Rebuilding Household Credit Histories: Slow Jobless Recovery from Mortgage Crises

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1 Rebuilding Household Credit Histories: Slow Jobless Recovery from Mortgage Crises Guannan Luo January, 2015 Abstract This paper demonstrates that credit reporting banks observing households default histories can cause slow recoveries of housing prices and employment from mortgage crises Comparing credit cycles with and without credit reporting and capturing the impact of mortgage default on employment by extending the Diamond-Mortensen- Pissarides search framework, I find that with credit reporting, banks offer higher loanto-value ratios on mortgages but the default risks on them are higher The recovery after a bank liquidity tightening is slower because: i) excluding defaulting borrowers from obtaining mortgages decreases demand for housing; ii) foreclosure tightens firms available credit; and iii) further default constrains banks liquidity These effects are absent without credit reporting JEL Classification Codes: E21; E24; E32; E44; G01; G11; G21; J23; J63 Keywords: credit history; jobless recovery; mortgage default; business cycles I would like to thank Dale Mortensen, Matthias Doepke, Jonathan Parker, Mirko Wiederholt for advice Comments from David Berger, Marianna Kudlyak, Asher Wolinsky, and seminar participants at Northwestern University, Vanderbilt University, Federal Reserve Bank of Kansas City, the University of Hong Kong, City University of Hong Kong, the Australian National University, 2013 Financial Management Association s Annual Meeting, and 2013 European Meeting of the Econometric Society are gratefully acknowledged Department of Economics and Finance, City University of Hong Kong guannluo@cityueduhk 1

2 1 Introduction Since the 2007 Great Recession, the economic recovery has been slow in the United States Housing prices and employment rates decreased dramatically and stayed lower than their ex-ante levels even seven years after the bust 1 However, productivity did not drop much after the crisis and returned to its ex-ante level quickly Could housing and the mortgage market not only be at the heart of the financial crisis but also be central to the slow jobless recovery from the mortgage crisis? Before I address this question, first note that the credit granted on household debt has increased dramatically since households credit histories became available to banks credit system in the United States was first established in the 1950s, but only local banks shared their credit records In the late 1980s, FICO scores were created, so any borrower was assigned a credit score that was observed by all banks According to the Federal Reserve Economic Data, the ratio of households outstanding debt to GDP grew from about 25% in 1950 to around 50% in the 1980s to over 90% in 2007 Banks began to provide larger loans on mortgages since they observed households credit histories A mortgage in the United States today may have a 90% loan-to-value ratio 10% down payment) By contrast, when banks did not observe households credit histories, such as in 1990s China, a standard mortgage usually had a loan-to-value ratio less than 70% This paper demonstrates that one important reason for the slow recovery of housing prices and employment from mortgage crises such as the 2007 Great Recession is credit reporting banks observing households default histories This causal effect is studied by setting up a quantitative general equilibrium model to compare credit cycles under two separate regimes: i) without credit reporting; and ii) with credit reporting The results show that with credit reporting, banks grant larger credit on mortgages, but the default risks are also higher The housing prices and employment rates recover more slowly from a bank liquidity crisis The paper is the first to propose that credit reporting contributes to the current slow jobless recovery It is also the first to investigate the effect of credit reporting on mortgage, housing, and labor markets in business cycles I find that information about households credit histories is a double-edged sword Credit reporting relaxes borrowers constraints and raises investors returns during normal times, but it also causes persistently high mortgage default rates, low housing prices, and low employment rates at credit crunches To investigate the effect of credit reporting on the slow recovery from a mortgage crisis, I first built a quantitative general equilibrium model with several types of agents: impatient 1 According to the Federal Reserve Economic Data, the seasonally adjusted S&P Case-Shiller 20-City Home Price Index was around 200 in May 2007, stayed below 150 from January 2009 to March 2013, and equaled 1655 in January 2014 According to data from the US Bureau of Labor Statistics, the unemployment rates was 44% in May 2007, stayed no less than 78% from January 2009 to January 2013, and equals 67% in February 2014 The 2

3 households borrowers), patient households investors), a house leasing company, firms, and banks; and all these agents interact in the housing sale, housing rental, labor, and credit markets I then compared the credit cycles under the two separate regimes without or with credit reporting) by: i) solving banks optimal mortgage contracts; ii) characterizing recursive equilibria; and iii) using a method similar to Krusell and Smith 1998) to simulate the model using calibrated parameters 2 In the housing markets, following Kiyotaki, Michaelides, and Nikolov 2011), renting houses is more costly than owning In the labor market, to capture the impact of credit friction on unemployment rates, I extend the Diamond-Mortensen-Pissarides 1982, 1985) search framework to include working capital channels: firms have to take out loans from banks to finance their recruiting costs before production The firms working capital channel is similar to that in Khan and Thomas 2013) and Petrosky-Nadeau 2014) I also use a modification of Postel-Vinay and Robin 2002) to simplify the relationship between interest rates and labor market tightness into one period In the credit market, the two credit reporting regimes are modeled by borrowers reentry rates to obtain mortgages after default Without credit reporting, any defaulting borrower can easily get another mortgage from other banks that do not know his default histories So the reentry rate equals one In contrast, with credit reporting, the reentry rate is low because banks punish defaulting borrowers by excluding them from obtaining mortgages in the near future 3 A mortgage default causes foreclosure loss for banks In contract to prevailing business cycle literature with firms collateral borrowing, such as Kiyotaki and Moore 1997) and Bernanke, Gertler, and Gilchrist 1999), my paper imposes collateral borrowing on the household side The default punishment with credit reporting has a more persistent negative effect on credit cycles than the punishment on firms, because defaulting borrowers have to stay in the credit market and get punished for a long time, but defaulting firms can exit the credit market and start over as new entrants The results show that credit reporting increases not only loan-to-value ratios but also default risks under optimal mortgage contracts Without credit reporting, banks cannot punish defaulting borrowers Then any borrower will default once his mortgage is underwater Anticipating this, banks will require a large down payment to ensure that borrowers will not walk away So there is no default at equilibrium The optimal mortgage rate equals the risk-free interest rate, and banks profit equals zero However, with credit reporting, banks can punish defaulting borrowers by not lending to them again in the near future Afraid of 2 The parameter values are similar to those used in the prevailing literature, such as Shimer 2005) and Krusell and Smith 1998) 3 For example, under the current credit system in the United States, households credit histories are reflected in FICO Fair Isaac Company) scores or similar measures by credit reporting bureaus Mortgages are the most important factor that affects one s FICO score If a borrower defaults on his mortgage, it takes about 7 years for him to rebuild his FICO score to borrow again 3

