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THE JOURNAL OF FINANCE VOL. LX, NO. 2 APRIL 2005 Optimal Life-Cycle Asset Allocation: Understanding the Empirical Evidence FRANCISCO GOMES and ALEXANDER MICHAELIDES ABSTRACT We show that a life-cycle model with realistically calibrated uninsurable labor income risk and moderate risk aversion can simultaneously match stock market participation rates and asset allocation decisions conditional on participation. The key ingredients of the model are Epstein–Zin preferences, a fixed stock market entry cost, and moderate heterogeneity in risk aversion. Households with low risk aversion smooth earnings shocks with a small buffer stock of assets, and consequently most of them (optimally) never invest in equities. Therefore, the marginal stockholders are (endogenously) more risk averse, and as a result they do not invest their portfolios fully in stocks. IN THIS PAPER, WE PRESENT A LIFE-CYCLE ASSET allocation model with intermediate consumption and stochastic uninsurable labor income that provides an expla- nation for two very important empirical observations: low stock market partic- ipation rates in the population as a whole, and moderate equity holdings for stock market participants. Our life-cycle model integrates three main motives that have been identi- fied as quantitatively important in explaining individual and aggregate wealth accumulation. First, a precautionary savings motive driven by the presence of undiversifiable labor income risk (Deaton (1991) and Carroll (1992, 1997)). Sec- ond, pension income is lower than mean working-life labor income, implying that saving for retirement becomes important at some point in the life cycle. The combination of precautionary and retirement saving motives has recently been shown to generate realistic wealth accumulation profiles over the life cy- cle.1 Third, we explicitly incorporate a bequest motive that has recently been Francisco Gomes is from the London Business School; Alexander Michaelides is from the London School of Economics and CEPR. We thank Viral Acharya, Orazio Attanasio, Ravi Bansal, Michael Brennan, Joao Cocco, Pierre Collin-Dufresne, Steve Davis, Karen Dynan, Bill Dupor, Lorenzo Forni, Joao Gomes, Rick Green, Luigi Guiso, Michael Haliassos, Burton Hollifield, Urban Jermann, Deborah Lucas, Pascal Maenhout, Monica Paiella, Valery Polkovnichenko, Ken Singleton, Nicholas Souleles, Harald Uhlig, Raman Uppal, Annette Vissing-J rgensen, Tan Wang, Paul Willen, Amir Yaron, Steve Zeldes, Harold Zhang, two anonymous referees, and seminar participants at Carnegie Mellon, Columbia, the Ente-Einaudi Center, the Federal Reserve Board, H.E.C., Leicester, LBS, LSE, Tilburg, UCL, UNC Chapel Hill, Wharton, the 2002 NBER Summer Institute, MFS and SED meetings for helpful comments. Previous versions of this paper have circulated with the title: “Life- Cycle Asset Allocation: A Model with Borrowing Constraints, Uninsurable Labor Income Risk and Stock Market Participation Costs.” We are responsible for any remaining errors. 1 See, for instance, Hubbard, Skinner, and Zeldes (1995), Carroll (1997), Gourinchas and Parker (2002), Dynan, Skinner, and Zeldes (2002), and Cagetti (2003). 869 870 The Journal of Finance shown to be important in matching the skewness of the wealth distribution (de Nardi (forthcoming) and Laitner (2002)). More recently, life-cycle models incorporating some (or all) of these motives have been extended to include an asset allocation decision, both in an infinite horizon2 and in a finite horizon, life-cycle setting.3 However, several important predictions of these models are still at odds with empirical regularities. First, low stock market participation in the population (Mankiw and Zeldes (1991)) persists. The latest Survey of Consumer Finances (2001) reports that only 52% of U.S. households hold stocks either directly or indirectly (through pension funds, for instance), while these models predict that, given the equity premium, all households should participate in the stock market as soon as saving takes place. Second, households in the model invest almost all of their wealth in stocks, in contrast to both casual empirical observation and to formal empirical evidence (see Poterba and Samwick (1999) or Ameriks and Zeldes (2001), for instance). We develop a life-cycle asset allocation model that tries to address these two puzzles. We argue that it is possible to simultaneously match stock market par- ticipation rates and asset allocation conditional on participation, with moderate values of risk aversion (between one and five), and without extreme assump- tions about