A. Budget Constraint and Earnings Process

We consider a dual-income couple composed of partners $ \left(A,B\right) $ of equal age. At every adult age $ t=1,\dots, T $ , the partners receive individual earnings ( $ {Y}_{At},{Y}_{Bt} $ ) and decide on individual consumption ( $ {C}_{At},{C}_{Bt} $ ) and the share of their joint savings ( $ {\alpha}_t $ ) that is allocated to the stock market. Following Deaton (Reference Deaton1991), cash-on-hand ( $ {X}_t $ ) is defined as the sum of financial wealth and household earnings, $ {Y}_t={Y}_{At}+{Y}_{Bt} $ , and evolves according to

(1) $$ {X}_t=\left({X}_{t-1}-{C}_{t-1}\right)\left({R}^f+{\alpha}_{t-1}\left({R}_t-{R}^f\right)\right)+{Y}_t. $$

Joint savings, the difference between cash-on-hand and household consumption, $ {C}_t={C}_{At}+{C}_{Bt} $ , are invested in a portfolio consisting of a risk-free asset with a certain real gross return, $ {R}^f $ , and the stock market with risky real gross return, $ {R}_t $ . The couple is assumed to be borrowing and short-sale constrained, which implies $ {\alpha}_t\in \left[0,1\right],\forall t $ . Using $ {r}_t=\ln \left({R}_t\right) $ and $ {r}^f=\ln \left({R}^f\right) $ , we assume that the log excess return on stocks is generated by

(2) $$ {r}_t-{r}^f=\mu +{\nu}_t, $$

where $ \mu $ is the unconditionally expected equity risk premium and $ {\nu}_t $ is an innovation that is $ \mathrm{IID}. $ normal with mean zero. Investment opportunities are constant.

We propose a model of earnings dynamics that extends the model of Kaplan (Reference Kaplan2012) to a dual-income household. Assume that the partners’ log earnings consist of a deterministic life-cycle component and a stochastic residual component, such that $ {y}_{At}=\ln \left({Y}_{At}\right)={d}_{At}+{e}_{At} $ and $ {y}_{Bt}=\ln \left({Y}_{Bt}\right)={d}_{Bt}+{e}_{Bt} $ , where

(3) $$ {e}_{At}={\omega}_A+{\eta}_{At}+{\varepsilon}_{At}\hskip2em {e}_{Bt}={\omega}_B+{\eta}_{Bt}+{\varepsilon}_{Bt} $$

(4) $$ {\eta}_{At}={\phi}_A{\eta}_{At-1}+{\zeta}_{At}\hskip2em {\eta}_{Bt}={\phi}_B{\eta}_{Bt-1}+{\zeta}_{Bt}. $$

The deterministic components, $ {d}_{At},{d}_{Bt} $ , include quartic polynomials in the partners’ respective ages. Log residual earnings for each partner in equation (3) are decomposed into a persistent component $ \left({\eta}_{At}=\ln \left({Y}_{At}^p\right),{\eta}_{Bt}=\ln \left({Y}_{Bt}^p\right)\right) $ , which follows the first-order Markov process in equation (4), a transitory component $ \left({\varepsilon}_{At},{\varepsilon}_{Bt}\right) $ , and an age-invariant random effect $ \left({\omega}_A,{\omega}_B\right) $ . We assume that transitory shocks and random effects are IID normal with zero means and variances $ \left({\sigma}_{A\varepsilon}^2,{\sigma}_{B\varepsilon}^2\right) $ and $ \left({\sigma}_{A\omega}^2,{\sigma}_{B\omega}^2\right) $ , respectively. We allow the innovations to the persistent earnings components to be correlated with each other, with a correlation coefficient $ \rho $ , and with the innovations to the log stock excess return in equation (2) with correlation coefficients $ \left({\rho}_{A\nu},{\rho}_{B\nu}\right) $ . Specifically, we assume the following multivariate normal distribution for these innovations:

(5) $$ \left(\begin{array}{c}{\zeta}_{A t}\\ {}{\zeta}_{B t}\\ {}{\nu}_t\end{array}\right)\sim i.i.d.N\left[\left(\begin{array}{c}0\\ {}0\\ {}0\end{array}\right),\left(\begin{array}{ccc}{\sigma}_{A\zeta}^2& {\rho \sigma}_{A\zeta}{\sigma}_{B\zeta}& {\rho}_{A\nu}{\sigma}_{A\zeta}{\sigma}_{\nu}\\ {}{\rho \sigma}_{A\zeta}{\sigma}_{B\zeta}& {\sigma}_{B\zeta}^2& {\rho}_{B\nu}{\sigma}_{B\zeta}{\sigma}_{\nu}\\ {}{\rho}_{A\nu}{\sigma}_{A\zeta}{\sigma}_{\nu }& {\rho}_{B\nu}{\sigma}_{B\zeta}{\sigma}_{\nu }& {\sigma}_{\nu}^2\end{array}\right)\right]. $$

Retirement is assumed to be exogenous and deterministic, with all households retiring at age 65, corresponding to adult age $ t=41 $ . Earnings in retirement $ \left(t>41\right) $ are given by $ \kappa $ times the last working period labor income, where $ \kappa $ is the replacement ratio.

B. Collective Model Under Full Commitment

In the collective model, the couple solves a Pareto problem, which consists of maximizing a $ \pi $ -weighted average of the individual life-time power utilities in nondurable consumption

(6) $$ \underset{{\left\{{C}_{At},{C}_{Bt},{\alpha}_t\right\}}_{t=1}^T}{\max}\left\{\left(1-\pi \right){E}_1\left[\sum \limits_{t=1}^T{\beta}_A^{t-1}{p}_{At}\frac{C_{At}^{1-{\gamma}_A}}{1-{\gamma}_A}\right]+\pi {E}_1\left[\sum \limits_{t=1}^T{\beta}_B^{t-1}{p}_{Bt}\frac{C_{Bt}^{1-{\gamma}_B}}{1-{\gamma}_B}\right]\right\}, $$

subject to the budget, borrowing, and short-sale constraints in Section II.A. The model allows for intra-household heterogeneity in the coefficients of relative risk aversion ( $ {\gamma}_A,{\gamma}_B $ ), subjective discount factors ( $ {\beta}_A,{\beta}_B $ ), and conditional survival probabilities ( $ {p}_{At},{p}_{Bt} $ ), given survival to adult age one. We assume that the partners fully commit to the outcomes of the Pareto problem. In this case, the Pareto weight, $ \pi $ , which describes the bargaining power of the partner $ B $ , remains constant over the life cycle.

The value function of the Pareto problem in equation (6) is defined as

(7) $$ {\displaystyle \begin{array}{l}{V}_{Ct}\left({X}_t,{Y}_{At}^p,{Y}_{Bt}^p\right)=\\ {}\underset{C_{At},{C}_{Bt},{\alpha}_t}{\max}\left\{\left(1-\pi \right)\left[\frac{C_{At}^{1-{\gamma}_A}}{1-{\gamma}_A}+{\beta}_A{p}_{At+1}{E}_t\left[{V}_{At+1}\left({X}_{t+1},{Y}_{At+1}^p,{Y}_{Bt+1}^p\right)\right]\right]\right.\\ {}\left.+\pi \left[\frac{C_{Bt}^{1-{\gamma}_B}}{1-{\gamma}_B}+{\beta}_B{p}_{Bt+1}{E}_t\left[{V}_{Bt+1}\left({X}_{t+1},,{Y}_{At+1}^p,{Y}_{Bt+1}^p\right)\right]\right]\right\},\end{array}} $$

where $ {V}_{At} $ and $ {V}_{Bt} $ denote the value functions of the individual optimization problems of the partners obtained from setting $ \pi =0 $ and $ \pi =1 $ , respectively, in equation (6). The state variables of the dynamic programming problem are cash-on-hand, $ {X}_t $ , and the persistent components, $ {Y}_{At}^p $ and $ {Y}_{Bt}^p $ , of the earnings of the two partners.

