Consumption and House Prices in the Great Recession: Model Meets Evidence

Transcription

1 Consumption and House Prices in the Great Recession: Model Meets Evidence Greg Kaplan, Kurt Mitman, and Giovanni L. Violante December 21, 2015 Abstract How much of the sharp drop in US nondurable consumption around the Great Recession was caused by the collapse of house prices? Existing empirical analyses combining geographical variation with the Saiz instrument place the elasticity of nondurable consumption to housing net worth in the range We provide additional evidence, based on entirely different data sources, that elasticities of this magnitude are empirically plausible. We then build an heterogeneous-agent incomplete-markets model of the US economy with various aggregate shocks (income, financial deregulation, and exuberance ) leading to fluctuations in equilibrium house prices. Through a series of counterfactual numerical experiments, we study whether this empirical methodology can properly identify the macro response of consumption to movements in house prices. Keywords: Consumption, Great Recession, House Prices. JEL Classification: E21, E30, E40, E51. We thank numerous seminar participants for useful comments. Princeton University, IFS, and NBER Institute for International Economic Studies, Stockholm University and CEPR New York University, CEPR, IFS, and NBER. 1

2 1 Introduction During the Great Recession, the US economy experienced its sharpest drop in household consumption expenditures in the postwar period (De Nardi, French, and Benson, 2012; Petev and Pistaferri, 2012). Consumption declined precipitously across all categories, not just durables. Figure 1 (left panel) shows that real detrended expenditures in nondurable consumption and services dropped by over 10 percent from their peak in early 2007 to their trough roughly five years later. This rapid fall occured at the end of a decade of markedly above-trend growth. The leading interpretation of these atypical aggregate consumption dynamics emphasizes the extraordinary swings in U.S. housing net worth that occurred since the end of the 1990s (Mian and Sufi, 2014). The right panel of Figure 1 shows that house prices grew 3 percernt per year above trend in the decade, and then collapsed with a cumulative fall close to 50 percent in the next five years. At the same time, however, the center panel of Figure 1 shows more than a 10% drop in real wages during the bust. To what extent is the plunge in housing net worth responsible for the decline in the consumption expenditures of US households in the aftermath of the Great Recession? How much of the drop in consumption was due to other shocks that induced the fall in house prices? The comovement displayed in Figure 1 suggests identification is challenging (a point stressed by Cocco and Campbell 2007). A credible answer to this question is crucial in shaping the way macroeconomists think about key issues such as consumption insurance, the sources of aggregate fluctuations, and the role of policy in mitigating the costs of business cycles. In an influential paper, Mian, Rao and Sufi (2013) hereafter referred to as MRS exploit geographical variation in the change in housing net worth (HNW, thereafter) and consumption expenditures over the period to estimate the elasticity of nondurable consumption expenditures to HNW. In order to identify the exposure of different geographical areas to the common aggregate housing shock, MRS use the local housing supply elasticity index constructed by Saiz (2010) as an instrumental variable. They obtain estimates in the range for the period : a 10 percent decline in HNW causes a reduction in nondurable consumption expenditures of percent (MRS, Appendix Table 2). Our first contribution is to verify the robustness of the MRS findings using different data on both consumption expenditures and house prices. The MRS analysis of non-durables uses pro- 2

3 0.1 Log Real Nondurable C 0.15 Log Real Wages and Salaries Log Relative House Price (2007:Q1=1) Boom Bust Year (2007:Q1=1) Boom Bust Year (2007:Q1=1) Boom Bust Year Figure 1: Left panel: real consumption expenditures in nondurables and services. Source: BEA (NIPA Table 1.1.5). Center panel: Real wages and salaries. Soure: BLS. Right panel: FHFSA national house price index deflated by the price index of expenditures in nondurable and services. All series are detrended by a linear trend estimated separately over the period prietary Mastercard data on credit-card expenditures. Instead, we use store-level sales from the Kilts-Nielsen Retail Scanner Dataset (KNRS). In MRS, data on housing net worth by geographical area are constructed using Corelogic house price indexes and Equifax data on household debt. Instead we use publicly available data from Zillow as our source of information on house prices, and the FRBNY Consumer Credit Panel to construct measures of household debt by geographical area. 1 We use the same empirical methodology as MRS to obtain identification (i.e., the Saiz instrument combined with geographical variation in consumption dynamics). However, we differ from MRS along three dimensions. First, we focus on the period because it better overlaps with the decline in both consumption and HNW (see Figure 1). Second, we use the information on local prices for each good in the KNRS data to construct a consumption quantity index in each location. What we want to capture is the drop in real consumption expenditures, and we are aware of the fact that different regions experienced different price dynamics. Third, since the KNRS bundle is not representaive of total nondurables and services, we rescale our estimate appropriately. Our preferred estimate of the elasticity of changes in nondurable consumption to changes in HNW is 0.40, and statistically different from zero, thus very much in line with the MRS estimates. 2 1 The underlying source of these latter data is Equifax too, so this portion of the data is very comparable. 2 When we repeat this exercise for the period, the same sample periodused by MRS, we obtain slightly 3

