Family labor supply and asset returns

European Economic Review 124 (2020) 103389

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European Economic Review journal homepage: www.elsevier.com/locate/euroecorev

Family labor supply and asset returnsR Claudio Daminato a,∗, Luigi Pistaferri b a b

Department of Management, Technology and Economics, ETH Zurich, Switzerland Stanford University, SIEPR, NBER and CEPR, USA

a r t i c l e

i n f o

Article history: Received 31 December 2018 Accepted 18 January 2020

JEL classification: D12 D91 J22

a b s t r a c t We study how shocks to wages and asset prices affect consumption and labor supply decisions of two-earner households in a life-cycle framework with housing and flexible preferences. We provide a micro-foundation of the wealth effects on consumption and labor supply. We find evidence for non-separability between non-durable consumption, housing services and hours of work. The results also highlight the importance of family labor supply in understanding how households respond to shocks to financial and housing markets. © 2020 Elsevier B.V. All rights reserved.

Keywords: Wealth effect Consumption Labor supply

1. Introduction Understanding how households respond to shocks to economic resources has a long history in economics (see Jappelli and Pistaferri, 2017, for an overview). A large body of recent research has considered the effects of idiosyncratic income changes on both consumption and family labor supply (see Blundell et al., 2016, BPS from now on, for a recent contribution). A different literature has investigated the wealth effect on consumption, using both aggregate and household level data (see Berger et al., 2018). Interest in the wealth effects stems partly from the wide variability in both stock prices and house prices observed in the last decades. The goal of this paper is to explore how shocks to wages and asset prices affect consumption and labor supply decisions of households. The traditional literature linking consumption and income shocks assumes, in the spirit of the Bewley model, that households hold only a risk-free bond and hence neglects the effect of asset price shocks. In contrast, the wealth effect literature considers stochastic returns to wealth but ignores labor supply and is based mostly on reduced form relations. Shedding light on the mechanisms linking wage and asset price shocks to consumption and labor supply choices is important. Households facing exogenous shocks to their resources may change the consumption of goods, the consumption of leisure, as well as their housing choices. In particular, if leisure is a normal good, one would expect households to increase labor supply to compensate for negative wealth shocks. Moreover, housing is both an investment and a durable good providing a flow of consumption services. Hence, depending on the structure of individual preferences, an exogenous variation in house prices might have different implications for consumption and labor supply decisions. R Thanks to Mike Keane (the Editor) and two anonymous referees for comments. This paper uses geocoded confidential information matched with publicuse PSID data. All errors are ours. ∗ Corresponding author. E-mail addresses: [email protected] (C. Daminato), [email protected] (L. Pistaferri).

https://doi.org/10.1016/j.euroecorev.2020.103389 0014-2921/© 2020 Elsevier B.V. All rights reserved.

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C. Daminato and L. Pistaferri / European Economic Review 124 (2020) 103389

A large and growing empirical literature studies the consumption response to house price movements, using aggregate data (Carroll et al., 2011 and Case et al., 2013) or household level micro data (Campbell and Cocco, 2007 and Mian et al., 2013). The focus on house price changes is mostly due to the fact that housing wealth plays a very large role on the total wealth of most households, at least compared to stock-holdings. In most of this literature, labor supply is assumed to be exogenous. However, labor supply could play an important role in our understanding of how households respond to shocks.1 Some studies use exogenous wealth changes (for instance Imbens et al., 2001) or a run-up in stock prices (such as Cheng and French, 20 0 0) and find evidence that labor supply declines in response to a positive shock to wealth (or an increase in stock prices). Most of the literature ignores the role of family labor supply (such as “added worker effects”, Lundberg, 1985). In this paper we consider a life-cycle model in which two-earner households choose consumption, labor supply and asset accumulation. We assume that hourly wages of both earners, house prices and the return from stock-holdings are stochastic. This extends the framework developed by BPS (who assumed households invest only in a free-risk bond) and provides a theoretical framework for thinking about the wealth effect. Assuming that the housing durable is costlessly adjustable, we derive analytical expressions, based on approximations, for the dynamics of nondurable consumption, housing services and earnings of the two spouses in a rich framework featuring non-separability between consumption, housing and hours, correlated wage shocks, house price changes, and stochastic risky returns. We show how this approach allows identification of both the responses of consumption and earnings to shocks as well as preference parameters. We estimate the model using data from the Panel Study of Income Dynamics (PSID) for the 1998–2014 period (augmented with confidential geo-location information). The contribution of the paper is twofold. First, it provides a micro-foundation for the response of households’ decisions to both shocks to risky returns and house price changes. Second, it highlights the importance of including endogenous family labor supply when estimating the effects on consumption of shocks to risky returns and house prices changes, and hence contributes to the vast “wealth effect” literature. The rest of the paper is organized as follows. Section 2 describes the dynamic life-cycle model and the strategy we use to link households decisions to wage shocks, house price changes and shocks to risky returns. In Section 3 we describe the data, while Section 4 discusses the empirical strategy. Section 5 presents the estimated structural parameters and the behavioral responses to shocks. Section 7 concludes and discusses avenues for future work. 2. The life-cycle model In this section we provide a micro-foundation for the wealth effect on consumption (which is much studied) and labor supply (which is less so). We extend the framework developed by BPS and show how to derive links between the decisions of a two earners’ household and shocks to wages, house prices, and risky returns. The household utility function is nonseparable in consumption, housing services, and hours. Hence, the marginal utilities of nondurable consumption and hours depend on housing services. We study the decisions of homeowners during their working lives. One limitation of this sample selection is that renters are a group whose consumption responds negatively to shocks to local house prices, and this may inflate estimates of the consumption response to such shocks (which use only homeowners). On the other hand, the focus on homeowners implies that we do not need to model collateral constraints at the point of house purchase. 2.1. Exogenous processes Households face exogenous processes for house prices, the return on risky assets, and hourly wages. h . Taking Housing wealth of household i at time t is given by the product of the units of housing Hi,t and their price Pi,t

h +  ln H . The evolution of the house value for household i depends on the logs, we can then write  ln(Pi,t Hi,t ) =  ln Pi,t i,t dynamics of the units of housing (which may change due to renovations, splits, etc.) and on the changes in house prices (which may reflect changes in the quality of housing, keeping the number of units constant, as well as local housing market developments). In particular, we assume that the log of house prices for household i evolves as a unit root process with drift: h h  βp + ξh ln Pi,t = ln Pi,t−1 + Xi,t i,t

(1)

h an exogenous shock to house prices. The observable characteristics in (1) inwhere Xi,t are observable characteristics and ξi,t clude variables that measure permanent differences in housing prices across geographical locations or housing characteristics, demographics (such as education, race, etc.) meant to capture the impact that residential segregation may have on house prices (given that our geographical controls are limited), and controls for the evolution of housing prices at the agh ) is the gregate level (due to, say, monetary policy). We assume that the economically-relevant shock to house prices (ξi,t interaction of state, year, and residence in an SMSA. This is meant to capture changes in local housing market conditions and variability in housing supply from one year to the next.2 We discuss more in detail the estimation of the shocks to house prices in Section 4.1.

1

An exception is Disney and Gathergood (2018), who consider the effect of housing wealth on labor supply using household panel data from the UK. This is consistent with the finding of previous studies (Mian and Sufi, 2016), documenting that the relevant house price variation in the US is at the county level. 2

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Non-housing wealth available for consumption At is given by the sum of a risk-free asset, with constant return rb , and a risky asset (which can be interpreted as a portfolio of stocks), with stochastic return rs . The distinction between the return from financial wealth and the return from housing wealth (given by the evolution of house prices) is important because housing is both an investment good as well as a durable good providing consumption services, while financial assets lack the second feature. Total asset holdings and portfolio choice differ across households. The return from a household’s portfolio can be written as: p s s ri,t = r b + αi,t−1 (ri,t − rb )

(2)

s αi,t−1

where is the share of liquid wealth that household i invests in the risky asset at time t − 1. We assume that the excess return on the risky asset evolves according to: s  βs + θ + ξ s ri,t − r b = Xi,t i i,t

(3)

Individual returns depend both on observable characteristics Xi,t (such as education and age of the household head which are meant to capture, in reduced form, potential differences in risk-taking), and unobservable fixed effects θ i (such as the ability to form and manage a risky portfolio, financial literacy, etc.). See Fagereng et al. (2020) for evidence on the importance of this fixed returns heterogeneity. In each period, households form expectations about future returns of risky assets given their characteristics. Using (3), deviations of realized returns from expected returns are given by: s s s ri,t − Et−1 [ri,t ] = ξi,t

s is a conditionally homoskedastic, normally i.i.d. error, which we interpret as the shock to risky asset returns. where ξi,t Section 4.1 below provides further details on how we measure shocks to these returns. Finally, and similarly to BPS, we adopt a standard permanent-transitory process for hourly wages, assuming that the permanent component evolves as a unit root process. The log of real wage of individual j ( j = 1, 2 ) can then be written as: 

ln Wi, j,t = Xi, j,t β W j + Fi, j,t + ui, j,t Fi, j,t = Fi, j,t−1 + vi, j,t where Xi,j,t are observed characteristics of earner j that affect wages, and ui,j,t and vi, j,t are transitory shocks and permanent shocks to wages, respectively. We also assume that the shocks ui,j,t and vi, j,t are serially uncorrelated with constant variances. However, we assume that transitory (permanent) shocks of husband and wife can be correlated, with covariance σu1 ,u2 (σv1 ,v2 ). We assume zero correlation between shocks to wages and shocks to risky returns and between transitory shocks to wages and shocks to house prices. This assumption is motivated by previous studies that have found weak evidence regarding the correlation between wages and returns from stocks.3 In contrast, we allow permanent wage shocks and house h )=σ prices to be potentially correlated (E (vi, j,t , ξi,t v j ,h ). Prior literature finds a positive correlation between house prices and earnings innovations, suggesting that a technology shock permanently influencing one’s marketable skills might as well permanently influence the local real estate market (see, e.g., Attanasio et al. 2011), perhaps through worker mobility choices. Finally, we assume zero correlation between idiosyncratic shocks to house prices and shocks to risky returns. This is consistent with data for the US (see Davidoff, 2007). 2.2. Household maximization problem The households’ maximization problem is given by:

max

{ C, H , L 1 , L 2 }

Et

T −t 

β s ut+s (Ci,t+s , Hi,t+s , Li,1,t+s , Li,2,t+s )

(4)

s=0

subject to: p h Ai,t = Ri,t Ai,t−1 + Wi,1,t Li,1,t + Wi,2,t Li,2,t − Ci,t − Ii,t

p

(5)

p

where Ai,t = Bi,t + Si,t , Ri,t = 1 + ri,t , and h h h Ii,t = Pi,t Hi,t − (1 − δ )Pi,t Hi,t−1

(6)

The utility function also depends on exogenous demographic characteristics which we omit to reduce notational clutter. This is a unitary framework. Households choose nondurable consumption C, housing services H,4 and the labor supply of both spouses L1 and L2 , given the allocation of liquid wealth A to risky assets S and risk-free assets B. We assume the utility 3 4

In particular, using data from the PSID, Cocco et al. (2005) do not reject the null of no correlation between labor income shocks and stock returns. We assume that housing services are a fixed proportion of the housing stock, a partial justification for the abuse of notation here and below.

