Efficient credit policies in a housing debt crisis.

Eberly, Janice ; Krishnamurthy, Arvind

ABSTRACT Consumption, income, and home prices fell simultaneously during the financial crisis, compounding recessionary conditions with liquidity constraints and mortgage distress. We develop a framework to guide government policy in response to crises in cases when government may intervene to support distressed mortgages. Our results emphasize three aspects of efficient mortgage modifications. First, when households are constrained in their borrowing, government resources should support household liquidity up-front. This implies modifying loans to reduce payments during the crisis rather than reducing payments over the life of the mortgage contract, such as via debt reduction. Second, while governments will not find it efficient to directly write down the debt of borrowers, in many cases it will be in the best interest of lenders to do so, because reducing debt is an effective way to reduce strategic default. Moreover, the lenders who bear the credit default risk have a direct incentive to partially write down debt and avoid a full loan loss due to default. Finally, a well-designed mortgage contract should take these considerations into account, reducing payments during recessions and reducing debt when home prices fall. We propose an automatic stabilizer mortgage contract which does both by converting mortgages into lower-rate adjustable-rate mortgages when interest rates fall during a downturn--reducing payments and lowering the present value of borrowers' debt.

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During the financial crisis and in its aftermath, those segments of the economy most exposed to the accumulation of mortgage debt have tended to fare the worst. Whether one measures the impact by industry (construction), by geography (sand states), or by household (the most indebted), the presence of greater mortgage debt has led to weaker economic outcomes (see, for example, Mian and Sufi 2009 and Dynan 2012). Moreover, research suggests that financial crises may be more severe or may be associated with slower recoveries when accompanied by a housing collapse (Reinhart and Rogoff 2009; Howard, Martin, and Wilson 2011; and International Monetary Fund 2012).

These observations lead to an apparently natural macroeconomic policy prescription: restoring stronger economic growth requires reducing accumulated mortgage debt. In this paper, we consider this proposal in an environment where debt is indeed potentially damaging to the macroeconomy and where the government and private sector have a range of possible policy interventions. We show that while debt reduction can support economic recovery, other interventions can be more efficient. We also show that whether debt reduction is financed by the government or by lenders matters for both its efficacy and its desirability. Hence, while the intuitive appeal of debt reduction is clear, its policy efficiency is not always clear, and the argument is more nuanced than the simple intuition.

Our results emphasize three aspects of efficient mortgage modifications. First, when households are borrowing-constrained, government support should provide liquidity up-front. This implies loan modifications that reduce payments during the crisis, rather than using government resources for debt reduction that reduces payments over the life of the mortgage contract. The reasoning behind this result is simple and robust. Consider choosing among a class of government support programs, all of which transfer resources to a borrower, but which may vary in the timing of transfers. Suppose the objective of the program is to increase the current consumption of the borrower. For a permanent-income household, only the present discounted value of the government transfers matters for current consumption. But for a liquidity-constrained household, for any given present discounted value of transfers, programs that front-load transfers increase consumption by strictly more. Thus, up-front payment reduction is a more efficient use of government resources than debt reduction.

Second, while governments will not find it efficient to directly write down borrower debt, in many cases it will be in the best interest of lenders to do so. Reducing debt is effective in reducing strategic default. Lenders, who bear the credit default risk, have a direct incentive to partially write down debt and avoid greater loan losses due to default. In cases where there are externalities from default that will not be internalized by the lender, government policy can be effective in providing incentives or systematic structures to lenders to write down debt. Finally, a well-designed mortgage contract should take these considerations into account ex ante, reducing payments during recessions and reducing debt when home prices fall. We propose an automatic stabilizer mortgage contract which does both by converting mortgages into lower-rate adjustable-rate mortgages when interest rates fall during a downturn--reducing payments and lowering the value of borrowers' debt.

We begin with a simple environment with homeowners, lenders, and a government. We start from the simplest case, namely one with perfect information where all households are liquidity constrained. We then layer on default, private information, heterogeneous default costs, endogenous provision of private mortgage modifications by lenders, and an equilibrium home price response.

Initially, homeowners may consider defaulting on their mortgages because they are liquidity constrained (that is, cash-flow constrained) or because their mortgages exceed the value of their homes (strategic default), or because both considerations may be present. The government has finite resources and maximizes utility in the planner's problem. We initially consider a two-period model with exogenous home prices and then allow for general equilibrium feedback. We ask, "What type of intervention is most effective, taking into account the government budget constraint and the program's effectiveness at supporting the economy?" We consider a general class of interventions that includes mortgage modifications, such as interest rate reductions, payment deferral, and term extensions, as well as mortgage refinancing and debt write-downs. We extend the model to include default with known, uncertain, and unobserved default costs, with dynamic default timing, and with lender renegotiation.

The model is abstract and simple by design, allowing us to focus on the minimum features necessary to highlight these mechanisms in the housing market. It omits many interesting and potentially relevant features of the housing market and of the economy more generally. For example, we generate a "crisis period" exogenously by specifying lower income in one period to disrupt consumption smoothing by households. We could, in principle, embed our housing model in a general equilibrium framework that would derive lower income and generate the scope for housing policy endogenously, as in the studies done by Gauti Eggertson and Paul Krugman (2010), Robert Hall (2010), Veronica Guerrieri and Guido Lorenzoni (2011), Emmanuel Farhi and Ivan Weming (2013), and others. For example, in the work by Eggertson and Krugman, the nominal values of debt and sticky prices, along with the liquidity-constrained households that we include, cause output to be determined by demand; hence there is scope for policy to improve macroeconomic outcomes when the debt constraint binds and the nominal interest rate is zero. Including our model in such a structure would also allow us to examine how housing policy feeds back into the macroeconomy from the housing market. While this would be an interesting route to pursue, our focus is on distinguishing between various types of housing market interventions, so the additional impact that may come from the macroeconomic feedback is left for further work.

Here the crisis period is defined by low income, which constrains consumption due to liquidity constraints. The household cannot borrow against future income nor against housing equity in order to smooth through the crisis. The government has a range of possible policy interventions and a limited budget; we focus on policies related to housing modifications, given the severity of the constraints and defaults experienced there. For simplicity, we begin with a case without default. The main result that comes from analyzing this case is that the need for consumption smoothing favors transfers to liquidity-constrained households during the crisis period. Optimally, such transfers will take the form of a payment deferral, granting resources to the borrower in a crisis period in return for repayment from the borrower in a noncrisis period. We then add the potential for default and show that optimal policies that concentrate transfers early in the crisis but require repayment later may lead to defaults.

These results suggest that payment deferral policies alone (which grant short-term reductions in home payments but are repaid with higher loan balances later), may generate payments that rise too quickly and generate defaults, suggesting that payment forgiveness to replace or augment payment deferrals may be optimal. That is, government resources should first be spent on payment forgiveness. Once the resource allocation is exhausted, further modifications should take the form of payment deferral. We also show that "debt overhang" concerns, that is, the possibility that debt inhibits access to private credit and reduces consumption, do not change our results. Even if loan modifications such as principal reduction reduce debt overhang, liquidity constraints can be directly and more efficiently addressed by front-loaded policy interventions, rather than through a reduction in contracted debt.

