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# Interest-Rate-Rules-and-Mortgage-Default-October-28--2012-Department-of-Economics--Boston-

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Interest Rate Rules and Mortgage Default ▯ Isaiah Hull October 28, 2012 Abstract This paper examines how a central bank’s choice of interest rate rule impacts the rate of mortgage default and welfare. I do this by constructing a quantitative equilib- rium (QE) model that incorporates incomplete markets, aggregate uncertainty, overlap- ping generations, and realistic mortgage structure. Through a series of counterfactual simulations, I demonstrate ▯ve things: 1) nominal interest rate rules that exhibit cycli- cal behavior increase the average default rate and lower average welfare; 2) welfare can be substantially improved by adopting a modi▯ed Taylor rule that stabilizes house prices; 3) a decrease in the length of the interest rate cycle will tend to increase the average default rate; 4) if the business and housing cycles are not aligned, then aggres- sive in ation targeting will tend to increase the mortgage default rate; and 5) placing a legal cap on loan-to-value ratios will lower the average default rate and lessen the intensity of extreme events. In addition to these ▯ndings, my model also incorporates an important mechanism for default, which had not previously been included in the QE literature: default spikes happen when income falls and home equity is degraded at the same time. Overall, my results suggest that the univariate time series properties of interest rates (i.e. wavelength, persistence, and variance) may play a substantial role in generating mass mortgage-default events. If a central bank wishes to avoid such crises, they should either adopt a rule that generates interest rates with slow-moving cycles or use a modi▯ed Taylor rule that also targets house price growth. JEL Classi▯cation: E50, E52, C63, C68 Keywords: housing, monetary policy, Taylor rule, mortgage default, aggregate un- certainty, incomplete markets ▯ Department of Economics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467. Email: hulli@bc.edu. Website: www2.bc.edu/isaiah-hull. I am especially grateful to Peter Ireland, Georg Strasser, Robert Murphy, and Andrew Beauchamp for their valuable comments and support. 1 1 Introduction In July of 2000, the Federal Reserve initiated a series of rate cuts that lowered the e▯ective federal funds rate (FFR) from a peak of 6.54% to 1% in late 2004, dropping it below 2% by the end of 2001. During this same period, house price growth jumped from an already high rate of 9.6% to 14.6.%; and grew at an average of 6.8 percentage points per year faster than it did from 1988 to 2000. In late 2004 to mid-2007, interest rates shot up once again to 5.25%; and, with a lag, house price growth peaked at 15.9% and then dropped to -4.6% by mid-2007, falling further as the recession deepened. Not long after the house price drop, the default rate on ▯rst mortgages rose from a historically stable 1% to a peak of 5.39% in 2009. After a casual glance at the data, we might conclude that the FFR cycle that spanned late 2000 to mid-2007 can safely be blamed for Great Recession{that is, it caused the twin housing and ▯nancial crises, bridged by a mortgage default spike. Indeed, when John Taylor addressed the Jackson Hole conference in 2007, that is 1 exactly what he argued. In this paper, I will not attempt to ▯nd the cause of the \Great Recession"; however, I will try to elucidate how monetary policy can play a role in preventing such crises. In particular, I ask: how does the central bank’s choice of interest rate rule impact the frequency and intensity of mortgage default? To give the reader a sketch of the mechanisms that connect mortgage default crises and monetary policy, consider what happens during a typical interest rate cycle. First, assume that the real, risk-free interest rate falls sharply and remains low; and that these changes are at least weakly transmitted to mortgage interest rates. This will increase demand for housing by reducing the cost to borrow. If the increase in demand also pushes up house prices, then household equity positions will improve, allowing homeowners to borrow more (i.e. withdraw additional equity), which could increase the amount of debt that homeowners hold{and possibly increase household leverage. Finally, a drop in the risk free rate could divert marginal bank depositors to instead invest in capital, increasing output (income). Now, assume that the ▯rst part of this cycle is followed by a sharp increase in interest rates{just as the prolonged period of low interest rates from 2001 to 2004 was followed by a sharp rise in the FFR. Suddenly, the e▯ects of the housing boom will be reversed. Demand will collapse, pushing house prices down, and leaving households who extracted equity or bought homes near the trough of the cycle with negative equity. This will prevent households from withdrawing equity to smooth consumption or to re▯nance into lower interest rate loans should they become available. High interest rates will also divert investment away from capital, lowering income, and will push up adjustable-rate mortgage payments with a lag. The combination of low incomes, high mortgage payments, and negative equity positions will push up default rates, which will deteriorate ▯nancial intermediary balance sheets, and may cause a credit crunch. Prior to the Great Recession, few DSGE models contained many of the aforemen- 1 See Figure 1 in the appendix for a comparison of the Taylor Rule-implied interest rate and the e▯ective federal funds rate leading up to the crisis. tioned elements. Additionally, economists had only started to consider how to model multi-period mortgage structure realistically in general equilibrium. And many of the macro housing models that did exist lacked heterogeneity, eliminating the e▯ect of a prolonged house price rise completely (Jeske 2005). In this paper, I attempt to contribute to the now-vibrant housing literature that has drawn inspiration from Aiygari (1993), and Krusell and Smith (1998). In particu- lar, I construct a quantitative equilibrium model that incorporates incomplete markets, aggregate uncertainty, overlapping generations, price stickiness, credit-scoring, optimal default, inter-sectoral productivity correlation, and realistic mortgage structure. I then calibrate the model to match the cross-sectional and time series dimensions in the data; and run a series of counterfactual simulations under di▯erent interest rate rules to de- termine how they impact default rates and welfare. In addition to considering popular classes of interest rate rules (e.g. the Taylor Rule), I also look at more fundamental components of interest rate behavior by testing rules that generate cycles, but do not endogenously respond to other macrovariables, such as autoregressive rules and sine wave rules. 