4 such default punishment, borrowers are willing to bear some loss to repay mortgages that are underwater So banks are willing to offer higher loan-to-value ratios on mortgages if a credit crunch happens infrequently The default risks on these mortgages will be higher When mortgages are underwater in a credit crunch, employed borrowers repay, but unemployed borrowers cannot afford to repay even if they sell off their houses So most unemployed borrowers default after a bank liquidity tightening, and further default happens even when banks liquidity constraints are relieved The optimal mortgage rates are higher than the risk-free interest rate Banks gain during normal times but lose at credit crunches The recovery of housing prices and employment rates are slower after a mortgage crisis with credit reporting than without credit reporting The mechanism works as follows: when banks liquidity is tightened as in Allen and Gale 1999)), the real interest rate increases, so the housing prices decrease The employment rate also decreases because it becomes more costly for firms to recruit new employees through the working capital channels Without credit reporting, no one defaults, so the housing prices and employment rate increase to a new steady state once banks accumulate enough funds to relieve their liquidity constraints But with credit reporting, most unemployed borrowers cannot afford their mortgage payment so have to default Then on one hand, defaulting borrowers are punished by not obtaining mortgages again, the demand for housing is low, so the housing prices stay low On the other hand, foreclosure decreases banks liquidity, so it takes longer for the real interest rate to go down Then through firms working capital channels, firms recruit fewer employees, and hence the employment rate stays low, which further increases the mortgage default rate because more unemployed borrowers default This feedback effect between mortgage default and unemployment implies a jobless recovery with credit reporting My findings highlight three important implications with credit reporting First, housing prices and employment rates can be persistently low even though banks liquidity constraints are relieved Further default decreases banks capital, so the housing prices and employment rates fluctuate and cannot recover Second, default punishment increases loan sizes but decreases defaulting borrowers value of housing, so credit reporting has its downside Third, I suggest three macroeconomic policies to either accelerate the recovery or increase borrowers welfare at credit crunches: i) injecting liquidity to banks; ii) requiring banks to provide safer mortgages, such as increasing the down payment requirement; and iii) restructuring the credit reporting system to distinguish voluntary and involuntary defaults, so banks can punish the borrowers who walk away from their mortgages strategically but forgive the borrowers who default because they cannot afford their mortgage payments The rest of this paper is organized as follows Section 2 sets up the model and describes the key components of the model Section 3 establishes each agent s optimization problem Section 4 characterizes the optimal mortgage contracts and compares the results in the recursive equilibrium under two separate regimes: i) without credit reporting; and ii) with 4

5 credit reporting Section 5 simulates and compares the recovery from a mortgage crisis under the two regimes Section 6 examines the macroeconomic policies that may accelerate the recovery of housing prices and employment after a bank liquidity crisis Section 7 checks robustness and Section 8 concludes The detailed proofs of propositions and the simulation procedures are provided in the appendix 2 The Model Consider a dynamic economy in discrete time with two types of expenditure: nondurable consumption goods and durable housing Goods are produced every period, but housing is of fixed supply and never depreciates There are five types of agents: impatient households borrowers), patient households investors), firms, a house leasing company, and banks They participate and interact in up to four markets: housing sale, housing rental, labor, and credit Credit Market Labor Market Patient Households Investors) 6 Firms Wages Y H HH Savings H HHHj Recruiting Cost External Funds Low Interest Rate?ɛ t = 0) High Exogenous Interest Liquidity Rate Shocks ɛ t = 1)? Banks One-Period Loans) Wages Mortgages?? Impatient Households Borrowers) Housing Rent - Housing Sale and Rental Markets Figure 1: Structure of the Model Purchase Cost House Leasing Company Figure 1 shows the structure of the model Households play a key role in the model: they consume, work for firms or at home, save or borrow, and own, finance, or rent houses There are two types of infinitely-lived households with aggregate measure one: patient households investors) and impatient households borrowers) Investors save and own houses Borrowers either finance their home ownerships or rent houses Firms borrow recruiting costs from banks to hire households to produce consumption goods The house leasing company rents out houses as a monopoly It can finance its purchase of rented housing through bank loans Banks are competitive financial intermediates that facilitate borrowing and lending among agents The firms, house leasing company, and banks are owned by investors 5