We study the borrower's incentive to "strategically" default in the crisis period. We find that in many cases borrowers will choose to service an underwater mortgage. They do so for two reasons. First, default involves deadweight costs which the borrower will try to avoid. Second, when borrowers can choose when to default--either during a crisis or later--they will value the option to delay default and instead continue to service an underwater mortgage. In this context, payment reduction will have a more beneficial effect than principal reduction in supporting consumption. We also show that payment reduction increases the incentive to delay a default and thus reduces foreclosures in a crisis.

While government resources are best spent on payment reduction, a lender may find it preferable to write down debt. Since lenders bear the credit default risk, effectively they fully write down the loan (and take back the collateral) if it defaults. Hence, renegotiating the loan, including partially writing down debt to avoid strategic default, can be in the lender's own best interest. However, lenders also tend to delay in order to preserve the option value of waiting, since the loan may "cure" without any intervention. Without liquidity constraints, lenders concerned about strategic default would optimally offer a debt reduction at the end of the period (defined as just prior to default) in order to preserve option value but avoid costly default. When there are externalities from default that will not be internalized by the lender, government policy can still play a useful role, in this case by providing lender incentives to write down debt.

Summarizing, our analysis of loan modifications produces two broad results. First, with liquidity constraints, transfers to households during the crisis period weakly dominate transfers at later dates and hence are a more effective use of government resources. These initial transfers could include temporary payment reductions, such as interest rate reductions, payment deferrals, or term extensions. This result is robust to including default, various forms of deadweight costs of default, debt overhang, and the easing of credit constraints through principal reduction. Generally, any policy that transfers resources later can be replicated by an initial transfer of resources, although the converse is not true. Second, principal reductions should be offered by lenders and not by the government. Principal reductions can reduce any deadweight costs due to strategic default. This conclusion is independent of whether or not liquidity constraints are present. Lenders have a private incentive to write down debt since they bear losses in default, so writing down debt can increase the value of the loan to lenders. With the potential for delay, however, lenders will find it privately optimal to delay debt write-downs until just prior to default.

Allowing for endogenous price determination in the housing market reinforces these results. We embed the consumption and policy choice problem in an equilibrium model of housing, with rental housing demand augmented by households defaulting on their mortgages and moving from homeownership to rental. The key result from this section is that foreclosures by liquidity-constrained households undermine home-purchase demand and hence prices more than strategic defaults do. Any default incurs the deadweight cost of default, so this (potentially large) cost is the same regardless of the cause of the default. However, liquidity-constrained households carry their constraint into the rental market, which constrains their housing demand and puts further downward pressure on home prices. Strategic defaulters, on the other hand, are not in liquidity distress by definition and hence have greater demand for housing than do the liquidity-constrained. For a policymaker concerned about foreclosure externalities and home prices, distressed foreclosures by liquidity-constrained households are more damaging.

Our results demonstrate that different types of ex post interventions in home lending solve conceptually distinct problems. For example, payment-reducing modifications, which steepen the profile of payments through payment deferrals, temporary interest rate reductions, or term extensions, address cash flow and liquidity constraints. Loan principal reductions are inefficient at addressing cash flow issues, because they backload payment reductions, but they are effective at addressing the risk of strategic defaults in the later period (though not the initial period) that lenders face.

These results on ex post modifications are suggestive of the ex ante properties of loan contracts that would ameliorate the problems that arise during a crisis with both borrowing constraints and declining home prices. Specifically, a contract should allow for lower payments when borrowing constraints bind and a reduction in loan obligations when home prices fall to reduce the incentive for strategic default. Such a contract fills the role of automatic stabilizers in the housing market by responding to economic conditions. An automatic stabilizer mortgage contract that includes a reset option--that is, that can be reset as a lower-adjustable-rate mortgage during a crisis period--is consistent with the ex ante security design problem. The cyclical movement of interest rates is key to achieving the state-contingency: if the central bank reduces rates during cyclical downturns and when home prices fall, the reset option allows mortgage borrowers to reduce their payments in a recession as well as their outstanding debt. This latter effect on debt reduction occurs because a reduction in contract interest rates via a reset option reduces the present value of the payment stream owed by the borrower. This present value of payments, rather than a contracted face amount of principal, is the critical variable that enters a strategic default decision. Finally, since it relies on a reset option that is quite similar to the standard refinancing option, such a contract is also near the space of existing contracts with pricing expertise and scale.

Various forms of home price insurance or indexing of contracts to home prices have been proposed (for example, see Mian and Sufi 2014) to address the problems posed by negative equity. These options also implement the intent to avoid strategic default in a downturn. Some contracts of this type have been implemented on a small scale, although measuring home prices at the appropriate level of aggregation and allowing for home improvements and maintenance incentives pose practical challenges. Indexing to interest rates, as suggested in the stabilizing contract, has the advantages of being observable and consistent and preserving monetary policy effectiveness. Contracts with this feature already exist and have been implemented and priced on a large scale. Moreover, effectively indexing contracts to interest rates makes them sensitive to a broader range of economic conditions than home prices alone.

The remainder of this paper is organized as follows. The next section lays out a basic two-period model in which a household takes on a mortgage to finance both housing and nonhousing consumption. We then shock the household's income in a crisis period and study the optimal form of transfer that smooths household consumption, showing that it takes the form of a mortgage payment deferral. In section II we introduce the possibility that the household may default on the mortgage at the final period because the mortgage is underwater (a strategic default). Since payment deferral increases the incidence of strategic default, the optimal mortgage modification includes more crisis-period payment reduction and less deferral. In section III, we study the case where the borrower may strategically default in the crisis period as well as the final period. We find that borrowers may delay defaulting in a crisis period because the option to delay is valuable. In this context, our earlier results concerning the merits of payment reduction over principal reduction are strengthened. In section IV, we study the lender's incentives to modify mortgages. We show that lenders, unlike the government, will find it efficient to reduce mortgage principal, but only just before the borrower defaults. In section V, we consider the question of why there were so few modifications in practice during the recession, and show that one reason may be adverse selection. With the possibility of private information, lenders will be concerned that a given modification will only attract types of borrowers that cause them to make negative profits. We show that this consideration can cause the modification market to break down. In section VI, we embed our model in a simple housing market equilibrium and show that the merits of spending government resources on payment reduction rather than principal reduction are strengthened by general equilibrium considerations. In section VII, we turn to the ex ante contract design problem and suggest that a mortgage contract that gives the borrower the right to reset the mortgage rate into a variable-rate mortgage goes some way toward implementing the optimal contract. Section VIII concludes.