2 Related Literature This paper contributes to three subliteratures. The ▯rst consists of papers that attempt to explain the Great Recession and ▯nancial crises in general. The second consists of macro papers that have attempted to integrate housing. And the ▯nal looks at the optimality of the Taylor rule and other monetary policy rules. I will review each of these literatures in order below. 2.1 Great Recession Causes Literature Mian and Su▯ (2010) ▯nd that household leverage yields considerable predictive power for the 2007-09 recession. They show that the household debt-to-disposable income ratio accurately predicts movements in aggregate variables, such as unemployment, consumer default, and house prices. They suggest that measures of household leverage could provide a well-grounded empirical basis for explaining macroeconomic uctua- tions. This ▯nding is consistent with my theme, which explores interest rate rules as the cause of household balance sheet deteriorations, which lead to increased default rates and related macroeconomic uctuations. In a separate paper, Mian and Su▯ (2009) consider the importance of the subprime loans and their securitization in the ▯nancial crisis. They perform their analysis at the county-level; and ▯nd three important results. First, they determine that counties that experienced aggressive growth in subprime lending tended to also experience the greatest default intensities. Second, they ▯nd that the credit expansion in 2002-2005 was negatively correlated with income growth{an unexpected reversal of normal credit expansion patterns. And third, they demonstrate that subprime loans tend to be 2See Chambers, Garriga, and Schlagenhauf (2009) for a multi-period treatment of mortgages. 2 more common in areas with decreasing income. While my paper does not explicitly incorporate subprime borrowers, it does provide substantial heterogeneity and permits a changing, endogenous asset distribution (aggregate uncertainty), which allows us to consider whether borrower quality declines prior to crises. Other plausible explanations for the Great Recession include Mayer (2009), who attempts to explain the crisis through a decline in underwriting standards and a large post-teaser rate jump on mortgage interest rates. Bucks and Pence (2008) argue that the large cluster of defaults was caused by a change in the composition of borrowers. That is, less ▯nancially literate individuals were induced to become homeowners. When they were hit with negative income shocks, they were less able to optimize budgeting correctly (often ignoring shocks entirely), which lead to a sharp rise in defaults. In line with the theme of this paper, Leamer (2007) argues that recessions in the U.S. have largely been driven (or preceded) by downturns in housing investment and consumer durables. He claims that monetary policy should explicitly incorporate hous- ing investment smoothing as an objective; and suggests that a modi▯ed Taylor Rule could incorporate such an objective. Similarly, Ahearne et. al (2005) draws the con- nection between monetary policy and house prices. In a study of 18 major industrial countries, they ▯nd that monetary easing typically precedes an increase in housing investment and prices. Hott and Jokipii (2012) further cement the relationship between interest rate move- ments and housing investment. Using a multi-country dataset, they show that devia- tions in interest rates from the Taylor Rule account for 50% of the housing overvaluation that occurred prior to the Great Recession. Assesnmacher-Wesche and Gerlach (2008) ▯nd a similar relationship between interest rates and house prices using a 17-country VAR that spans 1986-2006. They ▯nd that a 25 basis point increase in short term rates pushes down GDP by 0.125% and housing prices by 0.375% with a lag. However, in contrast to Hott and Jokipii, they argue against using interest rates to smooth house prices. Finally, Foote, Geraradi, Goette, and Willen (2008) provide an additional empirical nuance to to the debate. They show that borrowers do not simply default if they have negative equity or if they receive a negative income shock, but rather, they default if both conditions are present. They call this the \double trigger" condition for mortgage default{a term that originated in ▯nance. 3 To this author’s knowledge, this paper is the only rigorously-microfounded model of default in general equilibrium that replicates this empirical regularity. 2.2 Macro-Housing Literature This paper also contributes to the macro-housing literature, which has become increas- ingly focused on heterogeneity, incomplete markets, and aggregate uncertainty. Early papers in the literature, such as Yang (2006), demonstrated the importance of lifecycle elements in housing; and devised mechanisms for modeling them correctly. Li and Yao 3 See Abraham 1993. 3 (2007) showed that the impact of house price changes depend on the degree of hetero- geneity in the economy. Indeed, depending on how agents are modeled, price changes may have no e▯ect on the macroeconomy, but can cause inter-group wealth transfers. Ortalo-Magne and Rady (2005) contribute one of the foundational papers in this lit- erature. They incorporate life-cycle elements, property ladders, and credit constraints into an equilibrium model with many heterogenous agents. While they greatly sim- plify the life-cycle elements (by including only four periods) and build a highly stylized model with no aggregate uncertainty, they still introduce a housing market prototype that is reused elsewhere in the literature. In particular, they assume that property comes in two varieties: \ ats" and \houses"; and explicitly model the utility-based di▯erences between the two. As for credit constraints, they assume that wealth cannot fall below some fraction of the value of the property. Finally, they assume that the supply of both ats and housing is ▯xed. Chatterjee, Corbae, and Rios-Rull (2009) expand the literature further by incorpo- rating default and credit scoring into a large scale macro context. They