the level of background risk. Our model has three key features. First, we include a fixed entry cost for households that want to invest in risky assets for the first time. A large literature has concluded that some level of fixed costs seems to be necessary to improve the empirical performance of as- set pricing models.4 Since the excessive demand for equities predicted by asset allocation models is merely the portfolio-demand manifestation of the equity premium puzzle, introducing a fixed cost in the model seems to be a natural ex- tension. Moreover, recent empirical work suggests that small entry costs can be consistent with the observed low stock market participation rates (see Paiella (2001), Degeorge et al. (2002), and Vissing-J rgensen (2002b)). The other two key features are motivated by the (perhaps surprising) impli- cation of the model that participation rates are an increasing function of risk aversion, at least over a wide range of parameter values. Specifically, changing risk aversion generates two opposing forces for determining the participation decision. On the one hand, more risk-averse households optimally prefer to in- vest a smaller fraction of their wealth in stocks. On the other hand, risk aversion determines prudence and more prudent consumers accumulate significantly 2 See, for example, Telmer (1993), Lucas (1994), Koo (1998), Heaton and Lucas (1996, 1997, 2000), Polkovnichenko (2000), Viceira (2001), and Haliassos and Michaelides (2003). 3 See, for instance, Cocco, Gomes, and Maenhout (1999), Cocco (2000), Campbell et al. (2001), Hu (2001), Storesletten, Telmer, and Yaron (2001), Davis, Kubler, and Willen (2002), Dammon, Spatt, and Zhang (2001, forthcoming), Polkovnichenko (2002), Yao and Zhang (forthcoming), and Gomes and Michaelides (2003). Bertaut and Haliassos (1997) and Constantinides, Donaldson, and Mehra (2002) analyze three-period models where each period amounts to 20 years. 4 See, among others, Constantinides (1986), Aiyagari and Gertler (1991), He and Modest (1995), Saito (1995), Heaton and Lucas (1996), Luttmer (1996, 1999), Basak and Cuoco (1998), and Vayanos (1998). Optimal Life-Cycle Asset Allocation 871 more wealth over the life cycle. We show that the higher wealth accumulation motive dominates for moderate coefficients of relative risk aversion (RRA) (i.e., not greater than five). As a result, the less risk-averse investors have a weaker incentive to pay the fixed cost. This explains why previous attempts to match participation rates in the context of a life-cycle model were fairly unsuccessful. If we try to match asset allocation decisions by assuming high values of risk aver- sion, the implied participation rates are counterfactually high (e.g., Campbell et al. (2001)). Motivated by this result, we allow for preference heterogeneity in the population, the second key feature of the model. As argued before, since the less risk-averse investors accumulate less wealth over the life cycle, the majority optimally chooses not to pay the fixed cost. Therefore, endogenously stock market participants tend to be the more risk-averse investors, and con- sequently, even after paying the fixed cost, they do not invest their portfolios fully in equities. The final important feature of the model is the assumption of Epstein–Zin preferences, which allows us to separate risk aversion from the elasticity of intertemporal substitution (EIS). In the context of a life-cycle model with labor income, wealth accumulation is a crucial determinant of both the stock mar- ket participation and the asset allocation decision. Within the power utility framework, households with low risk aversion also have a high EIS. Given that the expected return from investing in the stock market is higher than the dis- count rate, a higher EIS increases savings. As a result, even though the less risk-averse agents would not save much for precautionary reasons, they would have a strong incentive to save for retirement (and for a potential bequest mo- tive). Thus, breaking the link between risk aversion and the EIS is crucial for delivering predictions that are consistent with the observed empirical evidence. Therefore, in our model, households with very low risk aversion and low (moderate) EIS smooth idiosyncratic earnings shocks with a small buffer stock of assets, and most of them never invest in equities (thus behaving as in the Deaton (1991) infinite-horizon model).5 This seems to describe adequately the behavior of a large fraction of the U.S. population that retires without signifi- cant financial assets (and does not participate in the stock market). Within the low EIS