I. Basic Model

Households derive utility from housing and other consumption goods according to the consumption aggregate, [C.sub.t]:

(1) [C.sub.t] [equivalent to] [([c.sup.h.sub.t].sup.[alpha]] [([c.sub.t].sup.(1-[alpha])]

where [c.sup.h.sub.t] is consumption of housing services and [c.sub.t] is consumption of non-housing goods. The household maximizes linear utility over two periods

(2) U = [C.sub.1] + [C.sub.2],

where we have set the discount factor to 1, since it plays no role in the analysis.

At date 0, that is, a date just prior to date 1, the household purchases a home and takes out a mortgage loan. At the date 0 planning date, the household expects to receive income of [bar.y] at both dates. For now, there is no uncertainty. Income is allocated to nonhousing consumption and to paying interest on a mortgage loan to finance housing consumption. A home of size [c.sup.h] costs [P.sub.0] and is worth [P.sub.2] at date 2. In the basic model, [P.sub.2] is non-stochastic. (Later we will introduce home price and income uncertainty; for now, we take these as given and known to the household.) The home price [P.sub.0] satisfies the asset pricing equation,

(3) [P.sub.0] = r[c.sup.h] + r[c.sup.h] + [P.sub.2],

where r is defined as the per-period user cost of housing. That is, if an agent purchases a home for [P.sub.0] and sells it in two periods for [P.sub.2], the net cost over the two periods is 2r[c.sup.h] (= [P.sub.0] - [P.sub.2)].

To finance the initial [P.sub.0] outlay, the household takes on a mortgage loan. A lender provides [P.sub.0] funds to purchase the house in return for interest payments of [l.sub.1] and [l.sub.2] and a principal repayment of D. For the lender to break even, repayments must cover the initial loan:

(4) [l.sub.1] + [l.sub.2] + D = [P.sub.0],

where we have set the lender's discount rate to 1, as well. Given choices of ([l.sub.1], [l.sub.2], D), nonhousing consumption is

(5) [c.sub.1] = [bar.y] - [l.sub.1] and, [c.sub.2] - [bar.y] + [P.sub.2] - D - [l.sub.2].

The household chooses ([l.sub.1], [l.sub.2], D) to maximize (2).

It is straightforward to derive the result that a consumption-smoothing household maximizes utility by setting

(6) [l.sub.1] = [l.sub.2] - [alpha][bar.y] and, D = [P.sub.2].

That is, interest payments on the housing loan are [alpha][bar.y], and the principal repayment is made by selling the home for [P.sub.2]. These choices result in consumption

(7) [c.sup.h.sub.t] = [alpha][bar.y]/r and, [c.sub.t] = (1 - [alpha])[bar.y].

Note that with Cobb-Douglas preferences, the expenditure shares on housing and nonhousing consumption are [alpha] and 1 - [alpha]. Since the effective user cost of housing, r, is constant over both periods, the household equalizes consumption over both dates. (1)

I.A. Crisis

A "crisis" occurs in the model by allowing an unanticipated negative income shock to hit this household, so that income at date 1 is instead [y.sub.1] < [bar.y], leaving income at date 2 unchanged. There are two ways the household can adjust to this shock. It can default on the mortgage, reduce housing consumption, and increase nonhousing consumption. (2) Alternatively, it can borrow from date 2, reducing future consumption and increasing current consumption. We first study the second option and assume that the household does not default on its mortgage (we will consider default in the next sections). If the household does not default, it will consume too little of nonhousing services at date 1, both because permanent income is lower and also because the household cannot smooth nonhousing consumption by borrowing against future income, so that

(8) [c.sub.1] < [c.sub.2].

That is, if the household could borrow freely at an interest rate of zero, it would increase date 1 consumption and reduce date 2 consumption.

A household with other assets, or one with equity in its home, can borrow to achieve this optimum consumption path. We instead will focus on a liquidity-constrained household. This household has no other assets, little to no equity in the home, and is unable to borrow against future income. (3) Hence this household can only adjust its nonhousing consumption beyond the liquidity constraint by defaulting on its mortgage, since it cannot borrow against future income or consume from other assets or home equity. (4) If it does not default, then date 1 consumption is constrained by the precommitted mortgage payment.

Let us suppose that a government has Z dollars that it can spend to increase household utility. The scope for government intervention arises in this setting directly because of the liquidity constraint, as in Eggertson and Krugman (2010) and Guerreri and Lorenzoni (2011), or due to other nominal rigidities, as in Farhi and Werning (2013) and others. It could also be reinforced by an aggregate demand shortfall, consumption externalities, other credit market frictions, or other considerations. We do not model these explicitly, since our focus is on the housing market (although we allow for additional considerations in the next sections). Hence, the government's budget allocation may result from its intention to ease liquidity constraints in period 1, or similarly, as a way of implementing countercyclical macroeconomic policies, since date 1 is the "crisis" period in the model.

Suppose the government chooses transfers to households ([t.sub.1], [t.sub.2)] in the first and second period, respectively, that satisfy the budget constraint (5)

(9) [t.sub.1] + [t.sub.2] = Z.

Various choices of [t.sub.1] and [t.sub.2] can be mapped into standard types of loan modifications. For example, setting [t.sub.1] > 0 and [t.sub.2] = 0 in our notation corresponds to a pure "payment reduction" loan modification, which temporarily reduces loan payments, for example through a temporary interest rate reduction. A "payment deferral" program offsets the initial payment reduction with future payment increases, setting [t.sub.1] > 0 and [t.sub.2] < 0, for example through a maturity extension or loan forbearance. A program with [t.sub.1] = [t.sub.2] > 0, so that payment reductions are equally spread over time, corresponds to a fixed-rate loan refinancing (since loan payments are lowered uniformly) and to principal reduction, that is, a reduction in the loan principal that results in reduced interest and principal payments at each date.

With these transfers, the household's budget set is now augmented by a transfer in period 1 to help overcome the liquidity constraint and a second transfer at date 2, resulting in household consumption of

(10) [c.sub.1] = [y.sub.1] - [alpha][bar.y] + [t.sub.1], [c.sub.2] = [bar.y] - [alpha][bar.y] + [t.sub.2].

Here we consider only policies related to modifying the mortgage; in the next section, we add default so that policies are more directly tied to mortgage payments. The planner maximizes household utility

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

subject to equations 9 and 10. Note that since we are considering the case where the household does not default and hence does not reoptimize housing consumption, the consumption values [c.sup.h.sub.1], [c.sup.h.sub.2] = [alpha][bar.y]/r, are invariant to the choice of [t.sub.1] and [t.sub.2].

We can rewrite the planner's problem thus:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

(13) v([c.sub.t]) [equivalent to] [([alpha][bar.y]/r).sup.[alpha]] [([c.sub.t]).sup.1-[alpha]]

and we note that v (*) is concave. This problem provides the minimal incentive to support household consumption, as it focuses only on the liquidity constraint of a single household and does not take into account aggregate demand externalities that may be present in a crisis, as emphasized by other authors.

