WORKING PAPER SERIES 2013-ECO-13

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1 June 03 WORKING PAPER SERIES 03-ECO-3 Te Value of Risk Reduction: New Tools for an Old Problem David CRAINICH CNRS-LEM and IESEG Scool of Management Louis EECKHOUDT IESEG Scool of Management (LEM-CNRS) and CORE (Université Catolique de Louvain) James K. HAMMITT Harvard University (Center for Risk Analysis), Cambridge Toulouse Scool of Economics (LERNA-INRA) IESEG Scool of Management Lille Catolic University 3, rue de la Digue F Lille Tel: 33(0) Fax: 33(0)

2 IESEG Working Paper Series 03-ECO-3 Te Value of Risk Reduction: New Tools for an Old Problem David Crainic a, Louis R. Eeckoudt b, and James K. Hammitt c* a CNRS and ISEG Scool of Management (LEM) Catolic University of Lille, Lille, France b IESEG Scool of Management (LEM) Catolic University of Lille, Lille, France CORE, Catolic University of Louvain, Louvain, Belgium c Harvard University (Center for Risk Analysis), Cambridge, MA USA Toulouse Scool of Economics (LERNA-INRA), Toulouse, France *: Corresponding autor. jk@arvard.edu, tel: Marc 03

3 IESEG Working Paper Series 03-ECO-3 Te Value of Risk Reduction: New Tools for an Old Problem Abstract Te relationsip between willingness to pay (WTP) to reduce te probability of an adverse event and te degree of risk aversion is ambiguous. Te ambiguity arises because paying for protection worsens te outcome in te event te adverse event occurs, wic influences te expected marginal utility of wealt. Using concepts of prudence (equivalently, downside risk aversion), we caracterize te marginal WTP to reduce te probability of te adverse event as te product of WTP in te case of risk neutrality and an adjustment factor. For te univariate case (e.g., risk of financial loss), te adjustment factor depends on risk aversion and prudence wit respect to wealt. For te bivariate case (e.g., risk of deat or illness), te adjustment factor depends on risk aversion and cross-prudence in wealt. Keywords: value per statistical life, mortality risk, risk aversion, prudence JEL classification: D8, I

4 IESEG Working Paper Series 03-ECO-3 Te relationsip between willingness to pay for prevention and risk aversion is complex and even counterintuitive. Paying for prevention may worsen te outcome if te adverse event occurs, even as it reduces te probability of tat event. Initiated by te work of Drèze (96) and Jones-Lee (974), wo studied willingness to pay (WTP) to reduce mortality risk, and of Erlic and Becker (97), wo studied WTP to reduce te probability or te magnitude of a loss, a vast teoretical and empirical literature as emerged about te value of reducing te probability of an adverse event. Unsurprisingly, tis literature uses concepts of risk teory tat ad been simultaneously developed by Arrow (965) and Pratt (964). Subsequently, new concepts suc as downside risk aversion (Menezes et al. 980) and prudence (Kimball 990) ave emerged. Tis paper provides new insigt about WTP for prevention, incorporating tese newer concepts. We examine te WTP (compensating variation) to reduce te probability of an adverse ealt or financial event under expected utility. We begin wit te univariate case, in wic utility depends on a single attribute (e.g., wealt or ealt). We extend our result to te bivariate case, in wic utility depends on bot ealt and wealt, and investigate WTP to reduce te risk of a bad ealt outcome. Tis analysis applies to te case of mortality risk, in wic te measure of WTP to reduce risk is described as te value per statistical life (VSL). In bot cases, we find tat WTP to reduce risk can be expressed as te product of te marginal WTP under risk neutrality and an adjustment factor tat depends on bot risk aversion and prudence wit respect to te attribute at risk.. Univariate case Let x be a continuous variable, representing wealt or ealt, and consider te binary lottery wit outcome x 0 < x, were te probability of te worse outcome x 0 is p and te probability of te better outcome x is ( p). Expected utility is given by EU = p u(x 0 ) + ( p) u(x ) (.) were u( ) is a continuously differentiable utility function wit u > 0 (primes denote derivatives). Te marginal rate of substitution between x and p is u x u x0 dx u 0. (.) dp pu x p u x Eu 0 If x represents wealt, dx/dp represents te marginal WTP to reduce te cance of te low-wealt outcome and increase te cance of te ig-wealt outcome. For example, one For a differentiable utility function, prudence and downside risk aversion are equivalent and require u > 0.