and low risk aversion group, only a small fraction owns stocks, and they do so only as they get close to retirement. On the other hand, investors with high prudence and high EIS are the ones who participate in the stock market from early on, since they accumulate more wealth and therefore have a stronger incentive to pay the fixed cost. Therefore, the marginal stockholders are (endogenously) more risk averse and as a result they do not invest their portfolios fully in stocks. The heterogeneous agent model can simultaneously match the stock market participation rate and the average equity allocation conditional on participa- tion, from the Survey of Consumer Finances (SCF). The life-cycle profile of the 5 It is important to point out that we do not need heterogeneity in the EIS to obtain our results. As we will show, the less risk-averse investors can have the same EIS as the more risk-averse, just as long as this value is not too high (hence the need for Epstein–Zin preferences). 872 The Journal of Finance participation rate is also very close to the one observed in the data. On the negative side, the model still counterfactually predicts that young households that have already paid the participation cost will invest most of their portfolio in equities.6 Finally, the degree of heterogeneity in the wealth distribution is quite comparable to the one observed in the data. The rest of the paper is organized as follows. Section I summarizes results from the existing empirical literature on life-cycle asset allocation, while Sec- tion II outlines the model and calibration. In Sections III and IV, we discuss the results in the absence and presence of the fixed entry cost, respectively. Finally, Section V concludes. I. Empirical Evidence on Life-Cycle Asset Allocation and Stock Market Participation In most industrialized countries, stock market participation rates have in- creased substantially during the last decade. Nevertheless, a large percentage of the population still does not own any stocks (either directly or indirectly through pension funds). Moreover, even those households that do own stocks still invest a significant fraction of their portfolios in alternative assets. Figures 1A and B summarize evidence reported in Ameriks and Zeldes (2001).7 The results are sensitive to the identifying assumptions regarding time versus cohort effects. Time effects can arise, for example, from changes in mar- ket structure (e.g., transaction costs or information) or because investors use past returns to forecast future expected returns. Cohort effects can be due to dif- ferences in lifetime earnings potential, or different institutional settings (e.g., the social security system). Since age (a), time (t), and cohort (c, birth year) are linearly dependent (a ≡ t c), when constructing age profiles, it is impossible to simultaneously identify time and cohort effects. Figure 1A plots the average life-cycle equity holdings for stock market par- ticipants (as a share of total financial wealth), based on the 1989, 1992, 1995, and 1998 samples of the SCF. Although the life-cycle profiles are very sensitive to the inclusion of time dummies versus the inclusion of cohort dummies, the average stock holdings are significantly below 100% in both cases.8 Figure 1B plots the corresponding stock market participation rate, obtained by running a probit regression on the same data. These results are less sensitive to the choice of time versus cohort dummies. As expected, a very large fraction of the population does not own equities. In both cases the participation rate gradually 6 Hu (2001) and Yao and Zhang (forthcoming) are able to reduce the equity demand of young households by considering models with an explicit housing allocation decision. 7 Guiso, Haliassos, and Japelli (2002) obtain similar conclusions using cross-sectional informa- tion for five different countries (United States, United Kingdom, The Netherlands, Germany, and Italy). 8 The OLS regression with cohort effects predicts a share of financial wealth invested in stocks above 100% for the oldest age groups. This is just the result of imposing the same cohort effects on the full sample as in fact, in every individual cross-section, these age groups never invest more than 60% of their wealth in equities. Optimal Life-Cycle Asset Allocation 873 0 0.2 0.4 0.6 0.8 1 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 Age Time dummies Cohort dummies Figure 1A. Equity holdings as a fraction of total financial wealth for stock market participants. The results are taken from Ameriks and Zeldes (2001), and they are obtained from OLS regressions with age dummies and either time or cohort dummies. The data includes the 1989, 1992, 1995, and 1998 samples of the SCF. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 Age Time dummies Cohort dummies Figure 1B. Stock market participation rate. The results are taken from Ameriks and Zeldes (2001), and they are obtained from probit regressions with age dummies and either time or cohort dummies. The data includes the 1989, 1992, 1995, and 1998 samples of the SCF. 874 The Journal of Finance increases until approximately age 50. When including cohort dummies, the profile is flat after age 50, while with time dummies it is decreasing. Ameriks and Zeldes (2001) obtain the same results after redoing the analysis using TIAA-CREF data from 1987 to 1996, and so do Poterba and Samwick (1999), using SCF data. We can summarize the existing evidence as follows.9 First, the stock mar- ket participation rate in the U.S. population is close to 50%. Using the latest numbers from the SCF, we compute it as 51.9% (details given in Appendix C). Second, participation rates increase during working life and there is some ev- idence suggesting that they might decrease during retirement, although this might also be due to cohort effects. Third, conditional on stock market partici- pation, households invest a large fraction of their financial wealth in alterna- tive assets. According to the latest numbers from the SCF, the average equity holdings as a share of financial wealth for stock market participants is 54.8%. Fourth, there is no clear pattern of equity holdings over the life cycle. II. The Model A. Preferences Time is discrete and t denotes adult age, which following the typical con- vention in this literature, corresponds to effective age minus 19. Each period corresponds to 1 year and agents live for a maximum of 81 (T) periods (age 100). The probability that a consumer/investor is alive at time (t + 1) conditional on being alive at time t is denoted by pt (p0 equal to 1). Households have Epstein–Zin utility functions (Epstein and Zin (1989)) de- fined over one single nondurable consumption good. Let Ct and Xt denote re- spectively consumption level and wealth (cash on hand) at time t. Then, the household’s preferences are defined by Vt = (1 βpt)C1 1/ψt + βEt [ pt [ V 1 ρt+1 ] + (1 pt)b(X t+1/b)1 ρ1 ρ ] 1 1/ψ 1 ρ 1 1 1/ψ , (1) where ρ is the coefficient of RRA, ψ is the EIS, β is the discount factor, and b determines the strength of the bequest motive.10 Given the presence of a bequest motive, the terminal condition for the recursive equation (1) is: VT+1 ≡ b(X T+1/b) 1 ρ 1 ρ . (2) 9 We must point out that several papers have contributed to this research. See, for example, Guiso, Jappelli, and Terlizzese (1996) (who focus mostly on the impact of background risk on asset allocation), King and Leape (1998), Heaton and Lucas (2000) and the papers in the volume edited by Guiso, Haliassos, and Japelli (2002). 10 For more motivation and details on the modeling of bequest motives in life-cycle models see Laitner (2002), or de Nardi (forthcoming). Optimal Life-Cycle Asset Allocation 875 B. Labor Income Process Following the standard specification in the literature, the labor income pro- cess before retirement is given by Yit = PitUit , (3) Pit = exp( f (t, Zit))Pit 1Nit , (4) where f (t, Zit) is a deterministic function of age and household characteristics Zit, Pit is a permanent component with innovation Nit, and Uit is a transitory component. We assume that ln Uit and ln Nit are independent and identically distributed with mean { 0.5 σ 2u , 0.5 σ 2n }, and variances σ 2u and σ 2n , respec- tively. The log of Pit evolves as a random walk with a deterministic drift, f (t, Zit). For simplicity, retirement is assumed to be exogenous and deterministic, with all households retiring in time period K, corresponding to age 65 (K = 46). Earn- ings in retirement (t > K) are given by Yit = λPiK , where λ is the replacement ratio (a scalar between zero and one). This specification, also standard in this literature, considerably facilitates the solution of the model, as it does not re- quire the introduction of an additional state variable (see Section II.E). Durable goods, and in particular housing, can provide an incentive for higher spending early in life. Modeling these decisions directly is beyond the scope of the paper, but nevertheless we take into account these potential patterns in life- cycle expenditures. Using the Panel Study of Income Dynamics, for each age (t) we estimate the percentage of household income that is dedicated to housing expenditures (ht) and subtract it from the measure of disposable