[FIGURE 1 OMITTED]

Figure 1 illustrates the solution for nonhousing consumption for Z = 0. The vertical axis graphs [c.sub.2], while the horizontal axis graphs [c.sub.1]. The initial point A after the shock has [c.sub.2] > [c.sub.1]. The bold diagonal line traces out the set of points that satisfy the budget constraint, [t.sub.1] + [t.sub.2] = 0 (that is, Z = 0). The optimum calls for full consumption smoothing, which is to set [t.sub.1] > 0 and [t.sub.2] < 0 until [c.sub.1] = [c.sub.2] (the dotted 45-degree line) at point B.

As Z rises, the bold diagonal line shifts outward, but for any given Z we see that payment deferral ([t.sub.1] > 0, [t.sub.2] < 0) is better than payment reduction ([t.sub.1] > 0, [t.sub.2] = 0) because it allows higher transfers in the first period, which is in turn better than principal reduction ([t.sub.1] > 0, [t.sub.2] > 0), where transfers continue beyond the crisis period. This finding is consistent with general results in public finance showing that transfers into liquidity-constrained states enhance utility, since the marginal utility of consumption is high in those states. A reduction in mortgage principal does not transfer liquid assets into those states since the household is by definition liquidity-constrained and cannot borrow against its higher wealth. The increase in wealth is implemented by a stream of lower mortgage payments over the life of the loan, which is likely to extend well beyond the crisis period. Hence, gathering those benefits together into a front-loaded transfer is more effective. We highlight this result in this simplest setting because it is robust throughout as we add additional features to the model: transfers in the initial crisis period at least weakly dominate policies that transfer resources later.

We have described the solution ([t.sub.1] and [t.sub.2]) as the solution to the planning problem. However, there is nothing in our setup thus far that precludes the private sector from offering a loan modification. If private lenders could offer contracts with [t.sub.1] > 0, [t.sub.2] < 0 they would find it profitable to do so. This would correspond to loan refinancing with term extension, for example, which might be desirable to households by reducing payments immediately but also profitable for lenders over the life of the loan. Nonetheless, there are several reasons why policy may still be desirable. While we have not modeled a government's preference for countercyclical policy, private lenders might not offer the socially optimal amount of modifications if there are credit market frictions, consumption externalities, or an aggregate demand shortfall. Hence, it may be optimal for the government to offer or subsidize modifications in addition to available private sector contracts. Moreover, later we will show that with asymmetric information, the market in private contracts may collapse due to adverse selection, which provides further scope for policy intervention.

II. Optimal Decisions and Default Risk at Date 2

Without default, the best transfer policy is to reduce payments as much as possible in the crisis period in order to support consumption. Given the government's budget constraint, a policy that reduces mortgage payments in the crisis period and defers the payments until date 2 is the most cost effective; that is, for a given budget, it allows the most payment reduction during the crisis. However, in practice such loans may induce default by front-loading the benefits and back-loading the costs of the program to households. Households, especially those with underwater mortgages, may use the payment deferral and then subsequently default on the loan. In this section, we study the case where agents can reoptimize and possibly default at date 2, allowing us to examine how policy interventions at date 1 affect subsequent date 2 default. In section III, we consider the case where agents can reoptimize and default at either date 1 or date 2, so that there is a timing element in the default decision.

II.A. Stochastic Home Price, Date 2 Decisions, and Default

Suppose that at the start of date 2 before the household consumes or makes interest payments on debt, the home price [P.sub.2] changes. Agents then have the opportunity to reoptimize their consumption and borrowing choices, possibly defaulting on their mortgage loan. The home price change is unanticipated from the date 0 perspective. We analyze decisions at date 2, taking previous decisions as given.

At the start of date 2, prior to any interest payments or default decisions, a household has wealth of

(14) [bar.y] + [P.sub.2] - D + [t.sub.2].

If [P.sub.2] - D + [t.sub.2] [not equal to] 0, the household will want to rebalance consumption. For example, if P2 - D + t2 > 0, the household will want to increase housing and nonhousing consumption given that its wealth is greater than the initially expected amount of [bar.y]. We suppose that at date 2 the household can readily sell the home, repay any debts, and be left with [bar.y] + [P.sub.2] - D + [t.sub.2]. The household uses these resources to purchase (or rent) a home for one period. Given Cobb-Douglas preferences and a one-period user cost of housing of r, it is straightforward to show that utility over date 2 consumption is linear in wealth,

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [psi] is the marginal value of a dollar at date 2 and will be a constant throughout the analysis. It a household defaults on its mortgage, it loses the home, which was the collateral for the loan, and loses any equity in the home. Since the household still requires housing services, it enters the rental market to replace the lost housing services. The household also suffers a default cost, which may represent restricted access to credit markets, benefits of homeownership or neighborhoods, match-specific benefits of the home, and so on. Thus, in default the household's wealth becomes

(16) [bar.y] - [theta],

where [theta] is a deadweight cost of default. Note that the household also loses the date 2 home-related transfer of [t.sub.2]. The household utility from this wealth is ([bar.y] - [theta])[psi]. The household defaults if

(17) [bar.y] - [theta] > [bar.y] + [P.sub.2] - D + [t.sub.2],

so that wealth after defaulting exceeds wealth of continuing to service the mortgage.

Define the equity in the home ([P.sub.2] - D) plus the default cost as

(18) [phi] [equivalent to] [P.sub.2] + [theta] - D,

which represents the total cost of default to the household. Then the default condition is expressed by the inequality

(19) [phi] < -[t.sub.2],

which determines whether the household defaults on its mortgage and incurs the deadweight cost of default. Otherwise the household continues to service the mortgage.

II.B. Optimal Date 1 Loan Modification with Date 2 Default Risk

We now solve for the optimal loan modification, accounting for the possibility that some borrowers will default on their loans. Our principal conclusion is that the payment reductions and deferrals still dominate principal reductions. Moreover, since default risk increases under payment deferral, because borrowers have to pay back more in the future, government resources are best spent first providing payment relief and only then shifting to payment deferral.

Suppose that [phi], which measures the incentive to default, is a random variable that is realized at date 2. For example, realizations of [P.sub.2] may vary across homeowners, leading to different realizations of [phi]. Moreover, suppose the possibility that home prices are uncertain only becomes apparent to borrowers and lenders at date 1. That is, continue to assume that this uncertainty is unanticipated at the date 0 stage, so that the date 0 loan contract is signed under the presumption that home prices are certain.

Default risk affects the planner's decisions over ([t.sub.1], [t.sub.2)] because the planner has to account for the possibility that setting [t.sub.2] < 0 (or requiring date 2 payments for borrowers) may induce default. Denote the CDF of [phi] as F([phi]). Since borrowers default when [phi] < -[t.sub.2], for any given [t.sub.2] we have F(-[t.sub.2]) borrowers defaulting on loans. We will assume the interesting case where ([t.sub.1], [t.sub.2)] are such that it is advantageous for every liquidity-constrained borrower to take the modification contract, but a fraction F(-[t.sub.2]) strategically default on their loans in the second period.