model agents who take out loans and repay them with an unknown probability. This leads to a system of credit-scoring, where lenders use a known credit-scoring function and pricing kernel to create loan terms. While this model provides a reasonable structure for in- corporating default into a model with heterogenous agents, it can stay little about the timing and cause of default, since default probabilities are determined by an agent’s type (good or bad). In contrast, I allow agents to default optimally and use decision rules to compute the probability of default. Other papers in this literature use the new class of rigorously-microfounded macro- housing models to run policy simulations. Jeske & Krueger (2005), for instance, ▯nd that mortgage interest rate subsidies tend to bene▯t and to increase homeownership rates among high-income and high-net worth individuals in general. Another important part of modeling default choices is determining the structure of mortgages, since this may play a large role in determining default behavior. While there are a number of continuous time ▯nance models that permit variation in pay- ment schedules and default timing in a partial equilibrium context, this is not true in general for DSGE models, which frequently do not even model housing. In the wake of the Great Recession, however, a number of authors have attempted to create a serious role for mortgage structure in DSGE models. Most notably, Chambers, Garriga, and Schlagenhauf (2009) have constructed a general equilibrium model that permits indi- viduals to choose between FRMs and ARMs{and even to obtain combo or piggyback loans. While I do not incorporate mortgage choice into my model, I do borrow model- ing devices from the paper, and work to make them tractable in an environment with default. Perhaps most closely related to this paper, Iacoviello and Pavan (2010) create a quantitative equilibrium model that accurately reproduces empirical co-movements in aggregate debt accumulation and housing investment. In their paper, they introduce (and borrow) several computational macro modeling devices that permit them to per- form simulations in a realistic environment, but without making the computational 4 stage intractable. In particular, they use deterministic life-cycle productivity pro▯les to generate income heterogeneity, small pension payments after retirement to gener- ate lifecycle asset accumulation motives, lump-sum taxes to simplify interactions with government, housing transaction costs that are proportional the to change in housing position (to make changes larger and infrequent), and a simple no-arbitrage condition to determine the price of housing. My paper borrows many of the modeling devices from Iacoviello and Pavan (2010), and adds optimal default, individual credit-scoring, inter-sectoral productivity correlation, and mortgage structure{and applies them to a di▯erent research question. Goodhart, Osario, and Tsomocos (2009) attempt to advance the literature by cre- ating a template for a new generation of macro-▯nancial models. While they use only a handful of representative agents (unlike the other referenced housing papers), they model both default and complex linkages between the ▯nancial sector and the macroe- conomy. They include corporate lending, interbank lending, and central bank lending into a model where most agents can \default." However, since only a handful of rep- resentative heterogenous agents are used, individual agents do not default in a strict sense{rather, they choose a repayment rate. Banks respond to this by using rational ex- pectations to predict repayment rates and to penalize default. Additionally, interbank and central bank lending is used to rescue banks in the event of mass defaults. 2.3 Optimal Monetary Policy Finally, this paper contributes to the part of the optimal monetary policy literature that examines the welfare properties of interest rate rules. Woodford (2001), a land- mark paper in the subliterature, considers whether the Taylor rule can be rationalized through a plausible central bank objective. He does this by constructing a simple model and testing its welfare properties. He ▯nds that Taylor-style rules perform well, but su▯er from two problems: 1) they rely on the output gap being measured correctly; and 2) they do not vary with the Wicksellian natural rate of interest, but instead assume a ▯xed natural interest rate. He suggests that future work should attempt to \analyze the consequences of inertial rules in the context of more detailed models." This is one of the primary objectives of this paper. Julliard et. al (2006) test the welfare properties of the Taylor rule by constructing a DSGE model, using Bayesian methods to estimate its parameters, and performing counterfactual simulations under di▯erent parameter values. They ▯nd that the stan- dard Taylor rule performs reasonably well. Ahrend (2010), on the other hand, performs an empirical investigation of the Taylor rule in practice; and ▯nds that central bank departures from the rule can lead to substantial increases in asset prices. This paper explores both of these topics in a theoretical context. Other papers, such as Giannoni (2012) ▯nd that simple Taylor-style rules may be inferior to Wicksellian rules, which permit some type of history-dependence. Giannoni (2012) claims that such rules are less prone to indeterminacy and are more robust to model mispeci▯cation; and argues that they are especially e▯ective when coupled with 5 a \high degree of interest rate inertia." Forlati and Lambertini (2011) also consider interest rate inertia, but do it in the context of richer model that includes housing. They ▯nd that inertial interest rate rules lead to larger contractions in output. In summary{I expand on the macro-housing literature by adding optimal default, credit scoring, inter-sectoral productivity correlation, and mortgage structure to a quantitative equilibrium model. I add to the largely empirical Great Recession and ▯nancial crisis literature by performing counterfactual simulations using a theoreti- cally consistent framework and a rigorously-microfounded model. And, ▯nally, I add to the optimal monetary policy literature by evaluating a variety of di▯erent interest rate rules to determine their impact on default and welfare in a detailed model. 