# Journal of Credit Risk

**ISSN:**

1744-6619 (print)

1755-9723 (online)

**Editor-in-chief:** Nikunj Kapadia and Linda Allen

# The economics of debt collection, with attention to the issue of salience of collections at the time credit is granted

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Need to know

- We consider the role of policies that restrict debt collection in a setting where borrowers may have an optimistic bias. We develop a two-period model of consumer borrowing with debt collection in which consumers are at risk of receiving a negative income shock that may make them unable to repay their loan. In this setting, optimists underestimate the probability of this shock which induces them to over-borrow in parameterizations we consider.

- We consider a rule change that restricts debt collection practices and describe how this affects the welfare of rational and optimistic consumers. Rational consumers take the rule into account and their demand for borrowing increases. Pure optimists do not take collection rules into account in making borrowing decisions, so their demand for credit is unaffected by the rule change. Since optimists over-borrow, the effect of the rule change is to shift demand by rational consumers closer to that of optimists. This makes demand by the two types of consumers more similar. One can think of the rule as introducing a distortion into the market that reduces the welfare consequences of being optimistic.

- In a partial equilibrium setting with the interest rate fixed, the welfare of both rational and optimistic consumers is increased by a rule that restricts debt collection practices as the rule decreases the utility cost of debt collection. In a general equilibrium setting, the sign of the welfare change depends on the sizes of the interest rate change and the changes to the utility cost of debt collection; the welfare change is less likely to be positive if the restrictions generate a large increase in the interest rate, and more likely to be positive if the restrictions substantially decrease the utility cost of debt collection.

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Abstract

This paper considers the role of policies that protect consumers from aggressive debt collection tactics. In general, rational consumers will select credit contracts based not only on price but also on the lender’s practices in the case of default and might prefer a contract that offers higher interest rates but more lenient collection tactics. We consider a model in which consumers can be optimistic in that they underestimate, at the time they borrow, the chance of a negative income shock that could cause them to default. This mindset induces consumers to perceive the welfare associated with borrowing to be greater than it is and to place less weight on the lender’s collection practices, which may cause them to prefer lower interest rate contracts even if they are accompanied by high collection effort in the event of default. We report supporting evidence that is consistent with consumers being optimistic when they borrow. We develop a two-period model of consumer borrowing that incorporates debt collection: consumers borrow in the first period, and repay, settle or default in the second. We show in both partial and general equilibrium settings that a restriction on debt collection effort can be welfare improving under some conditions. The restriction operates by reducing the gap between perceived and actual consumer welfare for optimistic consumers.

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Introduction

## 1 Introduction

Engaging in a credit transaction, by its very nature, is an act of faith and optimism for a consumer; it is a statement that I am creditworthy and I will be able to repay my debts. Events between the time of the transaction and the time of repayment, however, can conspire to undermine this optimism. A significant share of unsecured consumer lending is not repaid as agreed,^{1}^{1} 1 Federal Reserve Bank of New York (2018) reports that, as of 2018 Q1, about 4.72% of credit card debt was at least ninety days delinquent. The Bureau of Consumer Financial Protection (2019) reports that the third-party debt collection industry earned an estimated USD11.5 billion in revenue in 2018, which generally reflects a cost to creditors of recovering defaulted debt. and the business of collecting unpaid debts from consumers is a substantial one, affecting a significant fraction of consumers each year.^{2}^{2} 2 In a survey analysis, Consumer Financial Protection Bureau (2017) reports that 32% of consumers with a credit record indicated that they had been contacted by at least one creditor or collector trying to collect one or more debts during the year prior to the survey; 72% of these reported having been contacted about two or more debts.

Neoclassical economics would assume a rational consumer who takes into account the possibility of default and debt collections when making borrowing decisions. Such a consumer would choose a credit contract based not only on the interest rate but also on the steps the contract permits the lender to take in the event of nonpayment, and the consumer would generally accept somewhat higher prices in exchange for some level of insurance against the negative consequences of being unable to repay. However, the consumer credit contracts used in practice are not so complete. As Hynes (2004) notes, such a lack of complete contracts can provide a policy justification for debt relief regulation. We posit that one source of this incompleteness is the optimistic mindset consumers have when applying for credit. Optimism means that the possibility of being unable to pay is less salient to the consumer at the time of borrowing, so that consumers downplay the question of how debt would be collected until they are faced with the decision of whether to repay. As a result, consumers who are offered two possible contracts – one with low interest rates and high debt collection effort, the other with high interest rates and low debt collection effort – would tend to choose the low interest rate contract, as that contract term is more salient. Earlier work studying the impact of policies restricting creditor remedies in default assumed that borrowers were aware of and value such restrictions. Dye (1986), for example, assesses the value to debtors of lenient versus harsher bankruptcy statutes. He argues that increasing the leniency of statues has unambiguously positive effects for current debtors, but that welfare effects for future debtors are ambiguous. He assumes that all consumers are aware of the leniency of bankruptcy statutes when they engage in credit transactions and that increased leniency shifts out their demand for credit. This is likely true for consumers close to declaring bankruptcy – the spike in bankruptcy filings prior to US 2005 bankruptcy law taking effect supports this (see, for example, Razeto and Romeo 2019) – but is not likely true for the general population of consumers engaging in credit transactions.

There is relatively little work in economics that addresses the welfare implications of debt collection policies, and the work that has been done assumes rational consumers, for whom collections tactics play an important role in credit demand. Fedaseyeu and Hunt (2018) develop a model of the collections industry in which creditors choose whether to collect debt themselves or to use third-party collection agencies, and creditors and agencies choose whether to employ lenient or harsh collections practices. Lenient tactics reduce creditor reputational risk in collections but can expose them to adverse selection by consumers who are assumed to incorporate the creditors’ collections posture into their credit choices. In contrast to our approach, creditor collection practices are salient to consumers, although they show that it is possible for harsh collection tactics to harm a creditor’s reputation even when consumers do not take those tactics into account when borrowing. Zywicki (2015) bases his discussion of debt collection policy on the assumption that consumers are aware of the collections posture of firms and that they react to changes in debt collection rules. For empirical support, he relies on Barth et al (1986), who model how demand and supply of personal loan contracts respond to US state-level debt collection and wage garnishment rules. They find that firms react to rule changes, but consumers do not. This finding supports the view that firms’ collection posture and collection rules are not salient to consumers when they make purchases, although neither work makes this point.^{3}^{3} 3 Barth et al (1986, p. 379) argues that the results “imply that restricting the use of [creditor remedies] does not confer net benefits on the typical borrower but rather imposes net costs”. This conclusion is based on changes in creditor remedies not being valued by consumers rather than these changes not being salient to consumers. More recent empirical work has found that new rules restricting debt collection lead to increased cost or reduced supply of credit, but the evidence again suggests that this is driven by changes to the supply side and not increased credit demand.^{4}^{4} 4 See Romeo and Sandler (forthcoming), Fedaseyeu (2015) and Fonseca et al (2017), each of which finds supply effects associated with increases in state debt collection regulations. As discussed further in Section 3 of our paper, Romeo and Sandler (forthcoming) find no demand effect.

The literature on bankruptcy, which also concerns protections for consumers who are unable to repay debts, generally assumes that such protections are salient at the time of borrowing. However, that literature does not establish that consumer demand for credit is affected by the extent of bankruptcy protection. For example, the empirical analyses in Gropp et al (1997) and Severino and Brown (2017) both purport to show that demand for credit shifts out in response to increases in state bankruptcy exemptions. These authors, however, only observe debt loads, not demand. We would anticipate an increase in interest rates as supply shifts in in states where bankruptcy exemptions are increased. If this occurs, debt loads for consumers in these states will increase in response to this supply effect. Their data does not allow them to cleanly test whether demand for credit has also shifted.

Similarly, research on consumer protections in the mortgage market has looked at how such protections affect both the choices of rational consumers and loan origination. Pence (2006) finds that judicial foreclosure laws reduce mortgage loan sizes by 3–7%, and considers that one factor that might contribute to this result is that the insurance value of these laws might induce some borrowers to move to judicial foreclosure states.^{5}^{5} 5 This is not a testable hypothesis with Pence’s data, and this hypothesis relies on foreclosure laws being salient to consumers when they purchase their homes. Ghent and Kudlyak (2011) find that, in US states that do not permit recourse after mortgage default, consumers are more likely to default when they have negative equity, but they do not study whether recourse affects interest rates at origination.

Other authors have considered the effect of nonsalient terms of credit contracts in consumer credit markets. Kamenica et al (2011) and Vining and Weimer (1988) discuss models in which information asymmetry favoring creditors can catch consumers unawares. Others have studied cases in which firms structure credit prices to contain salient and nonsalient fees, keeping the salient fees low and shrouding nonsalient fees, and imperfectly rational consumers suffer from biases and misperceptions that overstate the perceived benefit relative to the actual expected benefit of a credit transaction (Gabaix and Laibson 2006; Bar-Gill 2013). That nonsalient fees can exist in equilibrium is indicative of market failure, as competition does not provide incentives to educate consumers. Using a survey instrument to assess behavioral tendencies, Stango et al (2017) find that “nearly everyone in [their] sample is behavioral on a few dimensions, if not more, even if one takes a conservative view of what ‘behavioral’ means”. The survey finds evidence of present bias, overconfidence, exponential growth bias, limited attention and mathematical biases,^{6}^{6} 6 Mathematical biases include failure to believe in, or possibly understand, the law of large numbers, and the gambler’s fallacy. all of which can manifest themselves as a lack of salience of the longer-term effects of a credit transaction.

The psychology literature presents evidence that overconfidence or unrealistic optimism is the norm for most of the population. Moreover, optimism is both pervasive and relatively unresponsive to reality. Sharot (2011), Sharot et al (2011) and Garrett and Sharot (2014) document that optimism bias is maintained in the face of disconfirming evidence, as people tend to revise their view more in response to positive information than negative information. Weinstein (1980) presents experimental evidence that people overstate the likelihood of experiencing positive events and understate the likelihood of negative events even when faced with the evidence of how people like themselves have fared in the past. Johnson and Fowler (2011) show theoretically that in many cases optimism bias serves humanity well, as it induces people to rise to challenges that would be shied away from if they held unbiased beliefs.