4 In the second part of the paper, we build a structural equilibrium model of the US economy to investigate further the interpretation of this micro elasticity. Our model economy is populated by overlapping generations of finitely-lived households who are subject to uninsurable idiosyncratic shocks to their efficiency units of labor, supplied inelastically. Households can save into a financial asset, a bond, whose price is set on the world market. They consume non-durable consumption the final good numeraire and housing services. Housing services can be obtained by either renting or buying houses that come in a finite number of sizes. Buyers have access to long-term mortgages and one-period home-equity lines of credit (HE- LOCs) priced competitively by financial intermediaries. Homeowners, every period, can either choose to make their mortgage payment, refinance their mortgage, or sell the house subject to a transaction fee, or default. A competitive construction sector supplies new housing every period. Several exogenous aggregate shocks can hit the economy every period: productivity in the final good sector, the degree of financial regulation in the mortgage market captured by fluctuations in a subset of model parameters that determine households borrowing limits and borrowing costs and an exuberance shock to household beliefs about future house prices. Finally, the economy is divided into regions. The only structural parameter that differs across regions is the elasticity of housing supply. The model is parameterized to match key cross-sectional features of US economy. This set-up allows us to replicate, within the model, the empirical methodology used by MRS to identify the micro elasticity. Our first experiment is to ask whether the model can replicate elasticities of the estimated magnitude. The answer is that it crucially depends on the underlying aggregate shock: the estimated elasticity is not shock invariant, its value depends on what is the underlying force moving house prices. We think this is an important finding for inference and for policy because it implies that future fluctuations in house prices due to a different composition of shocks will have different implications for consumption dynamics. Next, we ask: does this empirical elasticity measure what it is supposed to? We argue that the right theoretical interpretation of this elasticity is what we call the goldmine experiment, i.e., a counterfactual where aggregate house prices change for entirely exogenous reason, e.g., the discovery of a goldmine (or the discovery the original news were false) under each ownersmaller elasticities, but not statistically significant from the ones estimated for

5 occupied house in the economy. When we engineer this experiment in the model, we uncover a downward bias in the IV estimator of the elasticity. Finally, we analyze whether this micro elasticity (even in its unbiased version) can be applied directly to quantify how much of the drop in aggregate consumption was caused by the decline in HNW around the Great Recession. We therefore run a counterfactual experiment where the only shock hitting the economy in the Great Recession is the income shock (that is, we simulate an economy with no financial deregulation and no exuberance). We conclude that, in absence of the shocks that were responsible for the bulk of the collapse in house prices (over 95 pct in the model) aggregate consumption would have declined by less than a third, compared to the data. These magnitudes imply a macro elasticity of nondurable consumption to HNW, for the Great Recession episode, of XYZ. 1.1 Related Literature [To Be Completed] The rest of the paper is organized as follows. Section 2 contains our empirical analysis providing an alternative micro estimate of the elasticity of consumption to house prices. Section 3 outlines the model, the equilibrium concept, and our approach to numerical computation. Section 4 describes the model s parameterization. Section 5 presents the results from all our numerical experiments. Section 6 concludes the paper. The Appendix includes more details about the empirical analysis and the computational algorithm, including some accuracy tests. 2 Empirical Analysis In this section, we verify the robustness of the MRS findings using different data on both consumption expenditures and housing net worth. In contrast to MRS, all our data sources are from publicly available sources, accessible by most academic researchers. We then provide four additional analyses that provide important context for confronting our structural model with data. First, we show how the estimated elasticity varies across different time-periods and horizons around the Great Recession period. Second, we separate the price and quantity components in the drop of nominal consumption expenditures. Third, we demonstrate the sharp non-linearity 5

6 of the consumption drop with respect to the size of the drop in housing net worth. Fourth, since our measure of consumption covers only a subset of expenditures in nondurables and services, we rescale our estimate of the elasticity to the entire bundle of nondurable and services. 2.1 Data sources The measure of expenditure in the baseline analyses in MRS is from data on new vehicle registrations. Purchases of consumer durables, such as automobiles, are highly cyclical and significantly more volatile than non-durable expenditures. To address this concern, MRS conduct a limited analysis of non-durable expenditures using proprietary Mastercard data on credit-card expenditures from a 5% random sample of the population in each county. Instead, we exploit a new source of widely-available nondurable consumption expenditure data. The data is storelevel sales from the Kilts-Nielsen Retail Scanner Dataset (KNRS). The KNRS is a weekly panel dataset of total sales (quantities and prices) at the UPC (barcode) level for around 40,000 geographically dispersed stores. From this weekly-upc level data we construct an annual storelevel panel of total sales. We then aggregate sales across all stores in a county to obtain a statistic that is indicative of the expenditures of the households who live in that county. In MRS, data on housing net worth by geographical area are constructed using Corelogic house price indexes and Equifax data on household debt. Instead we use publicly available data from Zillow as a source of information on house prices, and the FRBNY Consumer Credit Panel to construct measures of household debt by geographical area. The underlying source of these latter data is Equifax itself, so this portion of the data is very comparable. The main difference between Corelogic and Zillow is that the former is a repeat-sale index, whereas the latter is a hedonic price index that includes sales of new homes as well. 2.2 Baseline estimates Our OLS specification follows closely the specification in MRS. We regress the four-year changes in annual store-level sales from 2006 to 2010, and 2007 to 2011, on our estimates of the change in county-level housing net worth over the same time-period. We focus on the four-year changes since these provide a convenient mappings to our structural model, which is cast in a two-year 6