3 The Model In order to answer the questions above, I start by constructing a macro-housing model in which agents default optimally. That is, default is not forced, arbitrary, or deter- mined by type, but emerges from optimization. In building this model, my primary goal is to attack the problem as simply as possible, but with enough detail to capture the important features of default and housing choices. For this reason, I construct a rigorously-microfounded model in the style of Aiyagari (1993), and Krusell and Smith (1998). The initial formulation will focus on a real model with capital and endogenously- determined house prices. However, I will later modify the model by removing capital and the endogenous component of house prices. I will also explain how sticky and exible prices are added to the model. In the model, there are in▯nitely many heterogenous households of measure 1, who consume non-durables, invest in capital, purchase housing, and choose whether or not to default on mortgage debt. Firms produce a non-durable good with Cobb-Douglas technology. There is a representative ▯nancial intermediary, which accepts deposits from households, and then uses those deposits to issue mortgages to households. The government collects lump-sum taxes, pays pension bene▯ts, insures deposits at ▯nancial intermediaries, and collects housing from deceased agents. Finally, a central bank implicitly determines the risk-free return on deposits by setting the interbank lending rate. 3.1 Firms The ▯rm side of the economy consists of 1) a consumption good producer who rents capital and labor services; and 2) a technology that permits all households to transform the consumption good into housing units. 3.1.1 Consumption Goods The consumption goods are produced using Cobb-Douglas technology: 6 At ▯ 1▯▯ Yt= e K t t ; (1) c where A t c +A▯A t▯1+ ▯At ▯ A ▯ N(0,▯ ▯, and ▯ A = 1▯▯. Firms maximize pro▯ts, yielding the familiar ▯rst order conditions: A K t ▯ wt= (1 ▯ ▯)e t( ) (2) N t N t R t ▯e At( )1▯▯ ▯ ▯K; (3) K t Note that capital is assumed to depreciate at a constant rate, ▯ . Furthermore, N K t is the mass of employed workers. The mass (or fraction) of workers employed in any given period is derived from the conditional Markov process for employment and the assumption that households supply labor inelastically. For simplicity, I have assumed that this process does not depend on housing investment. For computational purposes, I discretize t using the Rouwenhorst (1995) method,4 which Kopecky and Suen (2009) argue is more accurate than other approximation algorithms for highly persistent AR(1) processes. In particular, this method allows me to generate a discrete approximation of an AR(1) process by setting four parameters and the desired number of states: ▯A, qA, ▯A, and ▯A, where ▯ A is the probability of the highest state,Aq is the probability of the lowest stateA is the desired standard deviation of the process, and ▯A is the desired mean of the process. The algorithm generates the Markov chain and the associated transition probability matrix. To pin down the mass of employed workers, I use a conditional Markov process that depends on the technology shock. That is, roughly speaking, when the technology shock is high, the probability of transitioning into employment will be high (and vice versa). More speci▯cally, I will calibrate the conditional Markov process to generate unemployment statistics that match U.S. business cycle data. 0 I use Pr(▯Ej▯E;▯A) to denote the probability of transitioning to employment state 0 E , given last period’s employment state,E▯ , and this period’s technology shock state, ▯ . Note that the Markov chain will remain the same in all periods; however, the A transition matrix will change, depending on the state of the technology shock, which means that we will have a di▯erent transition matrix for each state oA ▯ . 3.1.2 Housing Investment The housing investment speci▯cation is similar to Glover, Heathcote, Krueger, and Rios-Rull (2011). In particular, I assume that households have access to a linear tech- nology that transforms the consumption good into housing. That is, if the household 4 See Appendix 1 for an explanation of the Rouwenhorst method. 7 builds with c itunits of the consumption good, it will yield h itnew units of housing: h = ▯(IH )c e ; U t (4) it t it where U =tu +▯H+▯ , HhereH▯ H ▯ N(0;▯ )H IH istaggregate housing investment, ▯(IH ) is an analogy to capacity utilization, and ▯ (IH ) < 0. t t Notice that the housing speci▯cation implies a relative price limit. No one will pay 1 more than ▯(IH )et for a unit of housing, since it is possible to generate one using t that many units of the numeraire good. Additionally, I assume that investment is reversible, so no one will sell housing for less than that price. This implies that house prices will rise when the economy is hit by a negative housing sector productivity shock or when housing investment demand increases. Note that total housing investment can be written as follows: X h Ut h U t IH t ▯(IH tc itei = ▯(IH )t et ; (5) i2I where ▯ is agent i’s mass and where IH and C are used to denote aggregates. Ad- i t t ditionally, the housing stock evolves as follows: H t+1 = H t IH ▯ t H ;H t (6) where ▯ is housing stock depreciation. H 3.2 Households Households are born at a=1 and work until a=T. After retirement, households receive R R a pension, x ,tfor T periods, and then perish with certainty at age T+T . Note that this generates a hump-shaped paper asset pro▯le for households, since they must accumulate assets in order to smooth post-retirement consumption. At any point in time, heterogeneity across households is driven by two mechanisms: 1) employment status (▯ =it or 0); and 2) age-speci▯c productivity, ▯ . Fallowing Heer and Maussner (2008), I assume that the former is generated by a conditional Markov process that depends on non-durables shocks (as given in the ▯rms section). For the latter, I follow Iacoviello and Pavan (2010) in adopting a single, deterministic pro▯le for age-speci▯c productivity, which is computed using CPS data. These two mechanisms for heterogeneity will drive di▯erences in asset holdings and default decisions. U Unemployed agents receive per period unemployment bene▯ts, x , for thetdura- 5That is, when housing investment is high, the capacity of the sector to produce an additional unit is increasingly strained, making it more costly. 6 While this is not a particularly realistic assumption, it should not have a qualitative impact on welfare and default results, since house prices in the model fall when investment is reversed. 