income.11 More details on this estimation are given below, when we discuss the calibration of the model. C. Financial Assets The investment opportunity set is constant and there are two financial assets, one riskless (Treasury bills or cash) and one risky (stocks). The riskless asset yields a constant gross return, Rf , while the return on the risky asset (denoted by R St ) is given by RSt+1 R f = μ + εt+1, (5) where εt ～ N(0, σ 2ε ). We allow for positive correlation between stock returns and earnings shocks. We let φN(φU) denote the correlation coefficient between stock returns and per- manent (transitory) income shocks. Before investing in stocks for the first time, the investor must pay a fixed lump sum cost, F Pit. This entry fee represents both the explicit transaction cost from opening a brokerage account and the (opportunity) cost of acquiring information about the stock market. The fixed cost (F) is scaled by the level of 11 A similar approach is taken by Flavin and Yamashita (2002) in a model without labor income. 876 The Journal of Finance the permanent component of labor income (Pit), as this significantly simplifies the solution of the model. However, this specification is also motivated by the interpretation of the entry fee as the opportunity cost of time. D. Wealth Accumulation We denote cash on hand as the liquid resources available for consumption and saving. We define a dummy variable IP that is equal to one when the fixed entry cost is incurred for the first time and is zero otherwise. The household’s next period cash on hand (Xi,t+1) is given by X i,t+1 = Sit RSt+1 + Bit R f + (1 ht)Yi,t+1 FIP Pi,t+1, (6) where Sit and Bit denote respectively stock holdings, and riskless asset holdings (cash) at time t, and ht is the fraction of income dedicated to housing-related expenditures. Since the household must allocate cash on hand (Xit) between consumption expenditures (Cit) and savings we also have X it = Cit + Sit + Bit . (7) Finally, we prevent households from borrowing against their future labor in- come. More specifically we impose the following restrictions: Bit ≥ 0, (8) Sit ≥ 0. (9) E. The Optimization Problem and Solution Method The complete optimization problem is then max {Sit ,Bit }Tt=1 E(V0), (10) where V0 is given by equations (1) and (2) and is subject to the constraints given by equations (5) to (9), and to the stochastic labor income process given by (3) and (4) if tK, and Yit = λPiK if t > K. Analytical solutions to this problem do not exist. We therefore use a numerical solution method based on the maximization of the value function to derive the optimal decision rules. The details are given in Appendix A, and here we just present the main idea. We first simplify the solution by exploiting the scale- independence of the maximization problem and rewriting all variables as ratios to the permanent component of labor income (Pit). The laws of motion and the value function can then be rewritten in terms of the normalized variables, and we use lowercase letters to denote them (for instance, xit ≡ X itPit ). This allows us to reduce the number of state variables to three: age (t), normalized cash on hand (xit) and participation status (whether the fixed cost has already been paid or not). In the last period, the policy functions are determined by the bequest motive and the value function corresponds to the bequest function. We can now Optimal Life-Cycle Asset Allocation 877 use this value function to compute the policy rules for the previous period, and given these, obtain the corresponding value function. This procedure is then iterated backwards. F. Computing Transition Distributions After solving for the optimal policy functions, we can simulate the model to replicate the behavior of a large number of households and compute, for ex- ample, the corresponding average allocations. Here we propose an alternative method of computing various statistics that is based on the explicit calcula- tion of the transition distribution of cash on hand from one age to the next. The computational details are given in Appendix B, but the intuitive idea is straight- forward. Once we have solved for the policy functions, we can substitute those in the budget constraint to obtain the distribution of xt+1 as a function of xt. Doing this for every possible xt, we are effectively computing the full transition matrix.12 Once we have these distributions, the unconditional mean consumption for age t can then be computed as13 cˉt = θt { J∑ j=1 π It, j cI(x j , t) } + (1 θt) { J∑ j=1 πOt, j cO(x j , t) } , (11) where J is the number of grid points used in the discretization of