7 OLS IV OLS IV OLS IV Panel A: County level HNW 0.219** 0.342** 0.198** 0.465** 0.207** 0.210* (0.024) (0.098) (0.031) (0.124) (0.025) (0.085) N Clusters R Panel B: CBSA level HNW 0.263** 0.491** 0.260** 0.543** 0.239** 0.405** (0.024) (0.106) (0.027) (0.131) (0.029) (0.089) N Clusters R Table 1: Elasticity of non-durable expenditures to housing net worth model period. Our dependent variable (sales) is measured at the store-level, while our independent variable is measured at the county level. We weight observations by store-level sales in the initial year of each time period (alternative weighting possibilities have little effect on the results), and we cluster by county when computing standard errors. Our IV regressions also follow closely the specification in MRS. We use the estimates of housing supply elasticities from Saiz (2010) to instrument for changes in housing net worth. This instrument is provided at the CBSA level, and is not available for all of the counties in which we observe store-level expenditure changes (both because some counties are not in a CBSA and because not all CBSAs are covered by the Saiz data). As a result, the OLS and IV samples differ. For the four-year period 2006 to 2010, we obtain a baseline elasticity estimate of 0.22 using OLS, and 0.34 using IV (Table 1, Panel A). For the period the elasticity is 0.20 using OLS and 0.47 using IV. All these estimates are significant at the 1% level. Using non-durable expenditure data from Mastercard, MRS find remarkably similar estimates to ours. In Appendix Table 2 of their article, MRS report elasticities of housing net worth to non-durables of 0.34 to 0.38 using the same instrument but different data on house prices and consumption. There are two differences however between our sample and the one used by MRS: MRS focus on the three-year period , and MRS aggregate housing net worth to the CBSA 7

8 level. In Panel B of Table 1, we report corresponding elasticity estimates using housing net worth data at the CBSA level (we cluster by CBSA to compute standard errors in this case). At this level of geographic aggregation we estimate even larger elasticities: for the period these are 0.26 (0.49) using OLS (IV), and for the period these are 0.26 (0.54) using OLS (IV). Focussing on the three-year period 2006 to 2009, the estimates are slightly smaller using OLS, 0.41 using IV - and very close to what MRS find for the corresponding period. 2.3 Consumption vs expenditure Our baseline results have so far focused on nominal consumption expenditure. While this measure is of potential first order importance for understanding the transmission of house price changes into changes in aggregate activity, it provides an imperfect measure of the change in the real consumption of goods by households (e.g., in seminal work Aguiar and Hurst (2005) document that changes in food expenditures are not equivalent to changes in food consumption). Since it is real consumption that matters for the welfare of households, understanding the effect of changes in housing wealth on the quantity of nondurable goods consumed is equally important. Table 2 shows that when using consumption as a dependent variable the elasticity estimates are considerably lower than the estimates from the analysis based on the expenditure measure. For the period the IV estimates falls from 0.34 to 0.21 (a reduction of 38%), and for the period the IV estimates falls from 0.47 to 0.36 (a reduction of 23%). These findings suggest that a significant portion of the drop in consumption expenditures is due to equilibrium prices falling in response to the negative demand shock. This specification - county level housing net worth data and real consumption over four year periods - represents our preferred estimates of the elasticity of consumption to housing net worth. We thus summarize our findings as consistent with an elasticity for the Great Recession period of around

9 OLS IV OLS IV OLS IV Panel A: County level HNW 0.169** 0.212* 0.139** 0.358** 0.161** (0.025) (0.101) (0.028) (0.123) (0.026) (0.090) N Clusters R Table 2: Elasticity of real non-durable consumption to housing net worth Mean log change in store level total sales, Log change in Housing Net Worth, Figure 2: Mean log change in store level sales from Linear and non-linear fit lines. 2.4 Non-linearities Our findings suggest that, as in MRS, there is a nonlinearity in the relationship between changes in consumption and changes in house prices: larger declines in net worth are associated to smaller consumption responses. These non-linearities can be seen in Figure 2. [To Be Completed] 2.5 From the Kilts-Nielsen bundle to total nondurables expenditures A possible concern throughout our empirical analysis is that our measure of household consumption expenditure obtained from the Kits-Nielsen (KNRS) data may be rather narrow. One 9

10 may worry that these categories could display different dynamics than total nondurable expenditures, the object of interest in the model. In this section, we use the Consumption Expenditure survey (CE) to estimate the elasticity of total nondurable consumption to a subset of expenditures that is as close as possible to the KNRS bundle. This number will be then used to rescale the estimated elasticity (0.3) of the KNRS bundle to housing net worth shocks. Our starting point is the sample constructed from the CE by Aguiar and Bils (2015). 3 These data group expenditures into 20 categories (see their Table 2). Based on the composition of the KNRS data, described earlier, we define expenditures in the goods contained in the KNRS bundle for household i at date t ( cit KN ) as the sum of food at home (corresponding to dry grocery, frozen foods, dairy, deli, packaged meat, fresh produce in the KNRS data), alcohol, and personal care (corresponding to health and beauty aids and non-food grocery in KNRS data). The KNRS data also include a residual category called general merchandise (that includes computer/electronic, cookware, apparel, film/cameras, lawn & garden, etc.), whose closest groups in the Aguair-Bils data are clothing and appliances. Since appliances are really a durable good, we ll also repeat the exercise without it. Our definition of total nondurable/services ( c ND it ) consumption follows closely the one adopted in Blundell, Pistaferri, and Preston (2008): among the Aguiar-Bils categories, we include food at home and away, alcohol, personal care, clothing, tobacco, plus expenditures on transportation, recreation, and domestic services. In particular, with respect to the NIPA definition, we follow Blundell, Pistaferri, and Preston and exclude expenditures on healthcare, education, financial and insurance services because of their more durable nature that more closely assimilates them to investment and saving activities. Moreover, we exclude housing and utilities because they are a separate good in the model. When we include appliances to the KNRS bundle, we also add it to nondurable consumption. In our CE sample, mean annual KNRS expenditures without (with) appliances are $6,495 ($7,736) and mean nondurable expenditures are $13,339 ($14,580). Thus, the fraction of our strict definition of nondurables/services accounted for by the KNRS bundle is between pct, hence sizable. For comparison, the same calculation from NIPA Table (without appliances) yields 46 pct. 3 We refer the reader to their paper and their online appendix for details on sample selection. It suffices to remind here that its is a sample of urban households, aged with a full year of interview coverage for whom no income imputation was made. Moreover, certain households with expenditure outliers have been excluded. 10