To focus on the impact of intra-household heterogeneity in risk preferences and bargaining power on consumption and portfolio choice, we will assume in our simulations in Section IV that survival to $ T $ is certain, and that both partners have the same subjective discount factor, $ \beta ={\beta}_A={\beta}_B $ , as in Ortigueira and Siassi (Reference Ortigueira and Siassi2013) and Apps et al. (Reference Apps, Andrienko and Rees2014). Under these assumptions, we can simplify equation (7) to obtain a Bellman equation of the usual recursive form

(8) $$ {\displaystyle \begin{array}{l}{V}_{Ct}\left({X}_t,{Y}_{At}^p,{Y}_{Bt}^p\right)=\underset{C_{At},{C}_{Bt},{\alpha}_t}{\max}\left\{\left(1-\pi \right)\frac{C_{At}^{1-{\gamma}_A}}{1-{\gamma}_A}\right.\\ {}\left.+\pi \frac{C_{Bt}^{1-{\gamma}_B}}{1-{\gamma}_B}+\beta {E}_t\left[{V}_{Ct+1}\left({X}_{t+1},{Y}_{At+1}^p,{Y}_{Bt+1}^p\right)\right]\right\},\end{array}} $$

which is computationally less burdensome because it does not require an evaluation of the partners’ individual value functions.

C. Intra-Household Risk Sharing

From setting equal the first-order derivatives of equation (6) with respect to $ {C}_{At} $ and $ {C}_{Bt} $ , we obtain the intra-household consumption-sharing rule:

(9) $$ \frac{\beta_A^{t-1}{p}_{At}{C}_{At}^{-{\gamma}_A}}{\beta_B^{t-1}{p}_{Bt}{C}_{Bt}^{-{\gamma}_B}}=\frac{\pi }{1-\pi }, $$

which shows that the ratio of appropriately discounted marginal utilities in consumption is constant, a standard characterization of efficient risk sharing (see Browning et al. (Reference Browning, Chiappori and Weiss2014)). Moreover, together with $ {C}_t={C}_{At}+{C}_{Bt} $ , equation (9) uniquely determines $ {C}_{At} $ and $ {C}_{Bt} $ for given preference parameters, bargaining weights, and conditional survival probabilities. Assuming again certain survival and identical discount factors, the consumption-sharing rule in equation (9) becomesFootnote ⁷

(10) $$ \frac{C_{At}^{-{\gamma}_A}}{C_{Bt}^{-{\gamma}_B}}=\frac{\pi }{1-\pi }. $$

In the collective model, risk sharing is achieved through the consumption-sharing rule, which determines how total consumption is distributed between the two partners. Total household consumption is affected by shocks to the partners’ earnings and the return on risky assets. The individual consumption shares of the partners are affected by their respective coefficients of relative risk aversion and their bargaining power.

From the first-order derivative of the log of equation (10) with respect to household consumption, $ {C}_t $ , Ortigueira and Siassi (Reference Ortigueira and Siassi2013) obtain the risk-sharing result:Footnote ⁸

(11) $$ \frac{dC_{At}}{dC_t}=\frac{\rho_C}{\rho_A},\hskip2em \frac{dC_{Bt}}{dC_t}=\frac{\rho_C}{\rho_B}, $$

where $ {\rho}_A=\frac{\gamma_A}{C_{At}} $ , $ {\rho}_B=\frac{\gamma_B}{C_{Bt}} $ , and $ {\rho}_C=\frac{\gamma_C}{C_t} $ denote the coefficients of absolute risk aversion of the partners and the couple. We will discuss the couple’s coefficient of relative risk aversion, $ {\gamma}_C $ , below. Assume that partner $ A $ is more risk tolerant than partner $ B $ ; $ \frac{1}{\rho_A}>\frac{1}{\rho_B} $ . In this case, if $ {C}_t $ increases, $ {C}_{At} $ increases by more than $ {C}_{Bt} $ according to equation (11). The consumption share of partner $ A $ , $ \frac{C_{At}}{C_t} $ , increases while the consumption share of partner $ B $ , $ \frac{C_{Bt}}{C_t} $ , decreases. Vice versa, if $ {C}_t $ decreases, $ {C}_{At} $ decreases by more than $ {C}_{Bt} $ . The more risk-tolerant partner bears most of the variation in household consumption. Intuitively, the more risk-tolerant partner provides partial downside consumption insurance for the more risk-averse partner, who, in return, gives up some of the upside potential in consumption.

The implementation of the consumption-sharing rule, which reflects the intra-household risk-sharing mechanism in the collective model, is the key deviation of our model from Love (Reference Love2010), Hong and Ríos-Rull (Reference Hong and Ríos-Rull2012), and Hubener et al. (Reference Hubener, Maurer and Mitchell2015), who assume that consumption is equally shared among partners. Equation (9) shows that equal sharing requires the partners to have identical discount factors, survival probabilities, coefficients of relative risk aversion, and bargaining power.Footnote ⁹

Starting with Lise and Seitz (Reference Lise and Seitz2011), several authors have shown that consumption is usually not shared equally within the household, with the male partner in heterosexual couples receiving a larger share of total consumption than the female partner. See Bargain, Donni, and Hentati (Reference Bargain, Donni and Hentati2022) and Blundell, Karjalainen, Lechene, and Pendakur (Reference Blundell, Karjalainen, Lechene and Pendakur2025) for more recent contributions. In terms of our model, this empirical finding is consistent with a higher bargaining power of the male partner. It is also consistent with the less risk-averse (male) partner obtaining a higher consumption share on average in compensation for providing consumption insurance to the more risk-averse (female) partner in bad times. Gu et al. (Reference Gu, Peng and Zhang2024) provide empirical evidence that male partners, on average, have higher bargaining power and are less risk-averse.

Moreover, this literature shows that changes in intra-household consumption inequality over time can be explained by factors that are related to changes in relative bargaining power. For example, Blundell et al. (Reference Blundell, Karjalainen, Lechene and Pendakur2025) show that changes in relative wages contribute to explaining the decrease in intra-household consumption inequality in the United Kingdom from 1978 to 2019. This is evidence that risk sharing in practice is achieved through consumption sharing, in line with model predictions.

D. Collective Relative Risk Aversion

To understand the portfolio choice implications of intra-household risk sharing, we will investigate the couple’s coefficient of relative risk aversion implied by the household value function in equation (8), as derived by Ortigueira and Siassi (Reference Ortigueira and Siassi2013):Footnote ¹⁰

(12) $$ {\gamma}_C=\frac{\gamma_A{\gamma}_B}{\gamma_A\frac{C_{Bt}}{C_t}+{\gamma}_B\frac{C_{At}}{C_t}}, $$

which varies with age unless the consumption shares in the denominator are constant. The couple’s risk aversion is lower than the arithmetic average of individual risk aversion if the weighted average of the individual relative risk aversion coefficients, $ {\gamma}_A\frac{C_{Bt}}{C_t}+{\gamma}_B\frac{C_{At}}{C_t} $ , exceeds their harmonic mean, $ \frac{2{\gamma}_A{\gamma}_B}{\gamma_A+{\gamma}_B} $ .

