Analyzing the Demand for Deductible Insurance


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1 Journal of Risk and Uncertainty, 18: Kluwer Academic Publishers. Manufactured in The Netherlands. Analyzing the emand for eductible Insurance JACK MEYER epartment of Economics, Michigan State Uni ersity MICAEL B. ORMISTON epartment of Economics, Arizona State Uni ersity, Tempe, Arizona Abstract This paper investigates aspects of insurance demand related to deductible insurance. In particular, an important issue concerning analysis of the optimal deductible level is resolved. A simple sufficient restriction on the pricing of insurance is given which ensures that the second order condition for choosing the expected utility maximizing deductible level is met for any risk averse decision maker. This restriction is stated and its sufficiency is demonstrated using the level of expected indemnification rather than the level of the deductible as the choice variable in the decision model. Key words: deductible insurance, second order conditions JEL Classification: 81, G The demand for insurance has been the subject of extensive effort in recent years. One portion of this literature is concerned with the optimal form of indemnification for insurance policies, and has shown that, from the viewpoint of the insured, deductible policies are optimal under quite general circumstances. This has been accomplished both in the expected utility maximization model and in nonexpected utility decision models that preserve second degree stochastic dominance. 1 This being the case, research concerned with the quantity of insurance demanded has frequently focused on determining the optimal deductible level, and how this level changes in response to changes in wealth, the price of insurance, and the distribution of the random loss variable. While many interesting results concerning the optimal deductible level and its response to shifts in model parameters have been obtained, the analysis has typically been carried out assuming that second order conditions for the maximization are met, without precisely stating what this assumption implies about the decision model in question. An important unresolved question asks which assumptions on the decision model are sufficient to ensure that the second order condition for choosing the expected utility maximizing deductible level is met for all risk averse decision makers? The main contribution of this paper is an answer to this question. In particular, here it is shown that a simple assumption concerning the
2 4 MEYER AN ORMISTON pricing of insurance, that the price of deductible insurance is a convex function of expected indemnification, is sufficient to guarantee that the second order condition is satisfied globally when preferences exhibit risk aversion. 3 The paper proceeds as follows. Section 1 introduces the notation and assumptions used throughout the paper and reviews the standard specification of the deductible insurance demand model used and analyzed in the literature. Section proposes an alternative methodology for examining the demand for deductible insurance. The primary change from the standard model that is reviewed in Section 1 is that a different metric or parameter is used to identify a particular deductible insurance policy and to quantify a particular insurance level. The usual measure, the deductible level, is replaced by the level of expected indemnification. For deductible insurance these two measures are related to one another in a one to one fashion; thus, finding the optimal level for one is equivalent to finding the optimal level for the other. The main theorem of the section shows that if the price of deductible insurance is a convex function of the level of expected indemnification and the decision maker is risk averse, then maximizing expected utility by solving the first order condition yields a global maximum. 4 Finally, Section 3 summarizes the results and concludes with remarks concerning linear pricing models. 1. Assumptions and notation The decision maker is assumed to be endowed with initial wealth w and a risky asset whose value is M when no loss occurs. This risky asset is subject to random loss of size x, where x has support in, M and is distributed continuously. The cumulative distribution function describing x is denoted by FŽ x. and the density function by fž x.. An insurance policy, IŽ x., P 4, against this loss is composed of an indemnification function IŽ x. providing reimbursement of size IŽ x. when loss x occurs and a price or premium for insurance, P. IŽ x. is assumed to satisfy IŽ x. x. With insurance, random final wealth, z, is given by: z w M x IŽ x. P Ž 1. The deductible form of insurance is characterized by an indemnification function IŽ x. satisfying: if x IŽ x. ½ Ž Ž. x. if x where M is the level of the deductible. In this standard specification, the price for obtaining a policy with deductible is a function of and is denoted by the premium function, P Ž., which is assumed to be continuous, twice differentiable, and decreasing in. Thus, final wealth when insurance is of the