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function to be twice differentiable in all its primary arguments with uC > 0, uCC < 0, uH > 0, uHH < 0 and uL j < 0, uL j L j > 0, for j = {1, 2}. The utility function is potentially non-separable in consumption, housing services, and hours, which will depend on the sign of the cross-partials. s We consider αi,t−1 = Si,t−1 /Ait −1 (the share of risky assets out of total financial wealth) as reflecting predetermined portfolio choices made in the previous period. Since we are interested in studying the response of consumption and labor supply to shocks to financial returns realized in period t, we can take the portfolio allocation decision as pre-determined in the model. h in the durable good (housing). House value P h H The budget constraint (5) assumes that household can invest Ii,t will i,t i,t h H = ( 1 − δ )P h H h , where δ is the depreciation rate. This formulation assumes that then evolve according to Pi,t + Ii,t i,t i,t i,t−1 homeowners can liquidate their housing stock costlessly. Moreover, it assumes that borrowing against housing wealth is not constrained.5 While these are clearly strong assumptions, they allow us to derive a solution of the model using an approximation procedure similar to Blundell et al. (2008) and BPS, which has value since estimation of a structural model with housing transaction costs and the structure of shocks and preferences detailed above is particularly complex. We discuss the limitations of this approach when we present the estimation results and more broadly in Section 6. The advantages of the approximation procedure adopted here are that it does not require strong and unrealistic assumptions on the form of the utility function (see Hall, 1978 and Caballero, 1990), that it allows us to address the questions of interest while providing a transparent identification of the structural parameters (including the wealth effects for consumption and labor supply), and that it can be relatively flexible regarding the specification of household preferences and the structure of the wage process. The approximation behaves remarkably well in contexts very similar to ours (see Kaplan and Violante, 2010, Wu and Krueger, 2018 and Etheridge, 2019).

2.3. Linking individual choices to shocks to asset prices Our goal is to understand how shocks to wages, house prices and risky returns affect consumption, housing, and labor supply of both earners. In order to do this, we derive expressions for the growth rates of the endogenous variables as a function of the price shocks. Our solution method consists of a multi-step procedure, similar to the one used in BPS. However, we require an additional step designed to deal with the nature of housing as a durable good providing consumption services. The approximation procedure can be summarized as follows: 1. First, we use an application of the Envelope Theorem to link the marginal utility of housing services to the marginal utility of consumption and hours; 2. Second, we use a Taylor approximation to the first order conditions of the problem to obtain expressions for the growth rates of consumption, housing services and hours of work in terms of changes in wages, changes in house prices and the innovation in the marginal utility of wealth; 3. Finally, we take a log-linearization of the intertemporal budget constraint to express innovations to the (unobserved) marginal utility of wealth as a function of (statistically observable) shocks to wages, house prices and risky returns. We now describe these three steps more in detail. Step 1. The inclusion of housing in a life cycle framework is not straightforward, as housing plays the dual role of being both a consumption good and an investment good. Following the idea in Carroll and Dunn (1997), we tackle this issue by using an application of the Envelope Theorem. In particular, omitting from now on the i superscript (unless otherwise noted), we rewrite the maximization problem (4) in Bellman equation form as:

Vt (At−1 , Pth Ht−1 ) =

max

Ct ,Ht ,L1,t ,L2,t

h u(Ct , Ht , L1,t , L2,t ) + β Et Vt+1 (At , Pt+1 Ht )

subject to (5) and (6). The value function has two state variables, financial wealth at the end-of-period t − 1 and the value of housing Pth Ht−1 . In Online Appendix A we show how an application of the Envelope Theorem for the two state variables yields the following relation between the marginal utility of housing services uH and the marginal utility of consumption uC :6



h uH = Pth uC Et ψt+1 + covt

h where ψt+1 = 1 − (1 − δ )

  Va h ψt+1 , t+1 a

h /P h ) (Pt+1 t p

Rt+1

Et Vt+1

is a (scaled) user cost of housing (see Diaz and Luengo-Prado, 2008) and

(7) a Vt+1

a Et Vt+1

is related to the innovation in the marginal utility of wealth (Eq. (9) below). This expression shows how households substitute at the margin between consumption and housing. 5 Household borrowing decisions are limited only by the terminal conditions on liquid wealth and housing wealth. Given that wages are uncertain, this generates a ”natural borrowing constraint” of the type highlighted by Aiyagari (1994). 6 The same result can be obtained by deriving the relationship between the marginal utility of housing services and the marginal utility of hours of either earner.

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Once we have a relationship between the marginal utility of housing services and the marginal utility of consumption, we can link the marginal utility of housing services to the marginal utility of wealth using the relationship between the marginal utility of consumption and the marginal utility of wealth obtained through the log-linearization of the first order conditions of the problem.7 Step 2. The approximation of the first order conditions of the problem (4) with respect to assets, consumption and hours, together with the condition (7) derived above, yields the following growth equations for household i’s consumption and housing services and for earner j’s labor supply (given a predetermined share of liquid wealth invested in risky assets αts ):

ct+1  (ηc,w1 + ηc,w2 + ηc,ph − ηc,p ) ln λt+1 + ηc,w1 w1,t+1 + ηc,w2 w2,t+1 + ηc,ph ξth ht+1  (ηh,p + ηh,w1 + ηh,w2 + ηh,ph ) ln λt+1 + ηh,w1 w1,t+1 + ηh,w2 w2,t+1 + ηh,ph ξth

(8)

l j,t+1  (ηl j ,p + ηl j ,w j + ηl j ,w− j + ηl j ,ph ) ln λt+1 + ηl j ,w j w j,t+1 + ηl j ,w− j w− j,t+1 + ηl j ,ph ξ

h t

where ct+1 , ht+1 and l j,t+1 are log consumption, log housing services and log hours of earner j ( j = {1, 2}), net of predictable taste shifters, and w j,t+1 and pht+1 have similar interpretations. The expressions in (8) generalize those derived in Etheridge (2019) to the case of endogenous labor supply.8 Note that in the Euler equations (8) we do not impose additive separability between consumption, housing services and hours. In particular, the parameters ηm,n measure the Frisch elasticities of good m with respect to a change in price n (where n = { p, ph , w j }, respectively for the intertemporal price of consumption, the price of housing, and the wage of earner j). Hence, ηc,p is the elasticity of intertemporal substitution (EIS) for consumption, ηl j ,w j is the EIS of earner j’s labor supply, and ηh,ph the elasticity of housing services with respect to changes in house prices. Moreover, the Frisch cross-elasticities measure the response of good m with respect to a change in the price of goods other than m. They can be used to test for separability between consumption, housing services, and the leisure of both earners. Their signs are informative about whether goods are Frisch substitutes (if the cross-elasticity is positive) or Frisch complements (if the cross-elasticity is negative) (for utility ”bads” such as hours of work, the sign is of course reversed). In our context, an important parameter is ηc,ph , measuring the Frisch elasticity of consumption to changes in house prices. As can be seen from (8), this is the structural equivalent of the reduced form “wealth effect” estimated in the literature in cases in which house price shocks leave the marginal utility of wealth unchanged. If consumers are very inelastic to house price changes, ηc,ph is small and the estimated wealth effect should be small. Wealth effects need not being confined to consumption responses. In principle, wealth changes may allow individuals to “buy” more leisure. Estimation of labor supply elasticities with respect to changes in house prices, ηl1 ,ph and ηl2 ,ph , sheds light onto the importance of considering endogenous labor supply responses. At one extreme, a small wealth effect in consumption may hide large increases in leisure time, changing the way we think about the welfare consequences of changes in house prices. Besides intertemporal changes in prices, the Euler equations (8) also depend on changes in the Lagrange multiplier on the sequential budget constraint. These can be written as:

 ln λi,t+1

 φt+1 + i,t+1

(9)

where i,t+1 captures revisions in the growth of the marginal utility of wealth, and the component φt+1 is a function of the variance of changes in the marginal utility of wealth V ar ( ln λi,t+1 ), the variance of the return p on risky assets V ar (Ri,t+1 ), as well as the covariance between innovations to the marginal utility of wealth and shocks to risky assets. We assume that the component φt+1 does not vary across individuals and can be adequately captured by observables.9 Step 3. The growth rate equations (8) cannot be estimated because we do not know how to measure innovations to the marginal utility of wealth t+1 . Following Blundell et al. (2008) and BPS, the third step of our empirical strategy is to log-linearize the intertemporal budget constraint. This allows us to build a direct link between the innovations to the marginal utility of wealth t+1 and (statistically measurable) shocks to wages, house prices, and risky returns. In Online Appendix C we show that the intertemporal budget constraint in our context can be written as:

Et

T −t  s=0

7

T −t h T −t h    Pt+s Ht+s ψt+ W j,t+s L j,t+s Ct+s (1 − δ ) h s+1 + Et = Ai,t−1 + Pt Hi,t−1 + Et s s p p p p R R R j=0 t+ j j=0 t+ j j=0 Rt+ j t s=0 j={1,2} s=0

s

(10)

