Consumption and Time Use over the Life Cycle

Transcription

1 Consumption and Time Use over the Life Cycle Michael Dotsey Wenli Li Fang Yang y February 213 Abstract We incorporate home production in a dynamic general equilibrium model of consumption and saving with illiquid housing and a collateralized borrowing constraint. We show that the model is capable of explaining life-cycle patterns of households time use and consumption of di erent categories (housing, home input, and market good) that we document from the micro data. A plausibly parameterized version of our model predicts that the interaction of the labor e ciency pro le and the availability of home production technology explain households time use over the life cycle. The resulting income pro les, the endogenous borrowing constraint and the presence of home production account for the initial hump in all three consumption goods. The consumption pro les in the second half of the life cycle are mostly driven by the complementarity of home hours, home input, and housing in home production. The increasing use of home production is also partially responsible for the pronounced di erence between the life-cycle pro les of consumption and expenditure. JEL Classi cation: D13, E21, J22 Key Words: Consumption, Home Production, Life Cycle We are indebted to comments received on earlier drafts and presentations from participants at various seminars and conferences. We thank, in particular, Satyajit Chatterjee, Jason Faberman, Jesus Fernandez- Villaverde, Eric French, John Jones, Loukas Karabarbounis, Dirk Krueger, and Richard Rogerson for their comments and suggestions. The views expressed are those of the authors and do not necessarily re ect those of the Federal Reserve Bank of Philadelphia or the Federal Reserve System. y Michael Dotsey and Wenli Li: Research Department, Federal Reserve Bank of Philadelphia, Ten Independence Mall, Philadelphia, PA 1916 ( michael.dotsey@phil.frb.org; wenli.li@phil.frb.org). Fang Yang: Department of Economics, University at Albany - SUNY, Albany, NY ( fyang@albany.edu). 1

2 1 Introduction This paper examines the joint pro le of di erent types of consumption goods and time-use of households over the lifecycle. It, thus, represents a departure from the existing literature that has focused largely on the life-cycle pro le of aggregate consumption. The paper is motivated by the recent empirical work of Aguiar and Hurst (29) where they document signi cant heterogeneity in the life-cycle pro le of various components of consumption and the systematic linkage between this heterogeneity and whether the good could be produced at home or whether the good has to be consumed directly from the market. The modeling strategy we use depicts the household as a joint consumption-production unit in an otherwise standard life-cycle model with precautionary savings motives and endogenous labor supply decisions. An immediate implication of this strategy is that expenditure and consumption are not identical and that actual life-cycle consumption may behave very di erently from life-cycle expenditure. 1 Capturing this di erence obviously has important implications for the stringency of liquidity constraints and the desire for precautionary savings required to match expenditure patterns over the life cycle. In turn, an accurate model that endogenously captures the di erences between life-cycle expenditure and life-cycle consumption will in uence conclusions regarding how various scal policies and changing income inequality a ect economic behavior and welfare. A main contribution of our model is that it is the rst to account for the consistency between the life-cycle pro les of time use and the life-cycle pro les of consumption of various categories of goods. It, therefore, serves as a useful framework for examining key features of life-cycle behavior as well as policies that a ect this behavior. In modeling the process of home production, we follow the home production literature and introduce residential capital, which we term housing, into home production in addition to home hours. Doing so is desirable for a number of reasons. First, much of time use is spent performing activities associated with owning a home and we are interested in the joint relationship between how time is spent over the life cycle and how time-use decisions are related to consumption decisions. Second, as Fernandez-Villaverde and Krueger (211) point out, certain nancial constraints associated with durable good purchases can help account for life-cycle consumption pro les and housing serves as the durable good in our model. However, housing also possesses some features not usually associated with other durables, namely, the cost of altering the amount of housing as well as its slower depreciation, that turn out to be important in explaining consumption behavior later in life. Finally, it is also straightforward to discipline the model with respect to housing services as there exist accurate and detailed data regarding the housing decision. To summarize, the paper accounts for three important aspects of life-cycle behavior which we establish or con rm using micro data. The rst is that households labor supply 1 Aguiar and Hurst (27b) make a similar point regarding the di erence between consumption and expenditure. See section 8 of this paper for more discussion. 2