In this paper we develop a model of the demand and supply for unsecured credit that allows us to compare the outcomes of consumers, ranging from pure optimists to rational (unbiased) consumers. We then assess how rules restricting debt collection practices affect credit supply and demand and consumer welfare. Rational consumers incorporate income uncertainty and the possibility of not repaying in full along with the associated utility costs of collections into their utility functions; as consumers become increasingly optimistic, they place less and less weight on these factors and pure optimists completely disregard income risk and the possibility of default. In parameterizations we consider, optimistic consumers overborrow as they perceive consumer surplus from a credit transaction to be greater than it is. Credit demand by pure optimists does not respond to rules that reduce the harshness of collections tactics; in contrast, rational consumers take rule changes into account and react. In cases where demand by optimistic consumers is greater than that of rational consumers, rules that directly lower collection effort operate by shifting demand by rational consumers out, making them look more like optimistic consumers and reducing the negative effects of behaving optimistically. We can think of the rules as introducing a distortion into the market that reduces the welfare consequences of being optimistic.

We assume lenders choose either high or low collection effort and then consider a policy that prohibits high-effort collection practices. The restriction induces rational consumers to alter their borrowing, as it reduces the utility cost of default, but causes lenders to raise the interest rate in expectation of lower repayment rates. We show that, for certain parameterizations of the utility cost and interest rate effects, the restriction on collection effort will be welfare increasing for optimistic consumers. For other parameterizations the welfare impact of rules is negative.

Optimistic consumers incur a welfare cost from overborrowing that can be characterized as a marginal internality^{7}^{7} 7 “[C]osts that we impose on ourselves by taking actions that are not in our own interest” (Allcott and Sunstein 2015, p. 698). that produces a gap between the anticipated and experienced utility of purchases. Kahneman et al (1997) present psychological evidence of utility maximization mistakes. Allcott and Sunstein (2015) and Mullainathan et al (2012) model and propose nudges, choice restrictions and price changes as regulatory correctives. We characterize the lack of salience of debt collection to optimistic consumers as a form of marginal internality and we characterize its welfare cost as per Train (2015).

The paper proceeds with a discussion of the history of debt collection practice rules in Section 2. In Section 3 we present evidence from credit card contracts and from recent research that indicates collection effort is not salient to borrowers. In Section 4, we model the borrowers’ decision problems and show how optimism affects optimal borrowing and repayment, and we finish by showing the partial equilibrium effect on demand of a rule that limits collection effort. In Section 5 we turn to the lender’s problem and present a general equilibrium discussion in which rational consumers would prefer low collection effort, but consumers are sufficiently optimistic that lenders choose high effort. A welfare discussion follows in which we show that the marginal internality consumers bear depends on the relative shifts in the utility cost of collections versus the reduction in the interest rate. Section 6 discusses how results might change under different modeling assumptions, and Section 7 offers some concluding remarks.

## 2 A history of debt collection practice rules

Perhaps the most noted restriction on creditors’ ability to collect defaulted debt is the bankruptcy code, which enables consumers to seek protection from creditors and reduce or eliminate outstanding debt balances. However, as pointed out by Dawsey and Ausubel (2004) and others, formal bankruptcy is much less prevalent among consumers than simply defaulting on their debts without seeking bankruptcy protection. For such consumers, there are limitations on judicial remedies that still apply, most notably limitations on wage garnishment in many US states. Even outside of the judicial process, state and federal laws limit the ways in which creditors can seek to collect from delinquent consumers.

The 1977 Fair Debt Collections Practices Act (FDCPA) was the first US federal debt collection practices law. It was motivated in part by the introduction of low-cost long-distance calling and an associated increase in interstate debt collections. At that time, twenty-four states, with 80 million citizens, had little or no effective debt collection laws (US Government 1977). Stories of unscrupulous behavior by debt collectors motivated Congress to act. Among many stories detailing abuses by debt collectors, the Honorable Frank Annunzio reported in the Congressional Record that^{8}^{8} 8 See Congressional Record: Proceedings and Debates of the 94th Cong, HR 10191, E5404. “Regardless of your denial of the validity of the debt, you may then get the debt collector “treatment.” This may include obscene phone calls to your home, or the homes of your friends and neighbors at odd hours, and repeated harassment of your employer asking that you be pressured into paying.”

The FDCPA covers only third-party debt collectors and limits attention to debts contracted by consumers for personal, family or household purposes. It prohibits harassing, deceptive and unfair debt collection practices, it protects a consumer’s right to privacy and it has provisions that require the validation of debts.

Prior to the passage of the FDCPA, there was one federal law that limited the consequences of being unable to repay debts. Federal debtors prisons had been abolished in 1833 and replaced with bankruptcy laws. Doing away with debtors prisons eliminated the harshest sanction for debtors. The FDCPA’s limits on debt collection practices made a set of other harsh practices illegal.

## 3 Evidence on the salience of collection effort to borrowers

In this section we discuss empirical evidence of whether borrowers take into account anticipated collection effort when making borrowing decisions.

### 3.1 Evidence from credit contracts

Here, we ask what credit card contract clauses say about collection tactics as a way to assess whether or not collections clauses in credit card contracts are indicative of shopping. Schwartz and Wilde (1983) argue that consumers have a general understanding of what their contracts say, and that enough consumers shop on the contracts’ terms to get the market to respond. If this is the case, then we may observe variation in collections clauses in contracts. Some consumers, for example, may be willing to accept a higher interest rate, a lower cashback rate or fewer airline miles in return, say, for a cap on wage garnishment or on collection effort. Lenders may be willing to offer such contracts if consumers shop on collection terms, and if they can solve the adverse selection problem such contracts would generate.

To understand how clearly lenders commit to collections and whether there is variation in collections clauses indicative of shopping, we examined collections clauses in credit card contracts in the Credit Card Agreement Database maintained by the Consumer Financial Protection Bureau (CFPB). We limited our attention to contracts of the eight major US credit card issuers: American Express, Bank of America, Capital One, Chase, Citibank, Discover, General Electric and Wells Fargo. For 2014 Q3, the CFPB had 197 agreements on file for these eight issuers, for 197 different credit cards. The cards range from gas station cards, Sam’s Club and Costco cards to those associated with airlines, high-end retailers, automakers and financial services firms. The variation in card attributes and demographic focus is wide both within and across issuers. The collections clauses for all cards put out by a single issuer are the same, and collections clauses are very similar across all eight issuers. For example, the only collections-related clause for all forty-nine cards produced by American Express reads: “You agree to pay all reasonable costs including attorney’s fees that we incur to collect amounts you owe.” Other issuers use similar language, though some note that consumers are responsible for court costs or for attorney fees if the attorney is not a salaried employee of the issuer.

These are far from complete contracts detailing actions an issuer will take in response to unpaid debts. A complete contract would detail a commitment to a specific level of collections activity that would escalate with the level of indebtedness: for small unpaid debts, collections activity may be limited to furnishing a collections report to the credit bureaus; larger debts may generate letters and phone calls; and above a certain threshold unpaid debt may generate lawsuits. In addition, if collections clauses were a factor in consumer credit card choice, and if firms could solve the adverse selection problem that contract variations could cause, we might expect these clauses to contain trade-offs, such as an offer of no wage garnishment or a cap on collection calls per week for higher interest rates or late fees.^{9}^{9} 9 Ausubel (1991) presents evidence that a substantial subset of consumers choose cards with higher interest rates, acting as though they do not intend to borrow but doing so continuously. Ausubel characterizes this behavior as irrational. Unrealistically optimistic may be more on point, and if so, it suggests that such consumers would be inattentive to contract variations in collections clauses.

While this evidence does not prove that borrowers are not shopping, the lack of variation in collections clauses calls into question Schwartz and Wilde’s assessment that enough consumers shop on contracts terms to get the market to respond (Schwartz and Wilde 1983). In addition, the reality of collections is such that the original commitment is general and the actual level of collections activity is conditional on the amount of the unpaid debt and on characteristics of the debt that were set at the time credit was granted, and it does not vary with any group characteristics.

### 3.2 Evidence of demand effects from regulation of debt collection

Romeo and Sandler (forthcoming) estimate the demand for new credit card inquiries and the demand for new credit cards in response to state level debt collection rule changes. Four US states and New York City instituted new debt substantiation or disclosure requirements between 2009 and 2014.^{10}^{10} 10 North Carolina extended a series of restrictions to firms that buy debts through its Consumer Economic Protection Act, which took effect in October 2009. California’s Fair Debt Buying Practices Act took effect in January 2014. Arkansas enacted a state level Fair Debt Collection Practices Act in July 2009, and New York City and New York State implemented new debt collection regulations in April 2010 and December 2014. Romeo and Sandler use quarterly credit record data from the CFPB’s Consumer Credit Panel, a 1-in-48 sample of de-identified consumer credit records, to estimate the effects of these rule changes. Using this data, Romeo and Sandler are able to separate out two demand-side effects: demand for new credit card inquiries and demand for new credit cards. Credit inquiries are measured as the number of inquiries each consumer makes to obtain new credit, while credit demand is the count of consumers who seek to obtain new credit. An increase in new credit inquiries can reflect both demand- and supply-side effects; an increase in inquiries might reflect consumers seeking additional credit, or instead reflect additional inquiries for the same amount of credit because creditors tightened access to credit in response to the rule change. Demand for credit isolates the demand effect. An increase in demand will be observed as an increase in the number of consumers seeking new credit.

Romeo and Sandler find a small and statistically insignificant effect on the demand for credit card inquiries and no effect on the demand for credit cards in response to a tightening of debt collection laws. The lack of a demand effect is consistent with collections rule changes not being salient to the average consumer when they apply for credit.

## 4 A model of consumer borrowing with debt collection

This section describes consumer borrowing and repayment decisions and how optimism affects borrowers’ choices. Section 5 describes the supply of credit and explores welfare implications of debt collection policies.