6 In the housing sale and rental markets, borrowers finance or rent houses, and the house leasing company rents out houses In the labor market, households search for jobs and firms post vacancies to recruit new employees Any household either a borrower or an investor) is either employed by firms or unemployed Employed households receive wages and unemployed households obtain home production In the credit market, investors save and share profits or losses of the firms, house leasing company, and banks Besides investors savings, there are also external funds available for banks, which are sufficient and accept any prevailing low interest rate during normal times but are limited and require a high interest rate at credit crunches, as in Allen and Gale 1999) Banks provide one-period loans to finance firms recruiting costs, borrowers home ownerships, and the house leasing company s purchase cost of rented housing In each period, households can borrow at a high interest rate to cover part of their mortgage payments, which is the same as rolling over part of the mortgages into personal debts that require a high interest rate 21 Aggregate Uncertainty In any period, the economy is either in a good state normal time) or a bad state credit crunch) During normal times, banks can obtain sufficient external funds that accept any prevailing low interest rate in the economy At credit crunches, there are limited external funds, so banks have to pay high interest rates, following exogenous liquidity shocks in Allen and Gale 1999) Banks liquidity constraints will be relieved if the banks accumulate enough internal funds investors savings to satisfy the demand for loans in the economy Define ɛ t as the indicator function of a credit crunch at time t Then ɛ t = 1 if a credit crunch happens; otherwise, ɛ t = 0, or time t is a normal time Suppose that ɛ t follows a two-state Markov process: a credit crunch arrives during a normal time at rate ϕ and ends at rate ς, where ϕ, ς 0, 1] Whether a normal time or a credit crunch will arrive in the next period is realized after agents make their current decisions Let z t denote the probability that a credit crunch happens in the next period Then according to the setting: { ϕ, if ɛ t = 0; z t = Prɛ t+1 = 1 ɛ t ) = 1) 1 ς, if ɛ t = 1 The chance that a credit crunch will happen in the next period depends on the current state of the economy During normal times, a credit crunch arrives infrequently but has a relatively long duration, so ϕ and ς are relatively small 6

7 22 Labor-Search and Credit Frictions The labor market follows a search framework that modifies the studies in Diamond-Mortensen- Pissarides 1982, 1985) and Postel-Vinay and Robin 2002) The matching function between workers who look for jobs and firms that post vacancies follows a Cobb-Douglas form Job finding rates and vacancy filing rates are functions of labor market tightness, a higher value of which increases the probability of finding a job but decreases that of filling a vacancy Households are workers They have homogenous labor productivity and each exerts one unit of labor inelastically in each period They are either unemployed, newly employed, or retained employed Unemployed households produce home production and search for jobs Newly employed households receive bargaining wages from firms Retained employed households have outside options to work for other firms, so their wages increase up to the firms marginal productivity of labor at equilibrium, as in Postel-Vinay and Robin 2002) Both newly employed and retained employed households have an exogenous probability of losing their jobs Firms recruit workers It is costly for them to post vacancies to recruit new employees, but they are free to make offers to retained employees of other firms Firms new hires equal their posted vacancies times the vacancy filling rate At equilibrium, firms offer two-tier wage contracts: a bargaining wage for a new employee and a wage equal to the marginal productivity of labor for a retained employee, following Postel-Vinay and Robin 2002) To capture the credit friction of mortgage defaults on employment, the model extends Diamond- Mortensen-Pissarides 1982, 1985) search framework by including working capital channels for firms, under which firms have to borrow their recruiting costs from banks before each production season The assumption on firms working capital channel follows Khan and Thomas 2013) and Petrosky-Nadeau 2014) Firms and new employees bargain over the excess production of employment the firms marginal productivity of labor less the new employees outside option: home production For simplicity, suppose that firms have a homogeneous linear production function in labor Then the bargaining wage of a new employee, defined as W, is: W = B + 1 τ) A B), where A is a firm s constant marginal productivity of labor, B is an unemployed worker s home production, and τ is a firm s bargaining power, A > B 0 and τ 0, 1) Figure 2 shows the timeline for the labor and credit markets At the beginning of each period, firms borrow from banks to recruit new employees, and banks grant one-period loans to finance firms recruiting costs, the house leasing company s purchase cost of leased housing, and borrowers home ownerships Then firms produce consumption goods At the end of the period, agents realize whether a normal time or a credit crunch will arrive in the next period Then firms and the house leasing company repay their loans, and borrowers decide whether to repay their mortgages or default Banks receive returns on investments Because there is 7

8 no uninsurable risk in firms production and the house leasing company always repays loans, loans to firms and the house leasing company are risk-free However, investing in mortgages is risky because of the default risk Investors receive returns on savings and share the profits or losses from firms, the house leasing company, and banks t-1 t t+1 ɛ t 6 6 Firms Produce ɛ t Firms Borrow Recruiting Costs Banks Offer One-Period Loans Firms Pay Wages and Repay Loan Banks Receive Return on Loans Including Mortgage: Repay or Default) Figure 2: Timeline for Labor and Credit Markets Banks profits or losses are determined by mortgages Mortgages function in various ways First, mortgages are simplified to one period loans, such as the adjustable-rate mortgages with continuous refinancing Banks refine mortgage terms in each period so they bear no long-term interest rate risk Borrowers are free to adjust their quality of financed housing so they have no prepayment penalty Second, any mortgage contract specifies a maximum loan-to-value ratio loan amount per housing value) and a scheme of mortgage rates Banks require any mortgagee to hold a certain amount of equity in their financed housing down payment) The loan-to-value ratio equals one minus the percentage of down payment on the mortgage The mortgage rate varies according to the borrower s down payment, current employment status verifiable by banks), and default histories if observable However, the borrower s total demand for financed housing is not contractible, because he can take out mortgages simultaneously from multiple banks to finance his home ownership Third, mortgages are secured but nonrecourse debts The financed houses serve as collateral If a borrower defaults, banks foreclose his home but cannot sue the borrower for any loss There is a foreclosure cost that equals a fraction of the housing value Banks may punish the defaulting borrower by excluding him from obtaining mortgages again in the near future if they observe his credit histories, or in the regime with credit reporting Credit reporting is modeled by the reentry rate to obtain mortgages again after default Without credit reporting, banks cannot observe a borrower s past default so grant him mortgages as if he is clear of default history A defaulting borrower can get another mortgage immediately from another bank with no punishment Thus, the reentry rate to obtain mortgages after default equals one in the regime without credit reporting In contrast, with credit reporting, banks can observe any borrower s past default They punish a defaulting 8