See Online Appendix B for details on the Taylor approximation to the first order conditions of the problem. The equations derived in Etheridge (2019) are a special case of the equations for consumption and housing in (8); in particular, they can be obtained by imposing separability between consumption and leisure of the two earners (ηc,w1 = ηc,w2 = 0) and between housing services and leisure of the two earners (ηh,w1 = ηh,w2 = 0). 9 In the model, households hold the same share of wealth in risky assets at the optimum (since there is no heterogeneity in risk aversion). However, since there are idiosyncratic returns to stock market investments, ex-post portfolios will be heterogeneous. 8

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In Online Appendix C, we further show how the log-linearization of (10) allows us to write the innovation in the marginal utility of wealth t+1 as a linear function of transitory shocks to wages (u1,t+1 and u2,t+1 ), h ), and shocks to risky asset returns permanent shocks to wages (v1,t+1 and v2,t+1 ), shocks to house prices (ξt+1 s (ξt+1 ). Using the growth rates obtained in step 2 and the expression for the marginal utility of wealth, we can write the following dynamics for consumption, housing services, and the labor supply of the two earners:

⎛

⎞ ⎛ kc,u1 ct+1 ⎜ ht+1 ⎟ ⎜ kh,u1 ⎜ ⎟ ⎜ ⎝l1,t+1 ⎠  ⎝kl1 ,u1 kl2 ,u1 l2,t+1

kc,u2

kc,v1

kc,v2

kc,ξ h

kh,u2

kh,v1

kh,v2

k hξ h

kl1 ,u2

kl1 ,v 1

kl1 ,v 2

kl1 ,ξ h

kl2 ,u2

kl2 ,v 1

kl2 ,v 2

kl2 ,ξ h

⎛ ⎞ u1,t+1 ⎞ ⎟ kc,ξ s ⎜ ⎜u2,t+1 ⎟ ⎜ ⎟ khξ s ⎟⎜ v1,t+1 ⎟ ⎟⎜ ⎟ kl1 ,ξ s ⎠⎜ v2,t+1 ⎟ ⎜ h ⎟ kl2 ,ξ s ⎝ ξt+1 ⎠ s ξt+1

(11)

Here km,n are complicated functions of the structural parameters of the model, reported in full in Online Appendix h C. They measure the response of variable m (ct+1 , ht+1 and l j,t+1 ) to shock n (u j,t+1 , v j,t+1 , ξt+1 and s ). Notice that the response matrix boils down to the one derived by BPS in the case of non-separability if ξt+1 we shut down the response of housing services to shocks (kh,u1 = kh,u2 = kh,v1 = kh,v2 = 0) and the responses of consumption and earnings to shocks to house prices and risky returns (kc,ξ s = kc,ξ h = kl1 ,ξ s = kl ,ξ h = kl2 ,ξ s = 1

kl ,ξ h = 0).10 2 Since one of the contributions of this paper is to have a structural interpretation of wealth effects (consumption and labor supply responses to exogenous changes in asset prices), we report here the relevant expressions for responses of consumption and labor supply to unanticipated changes in the prices of housing and stocks, respectively:

(ηc,w1 + ηc,w2 + ηc,ph − ηc,p )[HWt + (1 − HWt )(1 − πt )ηl,ph − (1 − zth )ηc,ph − zth (1 + ηh,ph )] t s (ηc,w1 + ηc,w2 + ηc,ph − ηc,p )[ (1 − HWt )πt ] = −αt t (ηl j ,p + ηl j ,w j + ηl j ,w− j + ηl j ,ph )[HWt + (1 − HWt )(1 − πt )ηl,ph − (1 − zth )ηc,ph − zth (1 + ηh,ph )] = ηl j ,ph − t (ηl j ,p + ηl j ,w j + ηl j ,w− j + ηl j ,ph )[(1 − HWt )πt ] s = −αt t

kc,ξ h = ηc,ph − kc,ξ s kl j ,ξh kl j ,ξs

(12) where:

t = (1 − zth )(ηc,p − ηc,w1 − ηc,w2 − ηc,ph ) − zth (ηh,p + ηh,w1 + ηh,w2 + ηh,ph ) + (1 − HWt )(1 − πt )(ηl,p + ηl,w1 + ηl,w2 + ηl,ph ) In these expressions, HWt =

Pth Ht Pth Ht +At +Human Wealtht

is the share of housing wealth over total lifetime wealth (the

At higher the HWt the higher the sensitivity of consumption to shocks to house prices), and πt = A +Human is Wealtht t the share of financial wealth over total non-housing wealth (the sum of financial and human wealth, with the implication that the higher π t , the lower the sensitivity of consumption to shocks to wages). A bar over a parameter indicates a weighted average of the two earners’ specific parameters, with weights given by their human

wealth shares. For example, ηl,p = s1,y ηl1 ,p + (1 − s1,y )ηl2 ,p , where s j,y =

Human Wealth j,t Human Wealtht

is the share of earner j’s

human wealth over family human wealth. Finally, zth measures the share of lifetime spending due to housing net depreciation costs (i.e., the cost of depreciation of the existing housing stock not “compensated” by future house price increases; see the actual expression reported in Online Appendix C). If house prices grow at the same rate as the depreciation rate, zth = 0, a useful benchmark. 2.3.1. Model responses to shocks to asset prices The interpretation of the responses of consumption and labor supply to transitory and permanent shocks to wages is very similar to that discussed in BPS in the case of non-separability and the interested reader is referred to their paper for more details. Here we focus instead on the interpretation of the responses of consumption and labor supply to shocks to house prices and stock returns. To make interpretation more transparent, we consider special cases adopted in the literature (such as separability, exogenous labor supply, etc.). 10

Shutting down the response of housing services to shocks is equivalent to the case in which housing services are exogenous in the model.

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7

Consider first responses to house price changes. The expression for kc,ξ h in (12) shows that only in very simple cases the Frisch elasticity ηc,ph is the only parameter driving the consumption “wealth effect”. In general, a change in house prices may affect all decision variables (labor supply, etc.). In particular, in our model the response of consumption to a change in house prices depends on: (a) the degree of substitutability or complementarity between consumption and housing services (as given by the signs of ηc,ph and ηh,p ), (b) the share of housing wealth out of household lifetime wealth (the higher HWt the more responsive consumption of household i is to a shock to house prices), (c) the consumption EIS (the higher ηc,p , the higher the response), and (d) the elasticity of consumption with respect to transitory shocks to wages (the response of consumption to house prices is higher if consumption and hours are substitutes, ηc,w j < 0, and lower if they are complements, ηc,w j > 0). The expression for kc,ξ h sheds light onto the mechanisms underlying the response of household consumption to changes in house prices. It provides an explanation for previous empirical evidence suggesting that house price shocks have a larger impact on the consumption of older households (see, e.g. Attanasio et al. 2011). This is because older households tend to have higher values of HWt due to a combination of higher home equity and lower (remaining) human wealth. Furthermore, they provide a rationale for consumption to react more to house price variations during an economic boom (i.e., when HWt is higher). To see this point clearly, consider the separable case (ηc,w j = ηc,ph = 0 for j = {1, 2}) with exogenous labor supply (ηl j ,p =

ηl j ,ph = ηl j ,w j = 0 for j = {1, 2}) and zth = 0. In this case, the response of consumption to changes in house prices reduces to: kc,ξ h = HWt , showing that nondurable consumption responds to a shock to house prices by an extent that depends exclusively on the share of housing wealth out of total wealth, as predicted by a simplified version of the permanent income hypothesis (where the term HWt captures a simple endowment effect). Given the random walk stochastic structure of house prices, an increase in the latter (absent adjustment on other margins such as labor supply) gets factored in higher permanent income, and therefore one-to-one in consumption.11 This is the simplest rationalization of a “wealth effect” on consumption. From Eq. (8), the response of earner j’s labor supply to shocks to house prices is the sum of a direct effect (measured by the parameter ηl j ,ph ) and an indirect effect (how shocks to house prices impact innovations in the marginal utility of wealth). The expression for kl j ,ξh derived above makes this separation explicit. If the marginal utility of wealth changes little in response to house prices shocks, then the labor supply “wealth effect” simply depends on whether hours and housing services are substitutes (ηl j ,ph < 0) or complements (ηl j ,ph > 0) in utility. If they are substitutes, labor supply is used as a buffer to house price shocks, while complementarity makes people work fewer hours when house prices collapse. However, when house price shocks shift the marginal utility of wealth in a non-negligible way, the effect is ambiguous and cannot be signed a priori. The parameter kc,ξ s measures the response of consumption to shocks to stock returns. In the special case considered

above (preferences are separable, labor supply exogenous, and zth = 0), the response of consumption to shocks to stock returns is simply (and intuitively) the product of the share of financial wealth invested in risky assets and the ratio of financial wealth to total wealth (financial, housing, and human), αts πt . Outside of this special case, the response increases with the consumption EIS ηc,p (since an increase in the return to financial assets increases the price of current vis-à-vis future consumption). For example, in the one-earner case with separable preferences with respect to leisure and zth = 0, ∂ kc,ξ s

s 2 −2 / ∂ηc,p = αt (1 − HWt ) t πt (1 − πt )ηl,w ≥ 0. In general, the effect of a shock to risky returns on consumption might also depend on whether consumption and hours are substitutes (ηc,w j < 0) or complements (ηc,w j > 0) in utility, and on the complementarity between consumption and housing services (ηc,ph < 0 or ηc,ph > 0). The signs of kl j ,ξ s are informative about whether labor supply is countercyclical (kl j ,ξ s < 0) or procyclical (kl j ,ξ s > 0)

(assuming a buoyant stock market during expansions, and vice versa during recessions). Our model predicts that labor supply is countercyclical, i.e., a negative shock to risky returns induces an increase in the labor supply of earner j (and thus provides a form of consumption insurance), only if the following condition is satisfied:12

ηl j ,w j + ηl j ,p + ηl j ,w− j + ηl j ,ph > 0 Hence, the model suggests that the importance of labor supply of earner j as a possible insurance mechanism against shocks to risky returns depends on labor supply being highly responsive to both own and other goods “prices” (the spouse’s wage, as well as the intertemporal price of consumption and the price of housing). In Online Appendix D we show that identification of some of the preference parameters (such as the consumption EIS) hinges heavily on allowing for hours responses. This remarks is important, since there is an extensive literature that considers the effect of changes in house prices on consumption (see e.g., Campbell and Cocco, 2007) but assumes that households receive exogenous labor income over the life cycle, thus neglecting the possibility that households may choose hours of work.