We develop a two-period model. Consumers receive income ${y}_{0}>0$, in period 0 and can borrow an amount $b\ge 0$ to consume ${y}_{0}+b$. In period 1, consumers receive uncertain income ${y}_{1}$, such that ${y}_{1}(\epsilon )=\alpha {y}_{0}-\epsilon $, where $\alpha >1$ is the income growth rate and is common to all consumers, and $\epsilon $ is a potential income shock. We assume that consumers suffer a shock to period 1 income with probability $p$ and that if it occurs, the shock is distributed uniformly on the support $[\underset{\xaf}{\epsilon},\overline{\epsilon}]$, $$. $\alpha $ and ${y}_{0}$ are known to both consumers and firms, as is the distribution of $\epsilon $, but individual $\epsilon $ draws are observed only by consumers, and not until period 1.

Consumer utility depends on consumption in each period as well as nonconsumption consequences of debt collection. Let ${c}_{t}$ be consumption in period $t$, and let utility from consumption in each period be represented by $u({c}_{t})$. We assume monotonicity and strict concavity of $u({c}_{t}):{u}_{\mathrm{c}}({c}_{t})>0$ and $$.

Consumer loan contracts are characterized by an interest rate $r$ and a level of collection effort $e$, both of which are set by lenders at the start of period 0.^{11}^{11} 11 As mentioned in the previous section, the FDCPA applies only to third-party debt collectors, who generally work as agents of lenders. As such, the law does not restrict the collection efforts of lenders directly but does restrict the methods their agents can employ to collect debt. We abstract from this distinction for the purposes of our analysis. Consumers take the set of offered contracts as fixed and choose the offered contract that they believe will yield the highest expected utility. Consumers who borrow $b$ in period 0 are expected to repay $(1+r)b\equiv Rb$ in period 1. The credit is unsecured. If they do not repay, they will experience the consequences of debt collection as described further below. Collection effort can be at one of two levels, characterized as “low” and “high”, so that $e\in \{{e}_{\mathrm{L}},{e}_{\mathrm{H}}\}$.

Consumers choose whether and how much to repay after observing period 1 income. If the consumer repays the debt in full, no collection efforts are undertaken. If the consumer does not repay the debt, the lender makes a settlement offer, $?\in [0,1]$, which the consumer can either accept or reject. If the consumer rejects the settlement, then through collection efforts the lender is able to collect some fraction of the amount owed by the consumer, represented by $\mathrm{\ell}(e)\in [0,1)$, with $\mathrm{\ell}({e}_{\mathrm{L}})\le \mathrm{\ell}({e}_{\mathrm{H}})$. A consumer who does not repay in full experiences reputational, psychic and possible lawsuit costs, $\mathrm{\Gamma}(e;\xi )$, that depend on the consumer’s period 1 repayment decision $\xi $: pay in full, settle with the creditor or default, so that $\xi =\mathrm{f},\mathrm{s},\mathrm{d}$ respectively. These collection costs are assumed to enter the utility function linearly, so period 1 utility is given by $u({c}_{1})+\mathrm{\Gamma}(e;\xi )$. We specify

$$\mathrm{\Gamma}(e;\xi )=\{\begin{array}{cc}0\hfill & \text{if}\xi =\mathrm{f},\hfill \\ {\gamma}_{1}\hfill & \text{if}\xi =\mathrm{s},\hfill \\ {\gamma}_{1}+{\gamma}_{2}(e)\hfill & \text{if}\xi =\mathrm{d}.\hfill \end{array}$$ |

A consumer who pays in full bears no reputational, psychic or legal costs from the collections process, so $\mathrm{\Gamma}(e;\mathrm{f})=0$. ${\gamma}_{1}>0$ captures the costs of not paying in full that do not depend on lender actions, such as reputational harm or stigma associated with not meeting debt obligations.^{12}^{12} 12 Stigma has been an important factor in consumer debt repayment decisions. Recent research presents conflicting evidence as to whether stigma has declined in recent years (Gross and Souleles 2002) or is unchanged (Athreya 2004). ${\gamma}_{2}(e)$ captures costs from the collections process and depends on the lender’s choice of collection effort.^{13}^{13} 13 Consumer Financial Protection Bureau (2017) reports the results of a debt collection survey, which finds that 15% of consumers who had debts in collections were sued by a creditor, and that lawsuit percentages were increasing with the number of debts in collections and with age, and decreasing with household income and credit score. We assume ${\gamma}_{2}({e}_{\mathrm{H}})\ge {\gamma}_{2}({e}_{\mathrm{L}})$. Below, we will consider the effect of government policies that prohibit ${e}_{\mathrm{H}}$, limiting the intensity of collection practices and thereby limiting the consequences of collections for consumers.

In deciding how much to borrow, consumers consider the likelihood of an income shock and possible repayment levels and collection costs following a shock. They will choose $b$ to maximize

$$u({c}_{0})+\frac{1}{1+{\delta}_{\mathrm{c}}}E[u({c}_{1})+\mathrm{\Gamma}(e;\xi )],$$ |

where ${c}_{0}={y}_{0}+b$ and ${c}_{1}=\alpha {y}_{0}-\epsilon Rb$ for a consumer who repays in full.

To solve for the consumer’s borrowing decision we work backward, first describing the consumer’s decisions about whether to repay the debt as a function of the amount borrowed and the realization of the income shock $\epsilon $. We then turn to the consumer’s borrowing decision in the light of expected period 1 utility as distorted by optimism.

### 4.1 Repayment and default in period 1

In period 1 there are three steps. First, consumers decide whether to repay the debt in full. Second, lenders make settlement offers to consumers who have not repaid. Third, consumers decide whether to accept these offers or to default and undergo collection efforts.

Starting with the third step, we consider the consumer’s decision whether to accept a settlement offer in light of $b$, $\epsilon $ and $?$, having already failed to repay the debt in full. Such consumers choose whether to accept a settlement offer by comparing its terms to the consequences of collections, accepting the offer if

$$u({y}_{1}(\epsilon )-?Rb)-{\gamma}_{1}>u({y}_{1}(\epsilon )-\mathrm{\ell}(e)Rb)-{\gamma}_{1}-{\gamma}_{2}(e)$$ | |||

or | |||

$$ |

This condition will always be satisfied if $?\le \mathrm{\ell}$; for $?>\mathrm{\ell}$ the inequality will depend on $?$ and $b$ and on collection effort through $\mathrm{\ell}(e)$ and ${\gamma}_{2}(e)$, ie, the additional consumption from defaulting rather than settling relative to the utility cost to consumers from collections. Note that the concavity of $u(\cdot )$ implies that the left-hand side of this expression is decreasing in period 1 income ${y}_{1}$ (or, equivalently, increasing in the severity of the income shock $\epsilon $). This implies that, for any $?>\mathrm{\ell}$ and borrowing $b$, if default is preferred to settlement for some income shock ${\epsilon}^{\prime}$, then default will also be preferred for any larger shock $\epsilon \ge {\epsilon}^{\prime}$. Therefore, we can describe the optimal decision for the consumer as a threshold value of $\epsilon $, $\ddot{\epsilon}(b,?,e)$, such that the consumer chooses default if and only if $?>\mathrm{\ell}$ and $\epsilon \ge \ddot{\epsilon}(b,?,e)$.

Lenders will choose settlement offers knowing that the offers will be accepted by consumers who did not repay but for whom $$. We defer the discussion of the lender’s optimal choice of $?$ until Section 4.2, except we note that it would never be optimal for a lender to choose $$, given that $?=\mathrm{\ell}$ would yield higher repayment from every consumer without changing the collection costs.

Moving to the second step, once consumers observe $\epsilon $, they must choose whether to repay their borrowing or to default and either settle or face collection actions. Although settlement offers are not observed until after consumers have chosen not to repay, consumers at this point are able to predict the profit-maximizing settlement offers that lenders will make if they fail to repay and know, based on their realized shock $\epsilon $, whether if they fail to repay, they will end up accepting the settlement offer or defaulting completely.

We compare the consumer’s utility when paying in full and when settling, and then consider the payoff when defaulting and undergoing collections. In maximizing expected utility, consumers will prefer paying in full to accepting a settlement offer if

$$u({y}_{1}(\epsilon )-Rb)>u({y}_{1}(\epsilon )-?Rb)-{\gamma}_{1}$$ | |||

or | |||

$$ |

The left-hand side is positive for all $$, but whether the inequality is satisfied depends on $?$, $b$ and ${\gamma}_{1}$. Relying again on concavity of $u(\cdot )$, the left-hand side of the expression is increasing in the severity of the income shock $\epsilon $. This implies that for a given level of borrowing if paying in full is preferred to settlement for some shock ${\epsilon}^{\prime \prime}$, then paying in full will also be preferred for all $\epsilon \le {\epsilon}^{\prime \prime}$. We describe this decision in terms of a threshold value of $\epsilon $, $\stackrel{~}{\epsilon}(b)$, such that the consumer prefers to pay in full if and only if $\epsilon \le \stackrel{~}{\epsilon}(b)$.

Appendix A.1 online shows that at $\stackrel{~}{\epsilon}(b)$, where the consumer is indifferent between paying in full or settling, the consumer prefers paying in full to default. This implies that we have $$ such that consumers pay in full if $\epsilon \in [\underset{\xaf}{\epsilon},\stackrel{~}{\epsilon})$, settle if $\epsilon \in [\stackrel{~}{\epsilon},\ddot{\epsilon})$ and default if $\epsilon \in [\ddot{\epsilon},\overline{\epsilon})$.

The above thresholds depend on consumers choosing not to pay in full for at least some realizations of $\epsilon $. Consumers will never default if, following the largest income shock $\overline{\epsilon}$, they still prefer to pay in full:

$$ | (4.1) |

This expression will always be satisfied if $Rb$ is small enough, since when the repayment amount is low the consumption benefit of default is too small to justify the nonpecuniary default costs. Throughout the rest of this section we focus on the case where the required repayment, $Rb$, is large enough that consumers choose not to repay in full under at least some realizations of $\epsilon $. We return to the conditions under which consumers choose not to default below when discussing the consumer’s choice of $b$.

In Appendix A.9 online we derive the signs of the derivatives of $\ddot{\epsilon}(b,?,e)$ and $\stackrel{~}{\epsilon}(b)$, showing that ${\ddot{\epsilon}}_{b},{\stackrel{~}{\epsilon}}_{b}\le 0$, ie, both thresholds are decreasing in the consumer’s debt level. Intuitively, greater borrowing increases the marginal utility of consumption after repayment, altering the trade-off between consumption and collection costs in favor of consumption. We also show that, as we would expect, $\stackrel{~}{\epsilon}({e}_{\mathrm{H}})-\stackrel{~}{\epsilon}({e}_{\mathrm{L}})\ge 0$ and $\ddot{\epsilon}({e}_{\mathrm{H}})-\ddot{\epsilon}({e}_{\mathrm{L}})\ge 0$. Increased collection effort makes default more costly to consumers, all things being equal, making consumers more likely to pay in full even after experiencing an income shock.