9 borrower by granting him mortgages again with a low probability 4 It takes a long time for the borrower to rebuild his credit history to obtain mortgage again after default 5 Thus, the reentry rate to the mortgage market after default is low in the regime with credit reporting 3 Agents Problems and Recursive Equilibrium In this section, I first describe and set up the optimization problems for the five types of agents in the economy: patient households investors), impatient households borrowers), firms, the house leasing company, and banks I then establish the aggregate conditions in the housing, labor, and credit markets I finally define the recursive equilibrium 31 Patient Households Investors) Patient households are investors in the economy They have a higher discount factor than impatient households borrowers) To focus on the mortgage market for borrowers, suppose that investors are risk neutral in consumption and each own a fixed average unit of housing There is no borrowing against housing Then investors optimization problem can be simplified to depend on consumption only and solved in aggregate Adding housing only increases the level of the investors aggregate value but has no effect on the optimal solutions of consumption or saving In each period, investors consume, save in banks, and share profits or losses of the firms, house leasing company, and banks They can also borrow from banks at high interest rates For simplicity, fix the high interest rate at the impatient households discount rate Investors wealth consists of their labor incomes, returns on savings cost of borrowing if negative), and profits or losses from the firms, house leasing company, and banks They choose consumption and savings borrowing if negative) to maximize their utility within their budget constraint Their aggregate optimization problem at time t is: V t = max C t 0,S t Ct + βe t Vt+1 ] subject to Ct + S t ny t + R t 1 + β 1 R t 1 ) ISt 1 <0]) St 1 + Π F t + Π L t + Π B t ; where C t is the investors aggregate consumption, S t is their aggregate savings borrowing if negative), β is their discount factor, n is the measure of investors, Yt 2) is the average labor income in the economy, R t is the gross risk-free real interest rate, β is the impatient households discount factor, I St 1 <0] is the indicator function that equals one if S t 1 < 0 and zero otherwise, Π F t is the profit or loss if negative) from the firms, Π L t is the profit or loss if negative) from the house leasing company, and Π B t is the profit or loss if negative) from 4 The Federal Reserve System regulates and supervises banks in the United States 5 It takes about 7 years for one s mortgage default not to have a significant effect on one s credit score 9

10 the banks, 1 > β > β > 0 and n 0, 1) 32 Impatient Households Borrowers) Impatient households are borrowers in the economy They have a higher discount rate than patient households Their discount factor is defined as β, where β 0, β) They either finance their home ownerships through mortgages offered by banks or rent houses Owning a house is preferable to renting one, following Kiyotaki, Michaelides, and Nikolov 2011) To make the model tractable, history-dependent effects on wealth are removed by assuming quasi-linear utility risk-neutral in consumption but risk-averse in housing) for borrowers: UC it, H it, H r it) = C it + α 1 ρ H it + 1 φ) H r it) 1 ρ, α > 0, ρ > 1, φ 0, 1), where α is the borrower s scale of housing in utility relative to consumption, ρ is his relative risk aversion in housing, φ is his disutility of renting a house relative to owning it, C it is his consumption, H it is the quality of his financed housing, and H r it is the quality of his rented housing The quality of housing broadly reflects all value associated with a house, such as its physical size, construction materials, indoor decorations, community facilities, etc The procedure in each period works as follows A borrower first receives his wealth endowment, which consists of his labor income and the value of his previous financed housing After realizing whether a normal time or a credit crunch happens, the borrower decides whether to repay his mortgages or default If he repays his mortgages, banks offer him mortgages again to finance his new home ownership Any mortgage specifies a maximum loan-to-value ratio and a scheme of mortgage rates The borrower then decides his consumption, amount of financed or rented housing, and loan-to-value ratios on mortgages If the borrower defaults, banks foreclose his home and may also punish him by excluding him from obtaining mortgages again in the near future The punishment is modeled by a low reentry rate to obtain mortgages again after default Without credit reporting, banks cannot punish defaulting borrowers, so the reentry rate equals one But with credit reporting, banks can provide mortgages again to defaulting borrowers with a low probability A defaulting borrower has to rent houses if he is excluded from obtaining mortgages There are two types of mortgage default: either involuntary or voluntary A borrower has to default involuntarily if he does not have enough resources to repay his mortgages Define σ it as borrower i s default decision on his mortgages at time t Then σ it = 1 if the borrower defaults; and σ it = 0 if he repays his mortgages The borrower defaults involuntarily if the total value of his labor income, previous financed housing, and roll-over limit on mortgages is less than his obligation on his mortgages The limit of mortgages that can be rolled over into personal debts at the high interest rate is proportional to the borrower s labor income It also equals a fraction of his housing value regardless of his current income Define b as 10