8 7 tion of their jobless spells. Similarly, retired agents will receive pension bene▯ts of x . Employed household i, which is age a at time t receives a wage, w ▯ , where PtT t a a=1 ▯ a a 1, where ▯ is ahe density of age cohort a. In the baseline model, households consume non-durable goods, c , and housitg service ows, which are assumed to be directly proportional to their housing stock, hit8 They choose how much to save in bank deposits, d , how mith capital to hold, kit how much collateralized debt to borrow in the form of mortgages, b , and itether or not to default on the mortgage they hold. Investment in housing is lumpy; that is, households tend to make large and in- frequent changes in housing size (i.e. by moving), rather than changing housing size frequently and in small increments. Here, I follow the standard assumption the lit- erature that lumpiness is generated by housing stock adjustment costs, ▯(h ;h ), it it▯1 which depend on the size of the new and old housing stock. Furthermore, I adopt Iacoviello and Pavan’s (2010) assumption that houses have a minimum size, h. While they principally use this assumption to match the empirical fact that younger households tend to be renters (and that, in fact, households cannot purchase very small houses), I use this device largely to generate household leverage. When young households enter the model, they must take out a large mortgage in order to purchase a house. The high degree of leverage will translate into interest rate risk exposure{and, thus, a higher probability of default. Additionally, I assume that all households have access to a small, ▯xed amount of non-housing shelter. This includes both defaulters and young households who have not yet purchased a home. This non-housing shelter can be interpreted as living with friends or relatives{or staying in a low-quality, but free apartment. 9 For simplicity, households are assumed to supply labor inelastically. Additionally, we may write the household’s instantaneous utility from non-durables consumption and housing service ows as follows: 1▯▯c 1▯▯ cit (hit h u(c it )it + (1 ▯ ) (7) 1 ▯ ▯ c 1 ▯ ▯ h This speci▯cation is compatible with Jeske’s (2005) ▯nding that the ratio of housing to consumption tends to be hump-shaped over the life-cycle. Following Chambers, Gar- riga, and Schlafenhauf (2009), we set the curvature parameters for the utility function to ▯c= 1 (log utility) and ▯ h = 3, which will give us a hump-shaped pro▯le for h over c the lifecycle that matches the data. Individuals have two sources of income: wages, net of taxes (▯ a;t, from working at the 7In the baseline speci▯cation, unemployed agents will receive bene▯ts and will not be taxed. In the original set of simulations, I considered a version of the model without unemployment bene▯ts. My ▯ndings did not di▯er qualitatively for the two versions of the model. 8The model is later extended to incorporate apartment rentals. 9 For a discussion of the impact that non-housing shelter has on the model, see the appendix. 9 consumption goods ▯rm and pensions (or unemployment bene▯ts): ( wtn a ▯ a;t if employed yit x if unemployed or retired (8) t Household i faces the following budget constraint: cit ▯(h ;it it▯1) + dit p ht+it + k ity +it1 + it t▯1)dit▯1 + pt (9) hit▯1(1 ▯ ▯H) + (1 + R tk it▯1+ b it h Note that p t is the relative price of housing, m it is the mortgage payment, b itis the unpaid balance on the mortgage, R is the return to capital, and k denotes i’s t it capital holdings. I depart from the current literature in applying a novel constraint that makes hold- ers of one-period mortgages behave as if they held long term debt instruments: 8 < ▯pth it if hit h it▯1> 0 H h bit▯ ▯pth it if bit▯1< ▯p t it& h ith it▯1 (10) : bit▯1 otherwise, where ▯ 2 (0;1) denotes the collateral constraint{that is, the maximum loan-to-value ratio. According to the equation above, if homeowners move (adjust their housing stock), then they face a collateral constraint in that period, since they must obtain a new mortgage. Similarly, if they do not move, but carry forward less debt from the previ- ous period than would be allowed by the collateral constraint in this period, then they have the option to borrow up to the constraint. Finally, if they did not move and have exceeded the collateral constraint, then they cannot borrow more, but do not have to reduce the size of their mortgage. The intent of these constraints is to achieve the following with one-period mortgages: 1. Avoid \forcing" default. In many endogenous default models, default is ultimately forced by a collateral constraint that is repeatedly applied to one-period mortgages. Empirically, this constraint only applies at origination{and not to existing loans. If it is forcefully applied to existing loans (i.e. by modeling them as repeated one-period loans), households will default whenever they are unable to borrow enough to repay last period’s debt (i.e. if house prices fall steeply). This generates a spurious channel for default (i.e. negative equity immediately triggers default), which is simply an arti- fact of one-period loan ▯nancing. 10 2. Allow mortgage equity withdrawal (MEW). Since we are interested in understand- ing how the path of interest rates impacts default, allowing for MEW may provide a critical channel for default. If households can borrow against the values of their homes, this may cause certain interest rate rules to generate a higher default rate.10 3. Allow negative equity. In one-period loan models with collateral constraints, house- holds typically cannot have negative equity, since it will violate the collateral constraint. However, in reality, if the price of housing falls, but individuals choose to remain in their homes, there is no constraint that forces them to maintain positive equity. Since negative equity is an important part of most default crises, this constraint is main- tained to generate more realistic household balance sheets. Overall, this constraint will make it possible to maintain the simplicity of a one-period loan framework, while simultaneously generating household balance sheets and default behavior that more closely approximate what we would observe if 30-period debt in- struments were available. In addition to this, I borrow a constraint from Iacoviello and Pavan (2010) that limits borrowing to a fraction, , of discounted, remaining lifetime earnings: TXa+j bit= E t ▯T▯a+t yij (11) j=t The purpose behind this constraint is to impose a feasibility condition on repay- ment. If households cannot reasonably be expected to repay a mortgage with their remaining income ows, then a ▯nancial intermediary will not be willing to originate it. Empirically, this is similar to the income and debt-servicing ratios that banks re- quire borrowers to meet; however, it is tied to expected, discounted future income, rather than just current-period income and assets. 