Equilibrium Default and Temptation

Transcription

1 Equilibrium Default and Temptation Makoto Nakajima December 1, 2010 First draft: May 23, 2008 TBD Abstract JEL Classification: D91, E21, E44, G18, K35 Keywords: Consumer bankruptcy, Default, Hyperbolic discounting, Temptation and selfinsurance, Heterogeneous agents, Incomplete markets, General equilibrium Research Department, Federal Reserve Bank of Philadelphia. Ten Independence Mall, Philadelphia, PA I thank participants at the 2008 Cowles Summer Conference on Macroeconomics with Heterogeneous Households and 2009 SED Annual Meeting (Istanbul), and seminar participants at the University of Tokyo, and Nagoya University. The views expressed here are those of the author and do not necessarily reflect the views of the Federal Reserve Bank of Philadelphia or the Federal Reserve System. 1

2 1 Introduction The purpose of the paper is to re-examine macroeconomic and welfare effects of various form of consumer bankruptcy reforms using a model with temptation and self-control preference. From 1980 to 2004, the number of personal bankruptcy filings in the U.S. increased more than fivefolds (White (2007)) and the number of personal bankruptcy filings reached 1.4 percent of the number of households in In response to this increasing trend of the number filings, the Congress introduced the Bankruptcy Abuse prevention and Consumer Protection Act (BAPCPA) in 2005, which made bankruptcy law much less debtor-friendly and much more creditor-friendly. In parallel with such developments, there have been developments in research which tries to investigate the macroeconomic and welfare effects of the realized reform as well as the desirable bankruptcy law. However, all the model-based research so far exclusively uses the standard exponential discounting preference, while some argue that hyperbolic discounting preference is more desirable to capture behavior of debtors and defaulters, but there is no systematic analysis so far. The current paper fills the gap by analyzing the consumer bankruptcy law using a model with temptation and self-control preference, which can be considered a generalized version of the (quasi-)hyperbolic discounting preference. Against the rapid increase in the number of consumer bankruptcy filings, there had been effort of reforming consumer bankruptcy law in the U.S. Since the U.S. traditionally has a debtorfriendly bankruptcy law compared with other developed economies, the focus of the discussion was how to change the bankruptcy law to a more creditor-friendly one and to what extent. 1 The effort resulted in the BAPCPA enacted in 2005, which has the following four key features: (1) Filers now have to pass the means-testing to file under Chapter 7 (debt discharge). Basically, in order to file for Chapter 7, bankruptcy, filer s recent household income must be below the median income. Filers who do not pass the means-testing can only file under Chapter 13 (debt restructuring). (2) Costs of filings was raised. For the case of Chapter 7, the typical costs increased from 600 dollars to 2, 500 dollars. For the case of Chapter 13, the costs went up from 1, 600 to 3, 500 dollars. (3) Duration of punishment increased. Minimum interval of Chapter 7 filings was raised from 6 years to 8 years. The interval for Chapter 13 bankruptcy filings was increased from 6 months to 2 years. (4) Under Chapter 13, the repayment plan is no longer made by the filers. Instead, the repayment plan is made by the court. The length of repayment obligation and the exemption levels of income and assets were also changed so that less debt is discharged upon Chapter 13 bankruptcy filings. In parallel to the discussion of consumer bankruptcy law reform, there has been an attempt to construct an equilibrium model of consumer bankruptcy filings which enables us to implement a model-based evaluation of bankruptcy law reforms. Athreya (2002) and Chatterjee et al. (2007) construct a model featuring an option of Chapter7 bankruptcy filings and investigate macroeconomic and welfare effects of introducing means-testing, which is arguably the most important aspect of the BAPCPA. Li and Sarte (2006) extend the model by introducing the choice between Chapter 7 and Chapter 13 bankruptcy filings and revisit the effect of introducing means-testing. 1 White (2007) compare the personal bankruptcy law in the U.S., France, Germany, England and Wales, and Canada and conclude that, even after the reform in 2005, the U.S. consumer bankruptcy law is the most debtor-friendly. 2

3 The equilibrium models of consumer bankruptcy is also used to evaluate other types of consumer bankruptcy reforms. Livshits et al. (2007) compare the U.S. consumer bankruptcy law which allows debt discharge (fresh start) with the continental European bankruptcy law which does not grant debt discharge but allows debt to be rolled over (non fresh start). Athreya (2008) studies the effect of abolishing consumer bankruptcy, by raising the cost of bankruptcy filing to infinity. In a related exercise, Chatterjee and Gordon (2010) compare macroeconomic and welfare implications between the current U.S. consumer bankruptcy law and the U.S. garnishment law, which is largely ineffective due to the bankruptcy law but prevails if bankruptcy law is eliminated. Although a lot insights have been learned using models with equilibrium bankruptcy, all existing models use the standard exponential discounting preference. On the other hand, recent research found that models with (quasi-)hyperbolic discounting consumers can replicate borrowing behavior of consumers better than models with the standard exponential discounting consumers. The pioneer work which introduces (quasi-)hyperbolic discounting preferences into the standard macroeconomic model is Laibson (1997). Angeletos et al. (2001) estimate a structural model with (quasi-)hyperbolic discounting consumers and find that the model does a better job in replicating a large amount of secured debt in an environment with housing and financial asset/debt. White (2007) argues that we need to explicitly consider hyperbolic discounting consumers since policy implications could be very different depending on whether consumers have the standard exponential discounting preferences or hyperbolic discounting preferences, and when hyperbolic and exponential discounting consumers coexist in an economy, hyperbolic consumers tend to borrow and default more often than exponential discounting consumers. However, her analysis is not model based; this is the gap that the current paper is intended to fill. I use temptation and self-control preference developed by Gul and Pesendorfer (2001, 2004) and applied to a special case which includes both quasi-hyperbolic discounting and exponential discounting by Krusell et al. (forthcoming). There are two important benefits of using the temptation and self-control preference rather than hyperbolic preference. First, the formulation provided by Krusell et al. (forthcoming) includes both quasi-hyperbolic discounting and the exponential discounting as two polar cases, and allows intermediate cases. In their formulation, quasi-hyperbolic discounting is interpreted as consumers completely succumb to temptation to consume more today rather than saving for future. The standard exponential discounting is the case where consumers can exert self-control perfectly so that they are not tempted to consume more today. Second, hyperbolic discounting preferences imply that the same consumer in different periods is considered as a sequence of differen consumers with different preferences. In other words, the preference is time-inconsistent. When one wants to implement normative analysis, the time-inconsistency naturally poses a nontrivial question of welfare of which self of thew same consumer should be used. On the other hand, temptation and self-control preference are time-consistent; there is no change in preferences within a same consumer over time. The preference only implies that consumers are tempted to choose more today. Therefore, it is straightforward to implement normative analysis under the preferences with temptation and self-control. Krusell et al. (forthcoming) conduct an analysis of the optimal capital income taxation using the framework. In this paper, I will compare macroeconomic and welfare implications of variety of bankruptcy 3