11 this limit per housing value for a retained employed borrower who receives income A, where b 0, 1) Then the roll-over limit on mortgages for a borrow who has H i,t 1 units of housing and receives Y it as income at time t is: b A H i,t 1 1 ) Y it, where Y it is the borrower s labor income at time t, which depends on the borrower s employment status: retained employed Y it = A), newly employed Y it = W ), or unemployed Y it = B) Then borrower i defaults σ it = 1) involuntarily on his mortgages at time t if and only if his total available fund is not enough to cover his mortgage payment That is: Y it + H i,t 1 + ) b A H i,t 1 1 Y it < M i,t 1 π i,t 1 1 H i,t 1 M i,t 1 π i,t 1 1 < b Y it A, where π i,t 1 is the borrower s loan-to-value ratio on the mortgages taken out at time t 1, M i,t 1 is the mortgage rate on it, and and 1 are the per-unit housing prices at time t current price) and time t 1 original purchase price) respectively The borrower can also default voluntarily walk away from his mortgages strategically He defaults voluntarily on his mortgages if his continuation value of repaying the mortgages is less than his value of default At time t, define Vit nd as borrower i s continuation value of repaying his mortgages not default), Vit d as his value of default and being excluded from obtaining mortgages, and Vit re as his value of default but being forgiven reenter the mortgage market) In each period, the borrower decides his consumption C it, housing either rented H r it or financed H it ), and loan-to-value ratio π it given the mortgage terms a maximum loanto-value ratio π it and a scheme of mortgage rates M it ) that depends on the loan-to-value ratio chosen by the borrower π it, where π it π it ) and reentry rate after default Then borrower i s decision problem if he repays his mortgages is: 3) V nd it = max C it,h r it 0,H it 0,π it π it UC it, H it, H r it) + βe t V i,t+1 ] subject to C it + r t H r it + 1 π it ) H it Y it + M i,t 1 π i,t 1 1 ) H i,t 1 ; the problem if he defaults and is excluded from the credit market is: Vit d = max C it,hit r 0 ] ] UC it, 0, Hit) r + β1 γ)e t V d i,t+1 + βγet V re i,t+1 subject to C it + r t H r it Y it ; 4) 5) and the problem if he defaults but reenters the credit market with probability γ) is: V re it = max C it,h r it 0,H it 0,π it π it UC it, H it, H r it) + βe t V i,t+1 ] subject to C it + r t H r it + 1 π it ) H it Y it ; 6) 11

12 where V it is the borrower s value function when he is offered mortgages by banks, r t is the housing rent, and γ is a defaulting borrower s reentry rate to obtain mortgages again, γ 0, 1] Then γ = 1 without credit reporting, but γ is low close to zero) with credit reporting Borrower i s loan-to-value ratio has to be less than or equal to the maximum loan-to-value ratio offered on mortgages That is, π it π it When borrower i has enough resources to repay his mortgages, he chooses the maximum value between paying off his mortgages and defaulting His value function at time t is: V it = max { } Vit nd, 1 γ)vit d + γvit re 7) Borrower i defaults voluntarily if and only if his value of default is larger than his value of repaying his mortgages Thus, besides involuntary default as shown in condition 3), he defaults σ it = 1) voluntarily if and only if: V nd it < 1 γ)v d it + γv re it 8) Without credit reporting, defaulting borrowers can be excluded from obtaining mortgages Define χ it as the indicator function of borrower i being excluded from the credit market at time t χ it = 1 if borrower i is excluded from the credit market at time t and χ it = 0 otherwise Then χ it is a random variable of current and previous default such that: { 0, with probability γ; χ i,t 1 = 1 or σ it = 1 χ it = 9) 1, with probability 1 γ In contrast, without credit reporting, borrowers can obtain mortgages from other banks immediately after default Their default decisions have no effect on their mortgage terms Observability of household credit histories affects borrowers mortgages offered by banks, but the case without credit reporting is the same as the case with credit reporting when the reentry rate of obtaining mortgages again after default equals one 33 Firms Firms hire workers in the labor market to produce nondurable consumption goods in each period They have a homogeneous linear production function in labor: Al t, where l t is the measure of total labor employed at time t Firms have to borrow recruiting costs from banks to hire new employees at the beginning of each production season They have linear vacancy posting costs, with κ defined as the constant cost of posting a vacancy They pay out wages and repay their loans at the end of each period Following the labor-market search framework in Diamond-Mortensen-Pissarides 1982, 1985) and Postel-Vinay and Robin 2002), firms profits equal their shares of the output produced by new employees Firms enter and exit the 12