3 exhibits strong life-cycle patterns. For homeowners, market hours and home hours are fairly stable early in the life cycle. At age 55, home hours begin to increase while market hours start to decline sharply. For renters, market hours decline consistently through the life cycle while home hours remained roughly constant. The second is the substantial heterogeneity across life-cycle pro les of individual consumption expenditures. We group consumption goods into three broad categories according to their relationship with home production: a) housing, rented or owned, which is an important input for home production, b) a home input consisting of goods purchased from the market, which serves as an intermediate input for home production, and c) goods that are purchased in the market and consumed without the need for any intermediate home production. We refer to the latter as the market good. We nd that consumption of the market good rises early in life and declines substantially late in life. The other two categories, by contrast, rise more but do not decline nearly as much. Third, our model produces choices over housing that are consistent with what has been termed the housing puzzle. Old age households home ownership rates and their house values do not decline much, if at all, toward the end of life. Of equal interest is that renters house consumption declines much less than market consumption. 2 Three key features of our model help account for these observations. First, households are subject to collateralized borrowing constraints; they can borrow only up to a fraction of their house value. This feature, together with the standard assumption regarding the labor e ciency pro le and the presence of uninsurable labor income risk, helps account for the increasing consumption pro les in the early part of a household s life cycle as in Fernandez-Villaverde and Krueger (211). Second, at each point in time, households divide their time between market hours, hours in home production, and leisure. As households age, their market labor e ciency declines and they devote more and more of their time to home production. As a result, consumption of the market good declines drastically. Consumption of the home input and housing services also decline, but the decline is much more muted due to their complementarity with home hours in home production. Third, we di erentiate between owner-occupied housing and rental housing. We assume that housing adjustment is costly for homeowners, but costless for renters. This assumption allows us to directly measure the predictions of our model concerning owners and renters consumption and labor supply against those from the data. We nd that housing adjustment costs contribute to the slow decline in the home ownership rate and the value of the housing stock owned as households age as in Yang (29). An important implication of our model is that the life-cycle pro le of consumption is smoother than the life-cycle pro les of expenditure. The di erence in the behavior stems 2 Housing puzzle typically refers to the observation that homeowners do not deplete their home equity as much as what a standard life-cycle model predicts as they age (Venti and Wise 22). The puzzle that this paper is after pertains to the related observation that households do not downsize their houses nor do they become renters as standard theories predict as they age. In other words, households continue to maintain a similar level of housing expenditure at old age, in contrast to most other goods. 3

4 from the insurance mechanism provided by home production. This insurance mechanism a ects the strength of the precautionary savings motive needed to match the behavior of consumption expenditure in the data. In turn, this will lead to di erent welfare consequences in response to changes in the economic environment. 3 The rest of the paper is organized as follows. In section 2, we discuss the related literature. We present our empirical analysis in which we construct households life-cycle pro les of time use and consumption in section 3 and the model in section 4. We discuss our calibration strategy in section 5 and results in section 6. Further investigation of the model s di erent channels in driving households behavior is presented in section 7. Section 8 discusses the di erence between expenditure and consumption. Section 9 conducts robustness analysis and section 1 concludes. 2 Related Literature The paper draws on two strands of the literature: the consumption literature and the home production literature. One of the most prominent observations in the consumption literature is that aggregate consumption is hump-shaped over the life cycle. Market incompleteness in the form of a borrowing constraint along with uncertain income leads to precautionary savings, which is the key mechanism of leading theories that account for this observation (Hubbard et al. 1994, Carroll 1997, and Gourinchas and Parker 22). 4 Fernandez-Villaverde and Krueger (211) add to this literature by documenting that the hump persists for consumption of both durables and nondurables when considered separately and they propose an incomplete markets model in which durables serve as collateral to explain these stylized facts. Separately, Bullard and Feigenbaum (27) incorporate leisure into the utility function and show that this additional feature helps explain the decline of consumption late in life. Huggett and Ventura (1999), Heathcote (22), and French (25) combine precautionary savings, leisure and consumption in their model to study retirement issues. Our paper shares many features with these papers, including the precautionary savings motive, liquidity constraint, and endogenous labor leisure decision. The durable good in our model takes the explicit form of owner-occupied residential housing. Our biggest innovation lies in the fact that we examine the heterogeneity of consumption pro les along the lines of Aguiar and Hurst (29) and that we are jointly 3 In a separate paper, Dotsey, Li, and Yang (212) study the e ects of eliminating Social Security provisions in a model based on the one constructed here. We nd that the presence of home production reduces the cost of uninsurable income risk and thus increases the welfare gain of eliminating the Social Security system. 4 There is also a literature suggesting that time-inconsistent or myopic preferences over consumption can play a role. See Laibson (1997). Several authors including Attanasio et al. (1999) and Browning and Ejrnaes (22) argue that variation in household size could account for why preferences over consumption by a household might change over the life cycle that is, consumption is highest when household size is largest. 4