11 Dependent variable: log cit ND log cit ND log cit KN log cit KN Regressor log cit KN log cit KN log cit ND log cit ND With appliances N Y N Y Other controls Y Y Y Y ˆβ S.E. (0.003) (0.003) ˆγ S.E. (0.003) (0.003) R N observations 33,761 33,761 33,761 33,761 Table 3: Estimation of the elasticity of household expenditures on nondurable and services to expenditures on the Kilts-Nielsen bundle. Source: CE data We use cross-sectional variation to identify the elasticity of nondurable to KNRS expenditures. Consistently with the boom-bust period of interest, we focus on the period. In the regression we control for year dummies, which capture changes in relative prices of the KNRS bundle to total nondurables, and a number of demographics: four dummies for marital status, gender of the reference person, an age polynomial, eight levels of education, and the square root of family size to capture the fact that economies of scale may differ across different goods. In sum, we estimate: log c ND it = β 0 X it + β 1 log c KN it + ε it, (1) where X it are the controls described above and the coefficient of interest is β 1. It is well known that consumption expenditures are measured with error in survey data, and the CE is no exception. In order to arrive at a plausible estimate of β 1, we therefore also run the inverse regression, with log c KN it as dependent variable and log c ND it as key regressor with associated coefficient γ 1. Then, we know that the true value for β 1 will be bracketed in the interval [ ˆβ 1, 1/ ˆγ 1 ]. 4 The results of these regressions and their inverse regression counterparts are displayed in Table 3. Overall, it appears that the elasticity of nondurable consumption to the KNRS bundle is bracketed between [0.84, 1.34] when we exclude appliances, and [0.90, 1.35] when we include 4 This result is true also when there is measurement error in both the dependent variable and the regressor. The lower bound nature of ˆβ 1 is due to sn attenuation bias, the upper bound nature of 1/ ˆγ 1 is due to an endogeneity bias. 11

12 them. The estimates are very precise. 5 Averaging across the mean values of these two intervals, gives an elasticity of 1.1. When we use this factor to rescale the elasticity of KNRS expenditures to house price shocks, we obtain a target elasticity of Model It is useful to succintly delineate the main features of the model, before providing a formal description. The economy is populated by overlapping generations of households whose lifecycle is divided between work and retirement. During the work stage, they are subject to uninsurable idiosyncratic shocks to their efficiency units of labor, supplied inelastically. Household can save into a financial asset, a bond, whose price is set on the world market. They consume non-durable consumption the final good numeraire and housing services. Housing services can be obtained by either renting or buying houses that come in a finite number of sizes. Buyers have access to long-term mortgages priced competitively by financial intermediaries. Homeowners, every period, can either choose to make their mortgage payment, refinance their mortgage or sell the house subject to a transaction fee, or default. Defaulting results in foreclosure by the intermediary which entails a utility loss, a temporary exclusion from borrowing, and a loss of value of the house. Owning a house also allows one to open HELOCs, modelled as one-period non defaultable debt-contracts. A competitive construction sector supplies new housing every period. Three types of exogenous aggregate shocks can impact the economy every period: aggregate productivity, the degree of financial regulation in the mortgage market, and beliefs about future hous prices. It is convenient to postpone the exact definition of these shocks to Section 3.5, after we have outlined the rest of the model in detail. Finally, the economy is divided into a finite number I of regions. The only structural parameter that differs across regions is the elasticity of housing supply. Inputs are not mobile across regions, and goods are non-tradable. Effectively, every region is isolated from the others, and 5 As a validation of our estimate, we also estimate the elasticity of food consumption to nondurable consumption, allowing for the same controls, an exercise similar to that conducted by Blundell et al. (2008) from PSID data. We estimate a value of (S.E.0.003) which is again very precise and very close to the one estimated by Blundell et al.,i.e., (S.E ). 12

13 thus in what follows we describe the model economy and define equilibrium for a generic region. It is only when at the simulation stage that we aggregate across regions and, as explained in the Introduction, we exploit geographical variation of equilibrium outcomes to replicate the MRS IV methodology within the model. In illustrating the model, we begin by outlining all the model primitives that are needed to describe the household decisions, and we lay out the household problems. Next, we present the financial intermediation sector and the production sides of the economy. Finally, we define the equilibrium. Throughout, we adopt a recursive formulation of the economic environment in discrete time. 3.1 Households Demographics: The economy is populated by a measure-one continuum of finitely-lived households. Age is indexed by j = 1, 2,, J. Households work in the first phase of their life cycle and, at age J ret, they retire. They die with certainty at the end of period J. Preferences: Expected lifetime utility of the household is given by: [ ] J E 0 β j 1 u j (c j, s j ) + β J v( ) j=1 (2) where β > 0 is the discount factor, c j > 0 is consumption of non-durables at age j, s j > 0 is the consumption of housing services. Nondurable consumption is the numeraire good of the economy. The expectation is taken over sequences of aggregate and idiosyncratic shocks that we specify below. The function v measures the felicity from leaving bequests > 0. 6 Specifically, for u we assume: u j ( cj, s j ) = [ ( cj ) 1 γ ( sj ) ] 1 σ 1 γ 1 γ (1 φ) e j + φ e j 1 1 σ, (3) 6 This bequest motive prevents households from selling their house and dissaving too much during retirement, which would be highly counterfatual. 13