13 market at no cost, so they have zero profit of posting an additional vacancy at equilibrium Competitive firms choose the number of vacancies posted, v t, to maximize their profits, Π F t, which equal their shares of the output of the newly employed workers minus their repayment of loans The number of newly employed workers equals the number of vacancies posted times the rate at which a vacancy is filled risk-free interest rates Then their optimization problem at time t is: The firms borrow recruiting costs at max v ΠF t subject to Π F t A W ) v t ηθ t ) R t κv t, 10) t 0 where R t is the risk-free interest rate, ηθ t ) is a firm s vacancy filling rate, and θ t vt u t labor-market tightness vacancy-to-unemployment ratio) is the 34 House Leasing Company The house leasing company is a monopoly in the housing rental market It is owned by investors It rents out houses to the borrowers who are excluded from obtaining mortgages The house leasing company has to take out one-period loans from banks to finance its purchase costs of leased housing They repay loans at the end of each period, so banks lend to them at risk-free interest rates Then the house leasing company s realized profit or loss if negative), Π L t+1, equals its value of housing in hand minus its repayment of loans: Π L t+1 +1 R t r t )) L t, 11) where R t is the risk-free interest rate, is the housing price, r t is the housing rent, and L t is the house leasing company s total supply of leased housing Borrowers have an outside option of renting houses from other borrowers Competitive borrowers expect zero profit in renting out houses, so they offer housing rent that equals βe t +1 ] Then the house leasing company maximizes its profit on behalf of investors subject to that borrowers are interested in renting houses from the house leasing company instead of other borrowers Then at time t, the house leasing company s optimization problem is: max E ] t mt+1 Π L t+1 subject to rt βe t +1 ], 12) r t 0 where m t+1 is investors inter-temporal discount on consumption and Π L t+1 is given by 11) 35 Banks Banks facilitate borrowing and lending in the credit market They are owned by investors In each period, banks receive funds from either investors savings or external funds such 13

14 as from foreign institutes/investors or reserves) Banks repay interests for sure, so funds are available to banks at risk-free interest rates During normal times, banks have sufficient funds The risk-free interest rate is so low that external funds crowd out investors savings In contrast, at credit crunches, banks have limited loanable funds and hence the risk-free interest rate is high Banks are competitive loan suppliers They offer one-period loans to finance firms recruiting costs, borrowers home ownerships, and the house leasing company s cost of purchasing leased housing Firms and the house leasing company borrow at the risk-free interest rate rate, because there is no uninsurable risk in firms production, and the house leasing company always repays loans But mortgages are risky Borrowers can either repay their mortgages or default Banks determine a maximum loan-to-value ratio and mortgage rates offered to a borrower If the borrower repays his mortgages, banks receive the mortgage payment that equals the mortgage rate times the loan amount If the borrower defaults, banks foreclose his home but lose a fraction of the housing value Mortgages are nonrecourse debts, so banks cannot sue the borrower for any loss of foreclosure Define Q i,t+1 as the realized return on borrower i s time-t mortgages at time t + 1 Then: Q i,t+1 = 1 µ) +1 π it σ i,t+1 + M it 1 σ i,t+1 ), 13) where at time t, is the housing price, µ is the fraction of the housing value that will be lost at foreclosure, π it is borrower i s loan-to-value ratio on mortgages, M it is his mortgage rate, H it is his financed housing, and σ it is his default decision on mortgages σ it = 1 if borrower i defaults on his mortgages at time t and σ it = 0 if he repays his loans Loans to either firms or the house leasing company are risk-free, so banks profit or loss are determined by mortgages Define Π B t defined as banks profit or loss if negative) at time t Then banks realized profit at time t + 1 equals their return on mortgages minus the cost of funds the real interest rate) times the loan amount That is: Π B t+1 = i Q i,t+1 R t ) π it H it di, 14) Banks are owned by investors proportional to investors wealth, so banks maximize investors aggregate utilitarian welfare over time Since investors are risk-neutral in consumption, banks maximization problem is equivalent to the decision problem of a representative investor who receives the aggregate labor income of investors Banks break even on loans at equilibrium, so their profit or loss, given in 14), satisfies: E t mt+1 Πt+1] B = 0, 15) where m t+1 is defined as investors inter-temporal discount on consumption 14

15 36 Clearing the Markets: Housing, Labor, and Credit In the housing market, the quality of total housing supply is fixed at measure h in the economy, where h > 0 There are two types of housing markets: sale and rental In the housing rental market, borrowers rent houses, and the house leasing company rents out houses The housing rental market clears if and only if the borrowers aggregate demand for rented housing equals the house leasing company s supply of leased housing That is: i Hr itdi = L t, t, 16) In the housing sale market, borrowers finance their home ownerships through mortgages offered by banks Any investor owns an average unit of housing, so investors own nh housing in aggregate, leaving 1 n)h housing to borrowers A borrower either rents or finances a house The housing market clears if and only if the borrowers aggregate demand of rented and financed housing equals the portion of housing supplied to them That is: i Hr it + H it ) di = 1 n) h, t 17) Assume that shorting is prohibited in the housing markets, so Hit, r H it 0 In the labor market, both borrowers and investors constitute the labor force, so the labor force is of measure one The total employment rate, consisting of the newly employed and retained employed households, at time t satisfies: l t = 1 u t 18) An unemployed worker finds a job at rate λθ t ), and an employee is laid off at the job separation rate δ t employment sustains inter-temporarily with probability 1 δ t ) Then the unemployment dynamics satisfy: u t+1 = 1 λθ t )) u t + δ t l t 19) In the credit market, banks receive funds from either investors savings or external funds They offer one-period risk-free loans to finance firms recruiting costs and the house leasing company s purchase cost of leased housing, and provide mortgages to borrowers using financed houses as collateral The credit market clears if and only if the demand for funds from firms, the house leasing company, and borrowers equals the supply of funds from investors savings and external funds That is, at time t: κv t + i π it H it di + r t ) i Hr itdi = S t + X t, 20) 15

