Mortality, Family and Lifestyles

Abstract

While there is a large empirical literature on the intergenerational transmission of health and survival outcomes in relation to lifestyles, little theoretical work exists on the long-run prevalence of (un)healthy lifestyles induced by mortality patterns. To examine that issue, this paper develops an overlapping generations model where a healthy lifestyle and an unhealthy lifestyle are transmitted vertically or obliquely across generations. It is shown that there must exist a locally stable heterogeneous equilibrium involving a majority of healthy agents, as a result of the larger parental gains from socialization efforts under a higher life expectancy. We also examine the robustness of our results to the introduction of parental altruistic concerns for children’s health and of asymmetric socialization costs.

Appendices

Appendices

Basic Model: Existence and Uniqueness

The transition function is such that $G(0)=0$ and $G(1)=1$. Hence, $G(q_{t})$ meets the 45${{}^\circ}$ line at the two extremities, which are stationary equilibria. Note also that the derivative of $G(q_{t})$ with respect to $q_{t}$ is

$$ G^{\prime }(q_{t})=1+\frac{\beta \left( \hat{\varphi}-\tilde{\varphi}\right) } {\delta }\left[ \pi ^{H}(1-q_{t})^{2}-\pi ^{H}q_{t}2(1-q_{t})-\pi ^{U}2q_{t}(1-q_{t})+\pi ^{U}q_{t}^{2}\right] $$

We have

$$ \begin{aligned} G^{\prime }(0) &=1+\frac{\beta \left( \hat{\varphi}-\tilde{\varphi}\right) }{\delta }\pi ^{H}>1 \\ G^{\prime }(1) &=1+\frac{\beta \left( \hat{\varphi}-\tilde{\varphi}\right) }{\delta }\pi ^{U}>1 \end{aligned} $$

Thus, given that the slope of $G(q_{t})$ is higher than 1 at $q_{t}$ equal to 0 and 1, it must be the case that $G(q_{t})$ is above the 45${{}^\circ}$ line in the neighbourhood of 0, and below the 45${{}^\circ}$ line in the neighbourhood of 1. Hence, by continuity, $G(q_{t})$ must intersect the 45${{}^\circ}$ line somewhere, for a level of $q_{t}$ between 0 and 1. Regarding the uniqueness of that intermediate equilibrium, substituting for $q_{t+1}=q_{t}=q$ in the transition function shows that this intermediate equilibrium takes a single value: $q=\frac{\pi ^{H}}{\pi ^{H}+\pi ^{U}}>\frac{1}{2}$.

Basic Model: Stability

Stability requires $\left\vert \frac{\partial G}{\partial q_{t}}\right\vert <1$. We have

$$ \begin{aligned} G^{\prime }(0) &=1+\frac{\beta \left( \hat{\varphi}-\tilde{\varphi}\right) }{\delta }\pi ^{H}>1 \\ G^{\prime }(1) &=1+\frac{\beta \left( \hat{\varphi}-\tilde{\varphi}\right) }{\delta }\pi ^{U}>1 \end{aligned} $$

Thus $q=0$ and $q=1$ are not stable.

Finally, at $q_{t}=\frac{\pi ^{H}}{\pi ^{H}+\pi ^{U}}$, we have:

$$ G^{\prime }(q_{t})=1-\frac{\beta \left( \hat{\varphi}-\tilde{\varphi}\right) }{\delta }\frac{\pi ^{H}\pi ^{U}}{\pi ^{H}+\pi ^{U}}<1 $$

Given $\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) }{\delta }<1$, that expression is smaller than 1 in absolute value, so that the equilibrium is locally stable.

Altruistic Model: Existence and Uniqueness

Note that $G_{1}(q_{t})$ is such that: $G_{1}(0)=0$ and $G_{1}(1)=1$. Hence, $G_{1}(q_{t})$ meets the 45${{}^\circ}$ line at the two extremities, which are stationary equilibria. The derivative of $G_{1}( q_{t})$ with respect to $q_{t}$ is

$$ \begin{aligned} G_{1}^{\prime }(q_{t}) &=1+\left[ 1-2q_{t}\right] \frac{\left[ \left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] (1-q_{t})-\left[ \beta \pi ^{U}\left( \hat{ \varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] q_{t}\right] }{\delta } \\ &\quad+q_{t}(1-q_{t})\frac{\left[ -\left[ \beta \pi ^{H}\left( \hat{\varphi}- \tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] - \left[ \beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] \right] }{\delta } \end{aligned} $$

We have

$$ \begin{aligned} G_{1}^{\prime }(0) =&1+\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta }>1 \\ G_{1}^{\prime }(1) =&1+\frac{\beta \pi ^{U}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) }{\delta }>1 \end{aligned} $$

