Saving and the Demand for Protection Against Risk

Saving and the Demand for Protection Against Risk David Crainich 1, Richard Peter 2 Abstract: We study individual saving decisions in the presence of an endogenous future consumption risk. The endogeneity arises because agents can buy insurance in our model. We show that the slope of the reaction functions depends on the slope of absolute risk aversion. If insurance is not Giffen, saving acts as a substitute for insurance in the Walrasian sense when absolute risk aversion is non-increasing. Conversely, if the substitution effect of changes in the interest rate dominates the wealth effect, insurance acts as a substitute (complement) for saving in the Walrasian sense when absolute risk aversion is decreasing (increasing). Our findings complement earlier work where wealth effects were absent and/or the temporal structure was different. Keywords: risk; saving; insurance; substitute; complement JEL-Classification: D11; D14; D81; D91; G22 Working Paper Version: March 2016 1 CNRS (LEM, UMR 9221) and IÉSEG School of Management, Lille; e-mail: d.crainich@ieseg.fr 2 The University of Iowa, Iowa City; e-mail: richard-peter@uiowa.edu

1 1. Introduction Risk management activities are usually analyzed in isolation in the literature. For instance, the propensities to purchase insurance contracts, to make precautionary savings or to perform prevention actions are in most cases dealt with separately. Ehrlich and Becker s (1972) classical contribution was the first analysis dedicated to the interaction between various instruments - insurance and self-protection on the one hand, insurance and self-insurance on the other hand - used to manage financial risks. This initiated a series of theoretical papers examining joint risk management decisions. Among these, Dionne and Eeckhoudt (1984) considered the relationship between saving and insurance decisions and Menegatti and Rebessi (2011) examined simultaneous saving and selfprotection efforts. Note that the analysis of joint actions undertaken in order to protect oneself against disease has been introduced the health economics literature (see for instance the analysis of self-protection activities and disease treatment in Henessy (2008) or in Menegatti (2014)). In this paper, we focus on the relationship between the demand for precautionary savings and for insurance. Taken in isolation, the demand for precautionary savings has been shown to depend in the expected utility model - on the sign of the third derivative of the utility function (see Leland (1968), Sandmo (1970) and Drèze and Modigliani (1972)) while Kimball (1990) indicated that the intensity of this demand was measured by the ratio of (minus) the third derivative to the second derivative of the utility function. In these contributions, the future risk is exogenous. Surprisingly, the presence of endogenous risks and their effect on saving has not been discussed much. This is despite the fact that individuals have a variety of instruments to address future consumption risks (insurance, self-protection, self-insurance etc.). We thus consider in this paper that individuals can buy insurance contracts in the future (i.e. when they will be exposed to the risk) so that the precautionary saving and the insurance decisions interact. Our contribution complements the one provided by Dionne and Eeckhoudt (1984). But instead of considering substitution or complementarity in the Hicksian sense (i.e. keeping the expected utility constant in order to cancel the wealth effect and isolating the price effect), we determine the

2 interaction of these two activities in the Walrasian sense. Besides, we also adopt a different temporal structure: instead of considering that saving and insurance decisions are contemporaneous, we assume that individuals make precautionary savings in the current period and purchase insurance contracts in the future. The main result of our paper is that the nature of the interaction between insurance and precautionary savings depends on the way the Arrow-Pratt index of absolute risk aversion changes with wealth. More precisely, if we assume that the demand for insurance falls when the price of insurance increases (i.e. insurance is not a Giffen good), we show that an increase in the price of insurance leads to more precautionary savings (saving acts as a substitute for insurance in the Walrasian sense) if the utility function is DA- RA. Similarly, if higher interest rates increase savings, insurance is a substitute (complement) for saving in the Walrasian sense when absolute risk aversion is decreasing (increasing). The paper is organized as follows. Section 2 describes the model and defines the precautionary savings and the insurance purchase at the equilibrium. Sections 3 and 4 analyze the way these equilibrium values are modified by changes in the price of insurance and in the interest rate respectively. Section 5 concludes. 2. The Model We consider two points in time, and. A decision-maker (DM) receives income in with certainty, 1,2. A consumption stream, is evaluated according to +, where is first-period utility of consumption, is second-period utility of consumption, and is the rate of pure preference for the present. We assume that and are increasing and concave which reflects risk aversion. Future income is subject to a random loss of size that occurs with probability. This risk at is endogenous due to the fact that the DM can purchase insurance to protect himself. Let 0,1 denote the level of coverage and λ 0 be the loading factor such that 1+λ is the per-unit price of insurance. Consequently, the premium is given by 1+λ. Besides insurance, the DM