It is interesting to compare the couple’s relative risk aversion coefficient in equation (12) with the one implied by Gu et al. (Reference Gu, Peng and Zhang2024):

(13) $$ {\gamma}_C=\frac{\gamma_A{\gamma}_B}{\gamma_A{\pi}_{Bt}+{\gamma}_B{\pi}_{At}}. $$

The couple in Gu et al. (Reference Gu, Peng and Zhang2024) solves a mean–variance model, in which the inverse of the coefficient of relative risk aversion is a bargaining-power-weighted average of the inverse individual coefficients of relative risk aversion of the two partners. The Pareto weights in equation (13), $ {\pi}_{At} $ and $ {\pi}_{Bt} $ , vary with age as functions of observable variables. If the Pareto weights are the same, then $ {\gamma}_C $ becomes the harmonic mean of $ {\gamma}_A $ and $ {\gamma}_B $ .

In our model, consumption shares replace the Pareto weights in the denominator of the couple’s relative risk aversion. These shares are not only functions of individual bargaining power but also of the individual risk aversion coefficients of the partners (see equation (11)). In bad times, the consumption share of the more risk-averse partner suffers less than the consumption share of the less risk-averse partner. In turn, the less risk-averse partner experiences higher consumption growth in good times than the more risk-averse partner. This intra-household insurance mechanism, which is the basis of intra-household risk sharing in the collective model, is absent in Gu et al. (Reference Gu, Peng and Zhang2024).

However, the couple’s coefficient of relative risk aversion in Gu et al. (Reference Gu, Peng and Zhang2024) can be seen as an approximation to the couple’s coefficient of relative risk aversion in the collective model with consumption. Both coincide if the consumption shares in equation (12) are equal, which requires both partners to have identical relative risk aversion and bargaining power (see equation (10)). Below, we will investigate whether a unitary model with relative risk aversion in equation (13) can approximate the collective model with relative risk aversion in equation (12) if the partners differ in risk preferences.

E. Unitary Model for a Representative Agent

As a benchmark for the collective model with full commitment, we consider a unitary model for a single agent with relative risk aversion $ \gamma $ representing the couple:

(14) $$ {V}_{Ut}\left({X}_t,{Y}_{At}^p,{Y}_{Bt}^p\right)=\underset{C_t,{\alpha}_t}{\max}\left\{\frac{C_t^{1-\gamma }}{1-\gamma }+\beta {E}_t\left[{V}_{Ut+1}\left({X}_{t+1},{Y}_{At+1}^p,{Y}_{Bt+1}^p\right)\right]\right\}. $$

To ensure comparability, we assume that the single agent in the unitary model earns the same income as the dual-income couple in the collective model.

For a case without portfolio choice, Mazzocco (Reference Mazzocco2004) shows that the unitary model can replicate the saving decisions of a couple in the collective model, provided that the partners have ISHARA (identically shaped (IS), harmonic absolute risk aversion (HARA)) preferences. They must have identical beliefs and discount factors (including conditional survival probabilities in our case), and individual preferences of the HARA type, which includes power utility, with identical curvature parameters.Footnote ¹¹

To investigate whether the introduction of portfolio choice affects Mazzocco’s results, we will compare the unitary model with a collective model in which partners have identical risk preferences, $ \gamma ={\gamma}_A={\gamma}_B $ . In this case, the couple’s relative risk aversion in equation (12) reduces to $ {\gamma}_C=\gamma $ , which implies age-invariant consumption shares, which are equal to 0.5 if, in addition, bargaining power is the same for both partners. Note that the general collective model in Section II.B, in which partners have different subjective discount factors and conditional survival probabilities, cannot be exactly replicated by a unitary model. The couple’s relative risk aversion and the partners’ consumption shares vary across the life cycle in this model.

A. Estimating the Earnings Process for Dual-Income Couples

We use the Survey Research Center sample of the PSID (Social Research Center (2019)) to estimate the joint earnings dynamics of partners in dual-income couples, which allows us to ignore sampling weights as in Kaplan (Reference Kaplan2012) and De Nardi, Fella, and Paz-Pardo (Reference De Nardi, Fella and Paz-Pardo2020). We use the 1999–2017 biennial waves because we condition on stock market participation, which can be observed in the PSID since 1999. We do this because our baseline model predicts universal participation in the absence of participation costs, which we deliberately decided not to model to focus on risk sharing. We further require that both partners (male heads and their spouses) are aged between 23 and 67 and earn a minimum of $1,000. All dollar values are converted to 2017 dollars using the CPI. We use the return, including distributions on the S&P500 stock market index from CRSP, to estimate correlations between persistent labor income shocks and the innovations to stock returns.

We estimate the parameters of the earnings process in two stages: In the first stage, we separately regress annual log earnings of both partners on a quartic polynomial in age and full sets of education and time dummy variables. In the second stage, we estimate the covariance parameters of the earnings process from the residuals of the log earnings regressions by GMM using the orthogonality conditions derived in Appendix A, which contains details about the estimation approach. To account for the first estimation stage, we obtain second-stage standard errors from a bootstrap routine for overidentified GMM estimators proposed by Hall and Horowitz (Reference Hall and Horowitz1996). The GMM estimator simultaneously solves the moment restrictions for both partners and employs an identity weight matrix. We restrict our sample to households that are observed for at least four consecutive waves in the PSID to identify all earnings process parameters. The resulting sample includes $ N=\mathrm{1,183} $ dual-income couples.

Panel A of Table 1 shows first-stage OLS estimates of the log earnings regressions for household heads and their spouses. We only report the results for the coefficients of the age polynomial. The baseline households consist of college graduates, observed in 2017. Figure 1 displays the resulting age-earnings profile estimates for both partners in the baseline households.Footnote ¹² We find that household heads of dual-income households on average experience a typical hump-shaped age-earnings profile that peaks just before age 50. Spouses of dual-income households, on average, have a much flatter age-income profile that peaks at around age 55. The maximum average earnings of male heads ($131,000) considerably exceed those of their spouses ($77,000).

Table 1Estimation of Earnings Process Parameters for Dual-Income Couples

Figure 1Estimated Age-Earnings Profiles for Dual-Income Couples

Figure 1 shows estimated age-earnings profiles in thousands of 2017 dollars for dual-income couples (consisting of the head and the spouse) who participate in the stock market. The data consist of the biennial 1999–2017 ( $ T=10 $ ) waves of the PSID. The profiles are generated from the OLS estimates in Panel A of Table 1 for the college graduate baseline education category and the 2017 baseline year. Heads are male.

Panel B of Table 1 shows second-stage GMM estimates of the earnings process covariance parameters. We find substantial intra-household heterogeneity in earnings processes within dual-income couples. Household heads face much more persistent shocks ( $ {\phi}_A=0.9812 $ ) than their spouses ( $ {\phi}_B=0.8898 $ ). Spouses experience much more volatile, persistent income shocks and much less volatile transitory income shocks than household heads. The correlations between persistent earnings shocks and innovations to the return on the aggregate stock market turn out to be insignificant. The same holds for the correlation between the persistent earnings shocks experienced by household heads and spouses, consistent with results obtained by Blundell et al. (Reference Blundell, Pistaferri and Saporta-Eksten2016).