4 law reforms mainly under two types of preference assumptions. 2 I investigate five bankruptcy law reforms. Naturally, there are five main findings, as follows. In sum... The rest of the paper is organized as follows. In Section 2 the model is constructed. The model encompasses hyperbolic and exponential discounting preferences and the intermediate cases. Section 3 describes calibration of the model. Appendix A.1 includes details of calibration. Section 4 briefly describes computational procedure. More details are found in Appendix A.2. Section 5 presents the main results. Comparison of properties of the baseline models with different preference specifications is shown at first. Then I move on to compare macroeconomic and welfare implications of various consumer bankruptcy reforms under different models. Section 6 concludes. 2 Model The key features of the model are overlapping generations, equilibrium default, and temptation preference. Livshits et al. (2007) features overlapping generations and equilibrium default, while Nakajima (2010) introduces temptation preference into an overlapping generations model. However, the current paper is the first one to combine all three features. 2.1 Demographics Time is discrete. In each period, the economy is populated by I overlapping generations of agents. In time t, a measure (1 + π) t of agents are born. π is the population growth rate. Each generation is populated by a mass of measure-zero agents. agents are born at age 1 and could live up to age I. There is a probability of early death. Specifically, ψ i is the probability with which an age-i agent survives to age i + 1. With probability (1 ψ i ), an age-i agent does not survive to age i + 1. I is the maximum possible age, which implies ψ I = 0. Agents retire at age 1 < I R < I. Agents with age i I R are called workers, and those with age i > I R are called retirees. I R is a parameter, implying that retirement is mandatory. 2.2 Preferences I use the preference that features temptation and self-controlling, which is developed by Gul and Pesendorfer (2001, 2004). In particular, I use the a specific formulation developed by Krusell et al. (forthcoming). The preference used by Krusell et al. (forthcoming) is useful for the purpose of the current paper as it includes both the standard exponential discounting preference and the (quasi-)hyperbolic discounting preference (Laibson (1997)) as two special cases. More specifically, the preference is characterized by an period utility function u(c), and three parameters; γ, β, and δ. I assume that u(c) is strictly increasing and strictly concave. δ is the standard discount factor. In order to distinguish from β, δ is called the long-term discount factor. β is the short-term discount factor, or the nature of temptation, following the terminology of Krusell et al. (forthcoming). γ represents the strength of the temptation. Agents future utility is discounted with the discount factor δ. However, agents are tempted to make consumptionsavings decision by discounting future value with βδ instead of δ. The parameter γ determines how strongly agents are tempted to discount future value with βδ instead of δ. In one extreme case where γ = 0, an agent does not feel tempted; agents discount future value 2 I will discuss the inter,mediate cases as well. 4

5 with δ and make consumption-savings decision based on the same discount factor δ. This is the standard preference with exponential discounting factor δ. On the other extreme, if the strength of the temptation is infinitely strong (γ = ), agents are succumbed to temptation; agents make consumption-savings decision by discounting future value with βδ although the actual value is based on the discounted factor δ. When β [0, 1), this has exactly the same representation as the (quasi-)hyperbolic discounting preference developed by Laibson (1997), although the interpretation is different in the hyperbolic discounting preference; in the case with the hyperbolic discounting preference, agents today have a different preference (discount factors) from that of their future selves. In the intermediate case where γ (0, ), and β [0, 1), agents are tempted to some extent to consume more in the current period. In other words, the agent is tempted to make the decision by discounting the future value with βδ instead of using the standard exponential discount factor δ. How much the agent is tempted is represented by γ. If γ is higher, the agent is more strongly tempted (closer to quasi-hyperbolic discounting). I present the formal representation of the preference when the recursive problem of an agent is presented. Before moving on, let me emphasize that using the temptation preference instead of (quasi- )hyperbolic discounting means more than the difference in the interpretation. If the preference described above is interpreted as the hyperbolic discounting, the preference exhibits timeinconsistency, in the sense that the preference of agents, in particular, discount factors, changes over time. In this case, it is not straightforward to conduct a welfare analysis since there is a problem of welfare of which selves within an agent to be used. On the other hand, the preference with temptation and self-control does not suffer the same problem as the preference and thus welfare within an agent does not change over time Technology There is a representative firm which has an access to the following constant scale production technology: Y = ZF (K, L) (1) where Y is output, Z is the level of total factor productivity, K is capital stock, and L is labor supply. Capital depreciates at a constant rate ν per period. When a credit card company makes a loan to an agents, it is assumed that there is a transaction cost ι that is proportional to the size of the loan. There is no transaction cost for saving. 2.4 Endowment Agents are born with zero asset. Each agent is endowed with one unit of time each period and inelastically supplies labor since leisure is not valued. Labor productivity of an agent e takes the following form: e(i, p, t) = e i exp(p + t) (2) e i captures the average life-cycle profile of labor productivity, and is common across all age-i agents. p is the persistent shock to productivity. p is drawn from an i.i.d. normal distribution 3 See Krusell et al. (forthcoming) for more discussion. 5