11 This constraint, coupled with the previous one, yields the ▯nal borrowing constraint: b ▯ minfb ;b g I (12) it it it That is, the maximum amount a household can borrow is the minimum implied by the two borrowing constraints. Furthermore, mortgage interest rates are adjustable and are given as follows: ▯ r ▯ r = t + ▯t (13) it 1 ▯ qit 10Certain classes of interest rate rules may make it favorable to borrow against the value of your home immediately prior to a hike in interest rates, which will lead to a high degree of interest rate risk. 11An alternative{and arguably more realistic{speci▯cation of this constraint might incorporate current asset holdings. The impact of including this modi▯cation would be to permit more borrowing later in the life cycle, since there is no within-cohort heterogeneity. For the purposes of this paper, I do not consider this constraint explicitly, but may add it as a future extension. 11 Note that r is the rate earned on deposits, q is household i’s probability of default t it (computed from household decision rules), 12and ▯ it the mortgage premium. Default, it, is captured by a binary variable. Defaulters are not able to re-enter c the mortgage market for a period of time, f . A spparate binary variable, it denotes whether an individual has defaulted recently enough to be excluded from the mortgage 13 market. In contrast to Chatterjee (2009) and other papers in this literature, I assume that the housing market exclusion period is ▯xed, rather than random, for the sake of tractability. Furthermore, when a household defaults, the model requires that h = 0 it and b it. That is, the housing stock (which serves as collateral) is transferred to the 14 ▯nancial intermediary and the mortgage debt is eliminated. Finally, note that some households in the model will perish with housing remaining. In the literature, it is common for the dying generation to either bequest the housing to the incoming generation or turn it over to the government. For simplicity, we assume that the government takes the housing when the outgoing cohort perishes. 15 Now that all of the pieces of the model in place, we may collect the individual- c d level state variables, z it = fd it▯1; it▯1; it;hit▯1;b it▯1;kit▯1;▯itag, the aggregate- level state variables, Z t fK ;A t t▯1 ;U t▯1;IH ;t ;At Ut ;cAg, and the parameters = f▯;▯ c▯ ;h;▯;▯ ;▯ A▯ ;U ;▯U;▯ ;A ;cK;▯ H A U A;U g to simplify notation. The dynamic programming problem (DPP) for the household may now be written as follows: Vitz it ; t = max fcitditkithitbit itu(c it )it (14) X ▯ Pr(A )Pr(U )Pr(▯ j▯ ;A )VE 0 (z ;Z ; ) it+1 it+1 t+1 A ;U ;▯02f1;0g s.t. c + ▯(h ;h ) + d + p h + m + k = y + (1 + r )d + (15) it it it▯1 it t it it it it t▯1 it▯1 h p t it▯1(1 ▯ ▯ H + (1 + R )kt it▯1+ b it H I bit minfb ;b it it (16) c If it > 0;then b ;it= it (17) ( r▯ b if d = 0 m = it▯1 it▯1 it (18) it 0 if d = 1 it 12See the appendix for a shortcut for computing default probabilities without using the decision rules. 13 Note that defaulters have access to non-housing shelter, which means that they will not receive negative in▯nity utility from defaulting. 14 It is important to note that 1) the foreclosure process is costly{and, thus, the amount recovered will be less than the value of the house prior to the foreclosure; and 2) the value of housing at default will not exceed the size of the mortgage. If it did, the household would simply sell it, rather than defaulting. 15Note that this will have a fairly insigni▯cant impact on government’s budget constraint, since each perishing cohort accounts for 1/60th of the population and will tend to draw down its housing position near the end of the lifecycle. 12 This problem is solved using a custom approximate dynamic programming (ADP) algorithm, which is described in the appendix. 3.3 The Financial Intermediary I adopt a largely novel speci▯cation for the ▯nancial intermediary that generates a number of desirable results related to mortgage pricing and solvency. In particular, I place structure on the ▯nancial intermediary’s objective in order to obtain simple decision rules without solving a dynamic programming problem. I assume the following: 1. Deposits made at ▯nancial intermediaries yield the risk free rate, r. t▯1 2. Households may obtain competitively-priced, one-period mortgages from ▯nancial intermediaries, which are subject to the constraints given in the housing section. 3. Financial intermediaries are risk-neutral. 4. There are in▯nitely many ▯nancial intermediaries, which are represented by a single ▯nancial intermediary with zero net cash ows. 5. Financial intermediaries use rational expectations (i.e. a household’s decision rules) to determine a household’s probability of defaultingitq . 6. Financial intermediaries add a state-contingent premium, ▯ t to the mortgage interest rate in order to generate zero net cash ows. 7. The foreclosure process is costly, leaving ▯nancial intermediaries with only a frac- tion, ▯ of the housing, which they liquidate in the same period at the market H rate, t . Using these assumptions, the intermediary sets the mortgage payment for household i, who obtained a loan in period t as follows: ▯ m it+1= (1 + ritbit (19) where bit is the size of the mortgage. Notice that the interest rate on the mortgage ▯ contains two components: 1) (1 + rit, which is speci▯c to the individual and accounts for idiosyncratic default risk; and 2) a spread component,t▯ , which is identical for all borrowers and clears the market. Furthermore, note that the intermediary will receive it▯iwhen a household repays a H loan originated at time t-1 and ▯pitit▯1, when it does not. Thus, in order to obtain zero net cash ows from period t loans and deposits, it must set ▯ to solve the following equation: X X X X X d H d (1 + rt) dit▯1+ bit= 1( it= 0)m +it t 1( it= 1)hit▯1(1 ▯ H ) + dit i i i i i | {z } | {z } Out ows In ows (20) 13 Notice that it▯1 it▯1, it▯1are all predetermined at time t. Thus, the intermediary 16 sets t to reduce or increase mortgage volume until net cash ows are zero. 3.4 The Government The government has one function in the model: to make transfer payments to retired and unemployed individuals using taxes collected. For simplicity, the government is assumed to use a constant replacement ratio, ▯. That is, it transters ▯w to retired and unemployed individuals. 