B. Calibration Choices for Dual-Income Households

In the baseline calibration, we set the risk aversion of both partners, $ A $ and $ B $ , of the dual-income household to $ {\gamma}_A={\gamma}_B=5 $ . Later, we will allow mean-preserving spreads in risk preferences and vary $ {\gamma}_A $ ( $ {\gamma}_B $ ) between 2 and 4 (8 and 6). This variation is motivated by Brooks et al. (Reference Brooks, Sangiorgi, Hillenbrand and Money2019) who find that men, on average, are more financially risk-tolerant than women, controlling for differences in age and employment status. The subjective discount factor is set to $ \beta =0.96 $ . The Pareto weight is assumed to be $ \pi =0.5 $ in the baseline specification. Later, we will consider cases with $ \pi =0.25 $ and $ \pi =0.75 $ .

The parameters of the earnings process are based on Table 1. Following Deaton (Reference Deaton1991), we assume that the estimated volatility parameters of the earnings process reflect, to some extent, measurement error. Correspondingly, we first reduce all estimated volatilities in Panel B of Table 1 by 50% for the baseline simulations. We then use the unadjusted estimated volatilities to investigate the saving and portfolio choice implications of a high-background-risk scenario. After age 65, we use $ 0.68 $ as the replacement ratio of retirement income to the last year’s income of working life. The maximum adult age is $ T=85 $ , corresponding to age 109.

There are two financial assets, one risk-free asset (cash) and one risky asset (stocks). In line with Cocco et al. (Reference Cocco, Gomes and Maenhout2005), the risk-free asset yields a constant real gross return, $ {r}^f $ , of $ 2\% $ , while the mean equity premium, $ \mu $ , is $ 4\% $ . The unconditional volatility of stock returns is $ 18\% $ .

A. Partners with Identical Relative Risk Aversion

Figure 2 shows average simulated life-cycle profiles for a unitary benchmark model with $ \gamma =5 $ and three collective models with $ {\gamma}_A={\gamma}_B=5 $ and $ \pi =\mathrm{0.25,0.50,0.75} $ , respectively. The representative agent in the unitary model receives the same earnings as the dual-income couple in the collective model. Graph A of Figure 2 shows the age profile of financial wealth, Graph B shows the profile of consumption, Graph C shows the profile of the share of wealth allocated to stocks, Graph D shows the profile of the couple’s relative risk aversion according to equation (12), and Graphs E and F show the profiles of the consumption shares for partners $ A $ and $ B $ , respectively.

Figure 2Varying the Pareto Weight When Both Partners Have Identical Risk Aversion

Figure 2 shows average simulated life-cycle profiles for financial wealth (in Graph A), consumption (Graph B), the share of wealth allocated to stocks (Graph C), the relative risk aversion of the couple (Graph D), the consumption shares of partners A and B (Graphs E and F), obtained from the unitary life-cycle model with $ \gamma =5 $ and three collective models with $ {\gamma}_A={\gamma}_B=5 $ and $ \pi =\mathrm{0.25,0.50,0.75} $ , respectively.

Given that both partners now have identical discount factors and individual preferences of the CRRA type with identical curvature parameters, the ISHARA conditions are fulfilled under which the collective model without portfolio choice can be replicated by the unitary benchmark model for an agent representing the dual-income couple (Mazzocco (Reference Mazzocco2004)). Graphs A and B of Figure 2 show that the unitary model and the three collective models with different Pareto weights yield identical life-cycle profiles of financial wealth and consumption. Importantly, we show that the same result applies to the share of financial wealth in stocks in Graph C, which decreases with the relative importance of human capital to financial wealth over the working life (see Cocco et al. (Reference Cocco, Gomes and Maenhout2005)) and stays relatively flat during retirement when earnings are risk-free.

Varying the Pareto weight does not affect saving and financial risk-taking in the collective model, provided both partners have the same risk aversion. The Pareto weight does, however, affect the distribution of consumption across partners as shown in Graphs E and F of Figure 2: the partner with higher bargaining power optimally receives a larger share of consumption. For example, the partner $ B $ in Graph F receives about 55% of consumption if $ \pi =0.75 $ , but only 45% of consumption if $ \pi =0.25 $ . The consumption shares of both partners are equal if the Pareto weight is $ \pi =0.5 $ . Reflecting the couple’s constant relative risk aversion across the life cycle in Graph D, the partners’ consumption shares are age-invariant as well.

B. Mean-Preserving Spreads in Relative Risk Aversion

We now depart from the ISHARA conditions and investigate the optimal behavior of a dual-income couple consisting of partners with different risk preferences. Figure 3 shows average simulated life-cycle profiles comparable to those in Figure 2, but obtained from collective models with $ \pi =0.50 $ and four different combinations of risk aversion within the couple: $ \left({\gamma}_A=5,{\gamma}_B=5\right) $ , $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ , $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ , and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ . Thus, we consider models with an increasing spread in the partners’ risk aversion around the same arithmetic average risk aversion across the two partners ( $ \overline{\gamma}=5 $ ). Recall from Section IV.A that the collective model with $ \left({\gamma}_A=5,{\gamma}_B=5\right) $ is identical to the unitary benchmark model with $ \gamma =5 $ .

Figure 3Mean-Preserving Spreads in Relative Risk Aversion

Figure 3 shows average simulated life-cycle profiles for financial wealth (in Graph A), consumption (Graph B), the share of wealth allocated to stocks (Graph C), the relative risk aversion of the couple (Graph D), the consumption shares of partners A and B (Graphs E and F), obtained from four collective life-cycle models with $ \left({\gamma}_A=5,{\gamma}_B=5\right) $ , $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ , $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ , and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ , respectively. The partners have identical bargaining power in all models ( $ \pi =0.5 $ ).

Figure 3 reveals the main result from simulating the calibrated life-cycle model: the more heterogeneous the partners are in risk preferences (for a given mean of their coefficients of relative risk aversion), the more they benefit from risk sharing, as evidenced by increased average financial wealth during retirement (Graph A) and increased average consumption across the life cycle (Graph B). This simultaneous increase in wealth and consumption is achieved by optimally allocating larger shares of financial wealth to stocks (Graph C). Financial risk-taking increases with the potential to share risk, and this potential is increasing with the mean-preserving spread between the partners’ individual coefficients of relative risk aversion. While a couple with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ on average allocates 1.6 percentage points (2.3%) more financial wealth to stocks than a couple with $ \left({\gamma}_A=5,{\gamma}_B=5\right) $ across the life cycle, this difference increases to a substantial 17.3 percentage points (24.7%) for a couple with $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ . The largest increase in financial risk-taking in the latter case is observed at age 60.

This increase in the share of wealth in stocks is reflected in the couple’s coefficient of relative risk aversion (Graph D), which falls below the partners’ average individual risk aversion coefficient if the partners have different risk preferences. The more heterogeneous the partners are in risk preferences (for a given mean), the more risk-tolerant the couple becomes. While the relative risk aversion of a couple with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ on average falls below the relative risk aversion of a couple with $ \left({\gamma}_A=5,{\gamma}_B=5\right) $ by 0.22 units across the life cycle, this difference increases to 1.84 units for a couple with $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ . Importantly, a unitary model with a constant coefficient of relative risk aversion below the average individual risk aversion of the two partners is unable to exactly replicate the consumption and portfolio choice decisions of the couple in the collective model. This is because the collective model implies a U-shaped age profile of the couple’s relative risk aversion (Graph D), while relative risk aversion is age-invariant in the unitary model. We will investigate in Section IV.D, whether a unitary benchmark model can at least approximate the solutions of the collective model.