Thus, under $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) >\alpha \left( \pi ^{H}-\pi ^{U}\right)$, the transition function lies above the 45${{}^\circ}$ line in the neighbourhood of 0, and below the 45${{}^\circ}$ line in the neighbourhood of 1. Hence, the existence of an intermediate equilibrium can be proved as in the basic model. Fixing $q_{t+1}=q_{t}=q$ in the transition function yields the intermediate equilibrium:

$$ q=\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\beta \pi ^{U}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) }> \frac{1}{2} $$

In the case where $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$, the transition function $G_{2}(q_{t})$ is such that $G_{2}(0)=0$ and $G_{2}(1)=1$. Hence, $G_{2}(q_{t})$ meets the 45${{}^\circ}$ line at the two extremities, which are stationary equilibria. Note also that

$$ G_{2}^{\prime }(q_{t})=1+\left[ -2(1-q_{t})q_{t}+(1-q_{t})^{2}\right] \frac{ \left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] }{\delta } $$

We have

$$ \begin{aligned} G_{2}^{\prime }(0)&=1+\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta}>1 \\ G_{2}^{\prime }(1) &=1 \end{aligned} $$

Thus, under $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$, the transition function lies above the 45${{}^\circ}$ line in the neighbourhood of 0, but does not lie below the 45${{}^\circ}$ line in the neighbourhood of 1. Actually, it is not difficult to show that there cannot be an intermediate equilibrium in this model. Indeed fixing $q_{t+1}=q_{t}=q$ in the transition function yields:

$$ q=q+(1-q)^{2}q\frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi} \right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] }{\delta } $$

which cannot be true for $0<q<1$. Thus there exists no intermediate equilibrium under $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$.

Altruistic Model: Stability

In the case where $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) >\alpha \left( \pi ^{H}-\pi ^{U}\right)$, stability requires $\left\vert \frac{\partial G_{1}} {\partial q_{t}}\right\vert <1$. We have

$$ \begin{aligned} G_{1}^{\prime }(q_{t}) =&1+\left[ 1-2q_{t}\right] \frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] (1-q_{t})-\left[ \beta \pi ^{U}\left( \hat{\varphi}- \tilde{\varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] q_{t} }{\delta } \\ &\quad +q_{t}(1-q_{t})\frac{-\left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] -\left[ \beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] }{\delta } \end{aligned} $$

Hence

$$ \begin{aligned} G_{1}^{\prime }(0) =&1+\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta }>1 \\ G_{1}^{\prime }(1) =&1+\frac{\beta \pi ^{U}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) }{\delta }>1 \end{aligned} $$

Thus, given $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) >\alpha \left( \pi ^{H}-\pi ^{U}\right)$, neither $q=0$ nor $q=1$ are stable.

Finally, at $q_{t}=\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) +\beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right)}$, we have:

$$ G_{1}^{\prime }(q_{t})=1-\frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}- \tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] \left[ \beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] }{\delta \left[ \beta \pi ^{U}\left( \hat{ \varphi}-\tilde{\varphi}\right) +\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) \right] }<1 $$

Hence, given that socialization efforts are between 0 and 1, the intermediate equilibrium must be locally stable.

When $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$, stability requires $\left\vert \frac{\partial G_{2}}{\partial q_{t}}\right\vert <1$. We have

$$ G_{2}^{\prime }(q_{t})=1+\left[ -2(1-q_{t})q_{t}+(1-q_{t})^{2}\right] \frac{ \left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] }{\delta } $$

Hence

$$ \begin{aligned} G_{2}^{\prime }(0) &=1+\frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}- \tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] } {\delta }>1 \\ G_{2}^{\prime }(1) &=1 \end{aligned} $$

so that $q=0$ is not stable. Regarding $q=1$, the equality $G_{2}^{\prime }(1)=1$ does not exactly coincide with what insures local stability (a strict inequality). However, it is clear that the transition function crosses the 45${{}^\circ}$ line from above at 1, and thus that equilibrium can be regarded as locally stable.

Asymmetric Costs: Existence and Uniqueness

Note first that the transition function $G_{1}(q_{t})$ is such that: $G_{1}(0)=0$ and $G_{1}(1)=1$. Hence, $G_{1}( q_{t})$ meets the 45$^\circ$ line at the two extremities, which are stationary equilibria. Note also that:

$$ \begin{aligned} G_{1}^{\prime }(q_{t}) &= 1+\left( 1-2q_{t}\right) \left[ \frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] (1-q_{t})}{\delta ^{H}}-\frac{\left[ \beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] q_{t}}{\delta ^{U}}\right] \\ &\quad+(1-q_{t})q_{t}\left[ -\frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}- \tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] }{ \delta ^{H}}-\frac{\left[ \beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi} \right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] }{\delta ^{U}}\right] \end{aligned} $$