6 when an agent is born, and follows an AR(1) process with normally distributed innovation term. t is the transitory shock to labor productivity. t is drawn from an i.i.d. normal distribution. An agent also faces shocks to compulsory expenditure x 0. x is independent and identically distributed, but the distribution can depend on the type, in particular, age, of agents. 2.5 Bankruptcy Agents have an option to default on their debt or bills associated with expenditure shocks. The default option is modeled as in Chatterjee et al. (2007) and Livshits et al. (2007). The default option in the model resembles in procedure and consequences a Chapter 7 bankruptcy filing, in particular, before the reform of the Bankruptcy Law in Suppose an agent has a negative amount of asset (debt) or receives an expenditure shock with which the asset position becomes negative, and the agent decides to file for a bankruptcy, the following things happen: 1. The defaulters have to pay for a cost of filing. The cost is represented as a fraction ξ of the labor income or social security benefit of the defaulting agent. 2. The debt and the expenditure shock (think of a hospital bill) is wiped out and the agent does not have an obligation to pay back the debt or the expenditure in the future (the fresh start). 3. The agent cannot save during the current period. If the agent tries to save, the saving will be completely garnished. 4. Proportion η of the current labor income is garnished. Social security benefit is not subject to this garnishment. This is intended to capture the effort of agents to replay until they decide to file for a bankruptcy within a period. 5. The credit history of the agent turns bad. I use h = 0 and h = 1 to denote a good and bad credit history, respectively. 6. While the credit history is bad (h = 1), the agent is excluded from the loan market. In other words, the borrowing constraint is zero. 7. With probability λ, the agent s bad credit history is wiped out, or, h turns from one to zero. The benefit of using the default option is to get away from debt or bills. The default option is a means of partial insurance. The costs are (i) the income garnishment in the period of default and (ii) temporary exclusion from the loan market. Agents in debt or with an expenditure shock weigh the benefits and the costs of filing for a bankruptcy, and file if it is optimal to do so or there is no other option. The former is called voluntary default and the latter is called involuntary default. It is possible that an agent with a bad credit history cannot consume a positive consumption when the agent is hit by an expenditure shock. Only in this case (involuntary default), default by agents with a bad credit history is allowed. An agent with a bad credit history cannot choose voluntary default. 6

7 2.6 Annuity Market There is a perfect annuity market which allows agents to insure against uncertain lifetime. Agents of the same type with the same positive amount of asset will optimally sign a contract among themselves so that the total wealth is distributed by the survivors in the next period. Practically, for agents of age i who face the survival probability of ψ i, they only need to save aψ i to receive a in the next period, in the case of survival. For agents with a negative amount of asset, debt of the deceased is assumed to be completely imposed on the credit card company that extended a loan to the deceased. 4 However, in this case, the credit card company pool debt of agents of the same type so that the risk of death will be shared by all the borrowers of the same type. At the end, even for borrowers, pooling of mortality risks by credit card companies virtually work as a working annuity market. 2.7 Government The government runs a simple pay-as-you-go social security program. The government imposes a flat payroll tax rate τ S to all workers, and use the proceeds to finance social security benefits b i of the current retirees. It is assumed that all retirees receive the same amount of benefits, and the government budget associated with the social security program balances each period. Naturally, b i = 0 for i O R and b i = b for i > I R. b is the constant amount of benefit. 2.8 Agent s Problem The problem of an agent is defined recursively. The individual state variables are (i, h, p, t, x, a), where i is age, h is credit history, p and t are persistent and transitory components of individual productivity shocks, x is the compulsory expenditure shock, and a is asset position. I will present the problem of an agent separating two parts, temptation problem and self-control problem. Let s start with the temptation problem. An agent with individual state (i, h, p, t, x, a) with h = 0 (good credit history) solves the following temptation problem: where W (i, 0, p, t, x, a) = max{w0 (i, 0, p, t, x, a), W 1 (i, 0, p, t, x, a)} (3) { if B(i, 0, p, t, x, a) = W0 (i, 0, p, t, x, a) = max a B(i,0,p,t,x,a) W 0 (i, 0, p, t, x, a a (4) ) if B(i, 0, p, t, x, a) W 0 (i, 0, p, t, x, a a ) = u(c 0 ) + βδψ i EV (i + 1, 0, p, t, x, a ) (5) W 1 (i, 0, p, t, x, a) = u(c 1 ) + βδψ i EV (i + 1, 1, p, t, x, 0) (6) B(i, 0, p, t, x, a) = {a R we(i, p, t)(1 τ S ) + b i + a = c + x + q(i, 0, p, t, x, a )a, c 0} (7) c 0 = we(i, p, t)(1 τ S ) + b i + a x q(i, h, p, t, x, a )a (8) c 1 = we(i, p, t)(1 τ S )(1 η ξ) + b i (1 ξ) (9) 4 In case a deceased held both asset and debt, the common thing to happen in many states in the U.S. is that the person who inherits the asset also inherits the debt. On the other hand, being in debt in the current model means the negative net asset position, and thus there is no reason that the debt is inherited by another person. 7