5 interested in the interaction of time use and consumption. Our paper is also closely related to the recent home production literature. This literature has shown that the introduction of home production in otherwise standard dynamic general equilibrium models is useful in understanding a variety of macroeconomic issues, including domestic and international business cycles, scal policies, and asset equilibrium puzzles (see, among many others, Benhabib, et al. 1991, Greenwood and Hercowitz 1991, Greenwood, et al. 1995, McGratten, et al. 1997, Canova and Ubide 1998, and Gomme, et al. 21). A key way in which models with home production di er from standard dynamic general equilibrium models (that do not include home production) is that home production allows households to substitute along additional margins, both in labor supply and in total consumption. While many recent studies have used time-use data in order to understand household production (Aguiar and Hurst 27a and Ramey 28), much less attention has been paid to the other factors involved in the production of home goods (see Rupert, et al. 2 and papers cited therein). 5 Our paper lls in this gap by tying households labor supply (including leisure choice) and consumption more closely together and we show that home production a ects households life-cycle consumption of all three goods, both early and later in life. 3 Empirical Observations on Lifecycle Behavior Our data come from two sources, the American Time Use Survey (ATUS) and the Consumption Expenditure Survey (CEX). We separate households into owners and renters and deal explicitly with problems of household size, cohort e ect, and time e ect. 6 Although some of our empirical results have been documented in the literature, our contribution lies in merging and reclassifying the two data sets according to the home production literature. Our di erentiation between homeowners and renters is also novel and important in explaining older age households housing puzzle. 3.1 Life-Cycle Pro les of Time Use The ATUS from the Bureau of Labor Statistics measures the amount of time people spend performing various activities (see table 1). The data are strictly cross-sectional as respondents are interviewed only once. Households are top coded at age 8. 7 The survey started in 23, with the most recent one ending in 211. We include in our sample the 25 to 211 ATUS, since the ATUS started reporting 5 One exception is Heathcote (22), who studies the e ect of home production on retirement. 6 We follow most closely the linear regression strategy in Gourinchas and Parker (22) and Aguiar and Hurst (29). 7 Households between ages 8 and 84 are assigned age 8 and those that are 85 and above are assigned age 85. 5

6 households house tenure in 25. We focus on households whose head is between the ages of 25 and 8 (inclusive) but exclude those whose head is either in school or in the military at the time of the survey. Our nal sample consists of 75,11 households, about evenly split across the seven survey years. We follow the tradition of Reid (1934) and separate nonmarket time into pure leisure and home hours where home hours comprise time spent on activities performed at home to produce goods and services that can also be purchased in the market and are, for the most part, not enjoyable to produce. 8 According to Robinson and Godbey (1999, table ), in a 1985 enjoyment of activities survey, households ranked having sex, playing sports, shing, enjoying art and music, and going to bars and lounges at the top and doing yardwork, cleaning house/dishes, doing laundry, providing child health care, and going to the car repair shop at the bottom. Indeed, empirical studies of home production or homemaking, as in the earlier literature, typically classify food preparation, cleaning the house, care of family members living in the household, and shopping and managing the household as home production. Some also include gardening, care of others who are not in the household, and entertaining children. 9 We thus de ne home hours as time spent doing house work, shopping, pet care, car care, adult care, shop search, car care service, and professional service. We also include child care and child care service in home hours if they are performed during week days. We de ne market hours as the time the head of the household spends working, job searching and commuting. We treat the remaining time including child care and child care service performed during weekends or holidays as leisure. 1 That is, as discussed in Baxter and Rotz (29), home production activities are associated with disutility, whereas leisure activities provide utility. This decomposition strategy allows us to highlight the substitution of home-produced goods and market goods. For those households that were interviewed on Saturday or Sunday (holidays are viewed the same as Sunday in the ATUS), we approximate their weekday hours by the average hours of those interviewed on weekdays, in the same year, of the same education, and gender. Similarly, we approximate the weekend and holiday hours for households interviewed during weekdays. We adjust all hours by family size, marital status and survey year e ects. We also control for families that have young children (under the age of 6). Following Aguiar and Hurst (29), we identify life cycle from cohort e ects by using the multiple cross-sections in our model and use cross-sectional di erences in family size and 8 In particular, she de nes home production as those unpaid activities which are carried on, by and for the members, which activities might be replaced by market goods, or paid services, if circumstances such as income, market conditions, and personal inclinations permit the service being delegated to someone outside the household group (Reid 1934, p.11). 9 See Ramey (28) for a thorough discussion. 1 Our classi cation of child care and child care services re ects the view that there is some element of enjoyment associated with child care. We think this enjoyment most likely occurs on weekends and holidays when families are more likely to pursue leisure activities together. We thank Loukas Karabarbounis for pointing this out to us. 6

7 interview year, respectively, to identify family size and interview year e ects. Speci cally, we estimate the following equations, H k it = k + k ageage it + k y Y it + k f F it + k ycy C it + k mm it + " k it; where Hit k represents time use in category k (market work or home work) by household i at time t, AGE it is a vector of 55 one-year age dummies, Y it is a vector of 6 one-year interview dummies (26 is the omitted year), and F it is a vector of family structure dummies that include 9 family size dummies, 1 to 1, (household of size 3 is the omitted group; 1 includes families with 1 or more family members). Y C it is a dummy indicating whether the family has any children under the age of 6. M it is a dummy for marital status. The coe cients on the constant k together with age dummies, age, k capture the impact of the life cycle conditional on family size and interview e ects. Figure 1 charts the share of market hours and home hours by age for the household head. As can be seen, market hours by homeowners and renters have overall similar shapes with the young supplying more than the old. However, there are also signi cant di erences. Overall, homeowners supply more market hours than renters especially at young ages. Furthermore, homeowners do not reduce their market hours until age 55, but renters market hours experience a drop at age Life-Cycle Pro les of Consumption Expenditure The CEX also from the Bureau of Labor Statistics collects household demographic characteristics and consumption expenditure information. The data are a rotating panel with each household being interviewed from 2 to 5 quarters, and every quarter 25 percent of the sample is replaced by new households. Because of this short-panel dimension of CEX, we pool the data and treat it as one cross-section. We focus on the 23 to 21 CEX data. We include in our market good food away from home, alcohol, tobacco, apparel, other lodging, fees and admissions for entertainment, and related equipment such as televisions, radios, sound systems, pets, toys, and playground equipment, reading, and personal care. We also include education expenses and out-of-pocket medical expenses in the market good, but our results are robust to the exclusion of these categories. We include in our home input food at home, household operations, household furnishing and equipment, utilities, fuels, and public services. We prorate transportation expenses by travel time for home production, market production or leisure that we obtained from the ATUS. For housing expenditures, we use rental payments for renters. We use homeowners reported house value of owned residence because we believe 11 We conduct Hausman speci cation test to see whether the coe cients for the regressions using homeowners are statistically di erent from those using renters for both market hours and home hours. The analysis rejects the hypothesis that the regression results are the same for the two types of households for both market hours and home hours. 7