3 decides about his consumption allocation over time. Let denote savings with the nonrandom interest rate 0. With these specifications, the DM s objective function is given by max % +' +1+ 1+λ 1 #,$ +1 +1+ 1+λ () To compress notation, we use subscripts L and N to denote consumption in the loss and the no-loss state, respectively. First-order conditions are given by 1 1+λ, + 1 1+λ, - =0, 1 *, +1+, + +1, - =0. 2 The first equation describes that the optimal level of coverage is such that the marginal rate of substitution between consumption in the loss state and in the no-loss state must be equal to the slope of the line of insurance. The second equation implies that for optimal saving marginal expected consumption utility must be equal across points in time. The second-order conditions are satisfied, see the appendix. 3. Changes in the Price of Insurance First, we investigate how a change in the price of insurance affects the demand for insurance and the demand for saving. Note that we do not fix intertemporal expected utility as in Dionne and Eeckhoudt (1984). In this sense, we consider Walrasian demand rather than Hicksian demand to clarify whether and when there is gross substitution or complementarity between insurance and saving. The cross-derivative of expected utility is given by 0 #$ =1+1 1+λ +,, 1 1+λ -,,. With the help of the first-order condition (1), this can be rewritten as 0 #$ =1+1 1+λ -, 1-1 +, where 1=,,, denotes Arrow-Pratt risk aversion. 3 From this we make the following observation. 3 Recently, Bommier et al. (2012) address the conceptual difficulty of defining comparative risk aversion in certain x uncertain -type choice situations. We admit that the measurement of risk aversion is much more subtle in an intertemporal context. Still, we draw on standard terminology but try to point out caveats when needed.

4 Remark 1: Reaction functions are decreasing (constant, increasing) if absolute risk aversion is decreasing (constant, increasing). The intuition is as follows: When savings are increased, this implies an increase in second-period risk-free wealth. With decreasing absolute risk aversion, a given loss is less painful at high wealth levels than at low ones. Consequently, insurance coverage should be reduced to save on premium money. Conversely, when insurance coverage increases, this implies a certain reduction in wealth due to the higher insurance premium but also a contraction in future consumption risk. A reduction in second-period wealth stipulates more saving according to consumption smoothing whereas a contraction in future consumption risk implies less saving because there is not so much need for precaution. DARA is equivalent to the fact that the coefficient of absolute prudence exceeds the coefficient of absolute risk aversion. As a consequence, the second effect dominates the first one and optimal savings decrease upon an exogenous increase in insurance coverage. Overall, there is a non-trivial interaction between saving and insurance as soon as risk aversion is not constant across wealth. This indirect effect will have to be taken into account in the comparative statics analysis. Let us begin with the direct effects, however. For insurance, we obtain that 0 #λ = 1 1+λ,, + 1 1+λ,, -, + +1, -, which we rewrite with the help of first-order condition (1): 0 #λ = 1 1+λ1-1 +, + +1, -. As is well known, insurance can be a Giffen good such that an increase in price does not necessarily reduce the optimal level of coverage. The explanation is that a price increase impoverishes the consumer; more specifically, she has less wealth in the bad state of the world where higher insurance coverage increases her consumption. This is the origin of the positive effect undermining the common intuition that an increase in price should reduce the level of coverage. This was first discussed by Hoy and Robson (1981); later, Briys, Dionne and Eeckhoudt (1989) and Hau (2008) state the necessary and sufficient conditions for insurance to be non-giffen. For our purposes it is sufficient to note from above that 0 #λ is negative under CARA and IARA, whereas under DARA the sign is am-