14 where φ measures the relative taste for housing, 1/γ measures the elasticity of substitution between housing services and non-durables, and 1/σ measures the IES. The expenditures equivalence scale { } e j captures deterministic changes in household size and composition over the life cycle and explains why u is indexed by j in (2). The warm-glow bequest motive at age J takes the functional form: v ( ) = ψ ( + )1 ϑ 1, (4) 1 ϑ as proposed by De Nardi (2014). The term ψ measures the strength of the bequest motive, while reflects the extent to which bequests are luxury goods. If > 0, the marginal utility of small bequests is bounded, while the marginal utility of large bequests declines more slowly than the marginal utility of consumption. Endowments: by: Working households receive an idiosyncratic labor income endowment y w j given log y w j = w + χ j + ɛ j (5) where w is the aggregate wage rate per efficiency units of labor. Individual labor productivity has two orthogonal components: χ j is a deterministic age profile, and ɛ j E j is a stationary idiosyncratic stochastic process. Households are born with initial wealth endowment b 1 drawn from an exogenous distribution that integrates up to the overall amount of wealth bequeated in the economy by the deceased households. Housing: In order to consume housing services, households have the option of renting or owning a home. Houses are characterized by their size, whose number is finite. For owneroccupied housing, house size belongs to the set H = {h 0,..., h N }, where h 0 < h 1,..., h N 1 < h N. For rental units, size belongs to the set H = { h 0,..., h N }. Renting generates housing services one-for-one with the size of the house, i.e. s j = h j. To capture the fact there may be additional utility from home ownership, we assume that an owner-occupied house generates s j = ωh j units of housing services, with ω > 1. The rental rate of housing is denoted by ρ. The per-unit price of housing is denoted by p h. Owner-occupied 14

15 houses have a per-period maintenance and tax cost of (δ h + τ h )p h h, expressed in units of the numeraire good. Maintenance fully offsets physical depreciation of the dwelling δ h. When a household sells its home, it incurs a transaction cost κ h (p h h). Homeowners of age j are subject to an i.i.d. moving shock that occurs with probability θ j : when hit by the shock they must sell (or default, see below). Financial Instruments: Households can save in one-period bonds, b, at the price q b exogenously determined by the net supply of financial assets from the rest of the world. It is convenient to also define the interest rate on bonds r b := 1/q b 1. We do not allow unsecured credit in the baseline economy, but we permit homeowners to borrow up to a fraction λ b of the equity in their house at an interest equal to r b = r b (1 + ι b ), where ι b > 0 is an intermediation wedge. In what follows to lighten the exposition the notation, with a slight abuse of notation, we keep denoting the interest rate on liquid assets r b but it is implicit that it equals r b use a similar convention for q b. when b < 0. We Housing purchases can be financed by taking on mortgages. All mortgages are long-term and amortized over the remaining life of the buyer at the real interest rate r m equal to r b times a proportional intermediation wedge (1 + ι m ) > 1. Newly originated mortgages are subject to a fixed origination cost κ m. They must also respect a maximum loan-to-value (LTV) ratio limit: the initial principal balance must be less than a fraction λ m of the value of the home, m λ m p h h. Mortgage holders have the option to refinance, by repaying the principal balance remaining and originating a new mortgage. If a household chooses to sell its home, it is also required to pay off its remaining mortgage balance. Households have also the option to default on their mortgage debt. Upon default, mortgages are designated as the primary lien on the house, implying that the proceeds from the foreclosure are disbursed to the mortgage. We assume no recourse in case of foreclosure. Foreclosing reduces the value of the house to the lender because it is the lender who must pay property taxes τ h and maintenance and the foreclosed house depreciates more than regular houses, i.e. depreciation in case of default δ d h is larger {( ) } than regular depreciation δ h. Thus the lender recovers min 1 δ d h τ h p h h, (1 + r m ) m. A household who defaults must rent the smallest house size for the rest of the period. In addition, it also incurs a utility penalty ξ in the period of default. 15

16 A household of age j that takes out a mortgage with principal balance m receives q m m units of the numeraire good in the current period, with q m 1. Thus, the down payment required by the household at origination is p h h q m m. Going forward, the household makes J j equal mortgage payments π m that must exceed the minimum mortgage payment: π m = r m(1 + r m ) J j (1 + r m ) (J j) m, (6) 1 and the remaining principal evolves as m = m(1 + r m ) π m. Note that the principal due on a mortgage of size m is not equal to the funds received from the bank at the time of purchase (q m m) because the pricing of the mortgage accounts for the possibility of default. One can interpret this as so-called points or other up-front interest rate charges that households face when taking out their loans. Even though all households pay the same interest rate on the principal due, the heterogeneity in principals m maps into heterogeneous effective interest rates. In simulations, when a household originates a mortgage of size m, we can use q m and π m to solve for the effective interest rate r m on the mortgage through the relationship: π m q m m = r m(1 + r m) J j (1 + r m) (J j) 1. (7) This formula solves for the interest rate r m that would yield constant mortgage payment schedule π m on an outstanding balance of q m m (the funds received at origination). 7 Note also that, once a mortgage is originated, the household continues to make its mortgage payments and there is no further requirement that p h h < m. This realistic assumption, crucial to understand deleverage behavior and, thus, consumption response to house price shocks, sets our model apart from several notable contributions in this literature (e.g., Favilukis et al., 2014; Iacoviello and Neri, 2010). Section 3.2 below provides the exact expression for the equilibrium price q m. Government: The government spends an amount G on services that are not valued by households. It also runs a PAYG social security system. Retirees receive social security benefits 7 A richer model would allow the households to simultaneously choose the amortization interest rate r m and the principal m so that effectively they could choose q m m, as is common in the data. However, this formulation would add a state variable (the amortization rate), and for tractability we impose the fixed amortization interest rates. 16