A planner with Z dollars to spend solves thus:

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

The first line is the utility of the constrained borrowers with high default costs (that is, high [phi]) who take the modification and do not default. The second line is the utility of the constrained borrowers who will default.

The government budget constraint requires (6)

(21) Z - [t.sub.2] [1 - F (-[t.sub.2])] - [t.sub.1] = 0.

A fraction 1 - F (-[t.sub.2]) of borrowers make the repayment of -[t.sub.2]. This repayment plus the Z dollars must cover the initial payment of [t.sub.1].

Denote [mu] as the Lagrange multiplier on the budget constraint. The first-order condition with respect to [t.sub.1] gives

(22) v'([y.sub.1] - [alpha][bar.y] + [t.sub.1]) = [mu],

and with respect to [t.sub.2] gives

(23) [1 - F (-[t.sub.2])] [PSI] = [mu]{[1 - F (-[t.sub.2])] + [t.sub.2] f([t.sub.2])}.

Combining, we find

(24) v'[(l - [alpha]) [bar.y] + [t.sub.2]] [[PSI].sup.1] = 1 + [t.sub.2]f([t.sub.2])/1 - F (-[t.sub.2]).

The solution is easy to illustrate pictorially. Figure 2 graphs first and second period nonhousing consumption for various values of government transfers. The curves AB and AC in figure 2 illustrate the set of all transfers that satisfy the government's budget constraint. The key point is that this set is a "curve" for [t.sub.1] > Z. Starting from point A, where transfers are zero, along the dashed curve, as [t.sub.1] exceeds Z, -[t.sub.2] must become negative to satisfy the budget constraint. However, with negative date 2 transfers, a fraction of borrowers will default, and increasingly so as [t.sub.2] becomes more negative; this induces curvature in the government's budget set. We also graph the isoquants for the liquidity-constrained high-default-cost household. Taking only this household into account, we see that at the optimum point B, the planner sets [t.sub.1] > 0 and [t.sub.2] < 0. Accounting for the utility of the household that defaults increases [t.sub.1] further since this household places weight only on the date 1 transfer. As Z rises, the dashed (AB) curve shifts out to the AC curve, and at the tangency point C, the transfer [t.sub.1] becomes larger, while the required repayment [t.sub.2] falls. Thus the contract calls for payment reduction and payment deferral, with more reduction available as Z rises. (Later, we allow for the default cost [phi] to be unobserved to the policymaker and lender, so that adverse selection is an issue.)

[FIGURE 2 OMITTED]

The fact that the budget set becomes a curve when we allow for default underlies many of the results about the desirability of date 1 transfers. If the government promises future resource transfers to households but there is a recession or crisis today, households will want to pull those resources forward and consume more now. Liquidity constraints may bind and prevent them from doing so at all. Even if credit markets are available to do so, so that households could borrow from the future to consume today, the interest rate at which they could borrow has to allow for the possibility of default. So it is more expensive for households to rely on credit markets than to receive the equivalent payment reduction today. The curved budget line reflects the possibility of default and means that consumption bundles that could be achieved with transfers today ([t.sub.1] > 0) are not available if the government instead transfers resources in the future ([t.sub.2] > 0).

II.C. Principal Reduction with Default Risk

Above we considered the case where [t.sub.1] > 0 and [t.sub.2] < 0. In the case of principal reduction, both [t.sub.1] and [t.sub.2] are positive. In particular, since [t.sub.2] > 0, the planner transfers resources to the household and the budget constraint becomes

(25) Z - [t.sub.2] - [t.sub.1] = 0.

Suppose we solve the planning problem subject to the above budget constraint and restrict attention to solutions where [t.sub.1] and [t.sub.2] are non-negative. Figure 3 illustrates the solution. The shaded area illustrates the set of all points such that [t.sub.1] + [t.sub.2] = Z, [t.sub.1] > 0, and [t.sub.2] > 0. It is clear that the solution is a corner: set [t.sub.1] - Z and [t.sub.2] - 0 (point A in the figure). This implies that principal reduction (in which [t.sub.2] > 0) is not optimal, since the solution goes to the corner where the transfers are front-loaded, that is, for payment reduction focused in period 1. This occurs despite the fact that our problem allows for strategic default with default costs and that borrowers default less if [t.sub.2] > 0. For high enough Z, the transfer to date 1 is sufficient to ensure full consumption smoothing, and hence there is no need for further transfers.

In this setting, principal reduction is never optimal, even though default is costly and is accounted for by the planner, because the alternative of directly transferring the same resources to households in the first period raises utility more. It is optimal for the planner to use this strategy until the liquidity constraint no longer binds and consumption is completely smoothed. Until that occurs, principal reduction is suboptimal compared with payment deferral or reduction, and thereafter no policy intervention is needed to address liquidity constraints.

[FIGURE 3 OMITTED]

II.D. Principal Reduction to Alleviate Debt Overhang

The debt overhang from underwater mortgages is an additional macroeconomic consideration, since continuing to make mortgage payments prevents households from rebalancing their spending toward other forms of consumption, as emphasized by Karen Dynan (2012). Hence, in addition to reducing default, principal write-downs may also ease a debt overhang problem by easing a borrower's date 1 credit constraint. If the government would prefer to increase date 1 consumption, easing the credit constraint could be desirable. Does this change the calculus of government interventions to ease the liquidity constraint; that is, does debt overhang suggest that principal reduction is valuable over and above elimination of deadweight loss?

The answer is no. Suppose at date 1 the government offers a loan modification of [t.sub.2] > 0, [t.sub.1] - 0, to reduce principal by [t.sub.2]. (We structure the modification in this way to be clear that any increase in date 1 resources comes from easing the debt overhang and not from a direct government transfer at date 1.) Consider private lender transactions ([[tau].sub.1], [[tau].sub.2]) that at least break even for the lenders, that is, (7)

(26) -[[tau].sub.2] (1 - F(-[[tau].sub.2])) - [[tau].sub.1] = 0.

In figure 3, we represent the principal reduction of Z by moving from the zero transfer allocation to point B. The dashed curve in figure 3 represents the set of trades, ([[tau].sub.1], [[tau].sub.2]), that a private sector lender will make that allows the lender to break even. These trades allow agents to borrow against the future transfer Z in order to smooth consumption, solving the liquidity constraint problem at date 1. Again, the critical thing to note is that the borrowing constraint becomes a curve. Starting from point B, the household will trade to point C, which achieves less utility than point A. That is, the household will choose to borrow the Z back to increase date 1 consumption. However, since some borrowers default, the interest rate on the private loan will exceed 1, so that the government would do better by offering the transfer of Z at date 1, that is, a payment reduction rather than a principal reduction, to reach point A.

