Saving and perceived health risks

Abstract

When utility is health state dependent, saving decisions depend on how perceived future health will affect the marginal utility of consumption. In consequence, people’s failure to anticipate hedonic adaptation to adverse health changes leads them to save more if consumption and health are Edgeworth complements, but to save less if they are Edgeworth substitutes. When we add a zero-mean risk on future health, a new effect occurs. Since underestimating hedonic adaptation to health change also increases the individual’s sensitivity to health risk, cross-prudence, which corresponds to the willingness to accumulate wealth in the face of future health risks may, depending on the case in question, amplify but also limit the change in savings. These effects add another source of variability among observed saving behaviors. They also singularly complicate the estimation of the health state dependence of the utility function.

Appendix

1.1 Cross-prudence in wealth and saving without hedonic adaptation

• (a) In the presence of a zero-mean health risk at the second period, the individual saves more if u _chh  > 0 but less if u _chh  < 0.

Proof

Consider the program given by Eq. (5). Without hedonic adaptation (α = 0), we have:

$$ \mathop {Max}\limits_{s} \, \tilde{U}(s) \equiv u\left( {y_{1} - s,h_{1} } \right) + \delta \cdot {\text{E}}\left[ {u\left( {y_{2} + (1 + r)s,\tilde{h}_{2} } \right)} \right] $$

(12)

with $\tilde{h}_{2} = h_{2} + \tilde{\varepsilon },E\left[ {\tilde{\varepsilon }} \right] = 0\;{\text{and}}\;{\text{Var}}(\hat{\varepsilon }) = \sigma^{2} .$ Assume for simplicity that, $h_{1} = h_{2} = E\left[ {\tilde{h}_{2} } \right] = h$ and y ₁ = y ₂ = y. Without health risk $(\tilde{\varepsilon } = 0 ) ,$ the program of the individual is given by:

$$ \mathop {Max}\limits_{s} \, U(s) \equiv u(y - s,h) + \delta u(y + (1 + r)s,h) $$

(13)

And let us denote $s^{**}$ its solution (we assume an interior solution). It satisfies the first-order condition:

$$ u_{c} (y - s^{**} ,h) = \delta (1 + r)u_{c} \left( {y + (1 + r)s^{**} ,h} \right) $$

(14)

In the presence of health risk, the program becomes

$$ \mathop {Max}\limits_{s} \, \tilde{U}(s) \equiv u(y - s,h) + \delta E\left[ {u(y + (1 + r)s,h + \tilde{\varepsilon })} \right] $$

(15)

Assume again an interior solution, denoted $s^{*} .$ The derivative of $\tilde{U}(s)$ for $s = s^{**}$ is given by $\, \tilde{U}^{\prime}(s^{ * * } ) = - u(y - s^{ * * } ,h) + \delta (1 + r)E\left[ {u(y + (1 + r)s^{ * * } ,h + \tilde{\varepsilon })} \right].$ Replacing $- u(y - s^{ * * } ,h)$ by its expression in (14) gives:

$$ \, \tilde{U}^{\prime}(s^{ * * } ) \equiv \delta (1 + r)\left( {E\left[ {u(y + (1 + r)s^{ * * } ,h + \tilde{\varepsilon })} \right] - u_{c} \left( {y + (1 + r)s^{**} ,h} \right)} \right) $$

(16)

It follows that $s^{ * } > s^{ * * }$ if $\, \tilde{U}^{\prime } (s^{ * * } ) > 0$ and so if $E\left[ {u(y + (1 + r)s^{ * * } ,h + \tilde{\varepsilon })} \right] > u_{c} \left( {y + (1 + r)s^{**} ,h} \right),$ that is if u _c is convex in h, i.e. u _chh  > 0. On the contrary, $s^{*} < 0$ if u _chh  < 0.□

• (b) A mean-preserving increase in health risk leads the individual to save more if u _chh  > 0 but less if u _chh  < 0.

Proof

Start from the maximization program given by (15). And denote $\tilde{h} = h + \tilde{\varepsilon }.$ The first-order condition is given by

$$ - u(y - s^{ * } ,h) + \delta (1 + r)E\left[ {u(y + (1 + r)s^{ * } ,\tilde{h})} \right] = 0 $$

(17)

Consider now two random health variables $\tilde{h}_{1}$ and $\tilde{h}_{2}$ with $\tilde{h}_{2}$ corresponding to a mean-preserving increase in risk of $\tilde{h}_{1} .$ Denote $s_{1}$ the solution of the program of the individual when he faced the risk $\tilde{h}_{1}$ and $s_{2}$ the solution of the program with $\tilde{h}_{2} .$ We assumed again interior solutions. Following the same reasoning as in (a), we can straightforwardly assert that s ₂ > s ₁ if $E\left[ {u(y + (1 + r)s^{ * } ,\tilde{h}_{2} )} \right] > E\left[ {u(y + (1 + r)s^{ * } ,\tilde{h}_{1} )} \right].$From Ekern (1980), we know that if $\tilde{h}_{1}$ dominates $\tilde{h}_{2}$ via nth-order stochastic dominance but not for any order than n ($\tilde{h}_{2}$ is a nth increase of risk of $\tilde{h}_{1}$ in Ekern’s terminology), then $E\left[ {f(\tilde{h}_{1} )} \right] < E\left[ {f(\tilde{h}_{2} )} \right]$ for any arbitrary function f such that $sign\left( {{{d^{n} f(h)} \mathord{\left/ {\vphantom {{d^{n} f(h)} {dh^{n} }}} \right. \kern-0pt} {dh^{n} }}} \right) = sign( - 1)^{n} .$ With n = 2, the proposition is equivalent to saying that if $\tilde{h}_{2}$ is a mean-preserving increase of $\tilde{h}_{1} ,$ $E\left[ {f(\tilde{h}_{1} )} \right] < E\left[ {f(\tilde{h}_{2} )} \right]$ for any arbitrary function f such that $sign \, f^{\prime\prime}(h) > 0.$Define $f(h) = u_{c} \left( {y + (1 + r)s_{1} ,h} \right)$. It follows that $f^{\prime}(h) = u_{ch} \left( {y + (1 + r)s_{1} ,h} \right),$ $f^{\prime\prime}(h) = u_{chh} \left( {y + (1 + r)s_{1} ,h} \right)$ and $E\left[ {f(\tilde{h}_{i} )} \right] = E\left[ {u_{c} (y + (1 + r)s_{1} ,\tilde{h}_{i} )} \right]$ for i = 1, 2. Applying Ekern’s proposition, we thus conclude that $E\left[ {u_{c} (y + (1 + r)s_{1} ,\tilde{h}_{2} )} \right] \ge E\left[ {u_{c} (y + (1 + r)s_{1} ,\tilde{h}_{1} )} \right]$ and so s ₂ > s ₁ if and only if $u_{chh} \left( {y + (1 + r)s_{1} ,h} \right) > 0,$ that is if the individual is cross-prudent.□