3 ANALYZING TE EMAN FOR EUCTIBLE INSURANCE 5 deductible form is given by: w M x Ž. if x z ½ Ž 3. w M Ž. if x Let U denote expected utility from final wealth; that is, U Euz, where už. is the decision maker s twice continuously differentiable von Neumann Morgenstern utility function. When is selected to maximize UŽ,. the first and second order conditions for an interior solution are U Ž *. and U Ž *., respectively, where * is the optimal level of deductible and U and U are given by: U U u Ž w M x. dfž x. M u Ž w M.Ž 1. dfž x. u Ž w M x. dfž x. u Ž w M.Ž 1. Ž 1 FŽ.. Ž 4. u Ž w M. fž. u Ž w M x.ž. u Ž w M x. dfž x. u w M 1 1 F u Ž w M. Ž 1 FŽ.. Ž 5. It has proven difficult to determine reasonable, interpretable, conditions on the various components of the decision model under which U is indeed negative. Schlesinger Ž presents the most extensive and complete analysis of this issue and gives a rather complicated necessary and sufficient condition for determining the sign of U locally. Among other things, this condition involves the first and second derivatives of the price function, the first and second derivatives of the utility function, and the expectation of the derivatives of the utility function. Most research since 1981 simply assumes U holds locally at the stationary points and proceeds, referring the reader to Schlesinger s analysis. Eeckhoudt et al. Ž 1991., for instance, state that, The second order condition is far from trivial and assumed to hold. and refer the reader to Schlesinger.
4 6 MEYER AN ORMISTON. An alternate method for determining the optimal deductible level In this section an alternate method of finding the EU maximizing level of is presented. The change involves identifying a particular deductible insurance policy, and hence a quantity demanded, using Q, the expected level of indemnification, rather than. For deductible insurance, expected indemnification, Q, is given by M Q E IŽ x. Ž x. fž x. dx. Ž 6. This function is strictly decreasing in for all values of such that FŽ. 1 and, hence, has an inverse which we denote by Ž Q.. Because Q and are one to one, finding the EU maximizing level for Q and then using to determine, is an indirect and alternate method for determining the optimal level for. Later in the analysis, the following properties of are used. First, because dq d FŽ. 1 where 1 FŽ. 1, Ž Q. 1 FŽ Second, differentiating one more time implies that fž. FŽ Thus, is a decreasing and convex function. To restate the decision problem so that the level of expected indemnification, Q, rather than the level of the deductible is the choice variable, the price of this deductible insurance must be written as a function of Q rather than. efine to be price for deductible insurance policies expressed as a function of the level of expected indemnification Q. 5 Clearly, the two functions used to represent the price of deductible insurance, and Ž., are such that Q Ž.. This implies that and Ž.. As is typical in insurance demand models, it is assumed that Ž. and that 1. The latter restriction implies that the premium increases by at least as much as the increase in expected indemnification when the level of insurance is altered. This condition is necessary for an interior solution to the optimization problem. 6 Using this notation, final wealth can be written as a function of expected indemnification Q and is given by: w M x if x z ½ Ž 7. w M if x Proceeding as in the standard model, let VQ denote expected utility from final wealth; that is, VQ Euz, where z is now a function of expected indemnification. Note that VQ UŽ Ž Q... It is assumed that the decision maker chooses Q to maximize VQ. For this optimization problem the first and second order
5 ANALYZING TE EMAN FOR EUCTIBLE INSURANCE 7 conditions are V Ž Q*. and V Ž Q*. Q QQ, respectively, where Q* is the optimal level of expected indemnification and VQ and VQQ are given by: V Q M u Ž w M x. dfž x. u Ž w M.Ž. dfž x. u Ž w M x. dfž x. u Ž w M.Ž. 1 F Ž. Ž 8. V u Ž w M x.ž. u Ž w M x. dfž x. QQ u Ž w M.Ž. 1 F Ž. u Ž w M. f Ž Q. 1 F Ž Q. 9 Clearly, V U Ž Q., and because Ž Q. Q, VQ if and only if U. Thus, the first order equations resulting from maximizing U or VQ yield the same solutions; that is VQ and U, respectively, identify the same deductible insurance alternatives as potential maxima. Turning now to the second order condition for each optimization problem, observe that V QQ U Q U Ž Q..Atlocal optima, the first order condition U is satisfied, and hence it is the case that U if and only if VQQ at that point. Thus, at the values for or Q identified by the first order conditions, these second order restrictions are identical. What is less apparent, and is the basis for Theorem 1 which follows, is that under simple and reasonable assumptions, VQQ for all alues for Q and for all risk a erse decision makers; that is, V is globally concave in Q for all concave uz. This is the case even though U is not globally concave in. Obviously, when VQQ for all Q, then this ensures that solving VQ yields a global maximum for Q, and hence the associated value for the deductible given by Ž Q.,isa global maximum for U as well. Thus, even though U is not concave in, the first order condition for maximizing U does indeed yield a global maximum. The global concavity of V is formally demonstrated in the following theorem.