5 IESEG Working Paper Series 03-ECO-3 migt rent a safe-deposit box to protect valuables from teft. Alternatively, if x represents ealt, dx/dp represents te marginal willingness to compromise ealt to reduce te cance of a bad ealt outcome. For example, one migt coose radiation or oter treatments wit adverse side effects to reduce te cance tat a cancer proves fatal. Define x = E( x ) = p x 0 + ( p) x and x = variance ( x ) = p ( p) L, were L x = x x 0. Approximate te terms in te numerator of equation (.) using a second-order Taylor series expansion around x to obtain: plx u x u x plxu x u x and (.3) plx u x0 u x p Lxu x u x. (.4) Te denominator of equation (.) may be approximated by Eu u x u x u x, (.5) were is te prudence premium, wic may itself be approximated by (Kimball 990). x u x x u Substituting tese approximations into equation (.) yields Lxu x p Lxu x dx dp x u x u x Dividing te numerator and denominator by u x yields dx L dp x u x u x x x (.6). (.7) plx. (.8) u x u Equation (.8) sows tat te marginal rate of substitution between x and p is te product of two terms: te marginal rate of substitution if u is linear (i.e., te potential loss L x ) and an adjustment factor. Te numerator of te adjustment factor depends on te probability of te adverse outcome p, te possible loss L x, and te coefficient of absolute risk aversion evaluated at te

6 IESEG Working Paper Series 03-ECO-3 3 u expected value of x, u x x. Interestingly, te effect of risk aversion depends on te difference between te initial probability of loss p and te critical value /. Note tat if p = /, te numerator equals one, regardless of te potential loss and te degree of risk aversion. In contrast, if p < / te numerator is larger tan one and is increasing in bot L x and te measure of risk aversion. Tis result is intuitive: for p < /, a decrease in p reduces te variance of final wealt, wic is appreciated by a risk-averse decision maker, so e values te decrease by more. Te opposite effect occurs for p > /, for wic a decrease in p increases te variance of final wealt. 3 Notice tat te true value of dx/dp must be positive since a decrease in p induces a first-order stocastically dominant improvement in te decision maker s situation. Tis sift must be compensated by a decrease in x (a first-order stocastically dominant deterioration) to maintain welfare constant. To be useful, an approximation suc as te one in equation (.8) must at least ave te same sign as te true value. For a risk-averse decision maker, tis is definitely te case wen p /. 4 From tis point forward, we assume te initial probability of loss does not exceed /, wic is realistic for most applications. Te denominator of te adjustment ratio depends on te riskiness of te lottery (measured by its variance) and a measure of downside risk aversion, u u x x (see Modica and Scarsini 005 and Crainic and Eeckoudt 008). Tis coefficient can be interpreted as follows: te effect of a zero-mean risk on te marginal utility of wealt is measured by v x E u x u x, wic is te difference between te marginal utility of x wit and witout. Of course x u > 0 implies v(x) > 0; ence for a prudent/downside-risk-averse decision make, a zeromean risk increases te expected marginal utility. A second-order approximation of v(x) yields v x u x, (.9) For a similar explanation in te framework of self-protection, see Eeckoudt and Gollier (005). 3 Te effect of a cange in risk aversion depends on bot te numerator and denominator of equation (.8), because increasing risk aversion (e.g., by taking a concave function of te u) affects u as well. 4 For p > /, te approximation may but need not yield positive WTP.

7 IESEG Working Paper Series 03-ECO-3 4 wic, as we ave seen, measures a gain in marginal utility ( is te variance of ). To transform tis gain in marginal utility into its monetary equivalent, one divides, as usual, by te marginal utility of wealt downside risk aversion. 5 u u x so tat u x x is a measure of te intensity of Downside risk aversion (equivalently, prudence) is caracterized by a positive tird derivative (i.e., convex marginal utility). Te denominator of equation (.8) is increasing in bot te variance of te lottery and downside risk aversion; ence greater downside risk aversion yields a smaller willingness to pay to reduce te probability of te adverse outcome. Intuitively, greater downside risk aversion means tat marginal utility rises more as x declines, ence te utility cost of sacrificing x if te adverse outcome occurs is larger. Tis suppresses te willingness to sacrifice x to reduce p. For te case were x represents wealt, it is conventional to assume tat bot risk aversion and downside risk aversion are positive. Risk aversion increases te marginal willingness to pay to reduce p (for small probabilities of loss, i.e., p < /), but downside risk aversion decreases it. For te case were x represents ealt, te curvature of te utility function depends on ow ealt is measured. One possibility is tat x measures longevity. Empirical evidence suggests tat some individuals are risk neutral, some are risk averse, and some are risk seeking wit respect to longevity (Pliskin et al. 980). Moreover, te sign of risk aversion wit respect to longevity may vary wit age and longevity; e.g., it seems plausible tat young adults migt be risk seeking for longevities ranging over middle ages but risk averse over greater longevities. In tis case, te adjustment factor may depend on age and te values of x 0 and x. Alternatively, x may represent quality of ealt, wic is often conceptualized as ealt-related quality of life (HRQL) and combined wit duration to calculate qualityadjusted life years (QALYs), wic assume risk neutrality wit respect to HRQL (Pliskin et al. 980, Hammitt 00). HRQL is measured by various forms of ypotetical questions, including standard gambles between full ealt and deat. Te standard gamble form also 5 Interestingly, a similar presentation can be developed to explain te intensity of absolute u x risk aversion u x. Tere u x measures te loss of total utility generated by te presence of and it is divided by u x to obtain te monetary equivalent of tis utility loss.