11 Berger et al. (2018) and Etheridge (2019) derive a similar result. In the latter case, labor supply is assumed exogenous and the term zth neglected (owing to its likely small role and the inherent difficult in measuring it). 12 This neglects labor demand effects that may constrain the ability to work longer hours during economic busts.

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C. Daminato and L. Pistaferri / European Economic Review 124 (2020) 103389 Table 1 Descriptive statistics for data from PSID 1998–2014. PSID waves

Non-durable consumption Financial assets Stock market participation Stocks (conditional) House value Mortgage Total assets Earnings head Earnings wife (participating) Observations

1998

20 0 0

2002

2004

2006

2008

2010

2012

2014

20,652 148,833 0.348 100,508 165,957 77,467 287,346 58,920 27,368 1086

23,426 144,332 0.367 88,843 192,540 92,725 296,012 66,159 29,903 1259

24,474 125,210 0.327 87,160 213,365 107,757 279,862 68,526 32,518 1284

28,543 129,055 0.312 90,227 253,298 124,407 324,252 69,301 35,100 1316

30,877 131,819 0.311 90,228 269,594 150,032 339,733 78,204 38,407 1339

30,986 121,585 0.283 72,827 223,986 158,948 266,659 86,652 43,062 1372

33,699 121,114 0.225 77,789 211,093 156,096 248,814 78,539 42,370 1346

32,639 119,603 0.211 68,004 198,062 156,310 255,113 81,702 45,346 1263

30,702 133,705 0.178 79,432 202,375 154,892 277,131 84,510 45,957 1179

Note: The Table reports mean values in PSID data for selected variables from 1999–2015 PSID waves. Mean values are computed applying the sample selection used for the estimation of the structural variance and preference parameters and described in Section 3. Monetary values were reported to 20 0 0 dollars.

3. Data We use data from the 1999-2015 Panel Study of Income and Dynamics (PSID) (augmented with confidential geo-coded data) to estimate the model. The PSID collects detailed information on a representative sample of the US population (plus the SEO, which oversamples households from the bottom of the income distribution). The great advantage of using PSID data after 1999 is that it includes joint information about demographics, income, asset holdings and consumption. We focus on non-SEO households with participating and married male household heads aged between 30 and 60. We drop observations with missing values for state, education, race, labor earnings, hours, total consumption and total assets, and with wages that are lower than half the minimum wage in the state where the household resides. In addition, because of the focus on risky assets and housing in this work, we drop observations with missing values for stocks and self reported house value. We impose three additional sample selections: we drop renters (since our focus is on modelling homeowners’ decisions), households living in mobile homes, and those that moved out of state from one year to the next (since we ignore the date of the move and hence cannot assign the correct county-specific house price change). These sample selections drop around 24 percent of the sample. Table 1 reports descriptive statistics for non durable consumption, housing, asset holdings and earnings of the two earners. Non-durable consumption and housing services. As in BPS, we focus on nondurables and services as we do not model the household decision to purchase durables. Consumption expenditures data for the period 1999–2015 cover food expenditures, child care, health expenditures, utilities, gasoline, car maintenance, transportation and education. Our measure of household nondurable consumption is obtained by aggregating these expenditures categories. We do not include clothing, recreation, alcohol and tobacco because information on these categories is available only starting from 2005. Moreover, as we do model the decision of homeowners with respect to housing services, this category is considered separately from the other nondurables. In particular, the value of housing services for homeowners are assumed to represent 6% of the self reported home value (a value that agrees with the user cost estimated by Poterba and Sinai, 2008). Assets, Housing, Earnings, Wages and Hours. We use data on households’ assets, and in particular on stockholdings and housing for the estimation of HWi,t , the share of housing out of total wealth, π i,t , the share of assets out of total nons , the share of liquid wealth invested in risky assets. Starting from 1999, the PSID collects in each housing wealth, and αi,t wave data on holdings of cash, bonds, stocks, house value, the value of businesses, and of private pension funds. In addition, the survey collects information on household debt, with details about first and second mortgage debt. We measure nonhousing assets as the sum of cash, bonds, stocks, the value of any business, the value of pension funds and the value of any car, net of all debts (including mortgage debt). We use the self reported house value to measure housing wealth. The data also includes information about purchases and sales of stocks between two waves, which we use to estimate shocks to stock returns. We use data on annual labor earnings and on annual hours of work of both earners for the estimation of the model, and construct hourly wage by dividing annual earnings by annual hours. Fig. 1 shows how stock market participation, conditional risky share, and the unconditional share of wealth in risky assets evolve over the life cycle. 4. Empirical strategy In this section we describe the empirical strategy we use to pin down the structural parameters of the model. The empirical strategy can be summarized as follows (with more implementation details in the rest of the section): h , ξs ; 1. We first construct the residuals ci,t , hi,t , yi,1,t , wi,1,t , yi,2,t , wi,2,t , ξi,t i,t s , using cur2. Next, we estimate the weighting factors si,1,t , π i,t , and HWi,t , and the share of risky assets in portfolio, αi,t rent and projected earnings, the value of housing, total financial wealth, and current portfolio allocation information;

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Fig. 1. Portfolio allocation over the life cycle. (a) Stock market participation; (b) Conditional risky share; (c) Overall share of financial wealth in risky assets.

3. We then estimate the wage, house price and risky return variances and covariances using the second order moments h and ξ s ; for wi,1,t , wi,2,t , ξi,t i,t 4. Finally, given the pre-estimated values for HWi,t , π i,t and si,1,t and the estimated value of wages, house price and risky return variances and covariances, we adopt the same strategy of BPS and estimate the structural parameters by GMM using the restrictions that the model imposes on the behavior of the variances and covariances of (residual) consumption growth, earnings growth of earner 1, etc. (i.e., the dynamics expressed by Eq.(11)). Note that while (11) is

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written in terms of moments of labor supply, we estimate it in terms of moments of earnings, using the fact that earnings growth is the sum of hours growth and (exogenous) hourly wage growth. We allow for measurement error in consumption, wages, earnings, self-reported housing values, and stocks. 4.1. Residual measures and shocks to asset returns The terms ci,t , hi,t , yi,1,t , wi,1,t , yi,2,t and wi,2,t are obtained as the residuals of OLS regressions for the log difference of Ci,t , Hi,t , Yi,1,t , Wi,1,t , Yi,2,t , and Wi,2,t , respectively, onto observable characteristics.13 In the wage equations, these include the year of birth, education, race, state and large city dummies. For consumption, housing services and earnings we also add dummies for number of kids, number of family members, employment status, income recipient other than head or wife in the household and whether the couple has children not residing in the household. For time-varying observables we use both the level and the first difference. This estimation differs from BPS for one important detail. Because the focus of this paper is to provide a structural understanding of the time variability in the outcomes (through the inclusion of shocks to both house prices and risky asset holdings), we do not include year dummies and the interaction of year dummies with observables (such as education, race and SMSA). The effect of aggregate shocks is thus subsumed in the residual terms. h ) we use the house price dynamics from Eq. (1). We assume that To construct our measure of the house price shock (ξi,t the log of housing units Hi,t evolves as a random walk with drift plus a classical measurement error, so that we can identify h ) using the observable self-reported housing values (ln (P h H )). Our estimating equation is: shocks to house prices (ln Pi,t i,t i,t

 ln(Pi,th Hi,t ) = Xi,t β ph + ξi,th + ζi,th

(13)

The observable characteristics we include in (13) are state dummies, whether living in an SMSA, and the number of rooms and house type dummies (whether a detached house, condo, etc.). These variables measure permanent differences in housing prices across geographical locations or housing characteristics. We also include demographics (such as education, race, etc.) meant to capture the impact that residential segregation may have on house prices (given that our geographical controls are limited), as well as dummies indicating whether the household moved since the last wave, whether the household made any additions or improvements to her home totalling $10,0 0 0 or more, and changes in the number of rooms and house type (which are more likely to reflect endogenous changes in house size or quality). Finally, we control for the evolution of housing prices at the aggregate level (due to, say, monetary policy) by adding year dummies. We assume that h ) is the interaction of state, year, and residence in an SMSA. This is the economically-relevant shock to house prices (ξi,t meant to capture changes in local housing market conditions and variability in housing supply from one year to the next.14 h as a combination of haziness in beliefs or mis-perception errors in self-reports of one’s We interpret the residual of (13), ζi,t house value, rather than true surprises or shocks. Finally, we run a fixed effect regression for the log of changes in stock holdings: Rsi,t =  ln Stocksi,t , controlling for the age of the household head and education-age interactions. Because we do not have information on the price at which stocks are potentially purchased (or sold) across two consecutive waves, we separate the exogenous price changes from the endogenous active changes in portfolio allocation by controlling for the balance between purchases and sales of stocks at time t. The fixed effect in these regression is meant to capture persistent heterogeneity in wealth returns recently documented by Fagereng et al. (2020), among others. The time-varying residual of this regression represents our measure of the unexpected s . shock to risky returns, ξi,t 4.2. Weighting factors: si,j,t , π i,t , and HWi,t As discussed in section 2, computation of the weighting factors uses the formulas for si,1,t , π i,t , and HWi,t derived from the approximation of the intertemporal budget constraint. These expressions are reported in full in Online Appendix C.15 All three weighting factors require an estimate of human wealth, as well as information on financial wealth and housing wealth. Human wealth of earner j is given by:

Human W ealthi, j,t = Yi, j,t + Et

Yi, j,t+1 p Ri,t+1

+ ...