6 8 MEYER AN ORMISTON Theorem 1. VŽ Q. is strictly conca e ifu Ž z., u Ž z.,, Ž Q. 1, and. Proof: Recall that Ž Q. 1 FŽ. 1 and fž. FŽ Substituting these into Ž. 9 in the last term gives V u Ž w M x.ž. u Ž w M x. dfž x. QQ u Ž w M.Ž. 1 F Ž. u Ž w M. 1 F Ž. Ž 1. which is strictly negative under the conditions of the theorem. QE Theorem 1 indicates that for deductible insurance, any Q* which solves the first order condition for EU maximization is a global maximum. Because is a strictly decreasing function of Q, the corresponding * Ž Q*. also globally maximizes UŽ.. That is, when u is strictly concave and the price of deductible insurance is an increasing and convex function of expected indemnification, both U and VQ have a global maximum, and at the maximum, the objective function is locally concave. VQ displays the additional property that it is globally concave, while U need not display this property. It is the case, however, that U is strictly quasiconcave in the level of the deductible. The following Corollary makes this point formally. Corollary 1. UŽ. is strictly quasiconca e ifu Ž z., u Ž z.,, Ž Q. 1, and. Proof: Recall that U is strictly quasiconcave if U whenever U. This follows directly from Theorem 1 and the fact that V QQ U Q U Ž Q.. 3. Concluding remarks Researchers frequently assume that the price of all insurance policies offered by sellers, whether of the deductible form or not, is proportional to expected indemnification; that is, Q, where Ž 1. is the loading factor and this price function applies to insurance of any form. Clearly this is a pricing assumption with premium schedules which are convex in expected indemnification and hence satisfy the conditions in Theorem 1. This fact has not been recognized by the many researchers in that they assume risk aversion and that price is proportional to expected indemnification, but then proceed to add to these assumptions the restriction that the second order condition must be satisfied. Of course, Theorem 1
7 ANALYZING TE EMAN FOR EUCTIBLE INSURANCE 9 indicates that the latter assumption is redundant. 7 That is, Theorem 1 demonstrates that the second order condition for choosing the EU maximizing deductible level is automatically satisfied whenever risk aversion and proportional pricing are assumed. This paper demonstrates an advantage of formulating the deductible insurance demand model using expected indemnification as the decision variable. With Q as the decision variable, it is easy to show that when the decision maker is risk averse and the insurance premium schedule is increasing and convex in Q, then the first order condition for choosing Q to maximize EU identifies a unique global maximum. Furthermore, once this fact is known, the same result holds in the standard formulation. That is, risk aversion and convexity in Q is sufficient for the first order condition for choosing to maximize EU to identify a unique global maximum. This fact can be demonstrated directly by converting the restrictions in Theorem 1 concerning how the premium varies with expected indemnification, Ž Q., into the corresponding restrictions on how the premium varies with the deductible level, Ž.. To see this note that Ž Q. 1 implies that Ž. FŽ. 1, and implies Ž. Ž. fž. ŽFŽ. 1.. It is easily verified that given these restrictions on Ž., the second order restriction, U, holds whenever the first order restriction, U, is satisfied; that is, at *. ence, this first order condition, if satisfied, identifies a global maximum. 8 To interpret these complex restrictions on Ž., however, involves returning to the relationship between price and expected indemnification. Thus, the primary reason for conducting the analysis using expected indemnification as the decision variable is to allow reasonable and interpretable conditions to be stated as sufficient for the maximization of EU. Acknowledgments The authors thank Ed Schlee, an anonymous referee, and the participants of the 8th International Conference on the Foundations and Applications of Utility, Risk, and ecision Theories for their valuable comments and suggestions. This research was funded in part by the Arizona State University, College of Business, Summer Research Grant Program. Notes 1. Examples of papers demonstrating the optimality of the deductible form for insurance in an expected utility setting include Arrow Ž 1971, and Raviv Ž Gollier and Schlesinger Ž demonstrate the optimality of the deductible form for insurance in a nonexpected utility setting.. Papers examining factors affecting the demand for deductible insurance include Pashigan et al. Ž 1966., Mossin Ž 1968., Gould Ž 1969., Schlesinger Ž 1981., Eeckhoudt et al. Ž 1991., Eeckhoudt and Gollier Ž and Schlee Ž to name a few.