Graphs E and F shed more light on the intra-household consumption-sharing rule in the collective model.Footnote ¹³ In line with the discussion under equation (11), the average consumption share of the more risk-tolerant partner $ A $ increases with household consumption over the working life, while the average consumption share of the more risk-averse partner $ B $ declines. The slopes of these graphs are steeper the more the partners differ in risk preferences for a given mean relative risk aversion coefficient. In the special case of $ \left({\gamma}_A=5,{\gamma}_B=5\right) $ , the collective model becomes observationally identical to the unitary benchmark model with $ \gamma =5 $ , and the consumption shares are constant again as in Figure 2.

Figure 4 shows average simulated age profiles comparable to those in Figure 3, but investigates combinations of different Pareto weights, $ \pi =\mathrm{0.25,0.75} $ , with sets of individual risk aversions, $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ . The benefits from intra-household risk sharing documented earlier in Figure 3 for $ \pi =0.50 $ remain present. The more heterogeneous the partners are in risk preferences for a given mean relative risk aversion coefficient, the more they are able to increase average wealth in retirement (Graph A of Figure 4) and average consumption across the life cycle (Graph B). Risk-taking (Graph C) increases as well, but now reflects to a larger extent the relative risk aversion of the partner with larger bargaining power. The larger the Pareto weight reflecting the bargaining power of partner $ B $ for a given combination of $ \left({\gamma}_A,{\gamma}_B\right) $ , the more the couple’s relative risk aversion (Graph D) resembles the risk aversion of the more risk-averse partner $ B $ , and the lower the share of wealth in stocks. For a given combination of $ \left({\gamma}_A,{\gamma}_B\right) $ , the partner with more bargaining power on average receives a larger share of consumption early in life when household consumption is relatively low but a lower share of consumption later in life (Graphs E and F).

Figure 4Varying the Pareto Weight When Both Partners Have Different Risk Aversion

Figure 4 shows average simulated life-cycle profiles for financial wealth (in Graph A), consumption (Graph B), the share of wealth allocated to stocks (Graph C), the relative risk aversion of the couple (Graph D), the consumption shares of partners A and B (Graphs E and F), obtained from four collective life-cycle models with combinations of $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ , $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ , and $ \pi =\mathrm{0.25,0.75} $ .

C. Intra-Household Risk Sharing Versus Risk Diversification

Our main results in Figure 3 revealed a perhaps surprisingly large economic impact of intra-household risk sharing on portfolio choice. To further evaluate the economic importance of risk sharing, we now compare the quantitative implications of risk sharing with those of earnings risk diversification. Specifically, we solve the collective models with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ and $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and equal bargaining power for two different coefficients of correlation between the persistent earnings shocks of both partners, $ \rho =0.1 $ and $ \rho =0.9 $ . The parameter $ \rho $ directly affects the partners’ ability to diversify idiosyncratic earnings risk within the household. The solutions with $ \rho =0.1 $ are already known from Figure 3, but are repeated here for ease of comparison. Moving from a collective benchmark model with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \rho =0.1 $ to a model with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \rho =0.9 $ reduces the scope for risk diversification but leaves the risk-sharing channel unaffected. This change can be motivated by Shore (Reference Shore2010) who finds that couples are less able to diversify idiosyncratic earnings risk during expansions. Similarly, moving from the same collective benchmark model to a collective model with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ and $ \rho =0.1 $ reduces the scope for intra-household risk sharing but leaves the risk-diversification channel unaffected.

Figure 5 shows the resulting simulation results. Compared to the benchmark collective model with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \rho =0.1 $ , a collective model with the same risk preference parameters and $ \rho =0.9 $ on average leads to a $ 1.7 $ percentage point ( $ 2.2\% $ ) reduction in the share of wealth allocated to stocks (in Graph C) across the life cycle. A higher correlation in the earnings streams of the partners increases nondiversifiable background risk, which in turn reduces financial risk-taking. Consumption (Graph B) is decreased early in life to increase precautionary saving and financial wealth (Graph A). In terms of economic magnitude, however, these changes are moderate, given the substantial increase from $ \rho =0.1 $ to $ \rho =0.9 $ . This can also be seen from the collective coefficient of relative risk aversion of the couple (Graph D), which hardly changes.

Figure 5Varying the Potential to Share Risk and to Diversify Risk Within the Household

Figure 5 shows average simulated life-cycle profiles for financial wealth (in Graph A), consumption (Graph B), the share of wealth allocated to stocks (Graph C), the relative risk aversion of the couple (Graph D), the consumption shares of partners A and B (Graphs E and F), obtained from four collective life-cycle models which differ in their potential to share risk $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ versus $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ , or diversify risk within the household, $ \rho =0.1 $ versus $ \rho =0.9 $ . The partners have identical bargaining power in all models ( $ \pi =0.5 $ ).

Compared to the benchmark collective model with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \rho =0.1 $ , a collective model with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ and $ \rho =0.1 $ , on average, leads to a $ 5.4 $ percentage point ( $ 7.0\% $ ) reduction in the share of wealth allocated to stocks (in Graph C of Figure 5) across the life cycle. Thus, a relatively modest reduction in intra-household risk preference heterogeneity (leaving the average relative risk aversion coefficient unchanged) has a large economic impact on risk sharing and the share of wealth invested in stocks. This is reflected in the wealth (Graph A) and consumption (Graph B) profiles, which show that substantially more financial wealth is accumulated if the couple is less able to benefit from risk sharing. The risk-sharing channel clearly dominates the risk-diversification channel in economic magnitude in Figure 5.

We know from Figure 3 that a mean-preserving spread in the partners’ coefficients of relative risk aversion from $ \left({\gamma}_A=5,{\gamma}_B=5\right) $ to $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ has less impact on savings and portfolio choice than an equally sized mean-preserving spread from $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ to $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ . In unreported results, we find that the risk-sharing and risk-diversification channels are similar in economic magnitude if Figure 5 is repeated for a benchmark collective model with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ and $ \rho =0.1 $ . This is still remarkable, considering that a modest change in the partners’ coefficients of relative risk aversion (leaving the average coefficient unchanged) is compared to a substantial change in the intra-household correlation of persistent idiosyncratic earnings shocks. In summary, the intra-household risk-sharing channel that affects all couples has larger portfolio choice implications than the risk-diversification channel that only affects dual-income couples.

D. Can a Unitary Model Approximate the Collective Model?

As a benchmark for the analysis of mean-preserving spreads in the partners’ coefficients of relative risk aversion in the collective model, we use a unitary model with a coefficient of risk aversion equal to the arithmetic average risk aversion coefficient of the partners in Section IV.B. Compared to this benchmark, we show that mean-preserving spreads in risk aversion have large economic effects on financial risk-taking. We now investigate whether a different unitary benchmark model can approximate these results from the collective life-cycle portfolio choice model when partners differ in relative risk aversion.

We already know that the consumption and portfolio choice decisions of the collective model can be exactly replicated by a unitary model if the partners have identical risk preferences. We also know that it is impossible for a unitary model to replicate the results of the collective model if there is intra-household heterogeneity in risk preferences because, in this case, the couple’s collective risk aversion varies across the life cycle as a function of time-varying individual consumption shares. However, the U-shaped profile of the couple’s risk aversion coefficient across the life-cycle is not very pronounced in Figure 3, which raises the question whether we can find a unitary model with an age-invariant relative risk aversion coefficient that yields a good approximation to the collective model for a couple. This is an important question because the unitary model is the workhorse model in the life-cycle portfolio choice literature.