17 y ret (z J w), where the argument of the benefit function proxies for average gross lifetime earnings. In what follows, we adopt the notation y j for income at age j, with the convention that if j < J ret then y j = y w j and y j = y ret otherwise. We also denote by Γ j y the age-dependent transition matrix for income. To finance these expenditures, the government levies a property tax τ h on the value of the house, a flat payroll tax τ ss and a progressive labor income tax τ y (y j ). Households can deduct the interest paid on mortgages against their taxable income. We denote the combined income tax liability function T (y j, m j ). A final source of revenues for the government comes from the proceedings of the sale of land permits for construction, as described in more detail in Section 3.3 below Household Decision Problems To simplify the notation, we let Ω O denote the vector of aggregate state variables, defined below. We begin by stating the problem of non-homeowners (renters and buyers). Next we state the problem of home-owners (sellers, keepers who repay, keepers who refinance, and households who default). Finally, we describe the problem of the retiree in its last period of life, when the warm-glow bequest motive is active. Renters and buyers: Let V n denote the value function of households who start the period without owning any housing. These households choose between being a renter and buying a house to become an owner by solving: V n (b j, y j ; Ω) = max { V r (b j, y j ; Ω), V o (b j, y j ; Ω) }, (8) where we let g o ( b j, y j ; Ω ) {0, 1} denote the decision to own a house. 8 8 It is implicit that, when this decision takes the value of zero, the household chooses to be a renter. 17

18 Those who choose to rent solve: V r (b j, y j ; Ω) = s.t. [ max u j (c j, s j ) + βe yj,ω V n (b j+1, y j+1 ; Ω ) ] (9) c j, h j,b j+1 c j + ρ (Ω) h j + q b b j+1 b j + y j T (y j, 0) b j+1 0 s j = h j H y j+1 = Γy j ( ) yj, Ω = Γ Ω (Ω) Those who choose to buy and become owners solve: V o (b j, y j ; Ω) = [ ] max u j (c j, s j ) + βe yj,ω V h (b j+1, h j+1, m j+1, y j+1 ; Ω ) c j,b j+1,h j+1,m j+1 (10) s.t. c j + q b b j+1 + p h (Ω) h j+1 + κ m b j + y j T (y j, 0) + q m (Ω) m j+1 m j+1 λ m p h (Ω) h j+1 b j+1 λ l max { p h (Ω) h j+1 m j+1, 0 } h j+1 H, s j = ωh j+1 y j+1 = Γy j ( ) yj, Ω = Γ Ω (Ω) where V h ( ) is the value function of a household that starts off the next period as a homeowner that we describe below. 9 Homeowners: A household that owns a home may be forced to move with probability θ j at the beginning of the period. 10 If the household isn t forced to move, it has the option to keep the house and make the appropriate mortgage payment, refinance the house, sell the house, or default (obviously, this latter option could only be optimal if the household has a mortgage). If 9 Note that the timing is such that a household can already cash out its home equity via a HELOC in the period of purchase. 10 We omitted this shock from the description of the renter s problem because it has no effect if we assume that renters can move across rental units costlessly. 18

19 it is forced to move, then it must either sell or default. V h (b j, h j, m j, y j ; Ω) = (1 θ j ) max }{{} Not moving + θ j }{{} Moving max Pay: V p (b j, h j, m j, y j ; Ω) Refinance: V f (b j, h j, m j, y j ; Ω) Sell: V n ( b j, y j ; Ω) Default: V d (b j, y j ; Ω) Sell: V n ( b j, y j ; Ω) Default: V d (b j, y j ; Ω) (11) It is also convenient to denote the refinance decision by g f ( b j, h j, m j, y j ; Ω ), the selling decision by g n ( b j, h j, m j, y j ; Ω ), and the mortgage default decision by g d ( b j, h j, m j, y j ; Ω ). All these decisions are dummy variables in {0, 1} and it is implicit that, when they are all zeros, the homeowner chooses to repay its mortgage during that period. We now describe all these four options one by one. A household that chooses to make a mortgage payment solves: V p (b j, h j, m j, y j ; Ω) = [ ] max u(c j, s j ) + βe yj,ω V h (b j+1, h j+1, m j+1, y j+1 ; Ω ) c j,b j+1,π m (12) s.t. c j + q b b j+1 + (δ h + τ h ) p h (Ω) h j + π m b j + y j T ( ) y j, m j m j+1 = (1 + r m ) m j π m π m π m b j+1 λ l max { p h (Ω) h j m j+1, 0 } s j = ωh j, h j+1 = h j y j+1 = Γy j ( ) yj, Ω = Γ Ω (Ω) Note that because the mortgage is long-term, there is no requirement that the principal outstanding on the mortgage be less than λ m times the current value of the home. If the aggregate house price had declined, the household could be underwater on its mortgage, but so long as it continues to make its mortgage payment it is not forced to deleverage. HELOCs because 19

20 they are refinanced each period are instead subject to a period-by-period constraint on the balance relative to the current value of the home equity. An homeowner who chooses to refinance its mortgage solves the following problem: V f (b j, h j, m j, y j ; Ω) = [ ] max u(c j, s j ) + βe yj,ω V h (b j+1, h j+1, m j+1, y j+1 ; Ω ) c j,b j+1,m j+1 (13) s.t. c j + q b b j+1 + (δ h + τ h ) p h (Ω) h j + (1 + r m ) m j + κ m b j + y j T ( y j, m j ) + qm (Ω) m j+1 m j+1 λ m p h (Ω) h j b j+1 λ l max { p h (Ω) h j m j+1, 0 } b j+1 0 s j = ωh j, h j+1 = h j y j+1 = Γy j ( ) yj, Ω = Γ Ω (Ω) A homeowner that chooses to sell its home solves the problem as if it started the period without any housing, i.e., with value function V n with financial assets equal to its previous holdings plus the net-of-costs proceeds from the sale of the home, i.e. b j = b j (1 + r m ) m j κ h ( ph h j ) + (1 δh τ h ) p h h j (14) The timing ensures that a household can sell and buy a new home within the period. 20