8 Equation (3) states that the agent can choose between defaulting and non-defaulting when the agent has a good credit history (h = 0). W0 and W1 correspond to the value conditional on not-defaulting and defaulting, respectively. Equation (4) states that the value is negative infinity when the choice set, defined by equation (7), is empty. Since the value conditional on defaulting is finite, the agent always chooses to default in case the choice set is empty. This case corresponds to involuntary default. Otherwise, the agent solves the consumption-savings problem conditional on not defaulting. The maximand is defined by the equation (5), with the consumption defined by equation (8). The income of the agent consists of labor income net of payroll tax (we(i, p, t)(1 τ S )), and social security benefit (b i ). w is wage rate. a is the asset (debt is a is negative) carried over from the previous period. x is the compulsory expenditure. c is consumption. a is the asset (or debt) carried over to the next period. q(i, h, p, t, x, a ) is the discount price for an agent of type (i, h, p, t, x) choosing the asset position a. Notice that the current asset holding of an agent, a, does not matter for the price of loans. To ease the notation, q(i, h, p, t, x, a ) is also used to capture the annuity contract used by agents with a positive a. The value conditional on defaulting is characterized by equation (6). In case the agent defaults, the debt (negative a) and the compulsory expenditure x are discharged, but the agent has to pay for the filing cost ξ, a fraction η is garnished away from labor income, and the agent will start the next period with zero saving. These consequences of default can be observed by the definition of consumption, defined by equation (9) If B and the agent chooses to default, it is called voluntary default. Notice that the future value is discounted by βδ. Think that δ is the standard long-term discount factor. When β [0, 1), β shifts the weight to the current utility relative to the future value. Or, loosely speaking, the agent is tempted to consume more rather than saving in the current period. The nature of temptation for the agent is that the agent is tempted to discount future value more. In case an agent has a bad credit history (h = 1), the temptation problem for the agent is formalized as follows: { W W1 (i, 1, p, t, x, a) if B(i, 1, p, t, x, a) = (i, 1, p, t, x, a) = max a B(i,1,p,t,x,a) W 0 (i, 1, p, t, x, a a (10) ) if B(i, 1, p, t, x, a) W 0 (i, 1, p, t, x, a a ) = u(c 0 ) + βδψ i E(λV (i + 1, 0, p, t, x, a ) + (1 λ)v (i + 1, 1, p, t, x, a )) (11) W 1 (i, 1, p, t, x, a) = u(c 1 ) + βδψ i EV (i + 1, 1, p, t, x, 0) (12) where B(i, h, p, t, x, a) is defined by equation (13) below, and c 0 and c 1 are defined by equations (8) and (9). B(i, 1, p, t, x, a) = {a R we(i, p, t)(1 τ S ) + b i + a = c + x + q(i, 1, p, t, x, a )a, c 0, a 0} (13) Notice three things. First, an agent with a bad credit history (h = 1) does not have a choice with respect to whether to default or not. Only involuntary defaults are allowed, when B(i, 1, p, t, x, a) =. See equation (10) above. Second, when an agent had a bad credit history, and does not (involuntarily) default in the current period, the agent s credit history is cleaned 8

9 up (h = 0) in the next period with a probability λ, and the bad credit history remains (h = 1) with a probability (1 λ). You can see this in equation (11) above. λ will be calibrated later to make sure that the average duration for which the bad credit history is kept matches the same statistics in the U.S. economy. This is a way to reduce the size of the state space and simplify an already complex model slightly. Finally, the budget set B(i, 1, p, t, x, a) is almost the same as in the case for the agent with a good credit history, but there is one additional constraint; a 0. Basically, this constraint excludes the agent with a bad credit history from the credit market. Now that we defined the temptation problem, we are ready to define the self-control problem. The temptation problem will be a part of the self-control that an agent solves since a key of the self-control problem is how successfully an agent can resist the temptation. V (i, 0, p, t, x, a) = max{v0 (i, 0, p, t, x, a), V1 (i, 0, p, t, x, a)} (14) { if B(i, 0, p, t, x, a) = V0 (i, 0, p, t, x, a) = max a B(i,0,p,t,x,a) V 0 (i, 0, p, t, x, a a (15) ) if B(i, 0, p, t, x, a) V 0 (i, 0, p, t, x, a a ) = u(c 0 ) + δψ i EV (i + 1, 0, p, t, x, a ) + γ{w 0 (i, 0, p, t, x, a a ) W (i, 0, p, t, x, a)} (16) V 1 (i, 0, p, t, x, a) = u(c 1 ) + δψ i EV (i + 1, 1, p, t, x, 0) + γ{w 1 (i, 0, p, t, x, a 0) W (i, 0, p, t, x, a)} (17) subject to equations (7), (8), and (9). Equations (14) and (15) are almost identical with the corresponding equations in the temptation problem (equations (3) and (4)). An agent chooses between defaulting and non-defaulting, and the value of non-defaulting is minus infinity if the feasible set is empty, meaning that the agent is forced to default (involuntary default). Equation (16) is where the two problems become different. There are two differences. First, the agent discounts future value only with the long-term discount factor δ. Second, there are two additional terms in the maximand, and they are multiplied by γ. W 0 (i, 0, p, t, x, a a ) is the value associated with the temptation problem conditional on the agent s current decision. W (i, 0, p, t, x, a) is the value of the temptation problem with the optimal decision associated with the temptation problem. γ is the parameter that determines the strength of temptation. If there is no temptation (γ = 0), the temptation problem doesn t matter, because the last two terms of the maximand disappear. As a result, the problem goes back to the standard Bellman equation with exponential discounting. Similarly, if β = 1, the discount factor used for the tempting problem is the same as the problem here. Therefore, the optimal decision associated with the current problem turns out to coincide with the optimal decision associated with the tempting problem. In other words, temptation doesn t harm the agent and thus doesn t need to be controlled. In short, the current problem goes back to the standard problem without temptation if either γ = 0 (strength of temptation is zero) or β = 1 (short-term discount factor plays no role). In case neither holds, the agent s problem has two dimensions. First, the agent wants to solve the standard problem with long-term discount factor δ. On the other hand, the agent wants to choose the action that is close to the one that would be chosen under the temptation problem so that W 0 (i, 0, p, t, x, a a ) W (i, 0, p, t, x, a) is brought to close to zero. Notice that, since W 0 (i, 0, p, t, x, a a ) W (i, 0, p, t, x, a), choosing decision that is different from the optimal decision under the temptation problem usually causes a lower utility. 9