As already indicated in Section II.D, we propose using a unitary model with the coefficient of relative risk aversion in equation (13) as a benchmark for approximating the collective model. This coefficient replaces the age-varying individual consumption shares in the denominator of the true coefficient of risk aversion in the collective model in equation (12) with the bargaining power weights of the partners. Gu et al. (Reference Gu, Peng and Zhang2024) parameterize these bargaining weights as functions of observable variables such as the relative earnings of the partners. This variable can also be generated from an assumed earnings process for a dual-income couple such as ours in Section II.A. Thus, unlike individual consumption shares, these bargaining weights can be generated in the process of solving a unitary life-cycle portfolio choice model.

For the following simulations, we assume that partners have identical bargaining power. In this case, the coefficient of relative risk aversion in equation (13) becomes the harmonic mean of the partners’ coefficients of relative risk aversion. The harmonic mean also results from the collective model if the partners share consumption equally within the household (see equation (12)) as assumed by Love (Reference Love2010), Hong and Ríos-Rull (Reference Hong and Ríos-Rull2012), and Hubener et al. (Reference Hubener, Maurer and Mitchell2015). However, equal consumption sharing is usually not an outcome of the collective model if partners differ in risk preferences. If the partners have identical bargaining power, risk sharing in the collective model instead implies equal marginal utility of consumption.

Figure 6 shows simulation results from two collective models with intra-household heterogeneity in relative risk aversion, $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ . All other parameters are set to their calibration values. These results are already known from Figure 3, but are repeated here for ease of comparison. Also shown in Figure 6 are the results from two unitary models with coefficients of relative risk aversion equal to $ \gamma =4.2 $ (the harmonic mean of 3 and 7) and $ \gamma =3.2 $ (the harmonic mean of 2 and 8), respectively. For the unitary models, we plot consumption shares of 0.5 in Graphs E and F of Figure 6 in line with assuming identical bargaining power. We deliberately choose collective models with relatively large differences in individual risk aversion, which lead to relatively pronounced U-shaped life-cycle profiles in the couple’s relative risk aversion, as can be seen from Figure 3. If the unitary benchmark model serves as a good approximation to these models, it will do even better when there is less heterogeneity in risk aversion.

Figure 6Collective Versus Unitary Model with Harmonic Mean Relative Risk Aversion

Figure 6 shows average simulated life-cycle profiles for financial wealth (in Graph A), consumption (Graph B), the share of wealth allocated to stocks (Graph C), the relative risk aversion of the couple (Graph D), the consumption shares of partners A and B (Graphs E and F), obtained from two collective life-cycle models with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ and two unitary models with $ \gamma =4.2 $ (the harmonic mean of 3 and 7) and $ \gamma =3.2 $ (the harmonic mean of 2 and 8), respectively. The partners have identical bargaining power in all models ( $ \pi =0.5 $ ).

In terms of portfolio choice (see Graph C of Figure 6), the unitary models provide very good approximations to the collective models. The difference in the average risky asset share across the life cycle is about 0.3 percentage points for $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and 1.1 percentage points for $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ . However, compared to the unitary benchmark model, the couple in the collective model enjoys higher average consumption across the life cycle (Graph B). Compared to the unitary model with $ \gamma =4.2 $ ( $ \gamma =3.2 $ ), the collective model with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ ( $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ ) increases average household consumption by $ 1.0\% $ ( $ 2.4\% $ ). The largest increase occurs during retirement. The consumption loss experienced by the couple in the unitary model with a harmonic average risk aversion coefficient can be interpreted as the loss resulting from sharing consumption equally as opposed to sharing consumption optimally. This interpretation is based on the collective coefficient of relative risk aversion in equation (12), which reduces to the harmonic mean of the partners’ coefficients of relative risk aversion if consumption is shared equally.

The unitary benchmark model in this analysis approximates the risk-sharing mechanism of the collective model. It is therefore interesting to see the differences between the two models if background risk increases. For this purpose, we repeat the simulations underlying Figure 6 with an earnings process that employs the full magnitude of the estimated volatilities in Panel B of Table 1.Footnote ¹⁴ Recall that we reduced these volatilities for our previous simulations by 50% to account for possible measurement error. We discuss the results of these simulations in Figure 7 in Section IV.E.

Figure 7Collective Versus Unitary Model—High Earnings Volatility

Figure 7 replicates Figure 6 for a high-earnings-risk scenario. The figure shows average simulated life-cycle profiles for financial wealth (in Graph A), consumption (Graph B), the share of wealth allocated to stocks (Graph C), the relative risk aversion of the couple (Graph D), the consumption shares of partners A and B (Graphs E and F), obtained from two collective life-cycle models with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ and two unitary models with $ \gamma =4.2 $ (the harmonic mean of 3 and 7) and $ \gamma =3.2 $ (the harmonic mean of 2 and 8), respectively. The partners have identical bargaining power in all models ( $ \pi =0.5 $ ).

E. Mean-Preserving Spreads in Idiosyncratic Earnings Risk

Before we compare the unitary and collective models in Figure 7, we note the substantial increase in financial wealth in Figure 7 (Graph A) compared to Figure 6 (Graph A). This is due to higher precautionary savings in response to increased background risk. This result is expected because the individual power utility in our model is characterized by prudence. Apps et al. (Reference Apps, Andrienko and Rees2014) show that precautionary saving increases in the collective model in response to a mean-preserving spread in earnings risk if individual preferences exhibit prudence, in line with the unitary case considered by Kimball (Reference Kimball1990). Compared to the couple with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ , the couple with $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ accumulates less precautionary savings in response to the mean-preserving spread in earnings risk, reflecting the larger potential to share risk within the household.

The increase in precautionary savings is accompanied by a substantial reduction in the share of wealth in stocks (see Graph C of Figures 6 and 7). Higher background risk crowds out financial risk-taking. The mean-preserving spread in earnings risk reduces the average risky asset share across the life cycle from $ 77\% $ to $ 67\% $ for $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and from $ 87\% $ to $ 78\% $ for $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ .Footnote ¹⁵

In this environment with increased background risk, the unitary models with harmonic average risk aversion coefficients provide worse approximations to the consumption and portfolio choice decisions in the collective models. While the average difference in the risky asset shares between unitary and collective models remains small over the life cycle (0.6 and 1.2 percentage points, respectively, for the collective models with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ ), there are now substantial timing differences, with the collective models generating smaller (larger) allocations to risky assets during working life (retirement) than the unitary models. For example, the average share of wealth in risky assets at age 60 is $ 65\% $ for the collective model with $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ , but $ 72\% $ for the unitary model with $ \gamma =3.2 $ .

This is due to an increase in the consumption share of the more risk-averse partner B early in life (when consumption closely follows income) in response to high earnings volatility, which increases the couple’s relative risk aversion (see equation (12)) and decreases the share of wealth in stocks relative to the unitary model. When more wealth is accumulated later in life, negative earnings shocks no longer affect household consumption to the same extent, and the share of consumption of the less risk-averse partner A increases. Correspondingly, the couple’s relative risk aversion decreases, and the share of wealth in stocks increases relative to the unitary model.

Moreover, the differences in average consumption between unitary and collective models are even more pronounced when background risk increases. Compared to the unitary model with $ \gamma =4.2 $ ( $ \gamma =3.2 $ ), the collective model with $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ ( $ \left({\gamma}_A=3,{\gamma}_B=7\right) $ ) increases average household consumption by $ 1.4\% $ ( $ 3.5\% $ ). The largest increase again occurs during retirement.