5 biguous. We will distinguish between the cases where insurance is Giffen and where it is not throughout the analysis. For saving we derive 0 $λ = 1+ +,, +1 -,, >0 due to the fact that marginal utility is diminishing. Here the reason is that a higher perunit price of insurance reduces the consumer s wealth in the second period so that the marginal benefit of saving is increased. As the insurance premium is paid in the second period, the marginal cost of saving is unaffected and consequently the net effect on saving is positive. 4 Let us now apply the implicit function rule to determine how direct and indirect effects play out together. We obtain that 7 8 58λ = 1 9 0 $$0 #λ +0 #$ 0 $λ, 6 5 8 48λ = 1 9 0 ##0 :λ +0 #$ 0 ;λ, where 9 is the determinant of the Hessian matrix. It is positive in any case as we know from the second-order conditions (see the appendix). The first line states the net effect of changes in the price of insurance on the demand for insurance whereas the second line gives the net effect of changes in the price of insurance on the demand for saving, i.e., the cross-price effect. From the discussion above we get the following Proposition. Proposition 1: If insurance is not Giffen, the optimal level of insurance decreases and the demand for saving increases when the price of insurance increases under non-increasing absolute risk aversion. The cases where absolute risk aversion is increasing or where insurance is Giffen are indeterminate. The intuition is quite simple. If insurance is not a Giffen good, the direct effect of a price increase on the demand for insurance is negative. The direct effect on the demand for saving is positive in any case. Now, a decrease in insurance exerts a non-negative substitution effect on saving when risk aversion is non-increasing (Remark 1). Likewise, an 4 When the premium is paid in the first period, an increase in the per-unit price of insurance reduces wealth in the first period so that the marginal cost of saving is larger, whereas the marginal benefit remains unchanged. Consequently, the direct effect on saving would be negative in this case. This underlines that our approach is quite different from the treatment in Dionne and Eeckhoudt (1984).

6 increase in saving exerts a non-positive substitution effect on the demand for insurance when risk aversion is non-increasing. As a result the net effect is negative for the demand for insurance and positive for the demand for saving. This extends the finding by Dionne and Eeckhoudt (1984) to the case of Walrasian demand. When preferences exhibit increasing absolute risk aversion, reaction functions are upward sloping (Remark 1). Consequently, the indirect effect from saving on insurance is positive, countervailing the negative direct effect, and the indirect effect from insurance on saving is negative, countervailing the positive direct effect. In this case, the net effects depend on the relative strength of both effects and are indeterminate. Similarly, when insurance is a Giffen good, preferences must necessarily exhibit DARA implying downward sloping reaction functions. Then, the negative substitution effect from saving on insurance countervails the positive direct effect of a price increase in the Giffen case, and the negative substitution effect from insurance on saving countervails the positive direct effect. Again, net effects are ambiguous. We can also analyze this graphically. In Figure 1, we can see the initial reaction functions and, which are downward sloping. This implies that we are in the case of DARA. An increase in the price of insurance λ is associated with a decrease in the level of coverage for a given level of saving. Therefore, insurance is not a Giffen good. Finally, the direct effect on saving implies an upward shift of the reaction function. As we can see, the new optimum is such that saving increases whereas the level of insurance coverage decreases. The reason is that the direct effect and the indirect effect for each decision variable move in the same direction here. The other cases can be analyzed with a similar graphical treatment.

7 4. Changes in the Interest Rate The indirect effects exposed in Remark 1 still hold. Let us now examine the direct effects of changes in the interest rate on the demand for: 1) savings and; 2) insurance coverage. As indicated in the expression below, the effect of a change in the interest rate on the demand for savings is ambiguous: 0 $= =, + +1, - +1+,, + +1,, - This expression corresponds to: 0 $= = β [ Eu '( w% + (1 + r) s) + (1 + r) seu ''( w% + (1 + r) s)] %. where w% = w2 + (1 + λ) α pl + L % with L = ( p(1 α) L,1 p ; (1 p)(1 α) L, p) Using the technique exposed by Gollier (2001) 5, we obtain: sr d [ '( % (1 ) )[1 ( % (1 ) ) % ( % (1 ) )]] EU = β Eu w + + r s R w + + r s + wa w + + r s 5 See section 16.2 page 239.

