Long-term health investment when people underestimate their adaptation to old age-related health problems

Abstract

This paper explores in a two-period model the economic implications of people’s tendency to underestimate their ability to adapt to age-related health problems. We model this misperception by assuming that the individual underestimates his future subjective health. Under standard assumptions, we show that, when people allocate their resources during their youth between present consumption, savings, and health investment, they invest more in health as long as the magnitude of the cross-marginal utility of health and consumption is not too negative.

Appendix: Comparative statics of the general model: proof of propositions 2

The individual’s optimization program in 3.2 is given by:

$$ \begin{gathered} \mathop {\text{Max}}\limits_{c,i} \, \hat{U}(c,i) \equiv u(c,h_{0} ) + \delta \cdot u\left( {c_{1} ,\hat{h}_{1} (i)} \right) \, \hfill \\ {\text{s}} . {\text{t}} .\quad c_{1} = y_{1} + (y_{0} - c - i) \hfill \\ \end{gathered} $$

(21)

In order to use standard shortcuts notations for derivatives, the discounted utility function of the second period is written v(c, h) instead of δu(c, h). The two (sufficient) first-order conditions are given by:

$$ \left\{ {\begin{array}{*{20}c} {u_{c} (c,h_{0} ) - v_{c} \left( {c_{1} ,\hat{h}_{1} (i^{*} )} \right) = 0 \, } \\ { - v_{c} \left( {c_{1} ,\hat{h}_{1} (i^{*} )} \right) + \hat{h}_{1}^{\prime } (i^{*} )v_{h} \left( {c_{1} ,\hat{h}_{1} (i^{*} )} \right) = 0} \\ \end{array} } \right. $$

(22)

The second partial derivatives of $\hat{U}(c^{*} ,i^{*} )$ are:

$$ \hat{U}_{cc} = u_{cc} + v_{cc} $$

(23a)

$$ \hat{U}_{ic} = v_{cc} - \hat{h}_{1}^{\prime } (i^{*} )v_{ch} $$

(23b)

$$ \begin{aligned} \hat{U}_{ii} & = v_{cc} - \hat{h}_{1}^{\prime } (i^{*} )v_{ch} + \hat{h}_{1}^{\prime \prime } (i^{*} )v_{h} + \hat{h}_{1}^{\prime } (i^{*} )\left( { - v_{ch} + \hat{h}_{1}^{\prime } (i^{*} )v_{hh} } \right) \\ = v_{cc} - 2\hat{h}_{1}^{\prime } (i^{*} )v_{ch} + \hat{h}_{1}^{\prime \prime } (i^{*} )v_{h} + \left[ {\hat{h}_{1}^{\prime } (i^{*} )} \right]^{2} v_{hh} \\ \end{aligned} $$

(23c)

The differentiation of the two first-order conditions with respect to i,c and m forms a system of two equations:

$$ \begin{gathered} \left( {\begin{array}{*{20}c} {\hat{U}_{cc} } & {\hat{U}_{ci} } \\ {\hat{U}_{ic} } & {\hat{U}_{ii} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {{\text{d}}c} \\ {{\text{d}}i} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{{\partial \hat{U}_{c} } \mathord{\left/ {\vphantom {{\partial \hat{U}_{c} } {\partial m}}} \right. \kern-0pt} {\partial m}}} \\ {{{\partial \hat{U}_{i} } \mathord{\left/ {\vphantom {{\partial \hat{U}_{i} } {\partial m}}} \right. \kern-0pt} {\partial m}}} \\ \end{array} } \right) \hfill \\ J \hfill \\ \end{gathered} $$

(24)

Cramer’s rule implies:

$$ \frac{{{\text{d}}i^{*} }}{{{\text{d}}m}} = \frac{A}{\det J}\quad {\text{with }}A = \left| {\begin{array}{*{20}c} {\hat{U}_{cc} } & {{{ - \partial \hat{U}_{c} } \mathord{\left/ {\vphantom {{ - \partial \hat{U}_{c} } {\partial m}}} \right. \kern-0pt} {\partial m}}} \\ {\hat{U}_{ic} } & {{{ - \partial \hat{U}_{i} } \mathord{\left/ {\vphantom {{ - \partial \hat{U}_{i} } {\partial m}}} \right. \kern-0pt} {\partial m}}} \\ \end{array} } \right| $$