(14)

In order to compute human wealth at time t, we need to form estimates of expected future earnings and future wealth returns. To reduce imputation error, we assume that discounting is at the risk-free rate (0.3%).16 As for the estimation of future earnings, we tackle it using a procedure similar to BPS. For men (who are assumed to be always working, at least up 13 In Online Appendix E we report some key statistics about these regressions (in particular, a detailed description of controls, sample used in estimation, and goodness of fit); we do the same for the regressions we use to extract shocks to house prices and returns to risky assets. 14 This is consistent with the finding of previous studies (Mian and Sufi 2016), documenting that the relevant house price variation in the US is at the county level. 15 h term and discuss its estimation. In Online Appendix C we also report the expression for the zi,t 16

This risk-free rate is the average 1-year Treasury constant maturity real rate for the period 1998–2014.

C. Daminato and L. Pistaferri / European Economic Review 124 (2020) 103389

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Fig. 2. Average s by the age of the household head. Note: The grey shaded area represents the 95% confidence interval. The Figure plots the average s computed by age.

Fig. 3. Average π by the age of the household head. Note: The grey shaded area represents the 95% confidence interval. The Figure plots the average π computed by age.

to age 65), we start by running a regression of earnings on pre-determined characteristics zi,j (education, race and cohort) and forecastable characteristics zi, j,t+s (a cubic in age and the interaction of age with pre-determined characteristics). We then use the estimates of this regression to predict men’s earnings at future ages in (14). For women, there is the issue of non-participation at certain ages. We thus repeat the procedure followed for men, but add in the earnings regression a Heckman-selection term, i.e. an inverse Mill ratio term obtained from a probit regression for employment on the same observables used in the earnings regression plus an exclusion restriction (which we assume to be the generosity of the local welfare system, as in Low et al., 2010). Hence in women’s case, the terms that appear in Eq. (14) is the product of the predicted value from the earnings regression (excluding the contribution of the selection correction term) and the probit estimate of being employed at a given age, education, etc.17 Once we have an estimate of human wealth for earner j (as detailed above), we simply obtain si, j,t = Human Wealthi, j,t H uman Wealthi,1,t +H uman Wealthi,2,t

. Fig. 2 plots si,j,t against the head’s age, and it can be interpreted as a representation of the

life cycle evolution of the distribution of earnings power within the household. Similarly to Blundell et al. (2016), the figure shows that the husband’s weight on total resources decreases with his age, mainly due to age differences within the households. We obtain an estimate of π i,t by simply taking the ratio of current net-of-debt financial wealth to the sum of current net-of-debt financial wealth and household human wealth. The estimated age profile of π i,t , reported in Fig. 3, shows a steep increase of π i,t with age.18 While young households display a negative value of π i,t due to the weight exerted by liabilities, 17 One caveat of this procedure is that expectations of human wealth may potentially depend on sources of information that are in people’s information set but are unobserved to the econometrician. Unfortunately, we do not have subjective expectation data which may help filling the gap. 18 The average value of the estimated E (πi,t ) = 0.01. This is lower than that estimated by BPS because our definition of liquid assets excludes housing (the largest component of wealth for homeowners) and include all debts, including mortgage debt.

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C. Daminato and L. Pistaferri / European Economic Review 124 (2020) 103389

Fig. 4. Average HW by the age of the household head. Note: The grey shaded area represents the 95% confidence interval. The Figure plots the average HW computed by age, that is the relative importance of housing wealth relative to life-cycle household resources.

the steep increase later in life is the result of a combination of asset accumulation due to precautionary savings (as well as the repayment of mortgage debt) and the decline of expected human wealth as earners get closer to retirement. This profile provides a quantitative explanation of the increased ability of households to smooth permanent shocks to wages and house prices (net of other factors) when earners approach retirement. Also, it suggests that shocks to wealth returns might have a larger impact on the decisions of older households. The computation of the share of housing wealth out of total wealth HWi,t (which is related to the ability of households to smooth permanent shocks to wages thanks to housing wealth) requires current values (for financial assets and house value) as well as an estimate of household human wealth. The estimate of HWi,t so obtained is plotted in Fig. 4 against the head’s age. The importance of housing wealth varies substantially over the life cycle, with young households having a much lower share of wealth in housing (around 10% for household heads younger than 35) than households close to retirement (when housing wealth represents around 30% of total wealth). This is due mostly to the decrease in human wealth as households age. These estimates of HWi,t provide a first insight into explaining the reduced-form evidence suggesting that older households react more to shocks to house prices: the relative weight of housing wealth impacted by the shock is relatively more important when the household is close to retirement than when the household is young. 5. Main results 5.1. Wage, house prices and risky returns variances Table 2 reports the estimates of wage, house prices and risky returns variances. We can compare the results for the variance of transitory and permanent shocks, and the correlation between transitory (and permanent) components of the two spouses with the estimates of BPS. Even though our sample uses two additional years and includes only homeowners who do not move out-of-state between two consecutive waves, the results are consistent with their findings. The estimated variance of the transitory shocks is larger than that of the more permanent component, especially for males. All the estimated wage variances are statistically significant. Finally, both transitory and the permanent components of the two spouses are positively correlated, with the correlation among the transitory components being more precisely measured. The estimated variance of risky returns (σs2 = 0.521) is high, suggesting that there is substantial heterogeneity in the composition of an individual’s portfolio, yielding a variance of the return to risky assets that is higher than what households would face if they were holding the market portfolio.19 Besides heterogeneous portfolios, another possible explanation for this large variance is the presence of substantial measurement error in reported stock-holding.20 Our estimate for the variance of house prices is σh2 = 0.021. This is in line with previous estimates of the variance of the house price process in the literature (see, e.g. Attanasio et al., 2011). In particular, our estimate for the standard deviation of house price shocks (σh = 0.14) is close to the value of the standard deviation in house price growth across counties in the US (0.13) estimated by Mian and Sufi (2016) using zip-code level data from 2006 to 2009. 19 Given the heterogeneity in the individual portfolios, our measure for the variance of individual stock returns cannot be readily compared to the historical variance of the market stock returns (e.g., SP&500 index). Using historical data for the 1998-2014 period, we find that the annualized standard deviation of the SP&500 return for the US is around 0.2, a value similar to that reported by Campbell (2003). If we take the (wave-specific) average return in our data and calculate its standard deviation (something akin to looking at the historical standard deviation of the SP&500 index), we find an annualized standard deviation of 0.09. 20 Below, we show that our results are robust to assuming that part of the estimated variation in stock returns is due to classical measurement error in stock-holding values.

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Table 2 Variance estimates. Variances of shocks to earnings Males

Females

Correlation of wage shocks

Transitory

σu21

Permanent

σv21

Transitory

σu22

Permanent

σv22

Transitory

σu1 ,u2

Permanent

σv 1 , v 2

0.034∗ ∗ ∗ (0.008) 0.021∗ ∗ ∗ (0.005) 0.024∗ ∗ ∗ (0.007) 0.024∗ ∗ ∗ (0.004) 0.191∗ ∗ (0.086) 0.124 (0.083)

Variances of shocks to asset returns Risky asset returns

σs2

House prices

σh2

Correlation of shocks to wages and house prices

Males

σv1 ,h

Females

σv2 ,h

0.521∗ ∗ ∗ (0.053) 0.021∗ ∗ ∗ (0.001) 0.026 (0.027) 0.017 (0.021)

Note: Parameters estimated using GMM. Baseline sample selection as described in Section 3. Selection correction for female non-participation is applied. Clustered standard error in parentheses. Three, two, and one stars indicate statistical significance at the 1, 5 and 10%, respectively.

In the final rows of the table we report a small and positive (though imprecise) correlation between house price shocks and permanent shocks to wages, which is more relevant for males. This suggests that the effects of technology shocks simultaneously making a worker’s marketable skills more valuable and land more expensive in a given location, are rather small once we control for observable characteristics. 5.2. Preference parameters Following BPS, we impose symmetry of the Frisch substitution matrix to increase efficiency of the preference parameters estimates. In our framework, one can show that the matrix of behavioral responses, in terms of consumption/housing services/hours elasticities, can be written as:

⎛

dc ⎜ dp ⎜ ⎜ dl1 ⎜ ⎜ dp ⎜ ⎜ dl2 ⎜ dp ⎜ ⎝ dh dp

dc dw1 dl1 dw1 dl2 dw1 dh dw1

dc dw2 dl1 dw2 dl2 dw2 dh dw2

⎞

⎛ c dc ηc,p p dph ⎟ ⎜ ⎟ l1 dl1 ⎟ ⎜ ⎟ ⎜−ηl1 ,p p dph ⎟ ⎜ ⎟=⎜ dl2 ⎟ ⎜ ⎜−ηl2 ,p l2 ⎟ p dph ⎟ ⎜ ⎝ ⎠ h dh ηh,p p dph

c w1 l1 −ηl1 ,w1 w1 l2 −ηl2 ,w1 w1 h ηh,w1 w1

ηc,w1

c w2 l1 −ηl1 ,w2 w2 l2 −ηl2 ,w2 w2 h ηh,w2 w2

ηc,w2

c ⎞ ph ⎟ l1 ⎟ −ηl1 ,ph ⎟ ph ⎟ ⎟ l2 ⎟ −ηl2 ,ph ⎟ ph ⎟ ⎠ h ηh,ph ph

ηc,ph

The symmetry of the Frisch substitution matrix implies: ηl2 ,w1 = ηl1 ,w2 w1 l1 , ηh,p = ηc,ph w l

2 2

−ηl j ,ph

wjlj ph h

pc , ph h

(15)

ηl j ,p = −ηc,w j wpcl and ηh,w j = j j

(for j = {1, 2}).

Our GMM estimator uses the inverse of the diagonal of the estimated variance-covariance matrix of the moments as a weighting matrix (i.e., moment conditions are weighted by a factor that is inversely proportional to their precision), as in Pischke (1995).21 Asset values entering the expressions for the weighting factors are averaged by age and education of the household head (to reduce the impact of measurement error), and in estimation we also apply a selection correction for female non-participation when estimating human wealth. The estimated preference parameters are reported in Table 3. In the first column we report the estimation results assuming separable preferences (i.e., setting all the cross-price elasticities ηc,w1 = ηl1 ,p = ηc,w2 = ηl2 ,p = ηl1 ,w2 = ηl2 ,w1 = ηc,ph = 21 Altonji and Segal (1996) show that using the optimal GMM weighting matrix creates a small sample bias due to the correlation between the second moments and their sampling variances, which are calculated from the same data. Both our baseline weighting matrix and the identity matrix overcome this problem. For robustness, we also estimate the model using an identity matrix as a weighting matrix.