8 homeowners have a better idea of how much their house is worth as opposed to how much their house can be rented for. Using reported rental value for homeowners does not change our results qualitatively. Table 2 summarizes the de nitions. We delete from our sample households that reported zero or negative consumption of the market good plus the home input, renters who reported less than $3 in annual rent, and homeowners who reported less than $1 in house value. All consumption data are adjusted by their respective 2 chained Consumer Price Index. Our nal sample consists of 74; 28 households of which about 66 percent are homeowners. We use the same strategy outlined earlier to identify life-cycle pro les of the three consumption categories. The results are presented in Figure 2 in log deviations from age 25. As one can see, for homeowners, the market good, home input, and housing services all move up substantially from age 25 to age 5. The hump in housing services, however, is the most pronounced. The increase is over 4 percent as opposed to about 2 percent in the home input and in the market good. Starting in a household s early 5s, consumption of the market good begins to decline. By comparison, consumption in the home good and housing services starts to decline much later (age 6) and less dramatically. For renters, the market good starts declining from age 25 while expenditures on home inputs and rental housing services start to decline at age 4. Similar to homeowners, renters experience a steeper decline in the market good than in either the home input or housing services. The di erences between renters and home owners are partially due to selection e ects in the sense that those households who remain renters as they age are generally ones that have had poor productivity draws and hence lower income and lower assets Model We consider a modi ed model of Fernandez-Villaverde and Krueger (211) and Yang (29). It is a discrete-time overlapping generations economy with an in nitely lived government. The government taxes labor income and provides pensions to retirees. The model has several key features. First, consumers value leisure and a composite consumption good that consists of a market good and a home-produced good. Second, households face uninsurable idiosyncratic shocks to their labor e ciency. Finally, we restrict intertemporal trade by borrowing constraints collateralized by housing. 12 As with labor hours, we conduct Hausman tests which suggest that there are statistically signi cant di erences between the regression coe cients for homeowners and renters for all three consumption goods. 8

9 4.1 Technology There is only one type of market good produced according to the aggregate market production function (1) F m (K; L) = K L 1 ; where K is the aggregate market capital stock and L is the aggregate market labor input. The nal good can be directly consumed, used as an intermediate input in home production, or invested in physical capital or housing. Market capital and housing depreciate at rates k and h, respectively. Home production requires an intermediate home input, housing, and labor. In particular, the home technology has the following nested CES functional form, 13 (2) c h = f H (d; h; n h ) = f! 2 [! 1 d (1! 1 )(h + (1 )s) ] (1! 2 )(n h ) g 2 ; where d denotes the home input, h denotes the housing stock owned by homeowners, s denotes the rental stock of renters, and n h the labor input in home production. 14 The parameter captures the discount of rental housing in home production, implying that owner-occupied housing is more productive than rental housing in home production, thus helping to generate a preference for owning relative to renting. There are numerous ways employed in the literature for generating a preference for owner-occupied housing. For the purpose of this paper it is not crucial which one is used. 15 The parameters! 1 and! 2 control the weights associated with housing, and the composite of the home input and housing in home production, 1 governs the intra-class substitutability between the home input and housing, and 2 governs the inter-class elasticity of substitution between the composite of home input and housing and home hours in home production. 16 Note that a household can either be a homeowner or a renter, but not both. Therefore, h and s cannot simultaneously take positive values. 13 Note that we use a lower case letter to represent the home production technology as home production takes place at the household level. 14 For simplicity, we have combined both nondurable expenditures such as raw food with consumer durables such as appliances into a composite durable good used in home production. We term this composite good the home input. 15 Models of the type we are using generally need some feature to help generate a relative preference for owning. Other possibilities include using tax bene ts of owning as in Gervais (22) and Chambers, Garriga, and Schlagenhauf (29), assuming higher depreciation of rental properties as in Chen (21), or putting a greater desire for owner occupied housing in the utility function as in Kiyotaki, Michaelides and Nikole (211). Our modeling strategy is closest to this last approach. 16 Following Sato (1967), we justify our aggregation by the fact that intra-class elasticity (between home input and housing) is potentially higher than the inter-class elasticity (between home input and home hours or housing and home hours) since home input and housing are more similar in techno-economic characteristics. 9

