Precautionary effort: a new look.
While the concept of precautionary saving is well documented, that of precautionary effort has received relatively limited attention. In this note, we set up a two period model in order to analyze the conditions under which the introduction (or deterioration) of an independent background risk increases effort.
Ehrlich and Becker's (1972) article initiated a very long research activity in economics around the notion of self-protection and its relationship with other risk management instruments. While many papers in economics still deal directly or indirectly on a regular basis with the topics, the discussion has spread over to other disciplines such as health management or environmental policy. Partly because of this transfer, other names for the concept of self-protection are currently used such as prevention or effort. Whatever the term adopted, self-protection activities reflect our attempt to reduce the probability of occurrence of "bad events" and these activities compete with other risk management tools such as market insurance, self-insurance, or diversification.
With only one very recent exception (see below), all the economic models of self-protection are developed in a monoperiodic context (1) where the effort and its impact on the probability of a bad event are simultaneous. In a very recent paper, Menegatti (2009) has convincingly argued that while the monoperiodic model makes sense in many situations, it turns out that in other ones, a long time period may elapse between the effort and its impact on the probability of a bad event. (2) In such a context, effort becomes for the decision maker (DM) a risk management tool that competes with saving and long-term insurance.
While much attention is paid to the impact of background risks in the literature devoted to insurance and saving, it is not the case for the self-protection problem. To the best of our knowledge, only Dachraoui et al. (2004) examine the impact of background risks on effort in a monoperiodic context. They conclude that the impact is ambiguous because both the marginal benefit and the marginal cost of efforts are increased by the background risk.
While investments in effort are currently made to protect oneself against a specific risk that may materialize in the future, the DM also faces future risks that are beyond his control (the background risk). In this article, we analyze the impact of the presence and/or deterioration of such risks on the choice of the ex-ante level of effort. Because--as is well known--such future background risks also induce extra current savings ("precautionary savings") we use by analogy the terminology of "precautionary effort", that is, the additional effort level chosen to face the background risk.
To achieve our goal, the article is organized as follows. In the next two sections, we first look at the impact of the introduction of a background risk on optimal effort and then consider the effect of a deterioration in a preexisting background risk. The section "An Extension" generalizes the previous analysis by extending the definition of the benefit of efforts along the lines developed in Jindapon and Neilson (2007). This section confirms the similarities between precautionary efforts and precautionary savings. A short conclusion summarizes the main results and indicates potential extensions.
THE BASIC MODEL
A DM avails upon an income [x.sub.o] that has to be allocated between current consumption and effort (e). Thanks to the effort, the DM faces in the next period a higher probability p(e) of a good outcome denoted x. With probability I - p(e), the DM receives the bad outcome x - l (l > 0) and his basic risk is thus represented by a binary random variable denoted [??].
Assuming that future utility is discounted at a rate [delta], the DM's optimization problem (3) is written:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
in which u and p(e) are increasing and concave functions. The first-order condition (FOC) is as follows:
[partial derivative]L/[partial derivative]e = -u' ([x.sub.o] - e) + 1/1+[delta] p' (e) [u (x) - u (x - l)] = 0. (2)
Let [e.sup.*] denote the optimal level of effort.
Now assume that besides the basic risk [IOTA], the DM has also to face in the future an independent zero-mean background risk [??]. Whenever this additional risk induces an increase in effort, one observes precautionary effort and Proposition 1 gives the condition that generates this behavior.
Proposition 1: The introduction of an independent zero-mean background risk generates more effort for any positive value of the loss l if and only if the third derivative of u is positive (u'" > 0).
Proof: With the new risk [??], the objective becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where E denotes the expectation operator. The FOC evaluated at [e.sup.*] is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Using standard arguments, it is easy to show that the new optimal level of effort denoted [??] will exceed [e.sup.*] if and only if Equation (4) is positive. From the FOC (2), this condition could be written as
u(x - 1) - Eu(x - l + [??]) > u(x) - Eu(x + [??]). (5)
Each of the two terms in the inequality (5) represents the utility cost of facing a zero-mean risk [??] when one avails upon wealth x - 1 (the left-hand-side difference) or wealth x (the right-hand-side difference). Since x > x - l, condition (5) holds true if and only if u'" > 0 as shown in Eeckhoudt and Schlesinger (2006).
Hence, prudence (or equivalently downside risk aversion in the terminology of Menezes et al., 1980) induces precautionary effort much in the same way as it induces precautionary savings.
Instead of looking at the introduction of a background risk, we now examine what happens when an existing background risk [??] deteriorates and becomes [??] where [??] is stochastically dominated by [??] at a given order m (m [greater than or equal to] 1). To illustrate, let e represent efforts made to improve the maintenance of an equipment during its early years of activity so that the probability of failure in the future (1 - p(e)) falls. The background risks [??] and [??] might then represent the DM's expectations about future economic conditions with [??] corresponding to a less favorable view of the future. For such situations, we can state Proposition 2.
Proposition 2: If the utility function has derivatives alternating in signs up to the order m + 1 and starting with u' > 0, the deterioration of the background risk induces more effort.
Proof: Before the deterioration of the background risk, the FOC in the presence of is written:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
Following standard arguments, the deterioration of the background risk that becomes [??] induces an additional effort if and only if:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
The condition is satisfied whenever
Eu' (x + [??]) > Eu' (x + [??]). (8)
Then Proposition 2 follows from the well-known results about stochastic dominance relationships and the fact that [??] dominates [??] at the order m.