10 The relative strength of the two considerations, or the strength of the temptation, is controlled by the parameter γ. In case γ =, the agent chooses the action as if the agent is solving the tempting problem. But the optimal value is based on the standard long-term discounting δ. This is exactly what is achieved in the so-called quasi-hyperbolic discounting model in Laibson (1997) and Angeletos et al. (2001). The current approach with temptation not only includes the quasi-hyperbolic discounting model as an extreme case, but has a very important advantage over the quasi-hyperbolic discounting model, as argued by Krusell et al. (forthcoming). How? Since the utility changes over time, the same agent in different periods can be naturally seen as different selves in the quasi-hyperbolic discounting model. This feature makes it non-trivial to define the welfare of agents. On the other hand, in the temptation model, utility of an agent does not change, and thus it is natural to define the welfare of agents. Equation (17) is similar to equation (16) in the sense that the future value is discounted only with δ and there are two additional terms associated with temptation. However, since (17) represents the value conditional on defaulting, there is no choice in terms of savings, since the agent is not allowed to save or borrow in the defaulting period. Finally, the self-control problem of an agent with a bad credit history (h = 1) can be characterized as follows: { V V (i, 1, p, t, x, a) = 1 (i, 1, p, t, x, a) if B(i, 1, p, t, x, a) = max a B(i,1,p,t,x,a) V 0 (i, 1, p, t, x, a a ) if B(i, 1, p, t, x, a) V 0 (i, 1, p, t, x, a a ) = u(c 0 ) + δψ i EV (i + 1, 1, p, t, x, a ) (18) + γ{w 0 (i, 1, p, t, x, a a ) W (i, 1, p, t, x, a)} (19) V 1 (i, 1, p, t, x, a) = u(c 1 ) + δψ i EV (i + 1, 1, p, t, x, 0) + γ{w 1 (i, 1, p, t, x, a 0) W (i, 1, p, t, x, a)} (20) subject to equations (8), (9), and (13). The optimal value function associated with the problem defined above is V (i, h, p, t, x, a). The optimal saving function is denoted as a = g a (i, h, p, t, x, a). The optimal policy rule for default decision is denoted as h = g h (i, h, p, t, x, a), where g h (i, h, p, t, x, a) = 1 and g h (i, h, p, t, x, a) = 0 denote defaulting and non-defaulting, respectively. 2.9 Credit Card Companies The only loans available in the model are unsecured loans. The unsecured loans are provided by competitive credit sector that consists of a large number of credit card companies. Free entry is assumed. Credit card companies can target any one type of agents with one level of debt. Since the credit sector is competitive, free entry is assumed, and each credit card company can target one specific level of asset, it is impossible in equilibrium to cross-subsidize, which is, offering one type of agents an interest rate which implies a negative profit while offering another type of agents an interest rate which implies a positive profit, so that, in sum, the credit card company makes a positive total profit. In this case, there is always an incentive for another credit card company to offer a lower interest rate for the second type of agents and steal the profitable customers away. In equilibrium, any loans to any type of agents and any level of debt make zero profit. 10

11 Suppose that a credit card company makes loans to type-(i, 0, p, t, x) agents who borrow a each. 5 Remember that the current asset position of agents a does not matter for pricing of loans. By making loans to a mass of agents of the same type, the credit card company can insure away the default risk, even if the loans are unsecured from individual point of view. In other words, credit sector provides a partial insurance, by pooling risk of default across different agents (but the same type). Now, assume the credit card company makes loans to measure m agents of the same type. Zero profit condition associated with the loans made to type-(i, 0, p, t, x) agents whose measure is m and who borrow a each can be expressed as follows: mψ i ( a ) + mψ i p p t t x I gh (i+1,0,p,t,x,a )=0f x (x )f t (t )f p (p p)dx dt dp x I gh (i+1,0,p,t,x,a )=1we(i, p, t)(1 τ S ) η( a ) x a f x(x )f t (t )f p (p p)dx dt dp = m( a q(i, 0, p, t, x, a ))(1 + r + ι) (21) where I is an indicator function which takes the value of one if the logical statement attached to it is true, and zero otherwise. f x, f t and f p are density functions associated with the three types of shocks. The first term on the left hand side is the sum of the income of the credit card company for the agents of type (i + 1, 0, p, t, x, a ) who repay the loans. The second term represents the sum of the income of the credit card company when the agents of type (i + 1, 0, p, t, x, a ) default. When an agent defaults, the fraction η of the labor income of the agent is garnished. If there is no compulsory expenditure shock (x = 0), all of the garnished amount is received by the credit card company as income. If the agent also receives a bill of a positive amount (x > 0), the garnished income is proportionally allocated between the credit card company and the issuer of a the bill depending on the relative size of the debt and the bill. represents the fraction that x a the credit card company receives. The right hand side of the equation is the total cost of loans. Notice that there is a transaction cost for loans ι in addition to the risk-free interest rate r. If the equation (21) is solved for q(i, 0, p, t, x, a ), we can obtain the formula for the equilibrium discount price of loans, as follows: q(i, 0, p, t, x, a ) = ψ i I p t x gh =0 + I gh =1 ηwe(i,p,t)(1 τ S) f x a x (x )f t (t )f p (p p)dx dt dp 1 + r + ι (22) where g h is a short-hand notation for g h (i + 1, 0, p, t, x, a ). Finally, I assume that there is a maximum limit on the interest rate charged by credit card companies, which is denoted by r. Since the price of bond q(.) is used instead of interest rate r(.) for loans, the upperbound of the interest rate r is converted into the lowerbound of the bond price by q = 1. In the U.S., since the Marquette decision in 1978, nationally operating credit 1+r card companies are no longer subject to the usury law of the states that they are operating in, which basically eliminated the usury law. 6 In other words, currently, there is no effective limit on the interest rate. Therefore, I will set r at an non-binding level in the baseline calibration, and 5 Notice that h = 0. We only need to consider h = 0 as agents with a bad credit history (h = 1) cannot borrow. 6 Supreme Court decision on Marquette Nat. Bank of Minneapolis v. First of Omaha Service Corp. 11