(25)

Since the determinant of J is positive, the sign of di ^*/dm is given by the sign of A. To compute A, remember first that $\hat{h}_{1}^{\prime } (i^{*} ) = (1 - \alpha + \alpha m)h_{1}^{\prime } (i^{*} )$and note that:

$$ \frac{{\partial \hat{U}_{c} }}{\partial m} = - \frac{{\partial \hat{h}_{1} }}{\partial m}v_{ch} = \alpha \left[ {h_{0} - h_{1} (i^{*} )} \right]v_{ch} $$

(26)

$$ \begin{aligned} \frac{{\partial \hat{U}_{i} }}{\partial m} & = - \frac{{\partial \hat{h}_{1} }}{\partial m}v_{ch} + \alpha h_{1}^{\prime } (i^{*} )v_{h} + \frac{{\partial \hat{h}_{1} }}{\partial m}\hat{h}_{1}^{\prime } (i^{*} )v_{hh} \\ &= \alpha [h_{0} - h_{1} (i^{*} )]\left( {v_{ch} - \hat{h}_{1}^{\prime } (i^{*} )v_{hh} } \right) + \alpha h_{1}^{\prime } (i^{*} )v_{h} \\ \end{aligned} $$

(27)

Using new shortcuts notations, h ₁ = h ₁(i*) and $\hat{h}_{1} = \hat{h}_{1} (i^{*} )$, replacing these values in B gives:

$$ \begin{aligned} A & = {{ - \partial \hat{U}_{i} } \mathord{\left/ {\vphantom {{ - \partial \hat{U}_{i} } {\partial m}}} \right. \kern-0pt} {\partial m}} \cdot \hat{U}_{cc} + {{\partial \hat{U}_{c} } \mathord{\left/ {\vphantom {{\partial \hat{U}_{c} } {\partial m}}} \right. \kern-0pt} {\partial m}} \cdot \hat{U}_{ci} \\ = \left( {\alpha [h_{0} - h_{1} ]\left( {\hat{h}_{1}^{\prime } v_{hh} - v_{ch} } \right) - \alpha h_{1}^{\prime } v_{h} } \right)(u_{cc} + v_{cc} ) + \left( {v_{cc} - \hat{h}_{1}^{\prime } v_{ch} } \right)(\alpha [h_{0} - h_{1} ]v_{ch} ) \\ \end{aligned} $$

(28)

$$ \begin{gathered} = u_{cc} \left( {\alpha [h_{0} - h_{1} ]\left( {\hat{h}_{1}^{\prime } v_{hh} - v_{ch} } \right) - \alpha h_{1}^{\prime } v_{h} } \right) + \alpha [h_{0} - h_{1} ]\hat{h}_{1}^{\prime } v_{cc} v_{hh} - v_{cc} v_{ch} \alpha [h_{0} - h_{1} ] - v_{cc} \alpha h_{1}^{\prime } v_{h} \hfill \\ + v_{cc} v_{ch} \alpha [h_{0} - h_{1} ] - \hat{h}_{1}^{\prime } \alpha [h_{0} - h_{1} ](v_{ch} )^{2} \hfill \\ \end{gathered} $$

$$ = \underbrace {{u_{cc} }}_{ < 0}\left( {\underbrace {{\alpha \left[ {h_{0} - h_{1} } \right]\hat{h}_{1}^{\prime } v_{hh} }}_{ < 0}\underbrace {{ - v_{ch} \alpha [h_{0} - h_{1} ]}}_{{ \le 0\;{\text{if}}\;v_{ch} \ge 0}}\underbrace {{ - \alpha h_{1}^{\prime } v_{h} }}_{ < 0}} \right) + \underbrace {{\alpha [h_{0} - h_{1} ]\hat{h}_{1}^{\prime } \left( {v_{cc} v_{hh} - v_{ch}^{2} } \right)}}_{ > 0} - \underbrace {{\alpha v_{cc} h_{1}^{\prime } v_{h} }}_{ > 0} $$