Figure 1 illustrates the thresholds, presenting the consumption utility associated with each of the three repayment regions. For large income shocks, $\epsilon \in [\ddot{\epsilon},\overline{\epsilon})$, it shows the utility of defaulting and not settling dominates that of settling or paying in full; in this range the marginal utility of additional consumption obtained through underpayment is greater than the marginal cost imposed by collections. For more moderate-income shocks, the marginal value of additional consumption decreases and for $\epsilon \in [\stackrel{~}{\epsilon},\ddot{\epsilon})$ the utility of settling dominates that of defaulting, while for $\epsilon \in [\underset{\xaf}{\epsilon},\stackrel{~}{\epsilon})$ the utility of repayment in full is highest. The envelope of the three utility functions, represented by the dotted line, shows the regions where defaulting, settling and paying in full are the consumer’s optimum response.

### 4.2 Repayment probabilities and settlement in period 1

Let ${\omega}_{\mathrm{f}}(b)$, ${\omega}_{\mathrm{s}}(b,?,e)$ and ${\omega}_{\mathrm{d}}(b,?,e)$ represent the probabilities a consumer repays in full, settles and does not settle, respectively, with ${\omega}_{\mathrm{f}}+{\omega}_{\mathrm{s}}+{\omega}_{\mathrm{d}}=1$. Given the assumption that the shock has a uniform distribution and the result that the consumer will pay in full for any $$, this becomes

$${\omega}_{\mathrm{f}}(b)=(1-p)+p\frac{\stackrel{~}{\epsilon}-\underset{\xaf}{\epsilon}}{\overline{\epsilon}-\underset{\xaf}{\epsilon}}.$$ |

Similarly,

${\omega}_{\mathrm{s}}(b,?,e)$ | $=p{\displaystyle \frac{\ddot{\epsilon}-\stackrel{~}{\epsilon}}{\overline{\epsilon}-\underset{\xaf}{\epsilon}}}$ | |||

and | ||||

${\omega}_{\mathrm{d}}(b,?,e)$ | $=p{\displaystyle \frac{\overline{\epsilon}-\ddot{\epsilon}}{\overline{\epsilon}-\underset{\xaf}{\epsilon}}}.$ |

We return to the lender’s choice of settlement offer $?$. Let $k(e)$ be the marginal cost of collections. Lenders choosing a settlement offer act to maximize

$$h(?)={\omega}_{\mathrm{s}}?Rb+{\omega}_{\mathrm{d}}(\mathrm{\ell}(e)Rb-k(e)b).$$ | (4.2) |

As shown in Appendix A.2 online, optimal $?$ is obtained at the point

$$?-\mathrm{\ell}=-{\left(\frac{\partial {\omega}_{\mathrm{s}}}{\partial ?}\right)}^{-1}{\omega}_{\mathrm{s}}-\frac{k(e)}{R},$$ | (4.3) |

or, equivalently,

$$\frac{?-\mathrm{\ell}}{?}=-{\sigma}_{{\omega}_{\mathrm{s}}?}^{-1}-\frac{k(e)}{?R},$$ |

where $(?-\mathrm{\ell})/?$ is the margin of $?$ to $\mathrm{\ell}$, and ${\sigma}_{{\omega}_{\mathrm{s}}?}$ is the elasticity of settlement probability ${\omega}_{\mathrm{s}}$ with respect to changes in $?$. This first term indicates that the optimal margin increases as the elasticity decreases; $?$ close to 1 will only occur if ${\omega}_{\mathrm{s}}$ is only minimally responsive to changes in $?$. In the second term, the lender weighs the marginal cost of collections against settlement return, $R$. If this cost is high enough, the lender may be induced to make a settlement offer that is close to or equal to what is eventually achievable through collections. As discussed above, we must have $?\ge \mathrm{\ell}$; any $$ gives lenders strictly higher profits because a slightly higher $?$ would be accepted by the same number of consumers. However, if $k(e)$ or ${\sigma}_{{\omega}_{\mathrm{s}}?}$ are high enough, it is possible that the lender will offer a settlement close enough to $\mathrm{\ell}$ so that all defaulting consumers will accept the settlement offer. Conditions for this to be the case are discussed in Appendix A.2 online.

### 4.3 Optimism and optimal borrowing

Given these thresholds, we can now formulate the two-period utility function that underlies consumer borrowing decisions. In formulating utility, we allow consumers to be optimistic, in that they may underestimate the likelihood of an income shock. As discussed above, we assume that consumers suffer a shock to period 1 income with probability $p$, and that the shock, if it occurs, is distributed uniformly on the support $[\underset{\xaf}{\epsilon},\overline{\epsilon}]$, where $$. Consumers know the distribution of shocks conditional on their occurring, but optimistic consumers underestimate the probability of a shock. Specifically, optimism is reflected by a parameter $\psi \in [0,1]$ such that consumers believe the probability of a shock is $\psi p$. Thus, $\psi =1$ would represent a realistic consumer and $$ represents an optimistic consumer.

From the perspective of period 0, the consumer’s perceived utility is represented by

$u({c}_{0})$ | $+{\displaystyle \frac{1-\psi p}{1+{\delta}_{\mathrm{c}}}}u(\alpha {y}_{0}-Rb)$ | ||

$+{\displaystyle \frac{\psi p}{1+{\delta}_{\mathrm{c}}}}[{\displaystyle {\int}_{\underset{\xaf}{\epsilon}}^{\stackrel{~}{\epsilon}}}u(\alpha {y}_{0}-\epsilon -Rb)\mathrm{d}\epsilon +{\displaystyle {\int}_{\stackrel{~}{\epsilon}}^{\ddot{\epsilon}}}u(\alpha {y}_{0}-\epsilon -?Rb)-{\gamma}_{1}\mathrm{d}\epsilon $ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{\ddot{\epsilon}}^{\overline{\epsilon}}}[u(\alpha {y}_{0}-\epsilon -\mathrm{\ell}(e)Rb)-{\gamma}_{1}-{\gamma}_{2}(e)]\mathrm{d}\epsilon ],$ |

where the integral bounds $\stackrel{~}{\epsilon}=\stackrel{~}{\epsilon}(b)$ and $\ddot{\epsilon}=\ddot{\epsilon}(b,?,e)$.

To make our notation more compact we define period 1 consumption for consumers who pay in full as

$${c}_{1\mathrm{f}}(\epsilon )=\alpha {y}_{0}-\epsilon -Rb,$$ |

with ${c}_{1\mathrm{f}}(0)=\alpha {y}_{0}-Rb$ used to represent consumers who do not incur an income shock; consumers who settle as

$${c}_{1\mathrm{s}}(\epsilon )=\alpha {y}_{0}-\epsilon -?Rb;$$ |

and consumers who default as

$${c}_{1\mathrm{d}}(\epsilon )=\alpha {y}_{0}-\epsilon -\mathrm{\ell}(e)Rb.$$ |

Consumers borrow to maximize two-period utility, equating the marginal utility of consumption in period 0, ${u}_{\mathrm{c}}({c}_{0})$, with their perceived expected marginal utility of period 1 consumption:

$u({c}_{0})$ | $={\displaystyle \frac{1-\psi p}{1+{\delta}_{\mathrm{c}}}}R{u}_{\mathrm{c}}({c}_{1\mathrm{f}}(0))$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{\psi p}{1+{\delta}_{\mathrm{c}}}}R[{\displaystyle {\int}_{\underset{\xaf}{\epsilon}}^{\stackrel{~}{\epsilon}}}{u}_{\mathrm{c}}({c}_{1\mathrm{f}}(\epsilon ))\mathrm{d}\epsilon +?{\displaystyle {\int}_{\stackrel{~}{\epsilon}}^{\ddot{\epsilon}}}[{u}_{\mathrm{c}}({c}_{1\mathrm{s}}(\epsilon ))]\mathrm{d}\epsilon $ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+\mathrm{\ell}(e){\displaystyle {\int}_{\ddot{\epsilon}}^{\overline{\epsilon}}}[{u}_{\mathrm{c}}({c}_{1\mathrm{d}}(\epsilon ))]\mathrm{d}\epsilon ]$ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{\psi p}{1+{\delta}_{\mathrm{c}}}}[{\displaystyle \frac{\mathrm{d}\stackrel{~}{\epsilon}}{\mathrm{d}b}}({u}_{\mathrm{c}}({c}_{1\mathrm{f}}(\stackrel{~}{\epsilon}))-({u}_{\mathrm{c}}({c}_{1\mathrm{s}}(\epsilon ))-{\gamma}_{1}))$ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\displaystyle \frac{\mathrm{d}\ddot{\epsilon}}{\mathrm{d}b}}({u}_{\mathrm{c}}({c}_{1\mathrm{s}}(\ddot{\epsilon}))-{\gamma}_{1}-({u}_{\mathrm{c}}({c}_{1\mathrm{d}}(\ddot{\epsilon})-{\gamma}_{1}-{\gamma}_{2}(e))))].$ | (4.4) |

The last two terms in (4.4) each equal zero by the definition of thresholds $\stackrel{~}{\epsilon}$ and $\ddot{\epsilon}$, respectively. Optimism causes consumers to underweight the risk to full repayment that the income shock may cause. As $\psi \to 0$, $\psi p\to 0$ and more weight in consumer perceptions is shifted to the marginal utility of full repayment with no income shock, $R{u}_{\mathrm{c}}({c}_{1\mathrm{f}}(0))$.