3.4.1 Transfers In order to cover payments to the unemployed and retired, the government must allo- cate the following amount to outgoing transfer payments: 0 1 TR+T TR+T X X ▯t= ▯w t@ ▯a+ (1 ▯ ▯at▯aA (21) a=TR+1 a=1 P R Note that T +R ▯a, the mass of retired individuals, is constant in this model, a=T +1 P R so it may be rewritten as ▯ . The mass of unemployed, T +T(1 ▯ ▯ )▯a, changes a=1 at over time, so it is denoted ty ▯ . This yields: R U t = ▯w t▯ + ▯ t (22) That is, the government must collect enough in taxes to pay the mass of retirRd, ▯ , and the mass of unemployed, t , ▯wtin transfers. 3.4.2 Revenue For simplicity, I assume the following about taxes: rates scale with productivity and unemployed agents do not pay taxes. With these assumptions, the tax for employed households in cohort a can be written as follows: ▯ + ▯ U hD ▯ = ▯w ▯ ▯ E R t ▯ t (23) at t a i(1 ▯ ▯ ) (1 ▯ ▯ ) t t Note that hD denotes the housing stock that households turn over to the government t in period t after perishing. Additionally, notice that this speci▯cation for taxes will 16There are two important things to note. First, in practice, we use a state-contingenttfunction to set ▯ , rather than setting it in all periods. This adds tractability; and is discussed more in the appendix. Second, ▯twill also have an impact on the default rate and deposits at time t, but the e▯ects will be substantially smaller than those on mortgage volume. 14 require the government to maintain a balanced budget at all times. That is, aggregate incoming transfer payments are equal to aggregate outgoing transfer payments: T+TR ▯ ▯ X ▯R+ ▯ U hD ▯t= ▯wt▯it t ▯ t ▯a▯ a h t = ▯w t▯R+ ▯ )t= ▯ t (24) (1 ▯ t ) (1 ▯ ▯t) a=1 P To see why this is the case, recall thatT ▯ ▯ = 1. Since ▯ = 0 and ▯ E = 0 a=1 a a aP R it when a > T (i.e. individuals are retired), it will also be the case that▯ a a 1. P R a=1 Thus, a=1 ▯a▯ita= (1 ▯ ▯ t, which gives us the above equation. Note that three components of the tax collected from households vary: 1) the wage; U 2) the age-speci▯c productivity component; and 3) the unemployment rate, t . When an individual’s productivity component is higher (i.e. the individual is earning more), she will pay more in taxes. This is also true if the wage is higher, which results in a generally progressive tax. However, an increase in unemployment will still tend to increase taxes, since the tax base will decline. However, if the wage also declines, this e▯ect may be limited. In order to test the magnitude of the these e▯ects, I used data from the autoregres- sive rule simulation with capital and endogenous house prices, which is described later in the paper. I found that the average tax rate was positively correlated with both the consumption goods sector shock and aggregate income, which suggests that any countercyclical tax behavior generated by a shrinking tax base (as described above) is dominated by the magnitude of wage changes. That is, in the model, aggregate tax revenue tends to fall when output falls and rise when output rises. 3.5 The Central Bank I assume that the central bank sets the interbank lending rate. Since each ▯nancial intermediary is indi▯erent between borrowing from other banks and from households, the interbank lending rate will also determine the risk-free rate earned on depotits, r . Since the model employs a representative ▯nancial intermediary, net interbank lend- ing is zero and is excluded. In the ▯rst set of exogenous interest rate rule simulations, I will assume that the central bank adopts an autoregressive interest rate rule. I will then test the properties of this rule by varying the autoregressive coe▯cient to determine how interest rate persistence impacts default and welfare. In addition to autoregressive rules, the central bank will also employ other exoge- nous interest rate rules, such as a sine wave rule, in di▯erent simulations. The purpose of this exercise will be to capture the impact of certain interest rate behaviors on default. The central bank will also employ a number of di▯erent rules that endogenously 1This might happen if the consumer goods sector is hit with negative productivity shock, since this will have a negative e▯ect on wage through the productivity decline, but a positive e▯ect through the increase in unemployment. 15 respond to the economy’s dynamics. In di▯erent simulations, these rules will incorpo- rate house price level targeting, in ation targeting, house price in ation targeting, and output gap targeting. The Taylor rule and modi▯cations of it will also be tested to determine the optimal coe▯cients for reducing default and maximizing welfare. Finally, it is important to note that in ation will have real e▯ects in my model, even in the absence of sticky prices. Since agents are heterogenous and the model incorporates realistic mortgage structure, an increase in in ation will reduce the real value of mortgage debt. Furthermore, in ation will tend to increase the market value of homes relative to the size of the mortgage contracts that were used to purchase those homes. This will improve homeowner balance sheets by increasing home equity positions. Ultimately, these e▯ects (and others) make it possible to perform all of the analysis in a exible price{rather than sticky price{framework. However, I will defer further discussion of sticky prices to later sections and the appendix. 