In summary, while a unitary model with a relative risk aversion coefficient in equation (13) can approximate the average share of wealth in stocks across the life cycle generated by a collective model with a relative risk aversion coefficient in equation (12), the unitary model has problems in replicating the timing of portfolio choice decisions across the life cycle if background risk is high. Moreover, the unitary model generates less household consumption than the collective model, in particular during retirement.

When is background risk high? In our case, the high-background-risk scenario results from taking literally the GMM estimates of the covariance parameters of the couple’s joint earnings process. The low-background-risk scenario attributes half of the estimated volatilities of the earnings process to measurement error. Thus, it is safe to assume that the unitary model will have problems in replicating the consumption and portfolio choice decisions of the collective model if earnings process parameters are not adjusted for measurement error, as in Deaton (Reference Deaton1991).

F. Stock Market Participation

In the absence of stock market participation costs, all households in our model optimally hold stocks. As a robustness check, we now add stock market participation costs to the model in line with Vissing-Jorgensen (Reference Vissing-Jorgensen2002), Gomes and Michaelides (Reference Gomes and Michaelides2005), Fagereng, Gottlieb, and Guiso (Reference Fagereng, Gottlieb and Guiso2017), Gu et al. (Reference Gu, Peng and Zhang2024) and others. To this end, we assume that a percentage $ F $ of the household’s earnings is paid as a nonrecurring fee when the household decides to enter the stock market for the first time. These costs may reflect pecuniary and nonpecuniary components related to setting up a trading account and learning how to trade. Afterward, a percentage $ f $ of the household’s earnings is paid out as recurring fees each period, as long as the household remains invested in the stock market.

Figure 8 shows simulation results from two collective models with intra-household heterogeneity in relative risk aversion, $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ . Both models are solved for $ F=5\% $ initial stock market participation costs and either $ f=2\% $ or $ f=1\% $ repeated stock market participation costs.Footnote ¹⁶ All other parameters are set to their calibration values. The choice of collective models is guided by the simulation results of Section IV.E, which establishes a higher precautionary savings motive for households that are less able to share risk within the household ( $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ relative to $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ ). Thus, the savings motive decreases with a mean-preserving spread in the partners’ individual risk aversion coefficients.

Figure 8Varying the Costs of Stock Market Participation

Figure 8 shows average simulated life-cycle profiles for financial wealth (in Graph A), consumption (Graph B), the share of wealth allocated to stocks (Graph C), the relative risk aversion of the couple (Graph D), the consumption share of partner A (Graph E), and stock market participation (Graph F), obtained from two collective life-cycle models with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ and $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ , respectively, and two combinations of initial ( $ F $ ) and recurring ( $ f $ ) stock market participation costs, ( $ F=5\%,f=2\% $ ) and ( $ F=5\%,f=1\% $ ). Graphs A–E refer to stockholders only. The partners have identical bargaining power in all models ( $ \pi =0.5 $ ).

Figure 8 is organized along the lines of previous figures. However, Graph F now shows the percentage of stockholders across the life cycle instead of the consumption share of the partner $ B $ (which still can be recovered as 1 minus the consumption share of the partner $ A $ in Graph E). The average stock market participation across the life cycle has a hump-shaped pattern. Early in life, some liquidity-constrained households optimally decide to stay out of the stock market. By age 40, almost all households participate in the stock market, regardless of risk preferences and the magnitude of stock market participation fees. Households remain invested in the stock market until late in life.

We find that a mean-preserving spread in the partners’ individual risk aversion coefficients decreases average stock market participation early in life. With recurring participation fees set to $ f=2\% $ ( $ f=1\% $ ), 65.5% (74.6%) of couples aged 40 and under with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ participate in the stock market, while only 63.6% (73.8%) of couples with $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ optimally decide to invest in stocks. This result is consistent with the reduced savings motive of couples that have better opportunities to share risk within the household.

Conditional on participation, the share of wealth in stocks (in Graph C of Figure 8) increases with a mean-preserving spread in relative risk aversion, confirming our earlier simulation results in Section IV.B. There are some differences at the end of life as stock market participation begins to drop, and the consumption share of partner A now rises at the end of life. This happens due to a selection effect; it is costly to participate in the stock market, and we are reporting the households that continue to participate. They tend to be wealthier than those who drop out, and therefore have higher consumption (Graph B). For these households, higher total consumption implies a higher consumption share for the less risk-averse partner (Graph E).

We emphasize that the opposite effects of a mean-preserving spread in relative risk aversion on stock market participation (negative impact) and the share of wealth in stocks conditional on participation (positive impact) do not contradict each other. Intra-household risk sharing reduces precautionary savings. Only households with a sufficient savings motive are willing to bear the participation costs. Thus, participation decreases with a mean-preserving spread in relative risk aversion. However, conditional on participation, the share of wealth in stocks increases with the potential to share risk within the household. Thus, financial risk-taking increases with a mean-preserving spread in relative risk aversion, conditional on participation.

A. Household Data

While we use U.S. data to calibrate our model, we use data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey in this section to investigate the relationship between the financial risk-taking of couples and intra-household heterogeneity in risk aversion. Based on a comparison of several U.S. and international data sets, Gu et al. (Reference Gu, Peng and Zhang2024) conclude that HILDA is the most suitable data source for analyzing intra-household financial decisions. HILDA is a nationally representative, annual panel of Australian households, which commenced in 2001.Footnote ¹⁷ For our purposes, the main advantage of HILDA compared to other data sets is the availability of a measure of individual risk aversion for both partners in couples over the full working life.

Information on financial wealth and portfolio choice is collected every 4 years in HILDA as part of the wealth module. We use the six waves of HILDA between 2002 and 2022 that include a wealth module for our analysis. We create an unbalanced panel of couples consisting of partners aged 25–65 whose individual risk aversion can be assessed (see below). There are 6,158 couples in our final sample with 14,029 couple-year observations. Thus, on average, each couple is observed for about 9 years (2.3 periods). The subsample of households holding stocks (stockholders, from now on) consists of 6,494 couple-year observations and includes 3,154 couples.

Individual risk aversion in HILDA is equal to the value 1 (2) [3] if a respondent reports a willingness to take substantial (above-average) [average] financial risks with their spare cash, and the value 4 if the respondent is not willing to take any financial risks. In line with Gu et al. (Reference Gu, Peng and Zhang2024), we remove missing values and exclude couples in which at least one partner responds with “never has any spare cash” (value 5).Footnote ¹⁸

Following Gu et al. (Reference Gu, Peng and Zhang2024), we first interpret the ordinal measure of individual risk aversion as if it were a cardinal measure. As a robustness check, we then create a cardinal measure of individual risk aversion based on a transformation of the ordinal variable proposed by Winship and Mare (Reference Winship and Mare1984) and Terza (Reference Terza1987).Footnote ¹⁹ This transformation has been applied in previous literature to create, for example, cardinal measures of bond ratings (Wu (Reference Wu1993)) and risk aversion (Botzen and van den Bergh (Reference Botzen and van den Bergh2012)). Based on these measures of individual risk aversion, we calculate the “Difference in risk aversion” as the absolute value of the difference in the partners’ individual risk aversion measures. We define “Average risk aversion” as the arithmetic average of the partners’ measures of individual risk aversion.

Table 2 shows summary statistics for all couples (in Panel A), and stockholders (Panel B). Forty-six percent of households hold stocks, either directly or indirectly through mutual funds. Financial wealth includes bank accounts, cash and bond holdings, direct and indirect stockholdings, trusts, and the cash value of life insurance. Based on this, we create a “Log financial wealth” variable. The “Share of wealth in stocks,” defined as the share of financial wealth in direct or indirect stockholdings, is the variable of interest for the following analysis. Conditional on participation in the stock market, the average share of financial wealth in stocks is 43%.