21 Finally, a household that has defaulted on its mortgage solves: V d [ (b j, y j ; Ω) = max u(c j, s j ) ξ + βe yj,ω V r (b j+1, y j+1 ; Ω ) ] (15) c j,b j+1 s.t. c j + ρ (Ω) h 0 + q b b j+1 b j + y j T (y j, 0) b j+1 0 s j = h 0 y j+1 = Γy j ( ) yj, Ω = Γ Ω (Ω) Recall that, in the period following foreclosure, the household must rent a house of size h 0 and is subject to a utility penalty ξ. Next period, the household starts without any housing and can either rent or buy another house. Bequest: In the last period of life, j = J, the warm-glow inheritance motive, apparent from preferences in (2), induces households to leave a bequest. For example, a retired homeowner of age J (who does not sell his house in this last period) would solve: V p (b J, h J, m J, y J ; Ω) = max c J,b J+1, u(c J, s J ) + βe Ω [v ( )] (16) s.t. c J + q b b J+1 + (1 + r m ) m J b J + y J T (y J, m J ) = b J+1 + (1 δ h + τ h ) p h ( Ω ) h J+1 κ h ( ph ( Ω ) h J+1 ) s J = ωh J, h J+1 = h J Ω = Γ (Ω) In other words, in the last period of life households pay off their residual mortgage and HELOC and take into account that their residual housing wealth contributes to bequests only as the expected net-of-costs proceedings from the sale, next period. 21

22 3.2 Financial Intermediaries The financial intermediation sector is perfectly competitive with free entry. Loans are therefore priced through a zero-profit condition that holds loan by loan. The pricing of the mortgage can be defined recursively as in Chatterjee and Eyigungor s (2012) long-term sovereign debt default model, adapted here to collateralized debt and finite lifetimes. Mortgage prices depend on the age j of the homeowner, all its choices of assets and liabilities for next period x h j+1 := (b j+1, h j+1, m j+1 ), on its current income state ( y j ), and on the current aggregate state vector Ω. Thus, we can write: q m (x h j+1, y j; Ω) = 1 {[ ] E (1 + r m ) yj,ω g n (x h j+1 m, y j+1; Ω ) + g f (x j+1, y j+1 ; Ω ) (1 + r m ) m j+1 j+1 ( + g d x h j+1, y j+1; Ω ) ( ) min 1 δ d h τ ( h p h Ω ) h j+1, (1 + r m ) m j+1 [ ] ]} + 1 g n ( ) g f ( ) g d ( ) [π m + q m (x h j+2, y j+1; Ω )m j+2 (17) Intuitively, if the households sells (g n = 1) or refinances ( g f = 1 ) its home, it has to payoff the mortgage, so the financial intermediary receives the full principal plus interest. If the household defaults on the mortgage ( g d = 1 ), the intermediary forecloses and recovers the minimum between the depreciated value of the home and the value of the residual mortgage debt. If the household makes a payment on the home ( g n = g f = g d = 0 ), the value to the intermediary is the contemporaneous value of the mortgage payment, plus the continuation value of the remaining balance of the mortgage going forward which is compactly represented by the pricing function. 11 Finally, one should note that, these zero-profit conditions hold in expectation only. Thus, strictly speaking, because of the aggregate risk, along the equilibrium path the financial intermediaries would be making profits and losses. We assume that the financial intermediaries (only) have access to a full set of Arrow securities that span the aggregate risk with the rest of the world and therefore make zero profits period by period. 11 Note that a lender who observes x j+1 can compute next-period decisions x j+2 for each possible future realization z j+1, Ω, and the moving shock θ j+1. 22

23 3.3 Production Production in the economy is divided between two sectors: the final good sector which produces non-durable consumption (the numeraire good of the economy), and a construction sector which produces houses. Labor is perfectly mobile across sectors. Final Good Sector: The final good sector operates a constant returns to scale technology C = ZN c (18) where Z is the productivity level, and N c are units of labor services. From the competitive firm problem the wage is simply w = Z, and thus exogenous. Construction Sector: The competitive construction sector operates with production technology I h = Z (N h ) α ( L) 1 α, with α (0, 1), where N h are units of labor services in this sector, and L is a fixed amount of new effective land that becomes available for construction each period. In other words, every period the government issues new permits equivalent to L units of buildable land, and follow Favilukis et al. (2015) in assuming that permits are sold at market price to developers, and thus all rents accrue to the government. The developer therefore solves the static problem: max N h p h Z (N h ) α ( L) 1 α wn h (19) which, after substituting the equilibrium condition w = Z, implies labor demand and housing investment functions: N h = ((1 α) p h ) 1 1 α, (20) I h = (p h ) α 1 α (α) α 1 α. (21) Note that the housing supply price-elasticity is α/ (1 α). This is what varies across regions. 23