3.6 Aggregate Consistency Conditions In addition to satisfying individual-level constraints, the economy is also subject to aggregate consistency conditions. Each constraint requires an aggregate-level variable to be equal to the weighted sum of the individual-level variables: TR+T X Kt= kat a (25) a=1 R TX+T N t ▯at a (26) a=1 R X +T C = c ▯ (27) t at a a=1 TX+T ▯t= ▯at a (28) a=1 TR+T X Bt= bat a (29) a=1 TR+T X Dt= dat a (30) a=1 R X +T Ht= hat a (31) a=1 R TX+T Ch = ch ▯ (32) t at a a=1 16 TR+T X ▯ t ▯(h at at▯1 )▯a (33) a=1 Yt= C +tIH + Kt▯ (1 t ▯ )K K t▯1 + ▯ t (34) 3.7 Price Stickiness The model was originally solved and simulated with sticky prices. After further evalu- ation, it became clear that sticky prices were not an essential feature of my model{and, thus, were removed to highlight the importance of other mechanisms. Below, I brie y describe the version of sticky prices that were originally incorporated into the model. It is important to note that their inclusion in the model did not result in qualitatively di▯erent results. Other than that, I will restrict further discussion of the sticky price case to the appendix. In order to generate sticky prices, I adopt an approach similar to the one taken in Chari, Kehoe, and McGrattan (2000) and Taylor (1979, 1980), but in a cashless economy. That is, there is a ▯nal goods producer who assembles intermediates with the following production function: ▯ Z ▯1 q Y t Yt(j)di ;0 < q ▯ 1 (35) Additionally, there is a continuum of intermediate goods ▯rms who use Cobb-Douglas technology to produce individual varieties: Yt(j) = e AtK tj) N (t) 1▯▯; (36) Intermediate good ▯rm j will choose P(j), K(j), and N(j) to maximize pro▯ts: ▯ (j) = P (j)Y (j) ▯ RK (j) ▯ W N (j) (37) t t t t t t I assume that only half of intermediate goods ▯rms are able to set prices in each 18 period. For instance, in period t, all j, such that j 2 [0;0:5) will set prices for periods t and t+1; and in period t+1, all j, such that j 2 [0:5;1] will set prices for periods t+1 and t+2. The price set by intermediate goods ▯rms in period t is denoted by P . Thts yields the following optimal price for intermediate goods ▯rms: ▯ ▯1 ▯ Et(P t t t (1 + r )t Pt+1V t+1 t+1) Pt= 1 1 ; (38) qE (P 1▯q+ (1 + r ) ▯1P 1▯qY ) t t t t+1 t+1 2▯q where ▯ = and V ts the minimized unit cost of production. Finally, the zero pro▯t 1▯q condition yields the following equation for the price index: ▯ ▯ 1 1 1▯q 1 q q P t P t▯1 + P▯t1▯q (39) 2 2 18 In the exible price speci▯cation, all ▯rms can set prices in all periods. 17 In each period, I solve fortP using an approximation scheme, which is outlined in the appendix. 4 Model Properties Now that all of the pieces of the model are in place, we may take a more careful look at the economy, starting with the characteristics of the individuals who default. Figure 2 shows cumulative density function (CDF) plots of the age, income, home equity, and capital holding for simulated households. The plots show defaulters in red and non-defaulters in blue. There are four useful things to take away from these plots. First, all defaulting households have negative equity positions, but not all non-defaulters have positive home equity positions. As outlined earlier, negative equity is a necessary{but not su▯cient{condition for default. Second, the income of defaulters is low relative to non-defaulters. In fact, 47.86% of defaulters are unemployed; whereas, only 5.7% of non-defaulters are unemployed. This should not be surprising, since income shocks are the second trigger. Third, defaulters tend to be substantially younger. In fact, all defaulters in this particular simulation were under age 50. Individuals over age 50 typically have su▯cient home equity and capital holdings to endure shocks to income and equity. Finally, the capital stocks of non-defaulters tend to be considerably higher. There are two reasons for this: 1) having more capital helps agents to smooth shocks that a▯ect wages and house prices, which makes them less likely to default; and 2) the types of agents who are unlikely to default (i.e. older agents with substantial home equity) have had more time to accumulate capital to smooth consumption during retirement. In addition to looking at default, I also consider other properties of individual-level variables in the model. The household shown in Figure 3 is drawn from the baseline model’s simulation and is used to demonstrate the degree of heterogeneity in the model. Note that the household shown experiences multiple unemployment spells, defaults at an early age, holds a mortgage until age 60, and never manages to accumulate a substantial amount of assets. As a result, they experience unusually low consumption in retirement. Figure 4 shows the average household lifecycle pro▯le for comparison. Note that Figure 4 replicates the stylized facts for U.S. homeowners according to Jeske (2005). That is, they accumulate housing from age 25 until age 50-59, but then stay in the same house or move into a smaller one thereafter. Additionally, non-housing consumption increases until about age 40, but then declines thereafter. Net paper assets are initially negative, but then become positive and grow as the individual pays o▯ debt and accumulates savings for retirement. Additionally, note that the age-income pro▯le, as shown in Figure 3 and 4 is cali- brated to match Consumer Population Survey (CPS) data. That is, income increases until the individual reaches her early 40s. It then levels o▯ and begins to decline. Fur- thermore, the individual in Figure 3 receives a random unemployment shock at age 62, which lasts a year. And, at age 65, the individual retires and accepts transfers from 18 the government at the replacement ratio, ▯. Some of the basic features of the aggregate economy are illustrated in Figure 5, which shows 100 periods of simulated data. One thing to note is that aggregate housing is substantially more volatile than consumption, as is also true in the historical data. Additionally, the simulated economy exhibits leverage cycles, as measured by the debt- to-income ratio. Finally, over the course of the 100 years of simulated data, we see signi▯cant changes in default rates, which range from 0% to 7%. This is roughly consistent with the U.S. housing market over the last 50 years. As far as the calibration of the unemployment rate is concerned{the simulated rate has a mean of 6.49% and a standard deviation of 1.47% over the full simulation period (not just the 50 years shown in Figure 6). In the actual data for the U.S. from 1957 to 2012, the mean unemployment rate was 6.03% and the standard deviation was 1.58%. Additionally, it is important to note that idiosyncratic unemployment shocks are based o▯ of shocks to consumption good production, which makes the unemployment rate move with the business cycle. In addition to examining default in the model, I also consider how macrovariables in the model respond to shocks by constructing impulse responses. I assume that the central bank uses the following standard version of the Taylor rule with ▯ = 1:5 an1 19 ▯2= :5: ▯ ▯ ▯ rt= r + ▯ (▯1▯ ▯t) + t (log(2 ) ▯ lot(Y )) + ▯ t t (40) Note that the impulse responses do not come from a linearized version of the model, but instead are constructed by solving the nonlinear version, simulating without shocks initially, and then introducing a single, one-standard d

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