(29)

So

$$ v_{ch} \ge 0\quad \Rightarrow \quad A > 0\quad {\text{and}}\,{\text{so}}\quad \frac{{{\text{d}}i^{*} }}{{{\text{d}}m}} > 0{\text{ (proposition 2)}} $$

(30)

It also follows straightforwardly from Eq. (29) that if v _ch is strongly negative, it is possible that A > 0 and di*/dm < 0. For some values or m, the individual may now underinvest in health.

Further comments: Cramer’s rule also allows computation of the effect of a change in m on c* and s*. In particular, we have:

$$ \frac{{{\text{d}}c^{*} }}{{{\text{d}}m}} = \frac{B}{\det J}\quad {\text{with}}\quad B = \left| {\begin{array}{*{20}c} {{{ - \partial \hat{U}_{c} } \mathord{\left/ {\vphantom {{ - \partial \hat{U}_{c} } {\partial m}}} \right. \kern-0pt} {\partial m}}} & {\hat{U}_{ci} } \\ {{{ - \partial \hat{U}_{i} } \mathord{\left/ {\vphantom {{ - \partial \hat{U}_{i} } {\partial m}}} \right. \kern-0pt} {\partial m}}} & {\hat{U}_{ii} } \\ \end{array} } \right| $$

(31)

which makes it possible to write $\frac{{{\text{d}}(s^{*} + i^{*} )}}{{{\text{d}}m}} = - \frac{{{\text{d}}c^{*} }}{{{\text{d}}m}}$ and so

$$ {\text{sign}}\frac{{{\text{d}}(s^{*} + i^{*} )}}{{{\text{d}}m}} = {\text{sign }}B $$

(32)

s* + i* is the amount of present resources that the individual chooses to allocate to the future, either to health or consumption. B is equal to ${{ - \partial \hat{U}_{c} } \mathord{\left/ {\vphantom {{ - \partial \hat{U}_{c} } {\partial m}}} \right. \kern-0pt} {\partial m}} \cdot \hat{U}_{ii} + {{\partial \hat{U}_{i} } \mathord{\left/ {\vphantom {{\partial \hat{U}_{i} } {\partial m}}} \right. \kern-0pt} {\partial m}} \cdot \hat{U}_{ic}$ or after replacing by:

$$ B = - \alpha [h_{0} - h_{1} ]v_{ch} \left( { - \hat{h}_{1}^{\prime } v_{ch} + \hat{h}_{1}^{\prime \prime } v_{h} + \left( {\hat{h}_{1}^{\prime } } \right)^{2} v_{hh} } \right) + \left( {v_{cc} - \hat{h}_{1}^{\prime } v_{ch} } \right)\left( { - \alpha \left[ {h_{0} - h_{1} } \right]\hat{h}_{1}^{\prime } v_{hh} + \alpha h_{1}^{\prime } v_{h} } \right) $$

(33)

If we develop then simplify, we get:

$$ B = \underbrace {{\underbrace {{\alpha \left[ {h_{0} - h_{1} } \right]\hat{h}_{1}^{\prime } \left( {v_{hh} v_{cc} - v_{ch}^{2} } \right)}}_{ > 0} +\, \alpha v_{h} v_{ch} \left( {(h_{0} - h_{1} )\hat{h}_{1}^{\prime \prime } - h_{1}^{\prime } \hat{h}_{1}^{\prime } } \right) + \underbrace {{\alpha v_{h} h_{1}^{\prime } v_{cc} }}_{ < 0}}}_{\text{undetermined\,sign}} $$

(34)

Therefore, as long as v _ch is not too strongly negative, the individual over invests in health; but without more restrictions on the utility function, we cannot be sure of the net effect of m on the amount of resources allocated to the future for health and consumption.