10 4.2 Financial Institutions Following Gervais (22), we assume there exists a two-period-lived nancial intermediary that pools households capital to supply mortgages and purchase rental housing. It purchases nal goods and uses them as housing services, which it then rents out to renters for use in home production. Speci cally, at the end of the rst period, the intermediary accepts deposits and buys residential capital. In the second period, it rents out residential capital to households at a price per unit and repays deposits with interest at rate r. At the end of the second period, the nancial intermediary sells the residential stock after depreciation to a new agency. The no-arbitrage condition implies that the rental rate on housing is (3) = r + h : 4.3 Consumer s Maximization Problem A continuum of consumers is born in each model period and immediately begins working and consuming. Each consumer at age t faces a positive probability of survival ( t ; t 1) that is independent of other household characteristics. The maximum life length is T. The demographic patterns are stable, so households at age t make up a constant fraction of the population at any point in time. Annuity markets are absent and accidental bequests are distributed to all households in the economy. Preferences Individuals derive utility from consumption of a composite good c that consists of a market-produced nondurable good, c m ; and a home-produced good, c h ; and leisure, l. Preferences are time separable with a constant discount factor. The period utility function from consumption and leisure exhibits constant relative-risk aversion given by (4) U(c; l) = [! 4c (1! 4 )l )] 4 1 ; where (5) c = [! 3 c m + (1! 3 )c h ] : The term! 4 represents the relative weight of the composite consumption good in utility, 4 represents the degree of substitution between the composite consumption good and leisure, denotes the relative risk aversion parameter,! 3 denotes the relative weight of the market good in the composite consumption good, and 3 measures the degree of substitution between the market good and the home-produced good. 1

11 Labor Productivity Labor productivity consists of two components. The rst is deterministic and age dependent with all consumers facing the same exogenous pro le, e t. The second is stochastic with each worker, i; at age, t, receiving a stochastic productivity shock " i t, which follows a Markov process (6) ln " i t = " ln " i t 1 + i t; i t s N(; 2 ). The total productivity of a worker at age t is then given by the product of the worker s age-t deterministic e ciency index and age-t productivity shock: e t " i t. After age T r, households begin to receive Social Security income. For simpli cation, we assume that the Social Security bene ts are functions only of households age. Housing Housing markets are characterized by large transactions costs. Following Yang (29), these costs are speci ed as (7) '(h; h ) = ( ; if h 2 [(1 1 )h; (1 + 2 )h]; 1 h + 2 h ; otherwise, where h denotes the next period s housing stock. In this speci cation, households are allowed to change their level of housing stock without moving by allowing depreciation up to a fraction 1 of the value of the house or by undertaking housing renovation up to a fraction 2 of the value of the house. If the house depreciates by more than a fraction 1 of the value, or appreciates by more than a fraction 2 of the value, we assume that the house has been sold. In that case, the household must pay costs that are a fraction 1 of its selling value. Buying a property incurs a fraction 2 of its purchase price. Finally, there is a minimum house size h (h > ) that can be purchased, i.e., (8) h h: Collateralized credit is the only form of credit in the economy. Further, the borrowing rate, mortgage rate, and deposit interest rate are all assumed to be equal. As a result, mortgages and deposits are perfect substitutes. To purchase a house, a household must provide a minimum down payment equal to a fraction of the value of the house. We use a to denote the net nancial asset position. At any given period the household s nancial assets must satisfy (9) a (1 )h : 17 To rule out negative bequests, we further assume that net worth is bounded below by 17 For a household without a house, the borrowing constraint reduces to the standard form a : 11

12 according to (1) (1 + r)a + (1 h )h : In a model where households di er only by age, wealth, and income, poor ones tend to be renters and wealthy households tend to be homeowners. In US, however, a fraction of rich households are renters. 18 These households choose to rent due to heterogeneity in preferences, house prices, job mobility, or family composition. To capture the impact of those factors, we assume households in each period face renting shocks. Let p t denote the probability of receiving a rental shock at age t. Let m t denote the renting shock at age t that takes the value 1 if the household receives a renting shock, and zero otherwise. Once a household receives a rental shock (m t = 1), it has to remain a renter for the next period or sell its house and become a renter. The exogenous shock is independent of other household characteristics. Timeline At the beginning of each period, after they observe their current idiosyncratic labor shocks, the next period s rental shock, and receive a bequest, households make their labor supply decisions and rent capital to rms. They also purchase the home input and rental housing for the current period. Then market production and home production take place. After production, households receive factor payments and make their consumption and asset allocation decisions. At the end of the period, market capital, and housing depreciate and uncertainty about early death is revealed. Accidental bequests from those who die early are distributed to new households next period to rst satisfy an exogenous beginning of period asset position, and if funds are leftover, they are distributed to the other households in the economy. Households also make rental versus owning decisions if they do not receive a rental shock, and in the event of owning, they choose their house size at the end of the period. The Household s Recursive Problem In a stationary equilibrium, the interest rate r and wage rate w per e ciency unit of labor are constant. The household s state variables are given by (m; t; a; h; "); which denote the agent s rental shock for the next period (m), current age (t), nancial assets (a), undepreciated housing stock (h); and labor productivity shock of the current period ("). If m = 1, then this household is not allowed to own a house in the next period. If h =, then the household is a renter for this period. We have V (m; t; a; h; ") = max fc m;d;s;a ;n m;n h ;h g n U(c; 1 n m o n h ) + t EV (m ; t + 1; a ; h ; " ) 18 For example, in 2, 12 percent of households whose income is in the top quintile are renters (Bureau of Labor Statistics). 12