As discussed in reference to (4.1), consumers will only fail to repay in full (ie, we will only have $$) if the repayment amount is great enough that default is preferable for at least some realizations of $\epsilon $. To determine conditions under which consumers will choose to default for at least some realizations of $\epsilon $, first define $\widehat{b}$ as the amount at which the consumer is indifferent between paying in full and default given the largest income shock $\overline{\epsilon}$; that is, $\widehat{b}$ is defined by

$$u({y}_{1}(\overline{\epsilon})-\mathrm{\ell}(e)R(e)\widehat{b})-u({y}_{1}(\overline{\epsilon})-R(e)\widehat{b})={\gamma}_{1}+{\gamma}_{2}(e).$$ |

We assume that in making the borrowing decision the consumer chooses to borrow more than $\widehat{b}$, which follows if

${u}_{\mathrm{c}}({y}_{0}+\widehat{b})$ | $>{\displaystyle \frac{1-\psi p}{1+{\delta}_{\mathrm{c}}}}R{u}_{\mathrm{c}}(\alpha {y}_{0}-R\widehat{b})$ | ||

$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{\psi p}{1+{\delta}_{\mathrm{c}}}}R[{\displaystyle {\int}_{\underset{\xaf}{\epsilon}}^{\stackrel{~}{\epsilon}}}{u}_{\mathrm{c}}(\alpha {y}_{0}-\epsilon -R\widehat{b})\mathrm{d}\epsilon $ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+?{\displaystyle {\int}_{\stackrel{~}{\epsilon}}^{\ddot{\epsilon}}}[{u}_{\mathrm{c}}(\alpha {y}_{0}-\epsilon -?R\widehat{b})]d\epsilon $ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+\mathrm{\ell}(e){\displaystyle {\int}_{\ddot{\epsilon}}^{\overline{\epsilon}}}[{u}_{\mathrm{c}}(\alpha {y}_{0}-\epsilon -\mathrm{\ell}(e)R\widehat{b})]\mathrm{d}\epsilon ].$ |

Taking the derivative of (4.4) with respect to $\psi $ can tell us how borrowing depends on optimism:

$\frac{\partial}{\partial \psi}}{u}_{\mathrm{c}}({c}_{0})$ | $={\displaystyle \frac{p}{1+{\delta}_{\mathrm{c}}}}R\{{\displaystyle {\int}_{\underset{\xaf}{\epsilon}}^{\stackrel{~}{\epsilon}}}[{u}_{\mathrm{c}}({c}_{1\mathrm{f}}(\epsilon ))-{u}_{\mathrm{c}}({c}_{1\mathrm{f}}(0))]\mathrm{d}\epsilon $ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{\stackrel{~}{\epsilon}}^{\ddot{\epsilon}}}[?{u}_{\mathrm{c}}({c}_{1\mathrm{s}}(\epsilon ))-{u}_{\mathrm{c}}({c}_{1\mathrm{f}}(0))]d\epsilon $ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{\ddot{\epsilon}}^{\overline{\epsilon}}}[\mathrm{\ell}(e){u}_{\mathrm{c}}({c}_{1\mathrm{d}}(\epsilon ))-{u}_{\mathrm{c}}({c}_{1\mathrm{f}}(0))]\mathrm{d}\epsilon \}.$ | (4.5) |

This expression could be positive or negative depending on the parameters. Intuitively, as consumers become more optimistic they attach less weight to the possibility of an income shock that could lead to default and collections. Optimistic consumers therefore expect higher income on average, which tends to increase borrowing, but they attach a lower probability to default, which tends to decrease borrowing because consumers who default do not fully repay the amount they borrow.

We will focus on cases in which the first effect dominates; that is, optimistic consumers borrow more than realistic consumers. The following proposition establishes sufficient conditions for this outcome.

###### Proposition 4.1.

Let $M\mathit{}U\mathit{}\mathrm{(}b\mathrm{)}$ represent the marginal utility of consumption as a function of $b$. Assume that

- (i)
$\underset{\xaf}{\epsilon}>(1-\mathrm{\ell}(e))\alpha {y}_{0}$ and

- (ii)
$\mathrm{\ell}(e){u}_{\mathrm{c}}(\mathrm{\ell}(e)c)\ge {u}_{\mathrm{c}}(c)$ for all values of $c$.

Assumptions (i) and (ii) are sufficient for

$$\frac{\mathrm{d}}{\mathrm{d}\psi}MU(b)>0$$ |

at the optimal borrowing amount.

###### Proof.

See Appendix A.3 online. ∎

Proposition 4.1(i) is an assumption about the severity of the shock relative to the additional consumption possible by defaulting; one implication is that a consumer who defaults following a shock will always consume less than a consumer who does not experience a shock. Proposition 4.1(ii) is an assumption about the curvature of $u(c)$, which is satisfied, for example, for log utility.

We next use (4.4) to assess the impact of collection effort on borrowing. We show conditions under which borrowing decreases in collection intensity, holding interest rates constant.

###### Proposition 4.2.

Define $b\mathit{}\mathrm{(}e\mathrm{,}R\mathrm{)}$ as the borrower’s optimal choice of $b$ given $e$ and $R$. If $\psi \mathrm{>}\mathrm{0}$ and

$$\frac{{u}_{\mathrm{c}}(\alpha {y}_{0}-\underset{\xaf}{\epsilon}-\mathrm{\ell}({e}_{\mathrm{L}})x)}{{u}_{\mathrm{c}}(\alpha {y}_{0}-\overline{\epsilon}-\mathrm{\ell}({e}_{\mathrm{H}})x)}\ge \frac{{\gamma}_{2\mathrm{L}}}{{\gamma}_{2\mathrm{H}}},$$ |

then $$. If $\psi \mathrm{=}\mathrm{0}$, then $b\mathit{}\mathrm{(}{e}_{\mathrm{H}}\mathrm{,}R\mathrm{)}\mathrm{=}b\mathit{}\mathrm{(}{e}_{\mathrm{L}}\mathrm{,}R\mathrm{)}$.

###### Proof.

See Appendix A.4 online. ∎

The assumption that

$$\frac{{u}_{\mathrm{c}}(\alpha {y}_{0}-\underset{\xaf}{\epsilon}-\mathrm{\ell}({e}_{\mathrm{L}})x)}{{u}_{\mathrm{c}}(\alpha {y}_{0}-\overline{\epsilon}-\mathrm{\ell}({e}_{\mathrm{H}})x)}\ge \frac{{\gamma}_{2\mathrm{L}}}{{\gamma}_{2\mathrm{H}}},$$ |

which is sufficient but not necessary for the result, is in part an assumption about the concavity of $u(\cdot )$ not being too great compared to the relative severity of collection costs under ${e}_{\mathrm{H}}$ and ${e}_{\mathrm{L}}$. The intuition for this result is relatively straightforward: an increase in collection effort makes default less attractive and increases the expected amount of borrowing that a consumer will repay following a shock; this in turn reduces the marginal benefit of borrowing. This effect will hold except in the case of purely optimistic consumers, who do not take into account the possibility of an income shock at all and are therefore indifferent with respect to collection effort. Of course, as discussed in the next section, an increase in collection effort will generally reduce lenders’ expected default losses and lower interest rates. Thus, consumers who are not purely optimistic will face a trade-off between high collection effort and low interest rates.

###### Definition 4.3 (Finite differences).

Given that collection effort is limited to two discrete values, an alternative way to express $$ is to form a finite difference. Below, we often write $$ as $$, or as ${\mathrm{\Delta}}_{e}b\le 0$ when not conditioned on $R$. More generally, using ${e}_{\mathrm{L}}$ as our base, we form ${\mathrm{\Delta}}_{e}x\equiv {\mathrm{\Delta}}_{e}x({e}_{\mathrm{L}})=x({e}_{\mathrm{H}})-x({e}_{\mathrm{L}})$ for all variables $x$ that are functions of the collection effort. In addition, we often use ${x}_{j}\equiv x({e}_{j})$, $j=\mathrm{L},\mathrm{H}$, to make our notation more compact.

We do not specify the form of the utility functions, and therefore cannot derive the demand for borrowing function. Propositions 4.1 and 4.2, however, provide us with key drivers of this function and parameterizations under which the sufficient conditions we specified are satisfied and the effects of these drivers on demand can be signed. We can specify $b\equiv b(R,e;\psi )$, with ${\mathrm{\Delta}}_{e}b=0$ for pure optimists, ${\mathrm{\Delta}}_{e}b\le 0$ under sufficient conditions specified above for more rational consumers, $\partial b/\partial \psi \le 0$, and, though we do not present results for $R$, we can show that $\partial b/\partial R\le 0$. Our results also indicate that $\partial ({\mathrm{\Delta}}_{e}b)/\partial \psi \ge 0$, as borrowing demand by less optimistic consumers will respond more strongly to an increase in collection effort.

### 4.4 Discussion: collection effort and consumer welfare

In Section 5 we will discuss lenders’ choices of $R$ and $e$. Here, having described consumer borrowing decisions, we discuss how the choice of $e$ affects consumer welfare when consumers are optimistic. Suppose now that in equilibrium creditors set $e={e}_{\mathrm{H}}$, and that rules are introduced that limit that harshness of collections by placing an upper bound on $e$: $e={e}_{\mathrm{L}}$. Holding the interest rate constant, our results above show that since pure optimists only respond to the decrease in collection effort through its effect on the interest rate, the change in $e$ leaves the demand by pure optimists unchanged. For rational consumers and nonpure optimists, the effect of a decrease in collection effort operates through its effect on marginal utilities and on the thresholds. Proposition 4.2 indicates that a decrease in collection effort shifts demand outward, such that $-{\mathrm{\Delta}}_{e}b\ge 0$.

To assess who borrows more on the spectrum from rational to purely optimistic consumers, we evaluated $\mathrm{d}MU(b)/\mathrm{d}\psi $ in (4.5) and provided sufficient conditions for it to be positive in Proposition 4.1, thereby indicating that borrowing increases optimism. Putting these results together, Figure 2 illustrates the demand effects of a rule-imposed reduction in collection effort. To simplify the figure, we compared the demand curve shifts for purely rational and purely optimistic consumers.