This is a general point: even if principal reduction is sufficiently generous to overcome individual borrowing constraints, direct payouts to borrowers are more efficient since the government avoids the default costs associated with borrowing against home equity. (8)

The key insight underlying these results is the constraint affecting date 1 consumption. Even if credit markets exist to transfer date 2 resources into date 1 consumption, default risk makes this approach more expensive than a direct date 1 transfer to households. Hence, even with default risk, we again find that transfers in the initial crisis period at least weakly dominate policies that transfer resources later. Government resources to reduce principal are better spent in engaging lenders to renegotiate mortgage loans than in engaging them to write down loans directly. (9)

III. Optimal Date 1 Decisions and Default Timing

The economic environment during a crisis is explicitly dynamic, however, so borrowers, lenders, and policymakers have to decide not only what to do, but when to do it. These considerations can be quite important, since conditions may change unpredictably over time. Therefore, we now study the case where the borrower can take action at either date 1 or date 2, and information becomes available along the way. In the last section, we restricted the borrower to default only at date 2 in order to keep our analysis simple and establish the intuition for the default decision. Timing makes the problem more interesting and adds some potentially surprising results about delay.

The problem is somewhat more complex to study, but it does not change our conclusions on the benefits of payment reduction/deferral over principal reduction. Government resources spent on principal reduction for a borrower who remains current on his mortgage still has lower consumption benefits than a payment reduction that increases the borrower's liquidity because of the liquidity constraint. Moreover, comparing equivalent payment and principal reductions, the payment reduction increases the borrower's incentive to remain current on his mortgage and thus reduces default in addition to increasing consumption. This is again due to the liquidity constraint, whereby the borrower places a high value on continuing to service a mortgage that has been modified to reduce current payments. Additionally, the analysis turns up a somewhat surprising result: borrowers who are underwater on a mortgage will typically continue to service it, because delaying the decision to default is a valuable option. Hence, borrowers need not be irrational or excessively optimistic when they continue to make payments on an underwater loan.

Suppose that at date I borrowers have information [E.sub.[phi]][equivalent to][E.sub.t=1][[phi]] (that is, their mortgage at date 1 is underwater). Given this information, we analyze the borrower's decision at date 1, accounting for how the date 1 decision affects the date 2 decision we analyzed in the previous section. If the borrower chooses not to default at date 1, then utility at date 1 is

(27) [([alpha][bar.y]/r).sup.[alpha]] [([y.sub.1] + [t.sub.1] - [alpha][bar.y]).sup.1-[alpha]].

If the household defaults at date 1, it can reoptimize its consumption plan to rebalance housing and nonhousing consumption, yielding a utility of

(28) [y.sub.1]([([alpha]/r).sup.[alpha]]) = [y.sub.1][PSI], where [PSI] [equivalent to] [([alpha]/r).sup.[alpha]][(1 - [alpha]).sup.1-[alpha]].

However, if the household defaults at date 1, it loses any value in the home as well as the option to delay default until date 2. Under default at date 1, date 2 wealth becomes [bar.y] - [theta], yielding a date 2 utility of

(29)([bar.y] - [theta]) [PSI].

With no default at date 1, utility at date 2 is

(30) {[bar.y] + E[max ([P.sub.2] + [t.sub.2] - D, - [theta])]}[PSI].

Hence, comparing values with and without a date 1 default, the default at date 1 occurs if

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

Rewriting, we obtain the condition under which default occurs at date 1 as

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

Figure 4 graphs the left- and right-hand side of (32) as a function of E[phi], which measures the degree to which a homeowner has equity ([P.sub.2] - D) (plus the default cost), or the inverse of "underwaterness." The gray curve graphs the value of the option to keep making mortgage payments and delaying default, on the right-hand side of equation 32. This value is uniformly positive, although low for low values of E[phi]. The dashed line is the benefit of defaulting, on the left-hand side of equation 32. This value is independent of E[phi]. For low values of E[phi]), the household chooses to default at date 1.

The borrower chooses to default when the benefit of defaulting (the bold line) exceeds the benefit of delay (the gray line), given his level of equity and default cost. In option terms, underwater borrowers have a call option on keeping the home, which is extinguished by default. Thus the choice to make the mortgage payment at date 1 is not just about whether the loan is underwater; it is a question of whether the cost of making this payment covers the value of the call option. When liquidity constraints are tight, the cost of making the payment is highest; this determines the height of the horizontal bold line in Figure 4. When the borrower is underwater, the value of the call option is lowest, as shown in the gray line, which rises as the household's equity in the home rises. (10) The intersection of the dashed and gray lines, at point A, determines the value of E[phi], or the degree of being underwater, that triggers default. This characterization is also consistent with the "double-trigger" model of default, as in Christopher Foote, Kristopher Gerardi, and Paul Willen (2008) and John Campbell and Joao Cocco (2014), for example: underwater, liquidity-constrained homeowners are the most likely to default.

[FIGURE 4 OMITTED]

We can now consider policy in this richer setting and revisit the planning problem of choosing ([t.sub.1], [t.sub.2]). A borrower with E[phi], to the right of point A--called the threshold E[[phi].sup.A] in figure 4--continues to pay his mortgage, allowing nonhousing consumption to adjust with the income shortfall. Note that for points just to the right of E[[phi].sup.A], the borrower is underwater on the mortgage. We can rewrite the condition for no default as

(33) E[[P.sub.2]] - D [greater than or equal to] E[[phi].sup.A] - [theta],

where we note that E[[phi].sup.A] < 0. Borrowers continue to service an underwater mortgage at date 1 both because of the deadweight cost of default, [theta] > 0, and because of the value of the option to delay a default decision, E[[phi].sup.A] < 0.

For the underwater borrower who continues to service his mortgage, the problem is the same as we have analyzed in the previous section. The optimal transfer sets [t.sub.1] > 0 and [t.sub.2] < 0 to support date 1 consumption. However, when default is a possibility, the government may choose to set transfers and intervene to prevent defaults, avoiding foreclosure externalities and further deterioration in the housing market. We examine these effects and potential equilibrium feedback in more detail in the next section, but begin by examining the effect of transfers on defaults here.

Borrowers with E[phi] considerably below E[[phi].sup.A] will default independent of any transfers. Consequently, these are cases where the transfers generate no economic benefits, so we set these cases aside. For borrowers with [phi] near but just below E[[phi].sup.A], transfers affect default incentives in interesting ways. Increasing [t.sub.1] shifts down the benefit to defaulting (dashed line) at all values of E[phi] to the thin horizontal line. Hence the trigger value falls from point A to point C; the household will be more deeply underwater before defaulting. Increasing [t.sub.2] increases the cost of defaulting, shifting up the solid gray curve to the dashed gray curve, and the trigger value falls from point A to point B. Note that this latter effect is strongest at higher values of E[phi], on the right-hand side of figure 4. However, this is the region for which default is dominated; the default option is out of the money. Hence, positive date 2 transfers move equity values most when households are least likely to default. This point can be seen clearly analytically. The derivative of the left-hand side of equation 32 with respect to [t.sub.1] is

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

The derivative of the right-hand side of equation 32 with respect to [t.sub.2] is

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

Thus a dollar increase in [t.sub.1] always decreases the benefit of defaulting at date 1 more than a dollar increase in [t.sub.2]. The difference in these effects increases as E[phi] falls, that is, as the mortgage is more underwater. Hence, the more underwater the loan is, the more effective an initial payment reduction is at avoiding default, relative to an equivalent transfer received at date 2. (11) The date I transfer supports consumption and reduces default, reinforcing our finding that date 1 transfers are more effective than flat or back-loaded transfers. Initially, this was clear with a date 1 liquidity constraint, but the same result obtains with date 2 default and now with the possibility of date 1 default and default timing on strategic default.