12 later investigates the macroeconomic and welfare implications of introducing a binding interest rate ceiling. In order to see the formula more clearly, let s see some of the special cases. Notice that, in case there is no default for the loan, the price of loan will be: q(i, 0, p, t, x, a ) = ψ i 1 + r + ι When there is no transaction cost (ι = 0), this is the equilibrium loan price for a 0 and the only loan price available for those with a bad credit history (who cannot borrow but save). Notice that there is a survival probability ψ i in the numerator. The credit card company is basically providing annuity among debtors. The way survival probability is in the formula implies that the loan price (implied interest rate) is lower (higher) for older agents, as they tend to have lower survival probabilities. This feature can explain why older agents cannot borrow as much as younger agents. In case all agents default on the debt in the next period, the price of loans will be: q(i, 0, p, t, x, a ) = ψ ηwe(i,p,t)(1 τ S ) i f p t x x a x (x )f t (t )f p (p p)dx dt dp 1 + r + ι Consider the special case where there is no garnishment, i.e. η = 0. In case the loan is defaulted with probability one: q(i, 0, p, t, x, a ) = 0 (25) This is because, when η = 0, credit card companies cannot receive anything from borrowers. In this case, assuming that q(i, 0, p, t, x, a ) is monotonic with respect to a, we can define a(i, 0, p, t, x, a ) which satisfies: a(i, 0, p, t, x, a ) = max a {a q(i, 0, p, t, x, a ) = 0} (26) a(i, 0, p, t, x, a ) is the endogenous borrowing constraint for agents of type (i, 0, p, t, x, a ). For an agent with a bad credit history, a(i, 1, p, t, x, a ) = 0. By construction, the constraint is less strict than the not-too-tight borrowing constraint by Alvarez and Jermann (2000). This is because the not-too-tight borrowing constraint is associated with no default in equilibrium, while the constraint here allows default in equilibrium. 7 See Chatterjee et al. (2007) for further characterization of the equilibrium loan price function. Finally, for agents with a positive a, or a bad credit history (h = 1), we can define the pricing function as follows: q(i, h, p, t, x, a ) = ψ i h = 1 or a 0 (27) 1 + r 7 Since the loan price q (interest rate) goes down (up) as the size of the debt increases, typically no agent borrows as much as a; a high interest rate (low loan price) effectively works as a borrowing constraint. Chatterjee et al. (2007) show that, although the endogenous borrowing limit a(.) is less strict than the natural borrowing limit of Aiyagari (1994), the largest size of debt held by an agent in the baseline equilibrium is smaller than the natural borrowing limit applied to the model. In this sense, the effective borrowing constraint in the model with bankruptcy can be more strict than the natural borrowing limit even if a(.) is less strict than the natural borrowing limit. (23) (24) 12

13 This equation takes into account the annuity contract signed among the agents with a positive asset Equilibrium I define below the recursive competitive equilibrium where the demographic structure is stationary, even though the size of population is growing at a constant rate π. In the equilibrium with stationary demographic structure, prices {r, w, q(i, h, p, t, x, a )} are constant over time. Let M be the space of individual state. (i, h, p, t, x, a) M. Let M be the Borel σ algebra generated by M, and µ a probability measure defined over M. I will use a probability space (M, M, µ) to represent a type distribution of agents. Definition 1 (Steady-state recursive competitive equilibrium) A steady-state recursive competitive equilibrium is a set of prices {r, w, q(i, h, p, t, x, a )}, government policy variable {τ S, b i }, aggregate capital stock K, labor supply L, value function V (i, h, p, t, x, a), optimal decision rules g a (i, h, p, t, x, a), g h (i, h, p, t, x, a), and the stationary measure after normalization µ, such that: 1. Given the prices, and policy variables, V (i, h, p, t, x, a) is a solution to the agent s optimization problem defined in Section 2.8, and g a (i, h, p, t, x, a) and g h (i, h, p, t, x, a) are the associated optimal decision rules. 2. The prices r and w are determined competitively, i.e., r = ZF K (K, L) ν w = ZF L (K, L) (29) 3. Bond price function q(i, h, p, t, x, a ) satisfies the zero profit conditions for all types. Specifically, the loan price functions are characterized as (22) and (27). 4. Measure of agents µ is consistent with the demographic transition, stochastic process of shocks, and optimal decision rules, after normalization. 5. Aggregate capital and labor are consistent with the individual optimal decisions, i.e.: K = 1 g a (i, h, p, t, x, a)q(i, h, p, t, x, g a (i, h, p, t, x, a))dµ (30) 1 + π M L = e(i, p, t)dµ (31) M 6. Government satisfies period-by-period budget balance with respect to the social security program, i.e., b i dµ = e(i, p, t)wτ S dµ (32) M M (28) 13