Assume that demand is for period 0 borrowing, $b$, at interest rate $r$. Optimists’ demand is represented by ${D}^{\mathrm{O}}$, while rational consumers’ demand prior to the rule change is represented by ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$. In accordance with our results above, we have ${D}^{\mathrm{O}}>{D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$, such that optimists anticipate benefits from borrowing that are greater, on average, than are actually experienced (Train 2015; Allcott and Sunstein 2015). Consumer surplus for rational consumers is the (blue) cross-hatched area under ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$. In period 0, optimists perceive their surplus to be the area under ${D}^{\mathrm{O}}$, so they overborrow; their actual, realized surplus is the cross-hatched area that rational consumers obtain minus the sum of the (red) cross-hatched and vertically lined areas below $S$. A rule that limits the harshness of collections will shift out demand for rational consumers from ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$ to ${D}_{{e}_{\mathrm{L}}}^{\mathrm{R}}$. Demand for optimists is unaffected because the rule change affects terms that are not salient to them. In this partial equilibrium context with no supply response, consumer surplus for both types necessarily increases. For rational consumers, surplus increases to the (blue) cross-hatched area under ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$ plus the horizontally lined area under ${D}_{{e}_{\mathrm{L}}}^{\mathrm{R}}$, while optimists enjoy this surplus minus only the cross-hatched (red) triangle below $S$. In precisely this sense, the value of the rule to optimistic consumers is that it reduces the utility cost of acting optimistically.

## 5 Lender profits, equilibrium contracts and restricting collection effort

In this section we first describe lenders’ credit supply and the loan contracts they offer in equilibrium and then turn to conditions under which debt collection policies that restrict loan contracts to those with low collection intensity will increase or decrease consumer welfare.

For consumer welfare to be improved by a rule that restricts collection effort to ${e}_{\mathrm{L}}$ it must be the case that consumer welfare is highest with $\{{R}_{\mathrm{L}},{e}_{\mathrm{L}}\}$ but that, in light of loan demand from optimistic consumers, lender profits are maximized with contract terms $\{{R}_{\mathrm{H}},{e}_{\mathrm{H}}\}$. In this section we first describe the relationship between $R$ and $e$ in equilibrium, and then present conditions under which a rule that restricts $e$ to ${e}_{\mathrm{L}}$ will improve consumer welfare.

### 5.1 Credit supply

Assume lenders offer loan contracts $\{{R}_{j},{e}_{j}\}$, $j=\mathrm{L},\mathrm{H}$, in period 0, and define ${\mu}_{\mathcal{F}}\in [\mathrm{\ell}(e),1]$ as the proportionate recovery rate, ie, the expected share of each dollar lent that is repaid or otherwise recovered by lender $\mathcal{F}$. We have

${\mu}_{\mathcal{F}}$ | $\equiv {\mu}_{\mathcal{F}}(\omega (b,?,e),\mathrm{\ell}(e))$ | |||

$={\omega}_{\mathrm{f}}(b)+{\omega}_{\mathrm{s}}(b,?,e)?+{\omega}_{\mathrm{d}}(b,?,e)\mathrm{\ell}(e).$ | (5.1) |

All lenders face the same opportunity cost of lending ${\delta}_{\mathcal{F}}$ and have access to the same collections technology. The collections technology is defined by $\mathrm{\ell}(e)$, $\mathrm{\Gamma}(e;\cdot )$ and $k(e)$, where $\mathrm{\ell}(e)$ and $\mathrm{\Gamma}(e;\cdot )$ are as defined above and $k(e)$ is the cost to creditors of pursuing collections with effort $e=\{{e}_{\mathrm{L}},{e}_{\mathrm{H}}\}$. We assume that $k(e)$ is increasing such that ${\mathrm{\Delta}}_{e}k\ge 0$.

Defining ${\mathrm{\Delta}}_{\mathcal{F}}=(1+{\delta}_{\mathcal{F}})$, the creditor’s profit function is given by

$$\pi (R,e)=\frac{1}{{\mathrm{\Delta}}_{\mathcal{F}}}[(R{\mu}_{\mathcal{F}}-{\mathrm{\Delta}}_{\mathcal{F}})b-{\omega}_{\mathrm{d}}k(e)b],$$ | (5.2) |

where $(R{\mu}_{\mathcal{F}}-{\mathrm{\Delta}}_{\mathcal{F}})b$ is the return on lending and ${\omega}_{\mathrm{d}}k(e)b$ is the cost of collecting defaulted loans.

Each lender offers a loan contract or set of loan contracts $\{{R}_{j},{e}_{j}\}$ in period 0. In equilibrium, lenders offer loan contracts such that

- (i)
consumers choose loan contracts that generate perceived utility that is at least as great as the offered loan contracts they do not choose;

- (ii)
no contract not offered by a lender would produce higher profits for the lender, given demand $b$ and the loan contracts offered by other lenders; and

- (iii)
there is free entry into the lending market.

As an implication of (i)–(iii), each lender makes zero profits in equilibrium:^{14}^{14} 14 The assumption of perfect competition is generally unrealistic for consumer credit markets (see Ausubel 1991; Calem and Mester 1995). We make this simplifying assumption primarily to simplify welfare comparisons; we expect that the basic trade-offs in the model would hold true with other market structures.

$$\pi (R,e)=\frac{1}{{\mathrm{\Delta}}_{\mathcal{F}}}[(R{\mu}_{\mathcal{F}}-{\mathrm{\Delta}}_{\mathcal{F}})b-{\omega}_{\mathrm{d}}k(e)b]=0.$$ | (5.3) |

Any equilibrium contract $\{{R}^{*},{e}^{*}\}$ must satisfy (5.3). Solving for $R$ yields

$$R=\frac{{\mathrm{\Delta}}_{\mathcal{F}}+{\omega}_{\mathrm{d}}k(e)}{{\mu}_{\mathcal{F}}}.$$ | (5.4) |

This specification of the supply appears to be perfectly elastic as interest rates do not depend directly on borrowing. However, $b$ does enter ${\omega}_{\mathrm{d}}$ and ${\mu}_{\mathcal{F}}$ through its effect on the thresholds for default and settlement: higher borrowing means borrowers are more likely to default in period 1, so higher interest rates are needed to compensate for lower expected repayment. As a result, supply is upward sloping in borrowing, as indicated by

$$\frac{\partial R}{\partial b}=\frac{1}{{\mu}_{\mathcal{F}}^{2}}\left(\frac{\partial {\omega}_{\mathrm{d}}}{\partial b}k(e)-({\mathrm{\Delta}}_{\mathcal{F}}+{\omega}_{\mathrm{d}}k(e))\frac{\partial {\mu}_{\mathcal{F}}}{\partial b}\right)\ge 0,$$ | (5.5) |

given that $\partial {\omega}_{\mathrm{d}}/\partial b\ge 0$ and $\partial {\mu}_{\mathcal{F}}/\partial b\le 0$, as shown in (A9) and (A15) in the online appendix, respectively.

Interest rates that satisfy (5.4) depend on collection effort $e$ through ${\omega}_{\mathrm{d}}$, $k(e)$ and ${\mu}_{\mathcal{F}}$, so we can think of (5.4) as specifying two potential supply functions, one each for ${e}_{\mathrm{L}}$ and ${e}_{\mathrm{H}}$. For a given level of borrowing, the difference between the interest rates with low and high levels of collection intensity can be expressed as

$${\mathrm{\Delta}}_{e}R=\frac{({\mathrm{\Delta}}_{e}{\omega}_{\mathrm{d}}k({e}_{\mathrm{L}})+{\omega}_{\mathrm{d}}{\mathrm{\Delta}}_{e}k){\mu}_{\mathcal{F}}({e}_{\mathrm{L}})-({\mathrm{\Delta}}_{\mathcal{F}}+{\omega}_{\mathrm{d}}k({e}_{\mathrm{L}})){\mathrm{\Delta}}_{e}{\mu}_{\mathcal{F}}}{{\mu}_{\mathcal{F}}({e}_{\mathrm{L}}){\mu}_{\mathcal{F}}({e}_{\mathrm{H}})}.$$ | (5.6) |

Whether the difference expressed in (5.6) is positive or negative will depend on parameters. Generally, we would expect higher collections effort to be associated with lower interest rates, because higher $e$ makes default more costly, which increases recovery rates and reduces the interest rate needed for nonnegative profits. However, the assumption that ${\mathrm{\Delta}}_{e}k\ge 0$ raises the possibility that the higher costs of more intense collections outweigh the benefit to lenders of higher recoveries. The first term in the numerator of (5.6) shows this tension between the savings from a decrease in defaults in response to an increase in collection effort, ${\mathrm{\Delta}}_{e}{\omega}_{\mathrm{d}}k({e}_{\mathrm{L}})\le 0$, and the increase in collection costs from the more intensive collections, ${\omega}_{\mathrm{d}}{\mathrm{\Delta}}_{e}k(e)\ge 0$. The second term, $-({\mathrm{\Delta}}_{\mathcal{F}}+{\omega}_{\mathrm{d}}k({e}_{\mathrm{L}})){\mathrm{\Delta}}_{e}{\mu}_{\mathcal{F}}$, is negative given ${\mathrm{\Delta}}_{e}{\mu}_{\mathcal{F}}\ge 0$ as shown in (A19) in the online appendix, which reflects how higher collection effort increases the recovery from defaulting consumers, reducing zero-profit interest rates. Clearly, we can construct a sufficient condition that yields ${\mathrm{\Delta}}_{e}R\le 0$. However, we note instead that the outcome ${R}_{\mathrm{H}}>{R}_{\mathrm{L}}$ is unlikely in practice and is not of interest to us here. We therefore assume a parameterization that yields ${\mathrm{\Delta}}_{e}R\le 0$, so that interest rates move inversely to collection efforts.

### 5.2 Collection effort in equilibrium

For a loan contract to be part of an equilibrium,

- (i)
it must satisfy the supply condition in (5.2);

- (ii)
borrowing decisions must be optimal given the loan contract as expressed in (4.4); and

- (iii)
it must be that, if in equilibrium $e={e}_{\mathrm{H}}$, consumers would not prefer any contract that satisfies (4.4) and has $e={e}_{\mathrm{L}}$.

We focus on conditions that determine whether equilibrium collection effort is ${e}_{\mathrm{H}}$ as opposed to ${e}_{\mathrm{L}}$.