1.2 Proof of proposition 2

Proposition 2

Consider an individual who faces a health risk such that he has no chance of maintaining his initial level of health. Suppose also that the individual underestimates his hedonic adaptation to health changes as specified in Eq. (3). Then, if u _ch  < 0, the individual saves more and the increase is amplified if he is cross-prudent. If u _ch  > 0, the individual saves less, but this decrease is mitigated if he is cross-prudent.

Proof

The subjective health capital is given by $\hat{\tilde{h}}_{2} = \tilde{h}_{2} + (1 - m)\alpha \left( {h_{1} - \tilde{h}_{2} } \right)$ or after rearrangement by:

$$ \hat{\tilde{h}}_{2} = h_{2} + (1 - m)\alpha \left( {h_{1} - h_{2} } \right) + (1 - \alpha )\tilde{\varepsilon } + m\alpha \tilde{\varepsilon } $$

(18)

Let us denote $H = h_{2} + (1 - m)\alpha \left( {h_{1} - h_{2} } \right)$ and $\tilde{H} = H + (1 - \alpha )\tilde{\varepsilon }.$ With these notations, Eq. (18) can be rewritten:

$$ \hat{\tilde{h}}_{2} = \tilde{H} + m\alpha \tilde{\varepsilon } $$

(19)

The term $m\alpha \tilde{\varepsilon }$ isolates the specific effect of the underestimation of hedonic adaptation to health risk on the variability of subjective future health. Without this effect, the individual maximizes:

$$ \mathop {Max}\limits_{s} \, \tilde{V}(s) \equiv u(y - s,h_{1} ) + \delta E\left[ {u\left( {y + (1 + r)s,\tilde{H}} \right)} \right] $$

(20)

Let us denote $s^{**}$ the solution of this program. Assume again an interior solution. The first-order condition is given by $- u_{c} (y - s^{ * * } ,h_{1} ) + \delta (1 + r)E\left[ {u_{c} \left( {y + (1 + r)s^{ * * } ,\tilde{H}} \right)} \right] = 0.$ Differentiating this equation with respect to h ₂, then rearranging implies that:

$$ \frac{{ds^{**} }}{dm} = \frac{{\alpha \delta (1 + r){\text{E}}\left[ {\left( {(h_{1} - h_{2} )} \right)u_{ch} \left( {y + (1 + r)s^{ * } ,\tilde{H}} \right)} \right]}}{{{\text{E}}\left[ {\tilde{V}^{\prime\prime}} \right]}} $$

(21)

Therefore, if $s^{ * }$ denotes the optimal level of saving when m = 0, then

$$ s^{ * * } > s^{ * } \quad {\text{if}}\quad u_{ch} < 0 $$

(22a)

$$ s^{ * } > s^{ * * } \quad {\text{if}}\quad u_{ch} > 0 $$

(22b)

In the general case where we take into account $m\alpha \tilde{\varepsilon },$ the individual chooses the saving level $s^{ * * * }$ solution of the program:

$$ \mathop {Max}\limits_{s} \, \hat{U}(s) \equiv u\left( {y_{1} - s,h_{1} } \right) + E\left[ {u\left( {y_{2} + s,\tilde{H} + m\alpha \tilde{\varepsilon }} \right)} \right] $$

(23)

Since that $m\alpha \tilde{\varepsilon }$ corresponds to a mean-preserving increase in health risk, “Cross-prudence in wealth and saving without hedonic adaptation” in “Appendix” shows that it implies that $s^{ * * * } > s^{ * * }$ if u _chh  > 0. As a consequence, given Eq. (22a), we can state that:

$$ s^{ * * * } > s^{ * * } > s^{ * } \quad {\text{if}}\quad u_{ch} < 0 $$

(24)

Thus, if u _ch  < 0 and m > 0, the individual saves more and the increase is amplified if he is cross-prudent. Furthermore, proposition 1 states that $s^{ * * * } < s^{ * }$ if the individual has no chance of maintaining his initial level of health when u _ch  > 0. Hence, it also follows, using Eq. (22b) that:

$$ s^{ * } > s^{ * * * } > s^{ * * } \quad {\text{if}}\quad u_{ch} > 0 $$

(25)

When u _ch  > 0 and m > 0, the individual saves less, but this decrease is mitigated if he is cross-prudent.