8 IESEG Working Paper Series 03-ECO-3 5 assumes risk neutrality wit respect to HRQL. If utility is risk neutral wit respect to x, ten te adjustment factor equals one.. Bivariate case Now consider a binary lottery on ealt and WTP to reduce te probability of te adverse outcome. Expected utility is given by EU = p u(w, 0 ) + ( p) u(w, ) (.) were u(w, ) is a continuously differentiable utility function for wealt w and ealt, 0 <, and p is te probability of te bad outcome. Te marginal rate of substitution of wealt for ealt risk is u w, u w, 0,, dw u 0, (.) dp pu w p u w Eu 0 were u j denotes te derivative of u wit respect to its j t argument. Define = E( ) = p 0 + ( p) and = variance ( ) = p ( p) L, were L = 0. As in te univariate case, approximate te terms in te numerator by pl u w, u w, plu w, u w, and (.3) pl u w, 0 u w, p Lu w, u w,. (.4) Te terms in te denominator of equation (.) may be approximated by pl u w, u w, plu w, u w, and (.5) pl u w, 0 u w, p Lu w, u w,. (.6) Substituting tese approximations into equation (.) yields, after simplification, u w, pl dw u w, u w, L. (.7) dp u w, u w, u w, As in te univariate case, dw/dp is te product of two terms: te marginal rate of substitution if utility is linear, u w, L, and an adjustment factor. Te first term can be u w,

9 IESEG Working Paper Series 03-ECO-3 6 interpreted as follows. Lu, w is te loss in total utility generated by te loss in ealt L, wic is converted into a monetary equivalent troug division by u, te marginal utility of wealt. Te numerator of te adjustment factor depends on te probability of te adverse outcome p, te potential loss L, and te coefficient of absolute risk aversion wit respect to ealt evaluated at te mean outcome u u w, w,. As in te univariate case, if p = / te numerator equals one. If p < /, te numerator increases in te magnitude of te potential loss L and te coefficient of risk aversion, and if p > / te numerator decreases in L and risk aversion. Te denominator of te adjustment factor depends on te riskiness of te lottery (measured by its variance) and a coefficient of cross-prudence in wealt, analogous to te Crainic and Eeckoudt (008) coefficient of prudence u u x x in equation (.8). As sown by Eeckoudt et al. (007), cross-prudence in wealt implies an individual will be more downside risk averse wit respect to wealt wen is ealt is risky rater tan certain, ence reducing WTP. As discussed in Section, te curvature of te utility function for ealt is uncertain. If represents HRQL, utility is risk neutral in and ence te adjustment ratio is equal to one. If represents longevity, utility may be risk averse, risk neutral, or risk seeking, wit peraps different signs for different values of. Te cross-partial derivative u (w, ) is often assumed to be positive, especially in te case were a low value of represents deat or severe disability (Dreze 96, Jones-Lee 974, Hammitt 00). Evidence of ow te marginal utility of wealt varies wit ealt state as been obtained from survey responses about WTP to reduce ealt risk. Viscusi and Evans (990) estimate tat te marginal utility of income is reduced by a factor of 0.77 to 0.9 by workplace injuries tat average one mont of work loss. Sloan et al. (998) estimate reductions by factors of 0. and 0.5 for multiple sclerosis. Finkelstein et al. (03) use selfreported well-being, income, and ealt data to conclude tat te marginal utility of income decreases by a factor of 0.75 to 0.90 for an increase of.3 cronic diseases. Domeij and Joannesson (006) find tat assuming te marginal utility of income is smaller wen ealt is worse is consistent wit observed savings patterns over te lifecycle. In contrast, Evans and Viscusi (99) cannot reject te ypotesis tat u (w, ) is constant or decreasing wit

