Information: Selection Effects Why do interest rates rise under full information? While High type continue to default less than Low type, High type default relatively more under full info while Low type default relatively less as High types try to maintain their reputation with asymmetric info. While the pricing menus reflect lower default probabilities for High types, High types select relatively more debt resulting in higher relative default rates and interest rates. Back

81 Appendix Welfare analysis Question: How much more consumption per period must an agent receive in the asymmetric info economy to be indifferent with the full info economy? Answer: Construct consumption equivalents: [ W FI (β, ω FI ] ) 1 1 γ λ(β, ω) = 1 W (β, ω) Aggregating, we find: λ (%) agg. a < 0 s = s s = s s = s, a < 0 s = s, a < 0 agg β H β L Note that Low type in debt actually benefit from asymmetric info. Back Details Across states

82 Appendix Consumption Equivalent Derivation For each (β, ω), define fraction λ(β, ω) by which consumption will have to be increased each period to be indifferent between the benchmark and full information economies Given benchmark value (up to shocks) W (β, ω) = V (ɛ, β, ω)dg(ɛ) and an analogous value W FI (β, ω FI ), we can write [ ] W FI (β, ω FI ) = E β,ω βt t u (ct (1 + λ(β, ω))), t=0 where ct is optimal consumption in the benchmark. solving for λ( ) yields the expression in the main text Back Across the state space

83 Appendix Welfare Across the State Space avg λ (%) debt, a assets, a 0 avg λ (%) high β low β type score, s Define λ(β, a) (λ(β, s)) to be the average λ for agents with β, a (s) Back

84 Appendix Benchmark Model: HH Policies β = 0.97, s = 0.05 β = 0.9, s = 0.05 β = 0.97, s = 0.9 β = 0.9, s = 0.9 e = , z = 0 mode, argmax (d,a ) σ (d,a ) (β,e,z,a) β = 0.97, s = 0.05 β = 0.9, s = 0.05 β = 0.97, s = 0.9 β = 0.9, s = a almost complete separation on β minimal differences across s for fixed β Back Type Scores Prices

85 Appendix Benchmark Model: HH Policies β = 0.97, s = 0.05 β = 0.9, s = 0.05 β = 0.97, s = 0.9 β = 0.9, s = 0.9 e = 1, z = 0 mode, argmax (d,a ) σ (d,a ) (β,e,z,a) β = 0.97, s = 0.05 β = 0.9, s = 0.05 β = 0.97, s = 0.9 β = 0.9, s = a almost complete separation on β minimal differences across s for fixed β Back Type Scores Prices

86 Appendix Benchmark Model: Type Scores e = , z = 0 a = s = 0.05 s = s = a = s = 0.05 s = s = asset choice, a low earnings: choice matters for reputation with low wealth Back Policies Prices

87 Appendix Benchmark Model: Type Scores 0.8 e = 1, z = 0 a = s = 0.05 s = s = a = s = 0.05 s = s = asset choice, a high earnings: deeper debt lowers reputation, more if already low deeper debts affect score more adversely

88 Appendix Benchmark Model: Prices a = s = 0.05 s = s = 0.9 e = , z = s = 0.05 s = s = 0.9 a = asset choice, a type score s seems only to matter for low a agents effect greater for agents whose (e, z) doesn t compensate Back Policies Type Scores

89 Appendix Benchmark Model: Prices a = s = 0.05 s = s = 0.9 e = 1, z = s = 0.05 s = s = 0.9 a = asset choice, a type score s seems only to matter for low a agents effect greater for agents whose (e, z) doesn t compensate Back Policies Type Scores

90 Appendix Full Information: HH Policies modal action, argmaxσ (d,a ) β = 0.97 β = 0.9 e = , z = mean action, E(a ) β = 0.97 β = current assets, a fairly strong separation modal (mean) action for β H weakly (strictly) above β L Back Prices

91 Appendix Full Information: HH Policies modal action, argmaxσ (d,a ) β = 0.97 β = 0.9 e = 1, z = mean action, E(a ) β = 0.97 β = current assets, a fairly strong separation modal (mean) action for β H weakly (strictly) above β L Back Prices

92 Appendix Full Information: Prices e = , z = β = 0.97 β = e =1, z = β = 0.97 β = asset choice, a q(a, β H ) uniformly above q(a, β L ), more so far in debt

93 Appendix Construction of Full Info vs. Benchmark Price Schedules in the benchmark, prices given by q (0,a ) (ω) under full information, prices given by q FI (0,a ) (ω FI ) Want to compare the average price schedule faced by each β type across all s in benchmark to average price schedule faced by each β type in full info case how to do this? fix the distribution µ from the benchmark model and compute average prices for each action according to q (0,a ) (β, s) = q (0,a ) µ (β, e, z, a, s) (e, z, a, s) e,z,a ê,ẑ,â µ (β, ê, ẑ, â, s) q FI (0,a ) (β) = ω q FI (0,a ) (ω) µ (β, ω) ˆω µ (β, ˆω) Back