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C. Daminato and L. Pistaferri / European Economic Review 124 (2020) 103389 Table 3 Preference parameters. Frisch parameter

Separability (1)

Non-separability (2)

Own-price elasticities

η c,p ηl1 ,w1 ηl2 ,w2 ηh,ph

0.350∗ ∗ ∗ (0.049) 0.475∗ ∗ ∗ (0.073) 0.823∗ ∗ ∗ (0.107) −0.188∗ ∗ ∗ (0.035)

0.271∗ ∗ ∗ (0.034) 0.509∗ ∗ ∗ (0.074) 0.923∗ ∗ ∗ (0.115) −0.326∗ ∗ ∗ (0.045)

Cross-price elasticities

ηc,w1

0.009 (0.026) −0.004 (0.010) 0.002 (0.0265) −0.002 (0.020) 0.059∗ ∗ ∗ (0.016) 0.118∗ ∗ ∗ (0.029) 0.131∗ ∗ ∗ (0.017) 0.247∗ ∗ ∗ (0.031) −0.008∗ ∗ (0.003) 0.038∗ ∗ (0.015) −0.022∗ ∗ ∗ (0.006) 0.056∗ ∗ ∗ (0.015)

ηl1 ,p ηc,w2 ηl2 ,p ηl1 ,w2 ηl2 ,w1 ηc,ph η h,p ηl1 ,ph ηh,w1 ηl2 ,ph ηh,w2 Observations GMM criterion

8,068 0.0237

8,068 0.0149

Note: Table reports preference parameters estimated using GMM. We use moment conditions characterizing the joint behavior of consumption growth, growth in housing services, husband’s hour growth, and wife’s hour growth. Sample selection is as described in Section 3. We add measurement error in consumption, hours, and earnings. Assets are averaged by age and education of the household head. We apply a selection correction for female nonparticipation. The inverse of the diagonal of the estimated variance-covariance matrix of the moments is used as weighting matrix. In the first column we report results for the model specified under the assumption of separable preferences. In the second columns we report preference parameters estimates for the model with nonseparability. Clustered standard error in parentheses. Three, two, and one stars indicate statistical significance at the 1, 5 and 10%, respectively.

ηh,p = ηl1 ,ph = ηh,w1 = ηl2 ,ph = ηh,w2 = 0). The second column reports the own-price elasticities and cross-price elasticities

for the version of the model that allows for non-separable preferences. The results strongly reject separability in preferences (a test of the joint significance of the cross-price elasticities yields the large value χ 2 (6 ) = 104).22 Moreover, the overall goodness of fit of the model improves substantially when we allow for nonseparability in preferences (the sum of GMM residuals decreases from 0.0237 to 0.0149). For these reasons, in what follows we comment only on the results of the model with nonseparable preferences. Table 4 reports the goodness of fit of the model for selected moments we use in estimation. The model with nonseparable preferences replicates well the majority of the moments we target, and deviations between implied and actual moments from the data are generally small (with lack of power for some moments being an important issue). Going back to the baseline results of Table 3, we find values for the own-price elasticities in line with previous findings. The estimate for the elasticity of intertemporal substitution in consumption ηc,p is 0.27, amply in the ballpark of previous estimates in the literature (see Attanasio and Weber, 2010 for a review). The estimates of intertemporal substitution in male and female labor supply also supports previous findings. In particular, we estimate a Frisch elasticity for males ηl1 ,w1 of 0.51 22

There are only 6 degrees of freedom due to the Frisch symmetry imposed.

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Table 4 Goodness of fit. Model

Data (95% Conf. Interval)

Selected target moments

var (ct ) var (y1,t var (y2,t ) var (ht )

0.116 0.240∗ 0.340∗ 0.041

0.132 0.236 0.327 0.064

0.142 0.280 0.406 0.088

cov (ct , ct−1 ) cov (y1,t , y1,t−1 ) cov (y2,t , y2,t−1 ) cov (ht , ht−1 )

−0.057∗ −0.101∗ −0.123∗ −0.011∗

−0.062 −0.116 −0.137 −0.014

−0.052 −0.082 −0.078 −0.007

cov (ct , y1,t ) cov (ct , y2,t ) cov (ct , ht ) cov (ht , y1,t ) cov (ht , y2,t ) cov (y1,t , y2,t )

0.004∗ −0.001∗ 0.004∗ 0.005∗ 0.007∗ 0.002∗

0.003 −0.001 0.001 0.002 0.004 −0.007

0.012 0.011 0.006 0.009 0.013 0.010

cov (ct , w1,t ) cov (y1,t , w1,t ) cov (y2,t , w1,t ) cov (ht , w1,t )

0.008∗ 0.187∗ −0.007∗ 0.005∗

−0.000 0.159 −0.007 0.003

0.009 0.188 0.009 0.009

cov (ct , w2,t ) cov (y1,t , w2,t ) cov (y2,t , w2,t ) cov (ht , w2,t )

0.006 0.005∗ 0.141∗ 0.006∗

−0.004 0.005 0.117 0.002

0.005 0.016 0.151 0.008

cov (ct , ξth ) cov (y1,t , ξth ) cov (y2,t ξth ) cov (ht , ξth )

0.006 0.000∗ −0.002∗ 0.029

0.002 −0.000 −0.002 0.013

0.006 0.004 0.004 0.024

cov (ct , ξts ) cov (y1,t , ξts ) cov (y2,t , ξts ) cov (ht , ξts )

0.006∗ −0.024∗ −0.044∗ −0.001∗

−0.015 −0.054 −0.069 −0.003

0.022 0.008 0.029 0.028

Note: Comparison of moments simulated by the model and in the data. ∗ indicates simulated moment lies within the 95% confidence interval for the mean of the moment in the data.

and a Frisch elasticity for females ηl2 ,w2 of 0.92. Our estimates sit in the upper range of estimates from MaCurdy (1981) and Altonji (1986) estimates (0.08–0.54). However, the literature surveyed by Keane (2011) finds an average estimate of the EIS for men of 0.83 and a median value of 0.17. The higher Frisch elasticity we find for women confirms previous findings by Keane (2011), BPS, and others. We find evidence of price inelastic demand for housing services, with the estimated own-price elasticity for housing services equal to −0.33. This estimate compares to the (−0.19, −0.63) range estimated by Hanushek and Quigley (1980) and the more recent (−0.20, −0.58) range estimated by Ioannides and Zabel (2003) for the US housing market. The estimated cross-price Frisch elasticities offer a number of interesting insights. First, we find evidence in favor of non-separability between: (a) the hours of the two earners, (b) consumption and housing services, and (c) hours and housing services, with the cross-price elasticities statistically significant at conventional levels of significance. In contrast, and differently from BPS, we do not find statistically significant evidence of nonseparability between consumption and leisure.23 Two results are particularly worth nothing. First, we estimate an economically large Frisch elasticity of consumption to changes in house prices ηc,ph = 0.131. Comparing this number to estimates from the literature is complicated because most papers estimate “uncompensated”, not Frisch elasticities. Below, we show that our ”uncompensated” numbers (i.e., our estimate of kc,ph ) are in a similar ballpark as estimates from Case et al. (2013) using aggregate data (an elasticity of non-durable consumption ranging between 0.03 and 0.18) and Mian et al. (2013) (who use household level micro data and estimate elasticities between 0.13 and 0.26).

23 Notice that, compared to BPS, we not only introduce housing and risky returns in the model, but also apply a different sample selection, keeping only stable homeowners. Moreover, we have a longer panel (1998–2014 vs. 1998–2008). Finally, BPS use a measure of nondurable consumption that includes nondurable goods and housing services, while we consider housing services separately from goods. Hence, a possible interpretation of the evidence is that leisure is separable from goods but non-separable with respect to housing services. Aggregating goods and services into a measure of nondurable consumption may miss this important source of preference heterogeneity.

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Table 5 Preference parameters − Robustness. Baseline (1)

Individual Assets (2)

Identity Matrix (3)

No part. correction (4)

zt = 0 (5)

50% m.e. risky returns (6)

σh2 = 0.014

μ = 0.75

μ = 0.5

(7)

(8)

(9)

0.270∗ ∗ ∗ (0.034) 0.510∗ ∗ ∗ (0.075) 0.927∗ ∗ ∗ (0.116) −0.325∗ ∗ ∗ (0.046)

0.283∗ ∗ ∗ (0.041) 0.502∗ ∗ ∗ (0.075) 0.923∗ ∗ ∗ (0.115) −0.279∗ ∗ ∗ (0.064)

0.242∗ ∗ ∗ (0.035) 0.514∗ ∗ ∗ (0.074) 0.925∗ ∗ ∗ (0.115) −0.303∗ ∗ ∗ (0.045)

0.225∗ ∗ ∗ (0.036) 0.517∗ ∗ ∗ (0.074) 0.922∗ ∗ ∗ (0.114) −0.283∗ ∗ ∗ (0.045)

Own-price elasticities

η c,p ηl1 ,w1 ηl2 ,w2 ηh,ph

0.271∗ ∗ ∗ (0.034) 0.509∗ ∗ ∗ (0.074) 0.923∗ ∗ ∗ (0.115) −0.326∗ ∗ ∗ (0.045)

0.286∗ ∗ ∗ (0.035) 0.500∗ ∗ ∗ (0.072) 0.896∗ ∗ ∗ (0.109) −0.322∗ ∗ ∗ (0.045)

0.350∗ ∗ ∗ (0.059) 0.540∗ ∗ ∗ (0.068) 0.980∗ ∗ ∗ (0.104) −0.0824 (0.063)

0.258∗ ∗ ∗ (0.030) 0.621∗ ∗ ∗ (0.078) 0.701∗ ∗ ∗ (0.096) −0.334∗ ∗ ∗ (0.041)