The option approach also illustrates the role of uncertainty, which raises the option value of waiting, or in terms of figure 4, shifts up the gray curve. The slanted straight line gives the payoff value under certainty (when [phi] is known); greater uncertainty shifts the gray curve up relative to the slanted line. Higher home price uncertainty is therefore associated with fewer defaults at date 1, as homeowners have a greater option value of waiting for home prices to rise. This illustrates the subtlety of arguments about the effect of uncertainty on the economy. Putting a floor under home prices (reducing the mass in the left tail) would reduce defaults, but reducing uncertainty, or trading off a floor with a commensurate ceiling on home prices, could increase defaults. (12)

Finally, we note that a borrower who does not experience an income shock, [y.sub.1] = [bar.y], never defaults at date 1. The left-hand side of equation 32 is zero in this case, because there is no benefit to reoptimizing date 1 consumption. Moreover, the right-hand side is strictly positive. Even in the case where c is expected to be negative, there is a positive value to waiting and exercising the option to strategically default at date 2, so that it is never optimal to default at date 1. This cleanly illustrates the intuition for strategic delay by unconstrained households.

We conclude from this analysis that payment reductions at date 1 are more effective than flat or back-loaded transfers in supporting consumption and preventing default at date I. Principal reductions at date 2 are most effective in preventing strategic default at date 2. This finding reinforces our earlier results for liquidity-constrained households. There, the binding liquidity constraint made it clear that for macroeconomic consumption purposes, date 1 transfers are the most effective use of government budget resources. Allowing for future default modified this finding: date I payments coupled with repayment at date 2 can induce default at date 2. Hence, payments should be flatter but still front-loaded. A flat or back-loaded transfer schedule is always dominated by date 1 payments until the liquidity constraint is fully relaxed. (13) With default and an option value of delay, we still obtain that policy transfers in the initial crisis period dominate policies that transfer resources later.

IV. Lender-Initiated Loan Modifications

We have shown that government resources aimed at supporting consumption and reducing default are better spent on payment reduction than on principal reduction. Because lenders directly bear the credit default risk, however, their incentives differ from the government's. Unlike the government, lenders may find it efficient to write down principal, because partially writing down principal may be cost effective compared to a default on the entire loan. We show that a lender's incentives to do so are highest when the borrower is underwater on his mortgage and the strategic default risk is therefore highest. Moreover, as with the borrower, when the lender can time a principal write-down, the lender will choose to delay doing so until the time that a borrower is about to default on the mortgage loan.

IV.A. Date 2 Principal Reduction

We first consider the lender's incentives at date 2 and then work backward to the dynamic problem at date 1. First consider a borrower whose home price exceeds the mortgage amount less the deadweight cost of default at the start of date 2. This borrower is expected to repay, and hence the lender receives the loan amount, D, plus the interest payment on the loan of [rc.sup.h]. On the other hand, a borrower with ([phi]) < 0 at the start of date 2 will be expected to default on his debt. In that case, the lender receives the home, which is worth [P.sub.2], and which the lender can rent out to receive [rc.sup.h]. Denote [V.sub.2] ([P.sub.2], D) as the value of the mortgage loan to a lender conditional on a given price [P.sub.2] and debt level D. Then,

(36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 5 OMITTED]

Figure 5 graphs [V.sub.2] (x) as a function of [P.sub.2] for two levels of debt, D and D' (where D' < D). The comparison illustrates that when [P.sub.2] < D - [theta] (that is, [phi] < 0), the lender can increase the value of its loan by reducing D to D'. (14) This occurs because [theta] is a deadweight cost of default. If the borrower defaults on his loan, the lender only collects [P.sub.2]. However, the borrower's value of keeping the loan and not defaulting is [P.sub.2] + [theta]. Thus, the lender can offer to write down the principal to D' < D and still increase the value of its loan. (15) Formally, the lender renegotiation solves

(37) [max.sub.D'<D] [V.sub.2] ([P.sub.2], D'),

with solution D' = [P.sub.2] + [theta] for [P.sub.2] < D - [theta].

We have identified a situation where principal reduction leads to better date 2 outcomes. Importantly, no government resources are required to implement the principal reduction, which is privately optimal since the lender benefits from avoiding default. As we point out later, though, the government may play an important role in encouraging and coordinating the renegotiation (for example, by standardizing the structure of modifications).

IV.B. Date 7 Principal Reduction

Now we move the lender's valuation forward to consider a possible principal reduction at date 1, where lenders (and borrowers) observe E[phi]. If E[phi] is to the left of point A on figure 4, then borrowers will immediately default. In this case, the analysis of lender incentives is the same as just discussed. The lender will choose to immediately write down principal to [P.sub.1] + [theta]. Consider next the case where E[phi] is to the right of point A, but still below zero. In this case, borrowers will not default. Interestingly, lenders will also choose not to write down principal immediately. As in the previous section, this effect can be understood in terms of an American option, though here from the lender's perspective. By waiting until t = 2 the lender can make the reduction contingent on the realization of future prices, and specifically on whether or not [P.sub.2] is below or above D - [phi]. By writing down principal, the lender extinguishes the option to write down later, and it is not optimal to exercise the option early. Formally, an early write-down has the lender maximizing:

(38) [max.sub.D'<D] E[[V.sub.2]([P.sub.2], D')|[P.sub.1]].

The lender loses value in doing so because [max.sub.D,<D] E[[V.sub.2]([P.sub.2], D')|[P.sub.1]] < [max.sub.D,<D] [V.sub.2]([P.sub.2], D')|[P.sub.1]], by Jensen's inequality. This delay effect is reinforced by any government transfer that decreases the incentive for default in figure 4, and hence reduces the incentive for a private lender renegotiation. Hence, a government write-down makes a private write-down less likely. (16)

In practice, there may be costs in waiting. It may take time to process the contractual requirements of reducing loan principal. Prices may move discretely and the borrower might default before the lender is able to implement the reduction. Such considerations may lead the lender to reduce principal preemptively, although the value of delay will always be balanced against those considerations.

While it is inefficient for the government to write down principal, there may be circumstances where the government will prefer the lender to immediately write down principal. For example, a householder may be the best match for a home he is already living in and continue to service an underwater mortgage, restricting his nonhousing consumption and depressing economic activity in the crisis period. If the lender were to write down principal at date 1, the borrower would be less constrained and thus increase his nonhousing consumption at date 1, which may have macroeconomic benefits. Our analysis shows that in order to incentivize lenders to write down principal early, the lender must receive a transfer equal to the value of the option to delay the write-down. That is, the government must "purchase" the option to delay the write-down from the lender in order to trigger an immediate write-down.