10 IESEG Working Paper Series 03-ECO-3 7 wen evaluating mild, temporary injuries associated wit ouseold use of toilet cleanser and insecticide. u Te denominator of equation (.7) contains an expression,, tat is not familiar. u Its detailed interpretation is given in te Appendix. It suffices to say ere tat te numerator u (cross-prudence) describes weter te value of u (i.e., te marginal rate of substitution between wealt and ealt) increases or decreases wit ealt. As sown in te Appendix, it is te bivariate equivalent of te term u appearing in te univariate model (equation (.8)). Note tat to be expressed in monetary units, u must be divided as usual by te marginal utility of wealt u. Eeckoudt and Hammitt (004) studied ow te marginal rate of substitution of wealt for mortality risk (VSL) depends on risk aversion wit respect to wealt. Teir analysis used te standard state-dependent expected utility model in wic is a binary variable representing te states living and dead (Drèze 96, Jones-Lee 974). Tey found tat VSL is independent of local risk aversion wit regard to wealt. Moreover, transforming a utility function to increase risk aversion wit regard to wealt cannot be accomplised witout altering oter critical factors, suc as ow te marginal utility of wealt varies wit wealt. Hence a transformation of te utility function tat increases risk aversion wit respect to wealt can increase, decrease, or leave uncanged VSL, depending on wic oter caracteristics are eld constant. 3. Conclusion WTP to reduce te probability of an adverse event depends on risk aversion and on ow te marginal utility of wealt is affected by spending for prevention. An increase in risk aversion induces a cange in ow te marginal utility of wealt varies as a function of wealt. We sow ow tese effects can be explained using te concept of downside risk aversion (prudence). For bot univariate and bivariate cases, we find tat WTP to reduce te probability of an adverse event can be approximated by te product of two terms: WTP under risk neutrality, and an adjustment factor tat depends on bot risk aversion ( u ) and prudence ( u ) in te univariate case or cross-prudence ( u ) in te multivariate case. Under te reasonable assumption tat te probability of loss is less tan or equal to one-alf, te adjustment factor is increasing in risk aversion and decreasing in prudence or cross-prudence.

11 IESEG Working Paper Series 03-ECO-3 8 Because an increase in risk aversion cannot be acieved witout simultaneously altering ow prudence or cross-prudence vary wit wealt, te effect of an increase in risk aversion is ambiguous, as previously demonstrated using alternative metods by Dionne and Eeckoudt (985) for te univariate case and by Eeckoudt and Hammitt (004) for te bivariate case of mortality risk. By distinguising te effects of risk aversion and prudence, our analysis provides a new perspective on te net effect of an increase in risk aversion on WTP to reduce te probability of a financial or ealt loss.

12 IESEG Working Paper Series 03-ECO-3 9 Appendix u Te purpose of tis appendix is to formally prove te interpretation of te term u tat appears in te expression for WTP under bivariate utility (equation (.7)). A preference for combining good wit bad (see Eeckoudt et al. 009) induces in te bivariate case a preference for lottery A over B: A x l, y x l, y / / B / / xy, xy, were is a zero-mean risk (a bad for a risk-averse decision maker) and l is a sure loss (a bad if marginal utility is positive). In lottery A, te two bad consequences never occur togeter: tey are spread between te two states of te world. In lottery B, on te contrary, eiter everyting is bad (in te first state of te world, were te decision maker faces bot l on x and on y, or noting is bad (in te second state). For an expected-utility maximize, a preference for A over B implies wic is equivalent to ux l, y E u x, y E u x l, y u x, y, (A.),,,, u x y E u x y u x l y E u x l y. (A.) Using te concept of utility premium (Friedman and Savage 948), it is sow in Eeckoudt et al. (007, p. ) tat inequality (A.) olds for risk-averse expected-utility maximizers if u > 0 and tis is termed cross-prudence in wealt. Wile tis result indicates a direction for preferences as in (A.) and relates it to te sign of a tird cross-derivative of u, one may also be interested in te intensity of tis preference. In order to do so, we define a positive cange in wealt m suc tat one obtains ux l, y E u x, y E u x l, y u x m, y. (A.3)