16 where S t is investors savings borrowing if negative), and X t is the external funds Further, the goods market clearing condition holds whenever the markets for housing, labor, and credit are cleared Specifically, the goods market clearing condition can be derived from the budget constraints of borrowers and investors and the profits of firms, the house leasing company, and banks, as shown in 10), 11), and 14), respectively That is: C t + C it di + κv t + µ σ it H i,t 1 di + R t 1 1 r t 1 ) L t 1 i = Y t + X t + R t 1 + β 1 R t 1 ) ISt 1 <0]) St 1 i 37 Recursive Equilibrium Observability of households credit histories affects the recursive equilibrium through borrowers default decisions and mortgage terms offered by banks Without credit reporting, any defaulting borrower can obtain mortgages again immediately after default In contrast, with credit reporting, defaulting borrowers are excluded from obtaining mortgages in the near future The difference between these two regimes can be modeled by the difference in reentry rate to the mortgage market after default, γ: γ = 1 without credit reporting, but γ is low closed to zero) with credit reporting Then in the recursive equilibrium: i) The investors aggregate savings S t solve their optimization problem given by 2), with m t+1 defined as the associated inter-temporal discount on consumption; ii) Any borrower s financed housing H it, loan-to-value ratio π it, and default decision σ it solve his optimization problem given by 4)-7) if he is offered mortgages by banks χ it = 0); his rented housing H r it solves the maximization problem given by 5) if he is excluded from obtaining mortgages χ it = 1) χ it is the indicator function of borrower i being excluded from obtaining mortgages, as shown in 9) The associated value function is V it, which is provided in 7) Borrower i defaults σ it = 1) involuntarily if 3) holds, and he defaults voluntarily if 8) is satisfied iii) The firms job vacancy posted v t solves their optimization problem given by 10); iv) The housing rent r t solves the house leasing company s optimization problem given by 12), with the realized profit or loss from leased housing Π L t given by 11); v) The maximum loan-to-value ratio π it and mortgage rates M it π it ) offered to borrower i solve banks optimization problem given by 15), with the realized profit or loss from banks Π B t given by 14); and the banks belief of borrower i s default decision coincides with the borrower s actual default behavior σ i,t+1 ; 16

17 vi) The housing price satisfies the market clearing conditions of the housing rental and sale markets given by 16) and 17); vii) The labor market conditions 18) and 19) hold; and the credit market clears with the banks external funds X t satisfying 20) 4 Results In this section, I first investigate the equilibrium interest rate and its effect on unemployment, investors optimal consumption and savings, borrowers demand for financed housing and loan-to-value ratio, the house leasing company s decision on housing rent, and banks mortgage contract offerings I then characterize and compare the results and household welfare under two separate regimes: i) without credit reporting; and ii) with credit reporting 41 Interest Rate Affects Unemployment In the credit market, during normal times, banks have access to sufficient funds The demand for loanable funds is always satisfied, and the credit market always clears In contrast, at credit crunches, banks have limited external funds that accept a low interest rate, defined as x, where x 0 If these funds plus investors savings are not enough to satisfy the aggregate demand for loans, banks have to borrow at a high interest rate the borrowers inter-temporal discount rate on consumption, β 1 The credit market is cleared at the high interest rate when banks liquidity constraints are binding Otherwise, the prevailing risk-free interest rate is determined by the investors inter-temporal discount rate on consumption, which is lower than the high interest rate because investors are more patient than borrowers Therefore, the risk-free interest rate equals the high interest rate, R t = β 1, if and only if: S max,t < 0, or ɛ t = 1 and κv t + i π it H it di + r t ) i Hr itdi > S max,t + x 21) where S max,t is the maximum amount of investors savings available for banks, which consists of the investors labor incomes and investment returns: S max,t ny t + R t 1 S t 1 + Π F t + Π L t + Π B t 22) Interest rates decrease once banks accumulate enough funds to satisfy the demand for loans In the labor market, firms choose the number of job vacancies posted to solve their optimization problem given by 10) They have zero profit of posting an additional vacancy 17

18 at equilibrium, so the labor market tightness is a function of the prevailing interest rate: θ t = η 1 R tκ τa B) Note that in the labor search and match framework, firms job vacancy filling rate is strictly decreasing in the labor market tightness, θ t Thus, firms optimal decision on recruiting implies that the labor market tightness, θ t, and firms vacancy posting rate, v t = u t θ t, are decreasing in the interest rate, R t Further, by the dynamics of unemployment rates given by 19), since the job finding rate is strictly increasing in the labor market tightness, the unemployment rate is strictly increasing in the interest rate With linear vacancy posting costs, firms have zero profit of posting an additional vacancy at equilibrium: Π F t = 0 Intuitively, a rise in the interest rate increases unemployment through firms working capital channel Firms have to borrow their recruiting costs from banks As firms have no uninsurable risk in production, they are offered the prevailing risk-free interest rate on loans If the interest rate increases, firms financial costs of recruiting new employees increase Then firms post fewer job vacancies, so the market tightness decreases It is more difficult for unemployed workers to find jobs, and hence the unemployment rate increases ) 23) 42 Investors Savings and Welfare Investors consume and save Their optimal behavior determines the prevailing risk-free interest rate Loanable funds are sufficient during normal times, but limited at credit crunches If banks liquidity constraints are binding, they have to borrow at the high interest rate the borrowers inter-temporal discount rate on consumption, β 1 Thus, the equilibrium risk-free interest rate equals the high interest rate when 21) holds and equals the investors inter-temporal discount rate on consumption otherwise Solving the investors optimization problem given by 2), the equilibrium interest rate and the investors optimal aggregate savings are shown in the following proposition: Proposition 1 At time t, the equilibrium risk-free interest rate is: Rt 1 = E t m t+1 ], where m t+1 = β 1+ζ t+1 1+ζ t, and ζ t is investors shadow price of consumption that satisfies the equilibrium condition: 1 + ζ t = β 1 βet 1 + ζ t+1 ] if 21) holds; otherwise, ζ t = 0 Investors save up to S max,t given by 22) if 21) holds; otherwise, their savings clear the credit market Proof: Appendix A1 provides a detailed proof of Proposition 1 The equilibrium risk-free interest rate is determined by the investors inter-temporal discount on consumption During normal times, banks have sufficient external funds that accept any prevailing interest rate in the economy, investors are indifferent between saving and consuming At credit crunches, banks have limited external funds that are not enough 18