8 where R( w ) denotes the coefficients of relative risk aversion evaluated in w. It is then sufficient that the coefficient of relative risk aversion is lower than unity for savings to be increasing with the interest rate. 6 The ambiguous relationship between savings and the interest rate can be explained as follows. The interest rate is the opportunity cost of consumption (as opposed to saving). As a consequence, raising the interest rate creates incentives to reduce consumption and, therefore, to increase savings (substitution effect). But an increase in the interest rate also makes individuals wealthier (as long as their initial saving is higher than zero) in period 2. Since the marginal utility of consumption is decreasing, this reduces the incentives to save at period 1 in order to consume at period 2 (wealth effect). In what follows, we will consider both cases (i.e. that savings are increasing or decreasing with the interest rate). We now evaluate the effect of a change in the interest rate on the demand for insurance. This effect is given by the following expression: EUα β Ls p λ p v p λ pv '' '' r = ( (1 (1 + ) ) L (1 )(1 + ) L) Using the first-order condition (1), we obtain: EU Ls p p A A ' αr = β (1 ) (1 + λ) vn[ N L] From this, we conclude that the demand for insurance falls when the interest rate rises when the Arrow-Pratt coefficient of absolute risk aversion is decreasing. This can be explained as follows: a higher interest rate makes individuals wealthier in period 2. As a result, their propensity to purchase insurance coverage falls (resp. rises) if their absolute risk aversion falls (resp. rises) with wealth. As in section 3, we now use the implicit function theorem to determine the overall effect of an increase in the interest rate on savings and on the demand for insurance. We have: 7 8 58 = 1 9 0 ##0 $> +0 #$ 0 #=, 6 5 8 48 = 1 9 0 $$0 #> +0 $$ 0 ;=. 6 Descriptively, relative risk aversion is often assumed to exceed unity. An alternative sufficient condition is that partial risk aversion is below one (Chiu et al., 2012), which finds much more support empirically (Binswanger, 1981; Bar-Shira et al., 1997).

9 From these expressions and the discussion above, we get the following Proposition. Proposition 2: If the demand for savings is such that the substitution effect dominates the wealth effect (so that the direct effect of the interest rate on savings is positive), the optimal level of saving increases when the interest rate increases. The optimal insurance coverage decreases (resp. increases; resp. is constant) when the interest rate increases under decreasing (resp. increasing; resp. constant) absolute risk aversion. The cases where the direct effect of the interest rate on savings is negative are indeterminate. The case of DARA utility functions is depicted in Figure 2. Figure 2 Since the direct effect on saving of an increase in the interest rate is supposed to be positive, an upward shift of s( α ) results from an increase in r. This shift leads to a reduction in the insurance coverage (indirect effect) since savings and insurance move in opposite directions when the utility function is DARA (see remark 1). In addition, an increase in the interest rate reduces the demand for insurance (direct effect) when the utility function is DARA. Again, this raises savings as an indirect effect (see remark 1). In this situa-

10 tion, the direct and indirect effects have the same impacts on the variables (increasing savings and reducing insurance coverage. Consider now the case of IARA utility functions depicted in Figure 3. s indirect effect direct effect direct effect indirect effect Figure 3 We suppose again that the direct effect on saving of an increase in the interest rate is positive, so that an upward shift of s( α ) results from an increase in r. Since insurance and savings now move in the same direction because the utility function is IARA (see remark 1), this shift raises the insurance coverage (indirect effect). The IARA assumption is also such that an increase in the interest rate raises the insurance coverage (direct effect). This shifts α ( s) curve to right and again, since saving and insurance go in the same direction, this increases the demand for saving (indirect effect). The case depicted represents once again a situation where the direct and indirect effects have the same positive impacts on saving and insurance coverage. When the direct effect of the interest rate on savings is negative, the direct and indirect effects move in opposite direction, so that the impact of an increase in the interest rate on both savings and the demand for insurance are indeterminate.