13 subject to (8), (9), (1), and (11) (12) c m + a + d + '(h; h ) + s1(h = ) + h 1(m = ) b + (1 + r)a + (1 h )h+ + (1 )[e t "wn m ] + pen(t); c m ; s ; a ; d ; n m ; n h 1; where 1(:) is an index function that takes a value of 1 if the statement inside the parenthesis is true and otherwise, denotes the Social Security income tax, and pen(t) is the pension after retirement for t T r. In any sub-period, an agent s resources depend on received bequests, b, asset holdings, a, housing stock, h, labor endowment, e t ", and pension, pen(t): Note that households receive a pension only after retirement and even after that, they can still work and are subject to the Social Security tax. The composite consumption good c is de ned as in equation (5), the home-produced good is de ned as in equation (2) using current period housing h or s, home input d; and home hours n h as inputs. Our online appendix A provides a formal de nition of a stationary equilibrium with market clearing conditions. Appendix B provides the detailed computation algorithm. 5 Calibration and Estimation We choose the parameters of our model in two steps. The rst step is a standard calibration exercise where we pick parameters individually that are based on economic statistics from the data as well as choosing parameters that are consistent with the literature. In the second step, we jointly choose a set of 11 parameters that minimize a loss function based on the di erence between certain model and data moments calculated o households time use and consumption. The calibrated parameters and the statistics that generate them are given in table 3 and the estimated parameters are given in table 4 column 3. Table 5 columns 2 and 3 indicate how close the model moments match the data moments. 5.1 Parameter Calibration The model period is two years. 19 Each person enters into the model at age 25 and leaves with certainty at 99. Panel b of Figure A1 in our online appendix shows the vector of 2-year conditional death probabilities. We use the mortality probabilities for people born in 196, weighted by gender from the Social Security Administration life tables. The parameter is the share of income that goes to the nonresidential stock of capital. In Appendix C, we show how we use the National Income and Product Account and adjust for housing and government capital to arrive at an estimate of :24 for. Because housing is not part of our capital stock, our estimate is lower than the :3 to :4 range typically 19 We report parameters at annual frequency, unless stated otherwise. 13

14 used in real business cycle calibrations. 2 We set k to :1 and h to :2; which are standard depreciation values used in the literature. 21 The rate, r; is the interest rate on capital net of depreciation and is set to :4. The deterministic age pro le of the unconditional mean of labor productivity, e t ; taken from French (25), is shown in the top panel of Figure A1. The labor-e ciency pro le is hump-shaped, with a peak at age 5. The persistence " and variance 2 of the stochastic productivity process are :96 and :45, respectively (Huggett 1996). The variance of the initial distribution of productivity is :38 (Huggett 1996). For simplicity, we assume the labor e ciency pro le for home production to be constant, but we relax that assumption in section 9. We set the Social Security tax to : We let pension pen(t) depend on the age of the household. The retirement age T r is 61. A household after age 65 receives the full pension payment. If he is at age 63-64, he receives 75 percent of the full pension, and if he is at age 61-62, he receives 25 percent of the full pension. The down payment rate is set to :2, a value commonly used in the housing literature. The probability of receiving a renting shock is from Li and Yao (27), who calibrated to average households migration rates for nonhousing-related reasons; these are shown in the bottom panel of Figure A1. Gruber and Martin (23) estimate that the median household in CEX spends 7 percent of a house s value to sell it and 2.5 percent to purchase it. We therefore choose transactions costs from sales 1 to be :7 and from purchases 2 to be :25: Davido (26) nds that homeowners over age 75 spend about.8 percent of home value less each year on routine maintenance, compared with younger owners of similar homes. We choose a big range and set 1 = h ; 2 = 2 h. We set the risk aversion parameter,, to be 1:5 (Attanasio et al. 1999, and Gourinchas and Parker 22). The initial distribution over state variables (wealth, house size and house tenure) for households of age 25 is calculated using data from the Survey of Consumer Finances (SCF) 21 to 21 for households whose heads are between ages 24 and Accidental bequests are rst distributed to new households to reproduce the distribution of capital endowments. The rest, if there are any, is distributed evenly to all households alive, which endogenously determines b. 2 Our capital income share is nonetheless very close to the.25 number calculated in Kiyotaki, et al. (211) where similar adjustment for housing capital stock is made. Gomme and Rupert (25) adjust labor income instead of capital and arrive at a number of.28. Both labor share and capital share are subject to measurement error and the small di erence in the estimates has little consequence for our results. 21 See Gomme and Rupert (26). We assume that the intermediate home input depreciates completely in two years given that household appliance and equipment accounts for less than 1 percent of total home input. 22 The Social Security payroll tax rate in the US is 1.4 percent after we take out the part of the bene ts due to Medicare and disability insurance. 23 Since the model does not allow negative wealth, negative wealth holdings in the data are treated as zero. Most households start with wealth endowments close to zero. 14