0.297∗ ∗ ∗ (0.033) 0.505∗ ∗ ∗ (0.074) 0.922∗ ∗ ∗ (0.116) −0.352∗ ∗ ∗ (0.044)

0.010 (0.026) 0.003 (0.027) 0.059∗ ∗ ∗ (0.015) 0.131∗ ∗ ∗ (0.017) −0.008∗ ∗ (0.003) −0.022∗ ∗ ∗ (0.006)

0.007 (0.026) 0.005 (0.026) 0.046∗ ∗ ∗ (0.015) 0.129∗ ∗ ∗ (0.017) −0.008∗ ∗ (0.003) −0.022∗ ∗ ∗ (0.006)

−0.012 (0.027) 0.016 (0.029) 0.044∗ ∗ ∗ (0.014) 0.129∗ ∗ ∗ (0.025) −0.024∗ ∗ ∗ (0.003) −0.035∗ ∗ ∗ (0.007)

0.006 (0.028) −0.005 (0.024) 0.099∗ ∗ ∗ (0.014) 0.131∗ ∗ ∗ (0.015) −0.008∗ ∗ (0.003) −0.022∗ ∗ ∗ (0.006)

0.006 (0.027) −0.000 (0.029) 0.056∗ ∗ ∗ (0.015) 0.147∗ ∗ ∗ (0.016) −0.008∗ ∗ ∗ (0.003) −0.024∗ ∗ ∗ (0.006)

0.010 (0.026) 0.002 (0.028) 0.061∗ ∗ ∗ (0.015) 0.131∗ ∗ ∗ (0.017) −0.008∗ ∗ (0.003) −0.022∗ ∗ ∗ (0.006)

−0.000 (0.026) −0.004 (0.029) 0.054∗ ∗ ∗ (0.015) 0.113∗ ∗ ∗ (0.027) −0.008∗ ∗ ∗ (0.003) −0.021∗ ∗ ∗ (0.006)

0.014 (0.026) 0.005 (0.027) 0.063∗ ∗ ∗ (0.015) 0.113∗ ∗ ∗ (0.017) −0.006∗ ∗ (0.003) −0.020∗ ∗ ∗ (0.006)

0.014 (0.026) 0.006 (0.025) 0.063∗ ∗ ∗ (0.015) 0.097∗ ∗ ∗ (0.017) −0.004 (0.003) −0.017∗ ∗ ∗ (0.006)

8068

8068

8068

8069

8068

8068

8068

8068

8068

Cross-price elasticities

ηc,w1 ηc,w2 ηl1 ,w2 ηc,ph ηl1 ,ph ηl2 ,ph Observations

Note: Parameters estimated using GMM. Model estimates in all columns allow for nonseparability in preferences over consumption, housing services and hours of work of the two earners. Column (1) reports the estimates of the baseline estimation as described in Section 5. Columns 2 to 9 report estimates obtained modifying the estimation in Column (1): Column (2) uses individual level assets. Column (3) uses an identity matrix as weighting matrix. Column (4) reports the estimates shutting off the correction for female non-participation. Column (5) reports the estimates obtained setting the budget share to housing depreciation costs (Zt ) to zero. Column (6) reports the estimates obtained assuming that 50% of the estimated variance in the shocks to risky returns is due to measurement error in the stockholding values, while Column (7) reports estimates when setting the variance of house price shocks σh2 = 0.014. Columns (8) and (9) report estimates obtained setting the share of housing wealth accessed by households to smooth shocks equal to 75 and 50%, respectively. Clustered standard error in parentheses. Three, two, and one stars indicate statistical significance at the 1, 5 and 10%, respectively.

Second, we find evidence of Frisch complementarity between housing services and leisure. Our estimate for the elasticity of female labor supply to changes in house prices ηl2 ,ph = −0.022 is larger than that of males (ηl1 ,ph = −0.008). These findings highlight the importance of properly considering labor supply when modeling the behavioral responses of households to changes in house prices. Indeed, because hours and housing services are substitutes in utility, these results suggest that an increase in the price of housing - absent changes in the marginal utility of wealth - induces a decline in the labor supply of both earners. In other words, households choose to spend a part of their wealth gains in increased leisure, and the effect of house price changes in consumption is mitigated because hours of work fall. As we shall see, once we allow for a marginal utility of wealth response, the effect is attenuated (but remains negative for most of the life cycle, with some interesting sources of heterogeneity). Robustness. We conduct several additional analyses to check the robustness of our main results. The results of these analyses are reported in Table 5. In the first column, for comparison, we reproduce the results of our baseline specification with nonseparable preferences (from Table 3, column (2)). In column (2) we estimate the model using individual assets instead of assets averaged by age and education. In column (3) the model is estimated using an identity matrix as weighting matrix instead of the inverse of the diagonal of the estimated variance-covariance matrix of the data moments. In column (4) we estimate the model dropping the correction for female non-participation (which requires re-estimating household human wealth, etc.). Column (5) reports the results obtained setting zt = 0, as in Etheridge (2019). To overcome potential concerns about the role of measurement error in stock-holding values, we also re-estimate the model assuming that 50% of the variation in risky return is attributable to measurement error (column (6)). Moreover, we estimate the model setting a value for the annual variance of house price growth equal to that computed using the Case-Shiller 20-City Composite Home Price Index for the period 20 0 0–2014 (σ2h = 0.014), column (7). Finally, we check (informally) that our finding of non-separability in preferences between consumption and housing services is not due to the assumption of absence of collateral constraints and transaction costs in extracting home equity. This is a strong assumption that may imply an excessively close link between consumption (and labor supply) growth and house prices changes. To perform this exercise, we add a “reduced form” parameter μ that multiplies HWt whenever it appears in the expressions for the response parameters, Eq. (11). We look at the sensitivity of estimates when imposing values for μ < 1. One way to interpret this exercise is that it approximates a situation in which changes in house prices

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Fig. 5. Response of consumption to a 10% positive shock to house prices. (a) By age of the household head; (b) By housing wealth shares.

Fig. 6. Response of consumption to a 10% positive shock to stocks returns. (a) By age of the household head; (b) By share of wealth in stocks.

cannot be fully converted in ”liquidity” available for consumption or leisure smoothing.24 Columns (8) and (9) report the estimates obtained when setting μ = 0.75 and μ = 0.5. The results of these alternative estimation exercise, as expected, attenuate the elasticities. However, the differences are contained and the finding of non-separability is unchanged. 6. Implications Response of consumption to shocks to asset returns over the life cycle. The responses of consumption to house price shocks and shocks to risky returns predicted by the model can be obtained plugging the estimated preference and weighting factors in the expressions for kc,ξ h and kc,ξ s from Eq. (12), respectively. The results are shown in Figs. 5 and 6 (the solid lines). The left panel in each figure shows how the response changes over the life cycle; the right panel how it changes by the share of housing wealth (the share of risky assets) out of total household wealth. Households respond to a 10% increase in house prices by increasing consumption, on average, by about 1.3%. Note that this is very close to the ηc,ph estimate from Table 3, implying that the average indirect effect from shifts in the marginal utility of wealth is small. Nonetheless, there is substantial response heterogeneity. Fig. 5 shows how the response of households to house price shocks changes at different stages of their life cycle and across the housing wealth distribution. Indeed, because of the larger relevance of housing wealth over total resources, older households increase consumption more than younger households in response to the same house price change. 24

This exercise is meant to be illustrative, since realistic adjustment costs do not induce “smooth” adjustments.

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Fig. 7. Response of labor supply to a 10% positive shock to house prices. (a) By age of the household head; (b) By housing wealth shares.

Fig. 8. Response of labor supply to a 10% positive shock to stocks returns. (a) By age of the household head; (b) By share of wealth in stocks.

From Fig. 6, a positive 10% shock to risky returns induces households participating in the stock market to increase consumption by about 0.04% on average. Also in this case, there is significant life-cycle heterogeneity. Since older households have a larger part of their financial wealth invested in stocks (a higher α s ), and have a higher share of total lifetime wealth on liquid assets (a higher π ), they change consumption more than younger households (see Fig. 6). The role of family labor supply. How important are labor supply wealth effects? Does an increase in wealth induce households to consume more goods as well as more leisure? As we discussed, the effect is ambiguous, since it depends on a number of factors (including non-separability between housing and leisure, importance of housing in total wealth, etc.). We start our analysis by computing the values for kl j ,ξh and kl j ,ξs (for j = 1, 2) implied by our estimated preference parameters and weighting factors. Fig. 7 plots the implied value of kl j ,ξh against age of the household head (left panel) and the impor-

tance of housing wealth over total household wealth, i.e., HWt (right panel). Fig. 8 does the same for kl j ,ξs , although the left panel is a plot against the importance of risky assets over financial wealth (i.e., αts ). Fig. 7 shows that the effect of house price shocks on labor supply changes over the life cycle and by the amount of housing wealth held relative to other forms of wealth. For most of the life cycle and for households where housing is the dominant asset, the effect is negative (implying a “wealth effect” for leisure). Furthermore, the figure shows that female labor supply is more elastic to changes in house prices than male labor supply, complementing previous findings in the literature. However, it is worth noting that the effect is positive for young households. This may in principle reflect a labor supply “insurance” effect in the presence of frictions. Young homeowners who face a positive house price shocks may be unable to change their consumption much, either because they are unable to borrow against home equity or because they

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do not have much non-housing wealth to draw from. One way to finance a consumption increase is hence to substitute labor intertemporally, i.e., working more when young and less when old, consistent with the pattern shown in Fig. 7.25 Fig. 8 shows that the response of hours of work of both earners to a positive shock to risky returns complement the results for the response to house price changes (with no ambiguity arising - most likely due to the fact that shocks to returns to financial wealth are more “iquid” that house price shocks). Hours fall in response to a positive shock to risky return and the effect is larger for older households, female earners, and for households with a large exposure to risky assets in their portfolio. Another useful exercise is to assess how family labor supply changes the quantitative role of the consumption “wealth effect”. To study this issue, we perform a simple decomposition exercise. In particular, we can use the intertemporal budget constraint to derive:



∂ y ∂ c ∂ y 2  1 = k ≈ s + ( 1 − s ) + savings response h c,ξ ∂ξh f lex l ∂ξh ∂ξh   

(16)

∂ y ∂ξ h

where the term “savings response” captures liquid asset reallocation as well as the response to changes in the value of housing wealth to house price shocks. We can assess the role of labor supply as an insurance mechanism against house price (or stock return) changes by comparing the simulated response of consumption to house price shocks with and without flexible labor supply, i.e., compare (16) with (17) below:

∂ c ∂ c ∂ y ≈ − ∂ξh f lex l ∂ξ h ∂ξ h f ixed l

(17)

Fig. 5 compares the actual response of consumption to house price shocks (given by Eq. (16), the solid line commented above) with the response obtained fixing female labor supply (the dashed line), and the response where labor supply adjustments of both earners are shut down (Eq. (17), the dash-dotted line). The latter is the case typically considered in the literature. Fig. 6 replicates the exercise for the shocks to risky returns. These results are informative about the importance of labor supply to compensate for house price shocks and shocks to risky returns. Later in life, when housing wealth plays a larger role (i.e., the relative weight of the wealth affected by the economic shock over total lifetime wealth is larger), a positive shock to housing prices increases consumption much more in the fixed labor supply case than in the flexible case. This is because, when labor supply is flexible, household prefer allocating some of the wealth effect to higher consumption of leisure and less work, dampening the consumption effect. Ignoring labor supply hence exaggerates the consumption wealth effect. House prices shocks and extensive margin of female labor supply. While the results above pertain to intensive margin responses, some may come from extensive margin responses (i.e., women deciding to leave/enter the labor market in response to positive/negative asset price shocks). The interest in extensive margin responses is partly because workers may not be able to change their working schedule freely. The problem with studying extensive margin responses is that they cannot be dealt with using the approximation framework we are using. Instead, we present some reduced form evidence, regressing changes in an employment indicator against house price shocks (controlling for demographics). Changes in the husband’s wage are instrumented using long-run wage growth. Similarly, we instrument house price changes with long-run house price growth. Results are reported in Table 6. As in the intensive margin case, house price shocks have an economically relevant effect on the probability of wife’s participation (women move into work when house prices fall and reduce their participation when wealth effects are positive). However, the effect is statistically important only for women aged 40 or more. Younger women, partly because they are in households with little home equity (and likely a larger mortgage), are more inelastic. In keeping with this intuition, we find a significant negative effect of shocks to house prices on the probability of wife’s participation when housing wealth represents a larger share of the household’s resources. House prices shocks and total hours response. With the estimated intensive and extensive margin responses to changes in house prices at hand, we can compute the total aggregate hours response. Since we do not consider extensive margin effects for men, we do this only for women. Assuming that the secondary earner supplies a number of hours equal to the average number of hours worked by employed women, we can write the change in hours in response to a house price shock as:

l2tot =

kl ξ h

2 

intensive margin response

+ ( 1 + kl2 ξ h )

Pl 2

(18)

extensive margin response

We calculate the total hours effects using the estimate of the intensive and extensive margin effects (kl ξ h and Pl2 ) 2 detailed above. We find that total hours of the secondary earner increase by approximately 1% on average following a 10% decrease in house prices. This effect is larger for households with a larger share of total wealth in housing. For example, in households with above-median HWi,t , a 10% decrease in house prices induces a 3% increase in total female hours. 25 As we shall see below, this is also consistent with the fact that - in the counterfactual fixed labor supply case - young households’ consumption wealth effect is attenuated, relative to the flexible labor supply case examined here.

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C. Daminato and L. Pistaferri / European Economic Review 124 (2020) 103389 Table 6 House price shocks and extensive margin of female labor supply.

House price shocks Other controls State and county FE Year FE Observations

(1) Full sample

(2) Housing wealth above median

(3) House over income above median

(4) Age of spouse above 40

−0.079 (0.086) YES YES YES 3761

−0.231∗ ∗ (0.111) YES YES YES 2058

−0.329∗ ∗ ∗ (0.114) YES YES YES 1940

−0.181∗ (0.100) YES YES YES 2834

Note: IV estimates of the impact of shocks to house prices on female labor market participation. Dependent variable is change in wife’s participation. Each regression controls for changes in husband’s wage, age, age squared, education and race of both spouses, household size, changes in household size, changes in the number of kids, changes in the age of the youngest kid, change in the presence of a one-year old, state of residence by metropolitan area effects and year effects. Changes in husband’s wage are instrumented using long-run wage growth. Similarly, we instrument house price changes with long-run house price growth. In column 1, we report results obtained using the whole sample. In Column 2, we restrict the sample to observations where the share of housing wealth out of total lifetime wealth HWt is above the median. Column 3 restricts the sample to observations where the ratio of house value over current husband’s income is above the median value. Finally, in Column 4 we report results for wives older than 40. Standard error clustered at the individual level in parentheses. Three, two, and one stars indicate statistical significance at the 1, 5 and 10%, respectively.

Limitations of our approach. Finally, it is worth discussing some of the paper’s limitations. First, the focus on homeowners means that we do not have much to say about determinants of the choice of owning vs. renting or the relevance of collateral constraints at the point of purchase. As we mention above, the consequence for our analysis of excluding renters is that we miss a group for whom house price increases may depress rather than stimulate consumption (if rents track house prices), implying that our average wealth effect is likely an upper bound for the aggregate or macro-elasticity.26 Second, we have assumed that households face no transaction costs when adjusting their stock of housing. Recent work (such as Kaplan and Violante, 2014) has studied how consumers respond to income shocks when some of their assets (housing or defined contribution pensions) are partly illiquid. The distinction between liquid and illiquid assets can rationalize the presence of “wealthy hand-to-mouth” consumers, i.e., consumers who are nominally wealthy, but have most of their wealth in illiquid form, and hence respond more to income shocks or anticipated income changes than individuals with similar wealth held mostly in liquid form. In general, frictions attenuate estimated responses (i.e., true elasticities are larger than estimated elasticities, see Chetty, 2012), with the bias being smaller for larger house price changes creating larger benefits from adjusting. Some of our findings (i.e., the smaller wealth effect estimated for young homeowners, who are most likely to have their wealth in illiquid form) suggest that the distinction between liquid and illiquid wealth may be important and possibly behind the heterogeneity we uncover in the data. Etheridge (2019) shows that the housing wealth share (which corresponds to our approximation for the house price elasticity of consumption in the case of separable preferences, exogenous labor supply and zero net housing depreciation) does in general a good job in approximating the house price elasticity obtained from a full benchmark model that includes transaction costs and collateral constraints. However, he shows that allowing for transaction costs and collateral constraints can decrease (some) house price elasticities, compared to the housing wealth share approximation, with the bias being larger for young households who are most likely to have their wealth in illiquid form. The results we obtain adding a “reduced form” parameter μ < 1 (multiplying the housing wealth share HWt ), while confirming our main findings, also suggest that allowing for transaction costs and collateral constraints may slightly decrease the estimates of the structural elasticities to changes in house prices, and then that the distinction between liquid and illiquid wealth may be important. Third, we assume consumers can freely borrow against their housing wealth, facing no collateral constraints. In practice, since there is wage uncertainty, consumers do face a natural borrowing constraints and may optimally decide not to borrow against increases in their housing wealth if there is enough uncertainty in wages (Aiyagari, 1994). Moreover, consumers in our model can overcome collateral constraints either by drawing from accumulated liquid wealth or from intertemporally substituting their labor supply (a channel that is absent in most of the literature). Future work should be directed towards incorporating realistic adjustment costs for the stock of housing and formal collateral constraints in a model with endogenous labor supply, which we have shown playing a non-negligible role in the overall response of households to changes in their total resources.

26 In a framework broadly similar to ours (without labor supply but with housing adjustment costs), Berger et al. (2018) show that the life cycle average elasticity of consumption to house price shocks increases from 0.23 to 0.39 when the rental option is eliminated (with most of the bias concentrated at young ages where fewer households own a house).

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7. Conclusions This paper presents and estimates a life-cycle model of consumption, housing and labor supply with two earners where households face shocks to hourly wages, house prices and returns from risky assets. We use approximations of the first order conditions and the intertemporal budget constraint to derive equations describing the response of consumption (of goods and housing services) and the labor supply of both earners to shocks to wages, house prices and risky returns. The solution of the model is obtained given flexible preferences, allowing for potential non separability between non-durable consumption, housing services and labor supply of the two earners. The response of consumption and labor supply to shocks to house prices and risky returns depend on preference parameters, such as own-price and cross-price elasticities, and terms reflecting the relevance of liquid assets, housing, and earnings to insure against shocks. In particular, we show that non-separability in preferences implies that the response of consumption to changes in house prices and shocks to risky returns also depends on the elasticity of consumption to transitory shocks to wages of both earners. We reject separability in preferences, providing evidence for the importance of considering labor supply dynamics when studying the response of consumption to shocks to the financial markets. This point is mostly neglected in the wealth effect literature. Even though both earners use labor supply to insure against shocks to the risky portfolio allocation, females are more responsive than males, suggesting that within household dynamics are important determinants of the household decision as a whole. Our estimate of the Frisch elasticity of consumption to changes in house prices (0.13) falls in the ballpark of previous estimates in the literature. Furthermore, we find evidence of complementarity between housing services and leisure, which is more relevant for females. This suggests that labor supply of both earners is an implicit consumption insurance mechanism against changes in house prices (especially when the household is young and presumably unable to borrow or draw from accumulated assets to finance a consumption increase).27 Moreover, it confirms the importance of family labor supply choices for understanding the broad response of households to shocks to resources. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.euroecorev.2020. 103389. References Aiyagari, S., 1994. Uninsured idiosyncratic risk and aggregate saving. Q. J. Econ. 109 (3), 659–684. Altonji, J.G., 1986. Intertemporal substitution in labor supply: evidence from micro data. J. Polit. Econ. 94 (3), S176–S215. Altonji, J.G., Segal, L.M., 1996. 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27 For older households the mechanism is potentially different. Given that they have a larger share of total resources in housing, large positive (negative) changes in house prices induce an increase (decrease) in the consumption of both leisure and goods.

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