Proposition 2 is in fact a natural extension of Proposition 1. Because the deterioration of a background risk is a more general notion than that of its introduction, more restrictions have to be imposed on u to generate the desired result.
Recently, Jindapon and Neilson (2007) proposed an extension of the approach originally developed by Ehrlich and Becker (1972). In Jindapon and Neilson, as in Ehrlich and Becker, the effort made in the current period increases the probability of good outcomes in the future. However, while in Ehrlich and Becker good outcomes dominate the bad ones by an elementary first-order dominance relationship (x > x - l), Jindapon and Neilson use a more subtle nth order stochastic dominance relationship. Similar to their setting, we assume that effort increases the probability of facing a risky prospect [??] that stochastically dominates another risky prospect [??] at any order n(n [greater than or equal to] 1).
If besides the DM faces an independent background risk [??], her objective is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
and the associated FOC is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where [e.sup.**] denotes the new optimal level of effort. Again, concavity of p(e) and u guarantee that the second-order condition for a maximum is satisfied.
Assume now that the distribution of the background risk [??] deteriorates and becomes [??] where [??] is dominated at the order m by [??]. Such a deterioration will induce more
current effort if and only if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
A careful analysis of Equation (11) reveals that on the left-hand side of the inequality, one "combines good with bad" (4) in the sense that in the first expected utility [??] (which dominates [??]) is combined with [??] (which is dominated by [??]) while in the second expected utility the "bad" [??]) is combined with the "good" ([??]). On the right-hand side of Equation (11), the "good" ([??] and [??]) jointly appear in the first expected utility while the second expected utility combines the two "bad"([??] and [??]). By applying Theorem 3 in Eeckhoudt et al. (2009), we conclude that if u has derivatives alternating in signs up to the order n + m and starting with u' > 0, the deterioration of the background risk induces more current effort. Without surprise, this finding is more demanding than Proposition 2 because now the basic risk (i.e., facing [??] or [??]) is more elaborate than in the previous section where it was a binary random variable. Notice finally, the alternation of the signs of the successive derivatives of u is justified in different but equivalent ways in the literature (see, e.g., Caballe and Pomansky, 1996; Menegatti, 2001; Eeckhoudt and Schlesinger, 2006).
DISCUSSIONS AND CONCLUSION
Saving, effort, and insurance are very different risk management tools. In order to face future risks one can either save to increase wealth in all states of the world, make efforts that increase the probability of the good states, or purchase an insurance to increase wealth in the bad states. Despite these differences, when background risk deteriorates ceteris paribus, more efforts are undertaken to reduce the probability of bad events, and under standard assumptions about the utility function both instruments (effort and saving) act simultaneously to increase protection. This result confirms the strength of the precautionary motive in the economy well beyond the usual precautionary saving motive.
At this stage, at least three potential extensions look promising. First, in many self-protection activities the cost of effort and its benefit are not in the same dimension. For instance, one can currently spend money to practice exercise in order to reduce the probability of future health deterioration. Hence, a bidimensional extension might be interesting. Furthermore, in our model, we have assumed an intertemporally separable utility function. Of course such an assumption might also be dropped. These two extensions would of course illustrate the role of the cross partial derivatives of the utility function in the choice of the effort level.
Finally while we have examined here effort in isolation, it would also be relevant to study the interdependence between effort, saving, and insurance in a model where these three risk management tools simultaneously compete for the use of current resources. For example, to hedge the risk of a potential nursing home stay in the future, the DM could either save more, purchase long-term care insurance, (5) or invest in prevention activities in the current period. A joint analysis of these three decisions seems a piori interesting.
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Caballe, J., and A. Pomansky, 1996, Mixed Risk Aversion, Journal of Economic Theory, 71: 485-513.
Chiu, W. H., 2000, On the Propensity to Self-Protect, Journal of Risk and Insurance, 67: 555-577.
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(1) For example, see Dionne and Eeckhoudt (1985), Briys and Schlesinger (1990), McGuire, Pratt and Zeckhauser (1991), Jullien et al. (1999), Chiu (2000), and Eeckhoudt and Gollier (2005).
(2) The reader is referred to the original article for many interesting and relevant examples.
(3) Since we want to concentrate on precautionary efforts, we do not consider here the saving decision.
(4) The definition of this principle and its implications in the expected utility model are documented in details in Eeckhoudt et al. (2009).
(5) Please see Zhou-Richter et al. (2010) for the discussion on the demand for long-term care insurance and risk perception.
Louis Eeckhoudt is Professor at IESEG School of Management, LEM, Lille, France. Rachel J. Huang is Associate Professor at Graduate Institute of Finance, National Taiwan University of Science and Technology, Taipei, Taiwan. Larry Y. Tzeng is Professor in Department of Finance, National Taiwan University, Taipei, Taiwan. Both Rachel J. Huang and Larry Y. Tzeng are research fellows at Risk and Insurance Research Center, College of Commerce, NCCU. Rachel J. Huang can be contacted via e-mail: firstname.lastname@example.org. The authors express their deep gratitude to Georges Dionne, Paan Jindapon, Mario Menegatti, Harris Schlesinger, Ilia Tsetlin, and Jenhung Wang who made stimulating comments on previous versions of the article.
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|Author:||Eeckhoudt, Louis; Huang, Rachel J.; Tzeng, Larry Y.|
|Publication:||Journal of Risk and Insurance|
|Date:||Jun 1, 2012|
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