94 Appendix Decisions Prices Selection Effects Back Asymmetric vs. Full Information: Moments def. rate int. rate med net worth med income frac. in debt debt income Data 0.54% 11.35% % 0.67% Bench agg β H β L Full info agg β H β L under full info, β L types who drive default rate get less debt selection affects important for interest rates (high types choose more debt)

95 Appendix Asymmetric vs. Full Information: Pricing Schedules q or q FI full info, high β benchmark, s benchmark, s full info, low β debt choice, a 0 more dispersion in price schedules with full info high type (i.e. high score) still face lower interest rates Construction Back

96 Appendix Value of Reputation across the State Space avg τ (%) debt, a assets, a 0 avg τ (%) high β low β type score, s Back

97 Appendix η default interest rates high willingness to pay for a high type score. removal of static costs makes reputation play a greater role τ increases by a factor of 14 Static vs. Dynamic Costs of Default Question: What happens if there are no static costs of default? Answer: Set η = 0, re-solve model: Moment Data η = 9.8% η = 0 Default rate (%) Average interest rate (%) Median net worth / median income Fraction of households in debt (%) Average debt-to-income ratio (%) τ (%)

98 Appendix Some Related Literature Quantitative Models of Bankruptcy: Full information, exogenous punishment via exclusion: Chatterjee et al. (2007), Livshits et al. (2007), Asymmetric info, anonymous markets assumption rules out credit scores, and exogenous off-path beliefs: Athreya et. al. (2012) Competitive Adverse Selection Models: With competitive search, principals post contracts with commitment, agents choose where to apply and match bilaterally: Guerrieri et. al. (2010) prove existence and uniqueness of perfectly separating equilibria in a static environment. Miyazaki (1977) and Wilson (1977) allow lenders to withdraw contracts after other lenders deviate. This makes deviations less profitable which can support pooling contracts. Back

99 Appendix How We Celebrate Birthdays in Wisconsin Back

100 Appendix Timing 1 HH begin period with state (β, e, a, s) 2 HH receive transitory earnings z drawn from H(z) and preference shocks ɛ = {ɛ (d,a ) } (d,a ) Y drawn from extreme value distn G(ɛ) 3 given price schedule q = {q (0,a ) (ω)} (d,a ) Y, agents choose (d, a ) 4 Intermediaries revise type scores from s ψ (d,a ) (ω) via Bayes rule 5 next period state β drawn from Q β (β β), e drawn from Q e (e e), and s drawn from Q s (s ψ)

101 Appendix HH Optimization Problem Taking price, type score functions f = (q, ψ) as given, HH solves V (ɛ, β, ω f ) = max v (d,a ) (β, ω f ) + ɛ (d,a ) (d,a ) F(ω f ) (9) where v (d,a ( ) is the conditional value function ( ) v (d,a ) (β, ω f ) = u c (d,a ) (10) +β ] [Q β (β β)q e (e e)q s (s ψ)h(z )W (β, ω f ) β,ω and W ( ) integrates over extreme value shocks W (β, ω f ) = V (ɛ, β, ω f )dg(ɛ). (11) subject to (d, a ) in the feasible set F(ω f ) defined by { c (d,a ) e + z + a q (0,a ) (ω) a > 0 for d = 0, a 0 (ω f ) = (e + z) (1 η) for d = 1, a = 0 (12)

102 Appendix HH Decision Rules Theorem Given f, there exists a unique solution W ( f ) to the individual s decision problem in (9) to (11) and W (f ) is continuous in f. Proof Following the discrete choice literature, ɛ i.i.d EV (α) = decision rule is given by the probability function: { } exp α v (d,a ) (β, ω f ) σ (d,a ) (β, ω f ) = { } (13) (ˆd,â ) F(ω f ) exp α v (ˆd,â ) (β, ω f ) The modal action has highest v (d,a ) ( ). With extreme value distribution, higher α implies lower variance of ɛ, so HH is more likely to take the modal action. Budget details Impact of α EqmBack

103 Appendix Type Scoring and Debt Pricing The intermediary updates the assessment of a HH s type given its actions and observable characteristics: ψ (d,a ) (ω) = Pr(β = β H d, a, ω) (14) using Bayes rule, then assigns a likelihood to each type s. Details Perfect competition, deep pockets = breakeven pricing: { p (0,a ) q (0,a ) (ω f ) (ω) = 1+r+ι if a < r if a 0 (15) where p( ) is the assessed repayment probability using both the type score ψ and decision rules σ: p (0,a ) (ω) = s Pr(s ω) (1 Pr(default on a ω, s )) (16) EqmBack

104 Appendix Parameterization and Model Fit PARAMETERS Notation Value Calibrated low type discount factor β L 0.89 low β to high β transition probability Q β (β H β L) 0.05 high β to low β transition probability Q β (β L β H) 0.10 exogenous default cost η 9.8% extreme value scale parameter α Selected high type discount factor β H 0.97 CRRA ν 3 risk-free rate r 3.0% intermediation costs ι 1.0% MOMENTS Data Model Targeted Default rate (%) Average interest rate (%) Median net worth / median income Fraction of households in debt (%) Average debt-to-income ratio (%) Back Parameter Details Moment Sources Moment Definitions HH Decisions