11 6. Discussion and Conclusion Risk management tools are usually analyzed in isolation. This is the case of precautionary savings which is defined as the extra saving due to risky future income. In the literature, the risk in question is usually considered as background since it is assumed that it cannot be modified through prevention actions, diversified or insured against. These paper thus deal with the way the introduction of an exogenous risk affects savings. Less attention has been dedicated to the interaction between savings and other economics decisions that could be used to deal with a future risk (that would be then endogenous). This is a question we address in this paper since we analyze the relationship between the demand for precautionary savings and insurance. Our contribution complements Dionne and Eeckhoudt (1984) who examined a similar question. However, we consider substitution or complementarity in the Walrasian sense and not in the Hicksian sense (as Dionne and Eeckhoudt did). Besides, we also assume as in it ususlly the case in the literature in risk theory - that individuals make precautionary savings in the current period and purchase insurance contracts in the future (while saving and insurance decisions are simultaneous in Dionne and Eeckhoudt (1984)). To determine the interaction between savings and insurance, we examine the effect of an increase in the price of insurance on the propensity to save and the effect of an increase in the interest rate on the demand for insurance. Assuming that saving and insurance are both normal goods, we show that these two risk management tools are substitutes if individuals utility functions are DARA (Decreasing Absolute Risk Aversion).

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13 References Bar-Shira, Z., R. Just, and D. Zilberman, 1997, Estimation of Farmers Risk Attitude: An Econometric Approach, Agricultural Economics, 17(2-3): 211-222. Binswanger, H., 1981, Attitudes Towards Risk: Theoretical Implications of an Experiment in Rural India, The Economic Journal, 91(364): 867-890. Bommier, A., A. Chassagnon, and F. Le Grand, 2012, Comparative Risk Aversion: A Formal Approach with Applications to Saving Behavior, Journal of Economic Theory, 147(4): 1614-1641. Briys, R., G. Dionne, and L. Eeckhoudt, 1989, More on Insurance as a Giffen Good, Journal of Risk and Uncertainty, 2(4): 415-420. Chiu, H., L. Eeckhoudt, and B. Rey, 2012, On Relative and Partial Risk Attitudes: Theory and Implications, Economic Theory, 50(1): 151-167. Dionne, G. and L. Eeckhoudt, 1984, Insurance and Saving: Some Further Results, Insurance: Mathematics and Economics, 3(2): 101-110. Gollier, C., 2001, The Economics of Risk and Time, Cambridge MA: MIT Press. Hau, A., 2008, When is a Coinsurance-Type Insurance Policy Inferior or even Giffen? Journal of Risk and Insurance, 75(2): 343-364. Hoy, M. and A. Robson, 1981, Insurance as a Giffen Good, Economics Letters, 8(1): 47-51.

14 Appendix We denote the objective function by 0 for expected intertemporal consumption utility. For the second-order conditions, note that 0 ## =?1 1+λ +,, +1 @1+λA -,, BC0 due to risk aversion of second-period utility. Furthermore, also 0 $$ =,, +1+ +,, +1+ -,, C0 because of risk aversion of first- and second-period utility. The cross-derivative of expected utility is given by 0 #$ =1+1 1+λ +,, 1 1+λ -,, and is ambiguous. To calculate the determinant of the Hessian matrix, we also need the square of the cross-derivative, which is 0 #$ = 1+ 1 1+λ,, + 21 1 1+λ1+λ,,,, + - +1 1+λ -,,. From there it is easy to see that the determinant of the Hessian is obtained as 0 ## 0 $$ 0 #$ =,, 0 ## + 1+ 1,, +,, - 1 21+λ >0, so that the objective function is globally concave in and.