14 3 Calibration 3.1 Demographics One period is set as one year in the model. Age 1 in the model corresponds to the actual age of 22. I is set at 79, meaning that the maximum actual age is 100. I R is set at 43, implying that the agents become retired at the actual age of 65. The population growth rate, π, is set at 1.2% annually. This growth rate corresponds to the average annual population growth rate of the U.S. over the last 50 years. The survival probabilities {s i } I i=1 are taken from the life table in Social Security Administration (2007). 8 Figure 1 in Appendix A.1 shows the conditional survival probabilities used. 3.2 Preferences For the period utility function, the following constant relative risk aversion (CRRA) functional form is used: u(c) = c1 σ 1 σ σ is set at 2.0, which is the commonly used value in macroeconomics. Discount factors β and δ and the parameter controlling the strength of temptation γ are calibrated differently for different economies. For the baseline model economy with exponential discounting consumers, γ is set to zero, and δ is calibrated primarily to match the aggregate capital stock in the steady state, which is 2.76 of the aggregate output. 9 For economies with agents who face temptation and self-control problem, I use the short-term discount factor β of 0.70 and The short-term discount factor of 0.7 corresponds to the discount rate of 40% which is the estimate obtained by Laibson et al. (2007). Discount factor of 0.55 corresponds to the 80% discount rate, which is twice the baseline discount rate. I use β of 0.55 for robustness check. As for the strength of temptation γ, I also use variety of values. In particular, I try γ of 1, 10, and. Notice that γ = implies the quasi-hyperbolic discounting preference, which Laibson et al. (2007) use. In this sense, γ = is the baseline value. In all cases with temptation and self-control problem, the remaining parameter δ is calibrated to match the same target for the aggregate capital stock. Of course, δ will be different for different economies, but all the models are calibrated to match the same set of targets so that all models with different preference parameters are observationally equivalent with respect to the chosen targets. 3.3 Technology The following standard Cobb-Douglas production function is assumed: (33) Y = ZK θ L 1 θ (34) Z is pinned down such that, in the baseline steady state, the output is normalized to one. θ is set at Capital depreciated at the constant rate of ν = per year. These values are computed using National Income and Product Account (NIPA). 8 Table 4.C6 of Social Security Administration (2007). 9 Remember that, with γ = 0, the short-term discount factor β does not matter. 14

15 3.4 Credit There are two parameters associated with credit industry but not directly related to bankruptcy law. First, the transaction cost for loans ι is set at 4 percent, which is the value used by Livshits et al. (2007) and reflects the average cost of loans in the U.S. economy. As you will see, ι = 0.04 is consistent with the average interest rate of unsecured loans, which is about 11 percent per year. Second, the upperbound of the interest rate charged by credit card companies r is set at 100 percent per year. Basically, this is a level which makes the usury rate non-binding. I will lower the maximum interest rate in one of the experiments. 3.5 Bankruptcy There are three parameters associated with the bankruptcy scheme; λ, which is associated with the average length of punishment, ξ, which represents the filing cost of bankruptcy, and η, which defines the amount of labor income garnished during the period of filing. λ is set at 0.1, implying that, on average, defaulters cannot obtain new debt for 10 years after filing for a bankruptcy. This average punishment period corresponds to a 10 year period during which a bankruptcy filing stays on a person s credit record according to the Fair Credit Reporting Act. According to White (2007), the average cost of filing for a Chapter 7 bankruptcy was 600 dollars before the BAPCPA was introduced in ξ is pinned down by converting 600 dollars into the unit in the model. I obtain ξ = , meaning 1.3 percent of the average household income. η is chosen such that the number of bankruptcies in the model matches the same number in the U.S. economy (0.84% of households per year according to Livshits et al. (2007)). However, notice that the parameter will be chosen jointly with other parameters. I will come back to the calibration of η, together with other parameters jointly calibrated. 3.6 Government The payroll tax rate for the social security contribution τ S is set at 10.7 percent. This is the OASI (Old-Age and Survivors Insurance) rate, which is a part of the Social Security payroll tax rate of 15.3 percent (Conesa and Krueger (1999)). This payroll tax rate yields a replacement ratio (defined as the ratio of the average social security benefits over the average labor income just before retirement) of 55 percent, which is close to its empirical counterpart. 3.7 Labor Productivity There are two steps for the calibration of the individual labor productivity. First, I use estimates from micro data to calibrate the average life-cycle profile of individual labor productivity and shocks to labor productivity. Second, I supplement the individual labor productivity shocks obtained in the first step by introducing the super-rich state. The super-rich state is meant to capture the top 1 percent income households in the U.S. The reason for the supplement is that the micro data used to estimate the individual labor productivity dynamics, Panel Study on Income Dynamics (PSID), is known to under-represent the richest part of the U.S. population. 10 In other words, if I use the individual labor productivity dynamics calibrated using PSID but at the same time calibrate the model to match the aggregate capital stock, majority of agents in 10 Budría et al. (2002) discuss the difference in the implied income and wealth inequality between PSID and Survey of Consumer Finances (SCF), which over-samples the rich. 15