15 5.2 Estimation We choose the parameters, ; ; h, i,! i ; (i = 1; 2; 3; 4) based on the following moments: capital-output ratio K=Y; the home ownership rate, the average net worth of homeowners relative to renters, the average home input of both renters and homeowners (d); the average house size of renters and homeowners (s; h); and the average market hours and home hours of both renters and homeowners (n m ; n h ). We normalize the average expenditure by economy-wide income. Thus, we simultaneously choose these 11 parameters to match the 11 selected moments as summarized in table It is important to note that although our procedure jointly uses 11 moments to identify 11 parameters, certain moments are relatively more responsible for pinning down the shares and elasticities in the CES aggregator. For example, is largely determined by K=Y: Also, and h are determined by home ownership rates and the relative net worth of homeowners and renters. Regarding the eight parameters in our CES aggregates, it is the relative di erences between homeowners and renters that allow us to identify these parameters. Given that we do not have permanent ex ante heterogeneity in the model, the di erences between homeowners and renters are driven by the di erences in the implicit price of housing (between renting and owning), and by the di erences in their wealth and income, which in turn are driven by the wages they face. Speci cally, according to our calculation, between ages 25 and 8 renters labor income on average is about 4 percent that of homeowners while their net worth is about 12 percent that of homeowners. 25 Thus, our identi cation through owners and renters is e ectively an identi cation through wages. In the robustness section, we reestimate the model using moments calculated according to household age. Since workers market wage varies strongly over the life cycle, this identi cation strategy is closely tied to wage changes. We show that our parameter estimates are not very sensitive to this alternative identi cation strategy. However, the estimates of 1 and! 1 can only be pinned down by di erential behavior of owners and renters, and for this reason we use moments across renters and homeowners as our benchmark estimation methodology. As mentioned earlier, the four elasticity parameters ( i, i = 1; 2; 3; 4) play crucial roles in determining households supply of labor to di erent activities and consumption of di erent goods. These parameters, along with the share parameters (! i ; i = 1; 2; 3; 4) are largely identi ed from the shares of house size (rental size) relative to income, the expenditure on home input relative to income, the relative amount of time worked at home and the relative amount of time worked in the marketplace. Although home ownership is more expensive than renting, owned housing is more productive than rental units. As a result, homeowners on average tend to have a higher ratio of h=d, and the di erence in this moment for the two types of households allows us to pin down 1 and! 1 : Using the relative amount of time 24 Though our model is exactly identi ed, because the problem is highly nonlinear, we are not able to match all the target moments exactly. 25 While the wealth ratio is one of our target moments, the income ratio is not and it is close to the.44 wage income ratio of renters to homeowners derived from SCF (21-21). 15

16 spent in home production across these two types of households helps to pin down 2 and! 2 : Similarly, the di erence across renters and homeowners regarding the relative time worked in the marketplace, because it in uences how much of the market good is purchased, helps to pin down 3 and! 3 : Finally, the di erence in d=y and h=y as well as the di erential use of time, because they help determine the di erences in c=y and l; is useful for identifying 4 and! 4. However, the estimation is a bit more complicated than that and is not totally driven by one set of moment di erentials driving one pair of elasticity and share parameters. Substitutability between various components in the CES aggregator a ects how productive a home is relative to renting, which in turn a ects choices of home ownership and, therefore via a selection e ect, all the relative moments of homeowners and renters. According to our calibration, the home input and housing are Hicksian substitutes in the production of the composite good. The composite of home input and housing and home hours exhibits strong complementarity in home production as both remain relatively elevated in the second half of the life cycle. 26 The market good and home good, on the other hand, are substitutes. This is evident as the consumption of the two goods moves in opposite ways after households retire from labor market. Finally, the nal composite consumption good made up of the market good and home good is substitutable with leisure in households utility suggesting that increases in leisure in the second half of the life cycle is also partially responsible for the marked decline in the consumption of the market good. The existing literature on home production has largely lumped home hours and leisure together into nonmarket hours, making the comparison with our estimates di cult. Nevertheless, there is some supporting evidence from the literature. For example, Abbott and Ashenfelter (1976) nd that housing, transportation, and other services tend to be complementary with nonmarket time. Barnett (1979) estimates a model of joint goods and leisure consumption and nds non-weakly-separable substitution between consumption and leisure. Greenwood and Hercowitz (1991) argue that to generate comovement in investment in durable goods in the market and at home one needs to have complementarity between durable goods and time in home production. The nding that the home input and housing are complements with home hours in home production explains why after a household moves from being a two-earner family to a one-earner family, home capital typically increases, as documented in Baxter and Rotz (29). The strong substitutability between market goods and home goods is consistent with the ndings in the literature, notably Rupert, et al. (1995). After we present household life-cycle pro les, we will conduct additional analysis to further understand the identi cation of these parameters. 26 The direct partial elasticities of substitution between home input and home hours and housing and home hours can be derived using formulas provided in Sato (1967). Using the economy-wide average consumption, we calculate that the elasticity of substitution between home input and home hours is.9 and the elasticity of substitution between housing and home hours is