13 IESEG Working Paper Series 03-ECO-3 0 In (A.3) te rigt and side of (A.) is increased troug te addition of m in te best state of te world until equality prevails. 6 Rearranging terms in (A.3), one obtains,,,, u x m y E u x y u x l y E u x l y. (A.4) Because ux m, y ux, y mu x, y, (A.4) becomes,,,,, u x y mu x y E u x y u x l y E u x l y. (A.5) Applying te metodology of Eeckoudt et al. (007) to te terms oter tan mu is left wit,,, mu x y u x y u x l y. Approximating u x l y to first order around, l,, u x y,,, x y, one mu x y u x y (A.6) so tat finally u x, y l m. (A.7) u x, y As a result, m measures in monetary terms te intensity of te pain induced by te misallocation of te losses in B as compared wit A. From (A.7), tis monetary intensity u depends upon u as intuitively claimed in te discussion of (.7). Acknowledgements JKH acknowledges financial support from INRA and te European Researc Council under te European Community s Sevent Framework Programme (FP7/007-03) Grant Agreement no A similar development for te univariate case is presented by Crainic and Eeckoudt (008).

14 IESEG Working Paper Series 03-ECO-3 References Arrow, K., Aspects of te Teory of Risk-bearing, Helsinki: Yrjš Jansson Foundation, 965. Crainic, D., and L.R. Eeckoudt, On te Intensity of Downside Risk Aversion, Journal of Risk and Uncertainty 36: 67-76, 008. Dionne, G., and L. Eeckoudt, Self-Insurance, Self-Protection and Increased Risk Aversion, Economics Letters 7: 39-4, 985. Domeij, D., and M. Joannesson, Consumption and Healt, Contributions to Macroeconomics 6(): article 6, 006. Drèze, J., L Utilitè Sociale d une Vie Humaine, Revue Française de Recerce Opèrationelle 6: 93-8, 96. Eeckoudt, L., H. Sclesinger, and I. Tsetlin, Apportioning of Risks via Stocastic Dominance, Journal of Economic Teory 44: , 009. Eeckoudt, L.R., and C. Gollier, Te Impact of Prudence on Optimal Prevention, Economic Teory 6: , 005. Eeckoudt, L.R., and J.K. Hammitt, Background Risks and te Value of a Statistical Life, Journal of Risk and Uncertainty 3: 6-79, 00. Eeckoudt, L.R., and J.K. Hammitt, Does Risk Aversion Increase te Value of Mortality Risk? Journal of Environmental Economics and Management 47: 3-9, 004. Eeckoudt, L.R., B. Rey, and H. Sclesinger, A Good Sign for Multivariate Risk Taking, Management Science 53: 7-4, 007. Erlic, I., and G.S. Becker, Market Insurance, Self-Insurance, and Self-Protection, Journal of Political Economy 80: , 97. Evans, W.N., and W.K. Viscusi, Estimation of State-Dependent Utility Functions using Survey Data, Review of Economics and Statistics 73: 94-04, 99. Finkelstein, A., E.F.P. Luttmer, and M.J. Notowidigdo, Wat Good is Wealt Witout Healt? Te Effect of Healt on te Marginal Utility of Consumption, Journal of te European Economic Association (S):-58, 03. Friedman, M., and L. Savage, Te Utility Analysis of Coices Involving Risk, Journal of Political Economy 56: , 948. Hammitt, J.K., QALYs versus WTP, Risk Analysis : , 00. Jones-Lee, M.W., Te Value of Canges in te Probability of Deat or Injury, Journal of Political Economy 8: , 974. Kimball, M., Precautionary Saving in te Small and in te Large, Econometrica 58: 53-73, 990. Menezes, C,G., C. Geiss, and J. Tressler, Increasing Downside Risk, American Economic Review 70: 9-93, 980. Modica, S., and M. Scarsini, A Note on Comparative Downside Risk Aversion, Journal of Economic Teory : 67-7, 005. Pratt, J.W., Risk Aversion in te Small and in te Large, Econometrica 3: -36, 964.

15 IESEG Working Paper Series 03-ECO-3 Sloan, F.A., W.K. Viscusi, H.W. Cesson, C.J. Conover, and K. Wetten-Goldstein, Alternative Approaces to Valuing Intangible Healt Losses: Te Evidence for Multiple Sclerosis, Journal of Healt Economics 7: , 998. Viscusi, W.K., and W.N. Evans, Utility Functions tat Depend on Healt Status: Estimates and Economic Implications, American Economic Review 80: , 990.