Borrowers will choose between $\{{R}_{\mathrm{L}},{e}_{\mathrm{L}}\}$ and $\{{R}_{\mathrm{H}},{e}_{\mathrm{H}}\}$ in period 0 based on perceived utility. Note first that if $$, then consumers will always choose $\{{R}_{\mathrm{L}},{e}_{\mathrm{L}}\}$, since regardless of optimism consumers will always strictly prefer a contract that offers both lower interest rates and lower collection effort. Given our interest in cases where collection effort is high in equilibrium, we focus on the cases in which ${R}_{\mathrm{L}}>{R}_{\mathrm{H}}$, which, as shown in (5.6), will be the case as long as $k({e}_{\mathrm{H}})-k({e}_{\mathrm{L}})$ is not too large. In general, consumers will choose $\{{R}_{\mathrm{L}},{e}_{\mathrm{L}}\}$ over $\{{R}_{\mathrm{H}},{e}_{\mathrm{H}}\}$ if

$u({c}_{0,\mathrm{L}})+{\displaystyle \frac{1-\psi p}{1+{\delta}_{\mathrm{c}}}}u({c}_{1\mathrm{f},\mathrm{L}}(0))$ | $+{\displaystyle \frac{\psi p}{1+{\delta}_{\mathrm{c}}}}[{\displaystyle {\int}_{0}^{{\stackrel{~}{\epsilon}}_{\mathrm{L}}}}u({c}_{1\mathrm{f},\mathrm{L}}(\epsilon ))\mathrm{d}\epsilon +{\displaystyle {\int}_{{\stackrel{~}{\epsilon}}_{\mathrm{L}}}^{{\ddot{\epsilon}}_{\mathrm{L}}}}[u({c}_{1\mathrm{s},\mathrm{L}}(\epsilon ))-{\gamma}_{1}]\mathrm{d}\epsilon $ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{{\ddot{\epsilon}}_{\mathrm{L}}}^{\overline{\epsilon}}}[u({c}_{1\mathrm{d},\mathrm{L}}(\epsilon ))-{\gamma}_{1}-{\gamma}_{2}({e}_{\mathrm{L}})]\mathrm{d}\epsilon ]$ | ||||

$\ge u({c}_{0,\mathrm{H}})+{\displaystyle \frac{1-\psi p}{1+{\delta}_{\mathrm{c}}}}u({c}_{1\mathrm{f},\mathrm{H}}(0))$ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{\psi p}{1+{\delta}_{\mathrm{c}}}}[{\displaystyle {\int}_{0}^{{\stackrel{~}{\epsilon}}_{\mathrm{H}}}}u({c}_{1\mathrm{f},\mathrm{H}}(\epsilon ))\mathrm{d}\epsilon +{\displaystyle {\int}_{{\stackrel{~}{\epsilon}}_{\mathrm{H}}}^{{\ddot{\epsilon}}_{\mathrm{H}}}}[u({c}_{1\mathrm{s},\mathrm{H}}(\epsilon ))-{\gamma}_{1}]\mathrm{d}\epsilon $ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{\ddot{{\epsilon}_{\mathrm{H}}}}^{\overline{\epsilon}}}[u({c}_{1\mathrm{d},\mathrm{H}}(\epsilon ))-{\gamma}_{1}-{\gamma}_{2}({e}_{\mathrm{H}})]\mathrm{d}\epsilon ].$ | (5.7) |

Proposition 5.1 says consumers that are “pure” optimists $(\psi =0)$ will always choose a contract with lower interest rates, so that the equilibrium contract is $\{{R}_{\mathrm{H}},{e}_{\mathrm{H}}\}$. For purely rational consumers, however, the choice of loan contract depends on the parameters.

###### Proposition 5.1.

For “pure” optimists $\mathrm{(}\psi \mathrm{=}\mathrm{0}\mathrm{)}$, the equilibrium contract is always $\mathrm{\{}{R}_{\mathrm{H}}\mathrm{,}{e}_{\mathrm{H}}\mathrm{\}}$. For purely rational consumers $\mathrm{(}\psi \mathrm{=}\mathrm{1}\mathrm{)}$, the contract chosen in equilibrium depends on the parameters, and in particular there are values of ${\mathrm{\Delta}}_{e}\mathit{}\mathrm{\ell}$ and ${\mathrm{\Delta}}_{e}\mathit{}{\gamma}_{\mathrm{2}}$ large enough and ${\mathrm{\Delta}}_{e}\mathit{}u\mathit{}\mathrm{(}{c}_{\mathrm{0}}\mathrm{)}$ small enough that purely rational consumers will prefer $\mathrm{\{}{R}_{\mathrm{L}}\mathrm{,}{e}_{\mathrm{L}}\mathrm{\}}$.

###### Proof.

See Appendix A.5 online. ∎

The first part of the proposition is intuitive: pure optimists do not consider $e$ when borrowing, so it is not hard to see that the equilibrium contract will be driven by interest rates alone. The result, for perfect optimists, that utility is strictly greater with $\{{R}_{\mathrm{H}},{e}_{\mathrm{H}}\}$ means that the inequality will also hold for $\psi $ sufficiently close to zero; that is, if optimism is high enough, the equilibrium will always involve $\{{R}_{\mathrm{H}},{e}_{\mathrm{H}}\}$.

Rational consumers take into account both $R$ and $e$, and moreover they take into account the effect of the contract on their borrowing. We show above that, under sufficient conditions in Proposition 4.2, ${b}_{\mathrm{H}}\le {b}_{\mathrm{L}}$. With ${R}_{\mathrm{H}}\le {R}_{\mathrm{L}}$, it follows that ${R}_{\mathrm{H}}{b}_{\mathrm{H}}\le {R}_{\mathrm{L}}{b}_{\mathrm{L}}$; therefore, ${e}_{\mathrm{H}}$ leaves consumers with higher period 1 consumption and lower borrowing costs but at the cost of lower period 0 consumption. The optimal choice will depend on the likelihood of an income shock as well as the shape of the utility function and collection parameters ${\mathrm{\Delta}}_{e}\mathrm{\ell}$ and ${\mathrm{\Delta}}_{e}{\gamma}_{2}$.

### 5.3 Welfare

This subsection focuses on consumer welfare, meaning consumers’ ex post payoffs that do not depend on optimism. The first part of Proposition 5.1 indicates that if consumers are sufficiently optimistic, then lenders will always use harsher collections practices in equilibrium, since optimism leads consumers to prioritize lower interest rates over more lenient collections practices. As the second part of Proposition 5.1 suggests, this will not always be the welfare-maximizing choice of collection intensity, which depends on how much interest rates are affected by the collection effort and the consumer’s ex post trade-off between consumption and collection costs $\mathrm{\Gamma}(e;\xi )$.

Generally speaking, whether the welfare of optimistic consumers is higher with a high or low collection effort depends on the model parameters, and we are not able to completely characterize conditions under which a policy that limits collection effort will increase or decrease welfare. We first examine a specific scenario in which it is unambiguous that contracts $\{{R}_{\mathrm{L}},{e}_{\mathrm{L}}\}$ offer consumers higher welfare, which establishes certain parameter values under which limiting collection intensity will increase welfare. We then turn to a more general discussion of conditions under which restrictions on collections would be expected to increase or decrease welfare.

The next proposition starts from the observation that collection effort can be inefficient, in the sense that the improved recovery that results from a higher collection effort can be outweighed by the greater cost of high collection efforts. If the additional cost of ${e}_{\mathrm{H}}$ is high enough relative to ${e}_{\mathrm{L}}$, the benefit to consumers in terms of lower interest rates will be smaller than the welfare cost of high collection effort.

###### Proposition 5.2.

Assume $\psi \mathrm{=}\mathrm{0}$, meaning that in equilibrium consumers will choose the contract with high collection effort $\mathrm{\{}{R}_{\mathrm{H}}\mathrm{,}{e}_{\mathrm{H}}\mathrm{\}}$. Fixing the other parameters, there is some $k\mathit{}\mathrm{(}{e}_{\mathrm{H}}\mathrm{)}\mathrm{>}k\mathit{}\mathrm{(}{e}_{\mathrm{L}}\mathrm{)}$ such that consumer welfare is greater with $\mathrm{\{}{R}_{\mathrm{L}}\mathrm{,}{e}_{\mathrm{L}}\mathrm{\}}$.

###### Proof.

See Appendix A.6 online. ∎

Turning to the general case, we specify a consumer welfare function for optimistic borrowers, ${\mathrm{CS}}^{\mathrm{O}}$, in which we focus on the gap between actual and perceived welfare. For any level of $b$ we can write

${\text{actualCS}}_{\mathrm{H}}^{0}(b)-{\text{perceivedCS}}_{\mathrm{H}}^{0}(b)$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}={\displaystyle \frac{(1-\psi )p}{1+{\delta}_{\mathrm{c}}}}[{\displaystyle {\int}_{\underset{\xaf}{\epsilon}}^{\stackrel{~}{\epsilon}}}u({c}_{1\mathrm{f}}(\epsilon ))\mathrm{d}\epsilon +{\displaystyle {\int}_{\stackrel{~}{\epsilon}}^{\ddot{\epsilon}}}[u({c}_{1\mathrm{s}}(\epsilon ))-{\gamma}_{1}]\mathrm{d}\epsilon $ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{\ddot{\epsilon}}^{\overline{\epsilon}}}[u({c}_{1\mathrm{d}}(\epsilon ))-{\gamma}_{1}-{\gamma}_{2}(e)]\mathrm{d}\epsilon -u({c}_{1\mathrm{f}}(0))],$ | (5.8) |

where ${\text{actualCS}}_{\mathrm{H}}^{0}(b)$ and ${\text{perceivedCS}}_{\mathrm{H}}^{0}(b)$ are the utilities for rational and optimistic consumers, respectively. For rational consumers, $\psi =1$, and the right-hand side equals zero. As $\psi $ decreases, the gap between perceived and actual consumer welfare increases.

We analyze this gap between actual and perceived consumer welfare algebraically in Appendix A.7 online. In doing so, we borrow the concept of marginal internalities, specified as $\mathrm{MI}({\mathrm{\Delta}}_{e};\psi )$, where $$ indicates that choice $\{{R}_{\mathrm{H}},{e}_{\mathrm{H}}\}$ is a cost consumers impose on themselves that is not in their own best interest (see, for example, Allcott and Sunstein 2015; Mullainathan et al 2012). Appendix A.7 shows that $$ becomes more likely as the utility cost of collections increases and if the interest rate response to a change in collection effort is small. To provide intuition as to the welfare effect of a policy that requires $\{{R}_{\mathrm{L}},{e}_{\mathrm{L}}\}$ we focus on the effects of ${\mathrm{\Delta}}_{e}R$, ${\mathrm{\Delta}}_{e}\mathrm{\ell}$ and ${\mathrm{\Delta}}_{e}{\gamma}_{2}$ and show that policy will be more successful if ${\mathrm{\Delta}}_{e}R$ is small and ${\mathrm{\Delta}}_{e}\mathrm{\ell}$ and ${\mathrm{\Delta}}_{e}{\gamma}_{2}$ are large.