The cost of this option is [max.sub.D,<D] [V.sub.2]([P.sub.2], D')|[P.sub.1]] - [max.sub.D,<D] E[[V.sub.2]([P.sub.2], D')|[P.sub.1]] > 0. In general, the value of this option will be less than the dollar amount of any principal write-down, so incentives for private writedowns may be effective even when the government does not write down the loan itself.

The last two sections of this paper demonstrate that delay can be desirable to both borrowers and lenders, who see default as extinguishing a valuable option to wait and possibly avoid costly foreclosure. The government may still intervene if it values the externalities associated with foreclosure or constrained consumption more than private agents do, and hence would prefer to move more quickly to address inefficient servicer delays, information problems, and capacity constraints. None of these actions involves principal reductions paid for by the government. (17)

V. An Adverse Selection Explanation for Lack of Modifications

In practice, lenders were not active in doing mortgage modifications, especially during the early period in the financial crisis. Later, lenders began to offer principal reductions as part of loan modifications; this was especially true of specialty servicers. Lenders identified other considerations, including reputational effects and incentives affecting a lender's whole portfolio of loans, rather than just individual borrowers. For loans not held on balance sheet by lenders, servicer incentives and capacity may also have reinforced delay and timing discreteness.

Our theoretical findings are consistent with the empirical work of Manuel Adelino, Kris Gerardi, and Paul Willen (2013), who document the reluctance of servicers to renegotiate mortgages and emphasize the presence of uncertainty arising from the risk of re-default and the "self-curing" of mortgage delinquencies. Other authors address administrative and structural frictions to loan renegotiation and recommend legal and policy changes to reduce them; for example, Christopher Mayer, Edward Morrison, and Tomasz Piskorski (2009) and John Geanakoplous and Susan Koniak (2011). The efficacy of these proposals is outside our present scope, although the challenges faced by servicers and the administrative structure of mortgages also point to the desirability of ex ante reforms (which we discuss in section VII) as opposed to ex post renegotiations.

In this section, we demonstrate one force arising naturally in our model that causes lenders to choose not to offer modifications. One disadvantage of intervening in a crisis is that participating in a modification program may create selection problems. For example, a classic problem in lending is that the borrowers most eager to take out a loan are those least likely to pay it back. That problem can also arise in mortgage modifications, and it can cause beneficial private modifications to collapse. To show this, we consider a setting in which the market unravels due to adverse selection.

Returning to our model with unknown default costs from section II.A, suppose now that [phi] is the private information of the borrowers. In addition, suppose that 1 - [lambda] fraction of the households are liquidity constrained as described, but [lambda] fraction are unconstrained. For these unconstrained households [y.sub.1] = [bar.y], so that they do not have to cut back on consumption at date 1 and have no need to borrow from future income.

Let us focus on a modification program with [t.sub.1] > 0 and [t.sub.2] < 0, where v' ([y.sub.1] - [alpha][bar.y] + [t.sub.1]) > [PSI]. That is, the terms of this program are such that all liquidity-constrained households find it beneficial to participate in the program. On the other hand, among unconstrained households, only those with low default costs, [phi] - [t.sub.2], will take the loan. For this household, the modification, or consumption loan, is a free transfer of [t.sub.1] since the household does not intend to repay the loan. For a high-default-cost household that is unconstrained, the loan is not useful; it does not increase utility because consumption is already smooth across periods and the terms of trade in the loan imply an interest rate above one. Then, within the population of households that accept modifications, the fraction of defaulters [F.sup.A] is

(39) [F.sup.A](-[t.sub.2]) = [lambda]F(-[t.sub.2]) + (1 - [lambda])F(-[t.sub.2])/[lambda]F(-[t.sub.2]) + (1 - [lambda]) > F (-[t.sub.2]).

The break-even condition under which a lender would offer the loan requires that

(40). Z - [t.sub.2](1 - [F.sup.A](-[t.sub.2])) - [t.sub.1] = 0.

Hence, the larger the fraction of defaulters, [F.sup.A], the smaller the initial transfer to support consumption, [t.sub.1], can be, for any given [t.sub.2]. As the share of unconstrained households ([lambda]) rises, [F.sup.A](-[t.sub.2]) goes to one, and the effective interest required for a lender not to lose money goes to infinity. In other words, the unconstrained strategic defaulters drive up the cost of the modification for liquidity-constrained borrowers. At higher interest rates, the liquidity-constrained borrowers also self-select: only low-default-cost households take the loan, so the fraction of defaulters in the population goes toward one. For sufficiently high [lambda], the modification market breaks down for standard "lemons market" reasons: the only contract offered is [t.sub.1] = [t.sub.2] = 0.

We can again write a planning problem to derive the optimal ([t.sub.1], [t.sub.2]). The solution calls for [t.sub.1] > 0 and [t.sub.2] < 0. following the same logic as the previous case. As [lambda] rises, there are more strategic defaulters in the pool, and the solution requires a smaller initial transfer [t.sub.1].

Can the private market reproduce this outcome? Suppose that modifications are offered by the private sector rather than by a government and that there is competition among lenders. Consider two lenders engaged in Bertrand competition. Fix a modification contract ([t.sub.1], [t.sub.2]) such that the lenders each break even. Now suppose that one of the lenders offers a contract

[[??].sub.1] = [t.sub.1] - [[epsilon].sub.1], with [[??].sub.2] = [t.sub.2]/[t.sub.1] [[??].sub.1] + [[epsilon].sub.2] for positive and small [[epsilon].sub.1], [[epsilon].sub.2].

The second contract involves a smaller date 1 loan, but also a smaller interest rate on the loan. The contract is not attractive to unconstrained borrowers because they will not repay, and hence care only about the size of the modification and not the effective interest rate. But we can always choose [[epsilon].sub.1] and [[epsilon].sub.2] such that the liquidity-constrained borrowers prefer the second contract over the first contract. That is, the interest-rate savings, [[epsilon].sub.2], can be chosen to be large enough to compensate for the reduction in loan size, [[epsilon].sub.1], to make this contract preferred by liquidity-constrained borrowers. In this case, the second contract is a profitable deviation by a lender. But as a result, the initial lender loses money, since this lender is left with a population of unconstrained strategic defaulters; he will therefore lose [t.sub.1]. The first lender will then have to match the second lender and reduce [t.sub.1], but this offer will also be undercut. Equilibrium can unravel in the sense elaborated on by Michael Rothschild and Joseph Stiglitz (1976).

This logic provides two insights. First, it offers one reason why modifications were not offered more widely. Competition and the fear of receiving an adverse pool of borrowers likely limited lender modifications. Only in clear cases where the lender could exclude likely strategic defaulters through screens and filters could a modification proceed. (18) Second, it offers a rationale for a standard government-supported modification contract. That is, if the government supported and subsidized a standardized contract for all modifications, then the unraveling problem disappears.