normalized cash on hand, and π It, j and π O t, j are the probability masses associated with each grid point at time t, for stockholders and nonstockholders, respectively. The participation rate at age t (θt) is given by θt = θt 1 + (1 θt 1) ∑ x j >x πOt, j , (12) where x is the trigger point that causes participation, which is determined endogenously through the participation decision rule. Finally, if we use αt to denote the share of liquid wealth invested in the stock market at age t, then the unconditional portfolio allocation is computed as: αˉt = θt { J∑ j=1 π It, j α(x j , t) ( x j cI(x j , t) )} θt J∑ j=1 [ π It, j ( x j cI(x j , t) )] + (1 θt) J∑ j=1 [ πOt, j ( x j cO(x j , t) )] . (13) 12 The results in the paper were computed both from the transition distributions and using Monte Carlo simulations. The results were found to be identical, as long as the number of simulations is not too small (2,000 or more). 13 Superscript I denotes households participating in the stock market, while superscript O de- notes households out of the stock market. 878 The Journal of Finance G. Parameter Calibration G.1. Preference Parameters We start by presenting results for a relatively standard choice, (risk aversion) ρ = 5, (EIS) ψ = 0.2, and (discount factor) β = 0.96. However, later on we report results for several different values of both the coefficient of RRA (ρ) and the EIS (ψ), as these parameters have very important implications for our results. We use the mortality tables of the National Center for Health Statistics to parameterize the conditional survival probabilities. The importance of the bequest motive (b) is set at 2.5. As we discuss below, this parameter choice is motivated by the desire to match the wealth accumulation profiles observed in the data, but we present some sensitivity analysis with respect to this parameter. G.2. Labor Income Process The deterministic labor income profile ( f (t, Zit) reflects the hump shape of earnings over the life cycle, and the corresponding parameter values, just like the retirement transfers (λ), are taken from Cocco, Gomes, and Maenhout (1999). With respect to standard deviations of the idiosyncratic shocks, the es- timates range from 0.35 for σu and 0.12 for σn (Cocco, Gomes, and Maenhout) to 0.1 for σu and 0.08 for σn (Carroll (1992)). We use numbers similar to the ones in Gourinchas and Parker (2002): σu = 0.15 and σn = 0.1. It is common practice to estimate different labor income profiles for different education groups (college graduates, high school graduates, households without a high school degree). In our paper, we only report the results obtained with the parameters estimated from the subsample of high school graduates, as the results for the other two groups are very similar. G.3. Asset Returns, Correlation and Fixed Cost The constant net real interest rate (Rf 1) is set at 2%, while for the stock return process we consider a mean equity premium (μ) equal to 4% and a stan- dard deviation (σε) of 18%. Considering an equity premium of 4% (as opposed to the historical 6%) is a fairly common choice in this literature (e.g., Yao and Zhang (forthcoming), Cocco (2001) or Campbell et al. (2001)). Even after hav- ing paid the fixed entry cost, the average retail investor still faces nontrivial transaction costs, mostly in the form of mutual fund fees. This adjustment is a shortcut representation for those costs, since the dimensionality of the problem prevents us from modeling them explicitly (as in Heaton and Lucas (1996), for example). The evidence on the magnitude of the correlation between stock returns and permanent labor income shocks is mixed.14 Davis and Willen (2001) and Heaton 14 Moreover, it has been argued that these estimations suffer from a small sample bias, since the time-series dimension is too short in micro data, and estimations using macro data usually yield larger and more significant correlations (see, e.g., Jermann (1999)). Optimal Life-Cycle Asset Allocation 879 and Lucas (2000) do not distinguish between the two components of labor in- come (permanent and transitory) when computing the correlation coefficients. For the purposes of calibrating our model, we need to know the magnitude of the correlation coefficient for these two shocks separately. Campbell et al. (2001) estimate the correlation between the permanent component of labor income shocks and stock returns, and obtain a correlation coefficient of 0.15.15 They do not estimate a correlation between transitory shocks and stock returns and just assume it to be equal to zero. We use these numbers (φN = 0.15 and φU = 0.0) for our benchmark calibration, and