16 the model end up holding too large wealth compared in the data. Below I will start by describing how estimates using PSID are used to calibrate the dynamics of individual labor productivity. Then I will describe how the stochastic process for the individual labor productivity shocks is modified to generate an observed concentration of earnings. The average life-cycle profile of the individual labor productivity {e i } I i=1 is taken from the estimates of Gourinchas and Parker (2002). Figure 2 in Appendix A.1 shows the life-cycle profile of the average labor productivity used in the model. Since mandatory retirement at the model age of I R, e i = 0 for i > I R. The stochastic component of the individual labor productivity consists of persistent shock p, and the transitory shock t. The persistent shock is initially drawn from N(0, σ 2 p) and follows an AR(1) process with the persistence parameter ρ p and the innovation ɛ is drawn from N(0, σ 2 ɛ ). The transitory component is i.i.d. drawn from N(0, σ 2 t ). I use the estimates obtained using PSID by Storesletten et al. (2004). Using PSID, they estimate σ 2 p = , ρ p = , σ 2 ɛ = , and σ 2 t = The persistent component is then approximated by first order Markov process with N p = 10 abscissas. I use the method developed by Tauchen (1986). The transitory component is also approximated by a discrete distribution, with N t = 10 abscissas, using the method proposed by Ada and Cooper (2003). The approximated stochastic process captures the life-cycle profile of the original stochastic process well; the process estimated by Storesletten et al. (2004) generates cross-sectional variances of log earnings of for age-22 (age-1 in the model) agents and for age-64 (age-43 in the model), and the discretized stochastic process used in the model generates variances of log earnings of for age-22 agents and for age-64 agents. Figure 3 in Appendix A.1 compares the variances of logearnings of the original process estimated by Storesletten et al. (2004) and the approximated process used in the model. 12 The second step of adding the super-rich is implemented by adding one more state for the persistent shock p. There are three parameters characterizing the super-rich state: the proportion of super-rich, m s, the persistence of remaining super-rich ρ s, and the level of productivity of the super-rich relative to the average productivity of non super-rich, p s. The proportion m s is set at m s = , because I use the super-rich to capture the top 1 percent richest households. 13 ρ s is set at 0.92, following the calculation by The Federal Reserve Bank of Dallas (1995). The productivity level of the super-rich p s is calibrated such that the top 1 percent earners in the U.S. earn 15.3 percent of the total earnings (Budría et al. (2002)). The calibration strategy yields p s = In order to pin down the transition probabilities associated with the super-rich state, 11 Storesletten et al. (2004) assume that there are permanent shocks and persistent shocks. The former is drawn from i.i.d. normal and does not change throughout agent s life and the latter is initially zero but follows an AR(1) process. Since the persistence is very high, meaning that the calibrated AR(1) process is close to a random walk, I combine the permanent and persistent and interpret the initial distribution of the permanent shock as the initial distribution of the persistent shock. In case the persistent shock follows a random walk, there is no distinction between the permanent and persistent shocks. 12 The approximation method by Tauchen (1986) can be controlled by a choice of the parameter which controls the size of the domain of approximating discrete stochastic process. I choose the parameter such that the approximating stochastic process generates a life-cycle profile of log earnings variances that is close to the data counterpart. 13 Notice that retirees have zero earnings. So the proportion of the super-rich is adjusted so that the total measure of the super-rich is exactly 1 percent of the total households, not only among the workers. 16

17 Table 1: Expenditure shock 1 State Probability Magnitude (dollars) No shock Small medical expense + divorce + unwanted birth ,584 Large medical expense ,866 1 Constructed based on Livshits et al. (2007) (Table 1). The probability and magnitude of shocks are adjusted since one period is three years in their calibration, while one period is one year in the current calibration. See Appendix A.1 for more details. I assume that (i) the probability for a non super-rich to become a super-rich does not depend on the current p, and (ii) when a super-rich ceases to be super-rich (with probability 1 ρ s ), the next state is drawn from the ergodic distribution of p excluding the super-rich. 3.8 Expenditure Shocks The compulsory expenditure shocks are taken from Livshits et al. (2007). Their expenditure shocks include major non-financial triggers of bankruptcies, namely, out-of-pocket (OOP) medical expenditures, divorces, and unwanted births. They compute the probabilities and magnitudes associated with large and small OOP medical expenditures, divorces, and unwanted births, and combine the last three since the magnitudes are similar among them, leaving large OOP medical expenditure shocks as a separate state. Table 1 summarizes the expenditure shocks constructed by Livshits et al. (2007). I adjust the probability and magnitude of each state (except for No Shock state) since one period is three years in their calibration, while one period is a year in mine. The magnitude of the expenditure shocks is converted into the unit in the model by dividing by income per household Simultaneously Calibrated Parameters As I mentioned, there are two parameters, δ and η, which cannot be pinned down independently from the model. I calibrate the two parameters such that two closely related targets capitaloutput ratio of 2.76 ( K = 2.76) and the proportion of defaulters each year is 0.84% are Y achieved in the steady-state equilibrium of the model. Notice two things. First, in order to find such parameter values, it is necessary to run the model many times trying different combination of (δ, η). Basically, what is done is a simulated method of moments with exact identification. Second, the values of (δ, η) are different depending on the model specification. For example, (δ, η) are different between the model with exponential discounting agents, and the model with quasi-hyperbolic discounting agents. However, the targets are the same across different versions of the model. Table 3 summarizes calibration for two versions of the model, one with exponential discounting agents, and the other with quasi-hyperbolic discounting agents. In the model with exponential discounting agents, the long-term discount factor δ is calibrated to be As for the model with quasi-hyperbolic discounting agents, δ is calibrated to be in the current model, which 14 More details are found in Appendix A.1. 17