We provide a graphical illustration in Figure 3, which shows demand for rational consumers shifting out from ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$ to ${D}_{{e}_{\mathrm{L}}}^{\mathrm{R}}$, and supply shifting up from ${S}_{{e}_{\mathrm{H}}}$ to ${S}_{{e}_{\mathrm{L}}}$, and compares this with demand from perfectly optimistic consumers ${D}^{\mathrm{O}}$. To reduce clutter, we focus on a special case where rational borrowing remains at the same level ${b}_{{e}_{\mathrm{H}},{e}_{\mathrm{L}}}^{\mathrm{R}}$ in response to the rule change; in reality, borrowing will increase or decrease depending on the relative shifts in the supply and demand curves. In Figure 3, the demand and supply shifts are similar in size and demand for optimists and rational consumers have similar slopes. We modify these shifts and slopes in Figure 4.

As discussed by Zywicki (2015), the welfare effects of the rule change are ambiguous for rational consumers. Consumer welfare prior to the rule change is represented in Figure 3 by the (blue) vertically lined triangle above ${S}_{{e}_{\mathrm{H}}}$ and below ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$; consumer welfare post rule change is denoted by the (blue) horizontally lined triangle above ${S}_{{e}_{\mathrm{L}}}$ and below ${D}_{{e}_{\mathrm{L}}}^{\mathrm{R}}$. Unlike the partial equilibrium case in Figure 2, where welfare among rational consumers necessarily increases because collection is less costly to consumers and supply does not shift, the change in consumer surplus can increase or decrease, and is assessed by the relative size of these triangles. If the post-rule-change triangle is larger, consumer welfare increases; otherwise, it decreases. If all consumers are rational, as assumed by Zywicki (2015) and by Barth et al (1986), then this completes the economic analysis. If, however, optimists participate in the market, we need to introduce the concepts of perceived and actual consumer welfare to complete the analysis.

For pure optimists, consumer surplus prior to the rule change is given by the area between ${S}_{{e}_{\mathrm{H}}}$ and ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$ minus the marginal internality, the vertically lined area bounded by ${S}_{{e}_{\mathrm{H}}}$, ${D}_{{e}_{\mathrm{H}}}^{\mathrm{R}}$ and ${b}_{{e}_{\mathrm{H}}}^{\mathrm{O}}$. After the rule change it is given by the horizontally lined area between ${S}_{{e}_{\mathrm{L}}}$ and ${D}_{{e}_{\mathrm{L}}}^{\mathrm{R}}$ minus the horizontally lined area bound by ${S}_{{e}_{\mathrm{L}}}$, ${D}_{{e}_{\mathrm{L}}}^{\mathrm{R}}$ and ${b}_{{e}_{\mathrm{L}}}^{\mathrm{O}}$. Given the slopes in Figure 4, it seems clear that pure optimists were made better off by the rule change, while the welfare of rational consumers was relatively unchanged.

As Figure 3 and the discussion in Appendix A.7 online indicate, overborrowing by pure optimists will decrease after the rule change, but welfare may not increase. It depends on the relative sizes of ${\mathrm{\Delta}}_{e}R$, ${\mathrm{\Delta}}_{e}\mathrm{\ell}$ and ${\mathrm{\Delta}}_{e}{\gamma}_{2}$. Suppose that ${\mathrm{\Delta}}_{e}R$ is large and ${\mathrm{\Delta}}_{e}\mathrm{\ell}$ and ${\mathrm{\Delta}}_{e}{\gamma}_{2}$ are small, and that attention to ${\mathrm{\Delta}}_{e}\mathrm{\ell}$ and ${\mathrm{\Delta}}_{e}{\gamma}_{2}$, risk aversion or income risk induces the demand for borrowing to be much more elastic among rational consumers. Then a rule that moves consumers to contracts $\{{R}_{\mathrm{L}},{e}_{\mathrm{L}}\}$ will induce a large interest rate increase but provide consumers with relatively little benefit from the decrease in collection effort. Figure 4 provides such an example; the welfare of rational consumers clearly declines in this case, as the post-rule, horizontally lined triangle is smaller than the prerule vertically lined one. Likewise, the welfare of optimists may decline as well; they overborrow a little less, but they pay a much higher interest rate.

## 6 Discussion: relaxing the assumptions of the model

The model described above is highly stylized. In this section we briefly discuss how outcomes might be affected if some of the assumptions were relaxed.

### 6.1 Borrower heterogeneity

The model assumes that borrowers are identical in their level of optimism and in their expected income path. One natural extension would be the case in which consumers vary in their degree of optimism. In the context of the other model assumptions, varying levels of optimism could mean different loan contracts being taken up by different types of consumers. More optimistic consumers would choose contracts with lower interest rates and higher collection effort, while less optimistic consumers, who attach greater importance to collections when borrowing, would choose contracts with higher interest rates and lower collection effort. Even if lenders cannot observe a borrower’s level of optimism, borrowers will naturally separate based on the contracts offered, since the model’s setup does not generate any reason for lenders to be concerned with adverse selection. The welfare analysis could be different in a model with varying levels of optimism, however. For example, it is possible that a rule restricting collection effort would be welfare-reducing for rational consumers, under conditions that lead rational consumers to prefer lower interest rates and greater collection effort, while the same intervention could be welfare-enhancing for optimistic consumers, because it might significantly reduce overborrowing when interest rates are low and collection effort is high.

A model in which borrowers’ optimism is correlated with default risk could lead to different results. If, for example, more optimistic consumers also have lower expected income in period 1, and consumer type is unobservable to lenders, then lenders would have an incentive to separate more- and less-optimistic consumers. This could distort markets in favor of contracts with higher interest rates and lower collection effort, which would be differentially attractive to less optimistic consumers. In a multiperiod model, in which lenders could observe consumer credit history, there would likely be additional opportunities for screening on the part of creditors that might mitigate this type of distortion.

### 6.2 Dynamics and welfare effects

In the model, any potential policy intervention takes place before any borrowing decisions are made. In reality, at the time of an intervention some consumers will already have debt outstanding and will be affected differently than those who have yet to borrow. The intervention will affect these groups differently. A restriction on collection intensity that applies to contracts already in place should be uniformly positive for existing debtors whose contract terms remain otherwise unchanged, but could be positive or negative for consumers who have yet to borrow given the resulting increase in interest rates.

### 6.3 Other credit terms

Our model assumes interest rates and collection intensity are the only features of credit contracts. In reality, lenders can alter other credit terms as a way of managing default risk, including through borrowing limits and underwriting criteria. Which quantities lenders adjust and the extent of any adjustment in response to the regulation of collection intensity would likely vary with the characteristics of their customer base. Romeo and Sandler (forthcoming) find that lenders respond to changes in state debt collection laws in ways that vary by consumer characteristics. For subprime borrowers, lenders respond to these restrictions by reducing both access to credit and the probability of zero initial annual percentage rates, while for prime borrowers, lenders reduce credit limits.

Restricting access to credit through underwriting could in principle be incorporated into the model above with little change: for example, if lenders can observe differences in the expectation or variance of consumer income growth, then lending terms would vary according to consumer type, and it might not be possible to profitably lend to consumers with income prospects that are low or more variable, leading them to be excluded from the market. Incorporating credit limits (or, more generally, interest rates that depend on the amount borrowed) would be more challenging in the context of the existing model, requiring us to reconsider how consumers choose loans and how lenders compete; however, we expect that the basic trade-offs and the intuition for the results would still hold.

## 7 Conclusions

We develop a model of consumer borrowing that incorporates the costs, both pecuniary and nonpecuniary, of debt collection effort in the case of default. The model allows consumers to be optimistic when engaging in credit transactions. Optimism means that the possibility of being unable to pay is less salient to the consumer at the time of borrowing, so that consumers do not fully consider the potential costs of debt collection until they are faced with the decision whether to repay. Whereas fully rational consumers would accept a loan contract with higher interest rates in exchange for a more lenient approach to debt collection, optimistic consumers attach less weight to debt collection practices. As a result, optimistic consumers who are offered two possible contracts – one with low interest rates and high debt collection effort, the other with high interest rates and low debt collection effort – would tend to choose the low interest rate contract, as that contract term is more salient.

In this setting, we consider a policy that restricts the intensity of debt collection effort. Such a policy will lead rational consumers to demand more borrowing at any given interest rate. Demand for credit by optimistic consumers will not be affected to the same degree, as they do not attach the same weight to collection effort when making credit decisions. A rule limiting debt collection effort can reduce the distortion caused by consumer optimism; intuitively, such a rule shifts demand by rational consumers closer to that of optimistic consumers, thereby reducing the difference between perceived and actual consumer surplus for optimists. In a partial equilibrium setting with a fixed interest rate, both types of consumers may experience welfare gains from the restriction on collection effort.

Of course, restricting the intensity of debt collection effort also reduces repayment. Creditors will respond by curtailing credit supply and increasing the interest rate. In a full equilibrium setting, we consider the resulting trade-off and describe conditions under which a restriction on collection effort can increase the welfare of optimistic consumers relative to the prerule equilibrium. In general, however, the welfare effects are ambiguous, as the reduction between perceived and actual consumer welfare on the demand side has to be weighed against a higher interest rate that results from a reduction in the supply of consumer credit.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper. The views expressed herein are entirely those of the authors and do not necessarily reflect those of the Consumer Financial Protection Bureau.

## Acknowledgements

We are grateful to Anderson Battin for excellent research assistance. We thank the editor and two anonymous referees, in addition to Brian Bucks, Paul Rothstein, Ryan Sandler and the participants in the 2018 Workshop on Credit Card Lending and Payments at the Federal Reserve Bank of Philadelphia for a number of helpful comments.

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