19 to cover the aggregate demand for loans The interest rate is high when investors savings are not able to clear the market but decreases when they have accumulated enough wealth to clear the credit market As given by 20), S t increases but X t decreases over time at consecutive credit crunches before banks liquidity constraints are relieved At optimum, investors consume when the interest rate is low and save when the interest rate is high Funds for consumption are also available at the high interest rate Investors optimal decision on savings borrowing if negative) solves 2) Because investors are more patient than borrowers, the interest rate at which they are willing to borrow is lower than the high interest rate Thus, investors do not borrow unless they have to borrow to meet their minimum consumption requirement, which happens when their labor incomes and returns on savings are less than the investment loss from banks or the house leasing company, that is, when S max,t < 0 43 Borrowers Housing and Loan-to-Value Ratios Borrowers either finance or rent houses They are more impatient than investors, so they prefer financing their home ownerships to owning houses A borrower chooses his financed housing and loan-to-value ratio given a maximum loan-to-value ratio and a scheme of mortgage rates offered by banks The borrower can default and walk away on his mortgage Banks can punish a defaulting borrower by excluding him from obtaining mortgages in the near future under the regime with credit reporting Then the borrower has to rent houses and suffers the disutility of renting relative to owning houses Proposition 2 A borrower who rents a house demands for the quality: H r it = 1 1 φ 1 α r t 1 φ and his utility of renting houses is: U d it = Y it r t H r it + α 1 ρ 1 φ) Hr it) 1 ρ, where the equilibrium housing rent in the economy is: r t = βe t +1 ] Proof: Appendix A2 provides a detailed proof of Proposition 2 The equilibrium housing rent equals the borrowers outside option of renting houses from other borrowers The optimal decision of the house leasing company solves its problem given by 12) The profit of the house leasing company is strictly increasing with the housing rent, so the zero profit condition on renting out houses for any borrower is binding at equilibrium Borrowers demand for financed housing differs under the two credit reporting regimes: without credit reporting the reentry rate to obtaining mortgages after default equals one) and with credit reporting the reentry rate is close to zero) ) 1 ρ, Without credit reporting, borrowers are able to get mortgages again immediately after default Banks cannot punish defaulting borrowers But with credit reporting, banks can exclude defaulting borrowers from obtaining mortgages again in the near future Afraid of such punishment, borrowers are less likely to default when their mortgages are underwater the mortgage payment is 19

20 larger than the housing value) Borrowers optimal decisions on financed housing and loanto-value ratios are explored in the following proposition: Proposition 3 At equilibrium, any borrower prefers financing to renting houses He borrows up to the maximum loan-to-value ratio offered by banks: π it = π it, i, t His demand for financed housing is H it = ω it α ) 1 ρ, where his shadow price of financed housing at time t is: ω it = 1 π it ) βe t +1 M it π it ) 1 σ i,t+1 )] 24) Proof: Appendix A3 provides a detailed proof of Proposition 3 Any borrower prefers financing his home ownership to renting a house His shadow price of rented housing is larger than his shadow price of financed housing If he defaults and os excluded from the mortgage market, he has to rent houses and bears the disutiltiy of renting relative to owning If the borrower is offered mortgages, he finances his home ownership by choosing the upper limit of the loan-to-value ratio offered, because he is more impatient than investors who own banks 44 Optimal Mortgage Contracts Banks offer a maximum loan-to-value ratio and a scheme of mortgage rates to a borrower Borrowers decisions on financed housing, loan-to-value ratio, and mortgage default affect banks optimal mortgage contracting problem As shown in Proposition 3, any borrower borrows up to the maximum loan-to-value ratio offered Then banks optimal contracting problem is reduced to choose a maximum loan-to-value ratio and a mortgage rate such that i) banks break even on the mortgage; ii) the borrower s continuation value is maximized; iii) banks beliefs about the borrower s default decision coincide with his actual behavior In a competitive lending industry, banks break even on optimal mortgage contracts at equilibrium The expected return on financing a borrower s home ownership equals the cost of banks loanable funds Investors are indifferent between receiving returns on mortgages and getting the interest rate on risk-free investments Then solving 15), the relationship between loan-to-value ratios and mortgage rates under the optimal mortgage contracts satisfies: π it = M it π it E t m t+1 1 σ i,t+1 )] + E t m t+1 1 µ)+1 σ i,t+1 ], 25) where m t+1 is investors inter-temporal discount on consumption, m t+1 β A borrower s total quality of financed housing is not contractible Under the optimal mortgage contract, the borrower is interested in financing his home ownership and has no incentive to take out a second mortgage that increases his probability of default 20