We have

$$ \begin{aligned} G_{1}^{\prime }(0) =&1+\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta ^{H}}>1 \\ G_{1}^{\prime }(1) =&1+\frac{\beta \pi ^{U}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) }{\delta ^{U}}>1 \end{aligned} $$

Thus, under $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) >\alpha \left( \pi ^{H}-\pi ^{U}\right)$, the transition function lies above the 45${{}^\circ}$ line in the neighbourhood of 0, and below the 45${{}^\circ}$ line in the neighbourhood of 1. Hence, the existence of an intermediate equilibrium can be proved as in the basic model. Fixing $q_{t+1}=q_{t}=q$ in $q_{t+1}=G(q_{t})$ allows us to derive the intermediate equilibrium:

$$ q=\frac{\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta ^{H}}}{\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta ^{H}}+\frac{\beta \pi ^{U}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) }{\delta ^{U}}} $$

When $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$, $G_{2}(q_{t})$ is such that: $G_{2}(0)=0$ and $G_{2}(1)=1$. Hence, $G_{2}(q_{t})$ meets the 45${{}^\circ}$ line at the two extremities, which are stationary equilibria. Note also that

$$ G_{2}^{\prime }(q_{t})=1+\left[ -2(1-q_{t})q_{t}+(1-q_{t})^{2}\right] \frac{ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta ^{H}} $$

We have

$$ \begin{aligned} G_{2}^{\prime }(0)&= 1+\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta }>1 \\ G_{2}^{\prime }(1) &= 1 \end{aligned} $$

Thus, under $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$, the transition function lies above the 45${{}^\circ}$ line in the neighbourhood of 0, but does not lie below the 45${{}^\circ}$ line in the neighbourhood of 1. Actually, it is not difficult to show that there cannot be an intermediate equilibrium in this model. Indeed fixing $q_{t+1}=q_{t}=q$ in the transition function yields:

$$ q=q+(1-q)^{2}q\frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi} \right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] }{\delta ^{H}} $$

which cannot be true for $0<q<1$. Thus there exists no intermediate equilibrium under $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$.

Asymmetric Costs: Stability

If $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) >\alpha \left( \pi ^{H}-\pi ^{U}\right)$, stability requires $\left\vert \frac{\partial G_{1}}{\partial q_{t}}\right\vert <1$. We have

$$ \begin{aligned} G_{1}^{\prime }(q_{t}) =&1+\left( 1-2q_{t}\right) \left[ \frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] (1-q_{t})}{\delta ^{H}}-\frac{\left[ \beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] q_{t}}{\delta ^{U}}\right] \\ &\quad +(1-q_{t})q_{t}\left[ -\frac{\left[ \beta \pi ^{H}\left( \hat{\varphi}- \tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) \right] }{ \delta ^{H}}-\frac{\left[ \beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi} \right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) \right] }{\delta ^{U}}\right] \end{aligned} $$

We have

$$ \begin{aligned} G_{1}^{\prime }(0) &=1+\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta ^{H}}>1 \\ G_{1}^{\prime }(1) &=1+\frac{\beta \pi ^{U}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{U}-\pi ^{H}\right) }{\delta ^{U}}>1 \end{aligned} $$

Thus, given $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) >\alpha \left( \pi ^{H}-\pi ^{U}\right)$, $q=0$ and $q=1$ are not stable.

Finally, at $q_{t}=q^{2}$, we have: $G_{1}^{\prime }(q_{t})<1$, so that, given that individual socialization efforts are between 0 and 1, the intermediate equilibrium, if it exists, must be locally stable.

If $\beta \pi ^{U}\left( \hat{\varphi}-\tilde{\varphi}\right) \,\leq\, \alpha \left( \pi ^{H}-\pi ^{U}\right)$, stability requires $\left\vert \frac{\partial G_{2}}{\partial q_{t}}\right\vert <1$. We have

$$ G_{2}^{\prime }(q_{t})=1+\left[ -2(1-q_{t})q_{t}+(1-q_{t})^{2}\right] \frac{ \beta \pi ^{H}\left( \hat{\varphi}-\tilde{\varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta } $$

Hence

$$ \begin{aligned} G_{2}^{\prime }(0) &=1+\frac{\beta \pi ^{H}\left( \hat{\varphi}-\tilde{ \varphi}\right) +\alpha \left( \pi ^{H}-\pi ^{U}\right) }{\delta }>1 \\ G_{2}^{\prime }(1) &=1 \end{aligned} $$

so that $q=0$ is not stable. Regarding $q=1$, the equality $G_{2}^{\prime }(1)=1$ does not exactly coincide with what insures local stability (a strict inequality). However, it is clear that the transition function crosses the 45${{}^\circ}$ line from above at 1, and, thus that equilibrium can be regarded as locally stable.