Precautionary Investment in Wealth and Health

• Date: 01 March 2019
• DOI: http://doi.org/10.1111/jori.12212
• Published date: 01 March 2019

©2017 The Journal of Risk and Insurance. Vol.86, No. 1, 237–255 (2019).

DOI: 10.1111/jori.12212

Precautionary Investment in Wealth and Health

Desu Liu

Mario Menegatti

Abstract

This article studies how health and wealth investments react to the pres-

ence of random returns, distinguishing the case where only the level

of health investment is chosen from the case where both health and

wealth investments are chosen. We show that this reaction depends mainly

on certain features of preferences: cross-prudence/imprudence in wealth,

cross-prudence/imprudence in health, and the value of the indexes of rel-

ative prudence in wealth and in health being larger or smaller than the

threshold of 2. We also show the role of Edgeworth–Pareto substitutabil-

ity/complementarity between wealth and health investments in determining

optimal choices.

Introduction

People often make investment decisions, with the aim of improving futurestatus at the

expense of current resources. The costs and benefits can occur in differentdimensions.

If a decision maker (henceforth DM) wishes to raise his future wealth level, he can

choose to move some wealth from the present to the future. When making this choice,

the DM increases saving by formulating a “wealth investment.” Similarly, the DM

may wish to improve his future health status. In this case, he may choose to make a

“health investment” by using part of present wealth and/or by bearing a reduction in

current health. Both wealth and health investments decisions are usually taken under

risk, which implies that the presence of different types of risk can modify a DM’s

decisions by pushing him to increase or decrease investment. Meanwhile, investment

can itself become an instrument to deal with risk and a major alternative to other tools

of risk management such as insurance and prevention.

Starting from these ideas, the effects of differentrisks on investment have been widely

examined in the literature. A first strand of research refers to financial investment in

Desu Liu is at the Department of Finance and Insurance, School of Business, Nanjing Univer-

sity, Jiangsu 210093, P.R. China. Liu can be contacted via e-mail: liudesu@nju.edu.cn. Mario

Menegatti is at the Dipartimento di Scienze Economiche e Aziendali, Universit`

a degli Studi di

Parma, I-43125 Parma, Italy. Menegatti can be contacted via e-mail: mario.menegatti@unipr.it.

The authors are grateful to a coeditor,two anonymous reviewers, Louis Eeckhoudt, Serge Mac´

e,

Jack Meyer,and Richard Peter for their very useful comments and suggestions. Desu Liu grate-

fully acknowledges financial support of the China National Social Science Fund under Grant

No. 15BJL093.

237

238 The Journal of Risk and Insurance

a context where agents’ utility only depends on wealth. In this strand, Leland (1968)

and Sandmo (1970) study how saving and investment decisions of a DM change,

given different features of his preferences, when the DM faces a risk on his future

income. Similarly, Sandmo (1969, 1970) and Rothschild and Stiglitz (1971) study the

same issue when the returns on the investment are risky. The direction and the size

of the extrasaving determined by the presence of risk in these problems (called, from

Leland, “precautionary saving”), are related to the concept of “prudence” in utility

theory in the seminal article by Kimball (1990). Finally,in the same strand of research,

several recent articles analyzed a framework where, in some specific cases and/or

under some conditions on the distributions of risks, different types of risk on wealth

are introduced together (see Li, 2012; Baiardi, Magnani, and Menegatti, 2014; Baiardi

et al., 2015).1

A generalization of the analysis described above is introduced by Eeckhoudt, Rey,

and Schlesinger (2007), who study, for the first time, investment decisions in a con-

text where utility depends not only on the wealth level but also on the health status

(i.e., considering a two-argument utility function). This generalized framework made

it possible to study other aspects of the effects of risk on investment decisions. First,

Eeckhoudt, Rey,and Schlesinger (2007) analyze how a risk on the health status changes

the optimal level of wealth investment, given different features of a DM’s preferences.

Second, they study, for the first time, the case of “tertiary prevention,” defined as a

situation where a DM sustains a cost in terms of present health in order to improve

future health status. Other results on the changes in wealth investment in the frame-

work studied by Eeckhoudt, Rey,and Schlesinger (2007) are derived by Courbage and

Rey (2007) and Menegatti (2009a, 2009b), who consider different combinations of a

risk on future health status and a risk on future wealth level.

In the same literature, Denuit, Eeckhoudt, and Menegatti (2011) introduce the idea

of health investment. Just as a wealth investment is a financial cost borne in order to

get a larger (expected) wealth level in the future, a health investment is defined as a

financial cost sustained in order to get a better (expected) health status in the future.

Many examples of this kind of investment can be found; it occurs every time a DM

suffering from a kind of illness pays the cost of purchasing a medicine or a therapy in

order to improve his future health status.

Given the literature described above, the analysis in this article examines some missing

pieces in the strand of research studying investment in wealth and in health. First,

the literature reviewed above considers risks on the level of wealth, saving returns,

and the health status, either alternatively or jointly. No literature has as yet studied

the effects of risk on the returns on health investment. It is however well known that

the effectiveness of a therapy or treatment in improving future health status is subject

to a wide variability that depends both on the availability of different alternative

treatments for the same purpose and on the specific individual reactionto each therapy

1Li (2012) assumes positive quadrant dependence between the risks. Baiardi et al. (2014) con-

sider small risks. Lastly, Baiardi et al. (2015) derive results depending on the sign of the

covariance between the total income and the interest rate.