Table 2Summary Statistics

Among stockholders, we find an average difference in the ordinal (cardinal) measure of risk aversion of 0.54 (0.75) in the sample (Panel B in Table 2). In 47.2% of these couples, the difference in risk aversion is positive. Including nonstockholders, this number decreases to 45.5% (Panel A). Gu et al. (Reference Gu, Peng and Zhang2024) report a slightly lower percentage of 43.4% using the 2006–2018 waves of HILDA with the wealth module only, while Mazzocco (Reference Mazzocco2004) finds a slightly higher percentage of 48.2% among elderly couples in the HRS. In any case, almost half of couples are not perfectly matched in terms of risk preferences. The average ordinal measure of risk aversion among stockholders is 3.1. As expected, this average is higher (3.25) for the whole sample.

Finally, we construct “Average age/10” and “Average age squared/100” variables that refer to the average age of partners in couples, scaled by 10. These variables are meant to pick up life-cycle patterns in financial risk-taking. The average age in our sample of couples with partners aged 25–65 is 44.3 (Panel A in Table 2). Stockholders, on average, are slightly older (46.1, Panel B) than nonstockholders.

B. Empirical Analysis

Using multivariate regression analysis, we investigate the relationship between the financial risk-taking of couples at the extensive and intensive margin and the ability of couples to share risk within the household. In line with our simulation results in Section IV, we focus on the difference in the partners’ coefficients of relative risk aversion as the main explanatory variable, while controlling for the average risk aversion.

Following Campbell (Reference Campbell2006), we separate the decision to participate in the stock market from the share of wealth invested in the stock market, conditional on participation. This share of wealth equals 100% for around 0.82% of couple-year observations. Correspondingly, we use a right-censored regression model (pooled Tobit) for the analysis of the intensive margin of risk-taking. For the analysis of the extensive margin, we use a pooled Probit model. By construction, we need to use linear regression models for both analyses if the key explanatory variables are based on the cardinal measure of individual risk aversion.

Table 3 shows the estimation results for the extensive and intensive margins of financial risk-taking and the three measures of risk aversion. The key explanatory variables in Panel A and Panel B of Table 3 are based on the ordinal and cardinal measures of individual risk aversion, respectively. All regressions include a full set of (unreported) time dummy variables. We present estimated, average partial effects (APEs) for all nonlinear models, which are comparable to the estimated coefficients in the linear models. The estimated standard errors are clustered at the household level to allow for within-couple serial correlation in error terms caused, for example, by unobserved, time-invariant, heterogeneity.

Table 3Estimation Results

In line with most of our simulation results in Section IV, which abstract from the stock market participation decision except in Section IV.F, we first discuss the results for the share of wealth invested in stocks, conditional on participation. Importantly, the APE for the intra-household “Difference in risk aversion” is positive and statistically significant at the 5% level in all regressions, irrespective of the method of construction of the measure of individual risk aversion. For the ordinal and cardinal measures in Panels A and B of Table 3, a one-unit increase in the difference in individual risk preferences increases the share of wealth in stocks by 1.3 percentage points.

For comparison, our simulation results in Graph C of Figure 3 show that a couple consisting of partners with individual risk aversion coefficients of 4 and 6, respectively, on average allocates 1.1 percentage points more financial wealth to stocks over their working life than a couple consisting of partners with identical risk aversion coefficients of 5. For larger mean-preserving spreads, we find more substantial increases. Thus, we find empirical support for the predicted positive relationship between mean-preserving spreads in risk aversion and financial risk-taking at the intensive margin. Quantitatively, the empirical findings are in the vicinity of model predictions.

An increase in the intra-household “Average risk aversion” has the predicted negative impact on financial risk-taking. A one-unit increase in average individual risk aversion decreases the share of wealth in stocks by more than 10 (8) percentage points in Panel A (B) of Table 3. The APE is statistically significant at the 1% level. This result gives us confidence that our measures of individual risk aversion indeed capture the risk preferences of the partners. In addition, we find a hump-shaped age pattern in the share of wealth allocated to stocks in both panels of Table 3. However, the age effects are statistically insignificant. The APE of “Log financial wealth” is negative but insignificant in both panels.

We next discuss the results for the stock market participation decision, which we investigated in the simulations in Section IV.F. The $ {R}^2 $ of the linear probability model for the extensive margin of financial risk-taking, 0.27 (in Panel B of Table 3), substantially exceeds the corresponding statistic of the linear model for the intensive margin, 0.07. Thus, it is less difficult to explain variation in stock market participation than variation in the demand for stocks, conditional on participation.

Contrary to our findings at the intensive margin, the intra-household difference in risk aversion usually has a negative impact on the extensive margin of financial risk-taking. A one-unit increase in the difference in individual risk preferences decreases stock market participation by 1.6 (1.1) percentage points for the ordinal (cardinal) measure of individual risk aversion in Panel A (Panel B) of Table 3. These effects are statistically significant at the 5% level. Recall that we find a 1.9% (0.8%) reduction in the stock market participation of couples aged 40 and under with $ \left({\gamma}_A=2,{\gamma}_B=8\right) $ compared to those with $ \left({\gamma}_A=4,{\gamma}_B=6\right) $ in Section IV.F for participation fees $ f=2\% $ ( $ f=1\% $ ). Thus, our empirical results resemble those generated from the simulations of our model.

There are at least two explanations for this finding: participation costs and intra-household cooperation. The first explanation is directly related to the model. Only households with a sufficiently strong savings motive are willing to bear the costs of stock market participation. The savings motive is higher for couples who are less able to share risk within the household, as shown in Section IV.E. Thus, a mean-preserving spread in relative risk aversion reduces stock market participation.

Alternatively, the negative impact of the difference in risk preferences on stock market participation could be the result of a failure to agree on participation when partners differ in risk preferences. The bargaining process may be complicated if the involved parties have very different appetites for risk. Indirect empirical evidence for this hypothesis is given by Serra-Garcia (Reference Serra-Garcia2022) who shows that intra-household heterogeneity in risk preferences is a predictor of future marital instability. In such a scenario, participation is a sign of intra-household cooperation in investment decisions. Correspondingly, one would expect to find a positive sign of the difference in individual risk aversion on the share of wealth in stocks, conditional on participation.

Average risk aversion has a strong negative impact on the decision to participate in the stock market. A one-unit increase in average risk aversion reduces participation by more than 15 (13) percentage points in Panel A (B) of Table 3, based on the ordinal (cardinal) measures of risk aversion. The variables controlling for the average level of risk aversion in the household are highly statistically significant at the 1% level.

We find a hump-shaped age pattern in the stock market participation decision of couples for both measures of individual risk aversion, which is statistically significant at the 1% level for the cardinal measure. While financial wealth does not significantly predict the intensive margin of financial risk-taking, it has a positive and statistically highly significant impact on the extensive margin of financial risk-taking in Table 3. Both findings are in line with life-cycle portfolio choice models that include stock market participation costs (see Fagereng et al. (Reference Fagereng, Gottlieb and Guiso2017) for a detailed discussion).

In summary, using HILDA data, we find supportive empirical evidence for the predicted positive (negative) relationship between mean-preserving spreads in the partners’ coefficients of relative risk aversion (reflecting improved opportunities to share risk within the household) and financial risk-taking at the intensive (extensive) margin.