8 3 MEYER AN ORMISTON 3. Meyer and Ormiston Ž show that these same pricing assumptions are necessary for the optimality of the deductible form of indemnification. 4. The assertions made by Schlee Ž in footnotes 8 and 15 yield a similar conclusion concerning a local maximum; however, no proof is given. 5. It is important to recognize that representing the price of deductible insurance as a function of Q in no way assumes that all insurance contracts are priced by this function of Q. Other indemnification functions could display a different relationship between price and Q than does the set of deductible indemnification functions. 6. It is also the case that Ž. and 1 together imply that expected profit from providing insurance is nonnegative. This is necessary to guarantee that a risk neutral insurer will, in fact, be willing to offer the insurance. 7. See, for example, Mossin Ž 1968., oherty and Schlesinger Ž 1983., Eeckhoudt, Gollier and Schlesinger Ž 1991., and Eeckhoudt and Gollier Ž Each assumes risk aversion and linear pricing. In addition, each goes on to assume that the second order conditions hold, which in light of Theorem 1 is redundant. 8. Substituting these inequalities into Ž. 5 and rearranging gives the desired result. References Arrow, K. Ž Essays in the Theory of Risk Bearing. Markham. Arrow, K. Ž Optimal Insurance and Generalized eductibles, Scandina ian Actuarial Journal, 1 4. oherty, N. A., and. Schlesinger. Ž The Optimal eductible for an Insurance Policy When Initial Wealth Is Random, Journal of Business 56, Eeckhoudt, L., and C. Gollier. Ž Risk. New York: arvesterwheatsheaf. Eeckhoudt, L., C. Gollier, and. Schlesinger. Ž Increases in Risk and eductible Insurance, Journal of Economic Theory 55, Gollier, C., and. Schlesinger. Ž Arrow s Theorem on the Optimality of eductibles: A Stochastic ominance Approach, Economic Theory. Gould, J. P. Ž The Expected Utility ypothesis and the Selection of Optimal eductibles for a Given Insurance Policy, Journal of Business 4, Meyer, J., and M. B. Ormiston. Ž The Pricing of Optimal Insurance Contracts, Proceedings from FUR 8, Mons Belgium, July 1997 Ž forthcoming.. Mossin, J. Ž Aspects of Rational Insurance Purchasing, Journal of Political Economy 76, Pashigan, B. P., L. Schkade, and G. Menefee. Ž The Selection of an Optimal eductible for a Given Insurance Policy, Journal of Business 39, Raviv, A. Ž The esign of an Optimal Insurance Policy, American Economic Re iew 69, Schlee, E. Ž The Comparative Statics of eductible Insurance in Expected and NonExpected Utility Theories, The Gene a Papers on Risk and Insurance Theory, Schlesinger,. Ž The Optimal Level of eductibility in Insurance Contracts, Journal of Risk and Insurance 48,
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