perform sensitivity analysis around these values. With respect to the fixed cost of participation we consider two limit cases: one where the cost is zero, and one where it equals 0.025 (2.5% of the house- hold’s expected annual income). This parameter reflects both the monetary cost associated with the initial investment in the stock market, and the opportu- nity cost associated with obtaining the necessary information for making such investment.16 G.4. Housing Expenditures We measure housing expenditures using data from the Panel Study of Income Dynamics from 1976 until 1993.17 For each household, in each year, we compute the ratio of annual mortgage payments and rent payments (housing-related expenditures—H) relative to annual labor income (Y): hit ≡ HitYit . (14) We combine mortgage payments and rent together, since we are not model- ing the housing decision explicitly. We identify the age effects by running the following regression on the full panel: hit = A + B1 age + B2 age2 + B3 age3 + time dummies + ζit , (15) 15 It is important to realize that in their tables, Campbell et al. (2001) actually report the cor- relation of the aggregate component of permanent labor income shocks with stock returns. This explains their high estimates: 45.6%. To obtain the correlation with the “total permanent shock,” we need to adjust for the standard deviation of the aggregate component relative to the total, which gives the 15% number. 16 Consider the average household that has an annual labor income of $35,000. If the time cost were zero, then this value of F would imply a monetary cost of $875. If instead the monetary cost were zero, then this would imply a time cost of 9.1 days (6.3 working days). More generally, any convex combination of these two costs is acceptable, for example, a time cost of 1 (2) day(s) and a monetary cost of $779 ($683). Paiella (2001) and Vissing-J rgensen (2002b) used Euler equation estimation methods to obtain implied participation costs from observed consumption choices. They find values in the $75–200 range, but these are per-period costs, so our number is quite reasonable when compared to their estimates. 17 Before 1976 there is no information on mortgage expenditures, and 1993 is the last year available on final release from the PSID. 880 The Journal of Finance Table I Regression of the Ratio of Housing Expenditures to Labor Income (heit), on Age Polynomials, and Time Dummies The data are taken from the Panel Study of Income Dynamics from 1976 until 1993. For each household, in each year, we compute the ratio of annual mortgage payments plus rent payments relative to annual labor income, and regress this ratio against a constant, a cubic polynomial of age (where age is defined as the age of the head of the household), and time dummies. We eliminate all observations with age greater than 75. Coefficient T-Stat Constant 0.703998 5.47 Age 0.0352276 3.70 Age2 0.0007205 3.17 Age3 0.0000049 2.84 Adj. R2 0.025 where age is defined as the age of the head of the household. We eliminate all observations with age > 75.18 The estimation results are reported in Table I. In the model we use ht = max(A + B1 age + B2 age2 + B3 age3, 0), (16) which, given our parameter estimates, truncates ht at zero for age80. III. Results without the Fixed Participation Cost A. Consumption and Wealth Accumulation Figure 2A plots mean normalized consumption (cˉt), mean normalized wealth (wˉt), and mean normalized income net of housing expenditures ((1 ht) yˉt). The preference parameters are ρ = 5 and ψ = 0.2, and the importance of the be- quest motive (b) is set at 2.5. Early in life, the household is liquidity constrained and saves only a small buffer stock of wealth. From approximately ages 30–35 onwards, she starts saving for retirement and bequests, and wealth accumu- lation increases significantly. During retirement, consumption decreases as a result of the very high effective discount rate (high mortality risk). Wealth does not fall towards zero due to the presence of the bequest motive.19 Table II shows the mean consumption to wealth ratio for different values of the preference parameters. We report results for values of risk aversion between one and five and for values of the EIS between 0.2 and 0.8, since this is the 18 There are several reasons for eliminating these households. First, there are very few observa- tions within each age group beyond age 75. Second, for most of these households, the values of hit are equal to zero. Third, this is consistent with the estimation procedure used for the labor income process. 19 Net income increases during the first years of re