17 6 Model Predictions This section compares the model-implied life-cycle patterns with those constructed in the data as discussed in section 2. The pro les in the model are calculated by integrating each variable (consumption or hours used) over the invariant distribution at each age. We show that the model does a good job of matching life-cycle pro les. 6.1 Home Ownership Figure 3 compares the model s prediction of the fraction of homeowners at each age with the data. Our model prediction tracks the data pro le with reasonable accuracy. In the model as well as in the data, most young households rent. As they age, many of them become homeowners. Home ownership rates continue to be very high late in life. In our model, renting has several advantages over owning. First, with no minimum size in rental housing poor households can rent small units that match their current income. Second, renters do not have to save for a down payment. Third, renters can adjust housing without transaction costs. Young households start with little wealth, face future income shocks, and on average earn less than middle-aged households. As a result, most young households nd renting a better choice than owning. Households, however, chose to own once they have accumulated su cient funds for down payments and when most uncertainty in labor income has been revealed. In other words, apart from those that receive a rent shock, older renters tend to be those that have drawn low labor productivity shocks. 6.2 Life-Cycle Pro les of Hours Figure 4 shows the life-cycle pro les of the average fraction of time spent in working and home production in the model. Notice that our model does a reasonably good job of capturing the overall life-cycle pro les of hours spent in market work and home production for both homeowners and renters with the exception of the very young homeowners and the middle-aged renters. Young households, all starting with little wealth, work relatively intensively to buy goods, to accumulate precautionary assets, and to save for future house purchases. As households age, they spend more time at home and decrease market hours as a result of the labor e ciency pro le, which peaks at around age 55. The distribution of Social Security bene ts starting at age 61 provides additional incentive for households to reduce their labor supply after age 61. Note that the increase of resources after age 61 from Social Security bene ts induces more reduction in market hours for renters in the model. This occurs because in our model pension distribution is a function only of age and thus is highly redistributive. Therefore, for old age renters whose labor productivity tends to be low, the pension payment may exceed their labor income substantially. Homeowners on average spend more time in home production than renters in the same 17

18 age group. This feature of the data is primarily responsible for our nding that home hours and houses are complements. Homeowners on average have more productive housing capital than renters and thus spend more time at home. 6.3 Life-Cycle Pro les of Consumption The left panel of Figure 5 shows, in percentage deviation from the corresponding value at age 25, the life-cycle pro les of average demand for market consumption, housing and market inputs for homeowners. Again our model prediction for the most part falls within the two-standard-deviation error band of the data estimation. Over the life cycle, average consumption of market goods for homeowners is hump-shaped. Market goods consumption at the peak (age 55) is about 3 percent more than that at age 25. After the peak, market goods consumption decreases dramatically with age. For instance, market goods consumption at age 8 is about 6 percent less than that at age 25. Expecting to have much higher earnings in the future, young households would like to borrow to nance their current consumption but they are borrowing constrained. This explains why early in life consumption of all categories increases. Later in life, households start to decrease overall consumption due to the fact that they are discounting future consumption by more as they age and face a lower survival rate. Additionally, older households substitute home consumption for market consumption. The demand for housing in the model matches well the empirical pro les, increasing rapidly early in life and downsizing slowly later in life. Households start with little housing stock. Because of the existence of borrowing constraints and the role of housing as collateral, however, they forgo other consumption and build their housing stock rather quickly. This trend continues until they reach age 65. After that, the housing stock starts to decline slowly because of transactions costs and the increase in home hours, which are complements to the home input and housing in home production. Further, older homeowners are less likely than young ones to move and incur the accompanying transactions costs, because they have a shorter expected life span and hence house tenure. 27 The home input and housing generally track each other over the life cycle as both are complements with home hours in home production. However, the down payment requirement in housing purchase and transactions costs in housing adjustment for homeowners make the track less than perfect. In particular, early in life, households substitute the home input with housing due to the down payment requirement for housing. This explains why the home input has a smaller increase than housing in the rst part of life cycle. Later in life, the home input, which is not subject to adjustment costs, declines faster than housing. The home input and housing, because they are complements with home hours in home production, decline much more slowly than market consumption does For detailed discussion about the e ect of transaction costs on housing, see Yang (29). 28 Using detailed ATUS, Baxter and Rotz (29) nd that when a wife leaves the labor force, home 18