24 3.4 Rental Sector: A competitive rental sector owns housing units and rents them out to households, subject to an operating cost per unit of housing equal to ϕ. Rental companies can buy and sell units frictionlessly on the housing market. The problem of the representatve rental company is: J( H; Ω) = max p h (Ω) [ H (1 δ h τ h ) H ] ( ) + (ρ (Ω) ϕ) H 1 [ + H 1 + r b E Ω J( H ; Ω ) ] Optimization implies that the equilibrium rental rate equals the user cost of housing, or: ρ (Ω) = p h (Ω) + ϕ ( ) 1 δh τ h [ 1 + r b E Ω ph (Ω ) ], which also establishes a relationship between equilbrium rent and equilibrium hous prices. 3.5 Aggregate Risk We assume that there are three sources of aggregate risk in our economy: (i) aggregate productivity Z (hence the wage), (ii) the world interest rate r b, and (iii) a set of parameters that characterize the degree of financial regulation: the maximum loan-to-value ratio at mortgage origination λ m, the mortgage rate intermediation wedge ι, and the mortgage origination cost κ m. We compactly denote the vector of exogenous shocks (Z c, r b, λ m, ι, κ m ) as Z. All the components of Z are assumed to follow a stationary Markov chain. Because of aggregate risk and incomplete markets, the equilibrium distribution of households µ is a state variable needed to forecast house prices, the only endogenous price in the economy. Thus the vector of aggregate states is Ω = (Z, µ). 3.6 Equilibrium To ease notation, in the definition of equilibrium we denote the vector of endogenous states for age-j homeowners as x h j := ( ) b j, h j, m j, l j X h j. Similarly, let x n j := (b j ) X n j denote the endogenous states for non-homeowners, respectively. Let µ := (µ o, µ n ) be the measure of these different types of households, with µ o + µ n = 1. 24

25 { ( ) ( ) A recursive competitive equilibrium consists of value functions V n x n j, z j; Ω, V r x n ( j, z j; Ω, ) ( ) ( ) ( ) ( )} V o x n j, z j; Ω, V h x h j, z j; Ω, V p x h j, z j; Ω, V f x h j, z j; Ω, V d x n j, z j; Ω, decision rules { ( ) ( ) ( ) ( ) ( ) ( ) g o x n j, z j; Ω, g n x h j, z j; Ω, g f x h j, z j; Ω, g d x h j, z j; Ω, c h j x h j, z j; Ω, c n j x n j, z j; Ω, ( ) ( ) b h j+1 x h j, z j; Ω, b n j+1 x n j, z j; Ω, h ) ) ( ) ( ) j (x n j, z j; Ω, h j+1 (x n j, z j; Ω, m n j+1 x n j, z j; Ω, m h j+1 x h j, z j; Ω, ( ) ( )} l n j+1 x n j, z j; Ω, l h j+1 x h j, z j; Ω, a wage function w (Ω), rental rate ρ, house price function { } p h (Ω), mortgage and HELOC price functions q m (x h j+1, z j; Ω), q l (x h j+1, z j; Ω), aggregate functions for construction labor, rental units stock, property housing stock, housing investment, and government expenditures { N h (Ω), H (Ω), H (Ω), I h (Ω), G (Ω) }, and a law of motion for the aggregate states Γ such that: 1. Household optimize, by solving problems (8)-(16), with associated value functions { V n, V r, V o, V h, V p, V f, V d} { and decision rules g o, g n, g f, g d, c h j, cn j, bh j+1, bn j+1, h j, h j+1, m h j+1 }, m n j+1, lh j+1, ln j Firms in the construction sector maximize profits, by solving (19), with associated labor demand and housing investment functions {N (Ω), I h (Ω)}. 3. The labor market clears at the wage rate w = Z c, and labor demand in the final good sector is determined residually as N c = 1 N h (Ω). 4. The financial intermediation market clears loan-by-loan with prices determined by conditions (17) and (??). Financial intermediaries makes zero profits in expectation, every period. 5. The rental market clears at price ρ and the equilibrium quantity of rental units satisfies: H (Ω) = J j=1 ˆ + [ˆ + h 0 ˆ X n j Z j O X h j Z j O X h j Z j O ) [ h j (x ( )] n j, z j; Ω 1 g o x n j, z j; Ω dµ n (22) h j ( bn j ( x h j, Ω ), z j ; Ω ( ] ) g d x h j, z j; Ω dµ h ) [1 g o ( bn j (x h j, Ω ) )] ( ), z j ; Ω g n x h j, z j; Ω dµ h 25

26 where the LHS is the supply of rental units and the RHS is the demand of rental units by continuing renters, households who sell and become renters, and households who foreclosed on their property.the notation b ) n j (x h j, Ω represents the financial wealth of the seller, after the transaction, see equation (14). 6. The housing market clears at price p h (Ω) and the equilibrium quantity of housing, measured at the end of the period after all decisions are made, satisfies: H (Ω) + I h (Ω) = J j=1 ˆ + ˆ ˆ [ˆ X h j Z j O X h j Z j O X h j Z j O X h J Z J O h j [(1 ( )) ( ( ))] g n x h j, z j; Ω 1 g d x h j, z j; Ω dµ h (23) ) h j+1 (x n j, z j; Ω ( ) g o x n j, z j; Ω dµ n h j [g ( ) ( ] )] n x h j, z j; Ω + g d x h j, z j; Ω dµ h ) h J+1 (x h J, z J; Ω dµ h with I h (Ω) 0. The first term represents continuing homeowners who pay their mortgage or refinance, the second term captures the demand for housing from new buyers, the third term the sale of houses and the foreclosed properties that are back on the market, and the last term the houses sold on the market when the wills of the deceased are executed. Recall that depreciation does not feature in the above equation because households maintenance expenditures offset it exactly Precisely, the term δ h H on the LHS would be offset by the term p 1 (δ h h p h ) h j (the maintenance cost in consumption units expressed in housing units) integrated across all homeowners. 26