Ghossoub, Mario (2010): Belief heterogeneity in the ArrowBorchRaviv insurance model.

PDF
MPRA_paper_37630.pdf Download (504kB)  Preview 
Abstract
In the classical ArrowBorchRaviv problem of demand for insurance contracts, it is wellknown that the optimal insurance contract for an insurance buyer – or decision maker (DM) – is a deductible contract, when the insurer is a riskneutral ExpectedUtility (EU) maximizer, and when the DM is a riskaverse EUmaximizer. In the ArrowBorchRaviv framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This paper argues for heterogeneity of beliefs in the classical insurance model, and considers a setting where the DM and the insurer have preferences yielding different subjective beliefs. The DM seeks the insurance contract that will maximize her (subjective) expected utility of terminal wealth with respect to her subjective probability measure, whereas the insurer sets premiums on the basis of his subjective probability measure. I show that in this setting, and under a consistency requirement on the insurer’s subjective probability that I call vigilance, there exists an event to which the DM assigns full (subjective) probability and on which an optimal insurance contract for the DM takes the form of what I will call a generalized deductible contract. Moreover, the class of all optimal contracts for the DM that are nondecreasing in the loss is fully characterized in terms of their distribution under the DM’s probability measure. Finally, the assumption of vigilance is shown to be a weakening of the assumption of a monotone likelihood ratio, when the latter can be defined, and it is hence a useful tool in situations where the likelihood ratio cannot be defined.
Item Type:  MPRA Paper 

Original Title:  Belief heterogeneity in the ArrowBorchRaviv insurance model 
Language:  English 
Keywords:  Optimal insurance, deductible contract, subjective probability, heterogeneous beliefs, vigilance, Agreement Theorem, Harsanyi Doctrine, Wilson Doctrine 
Subjects:  D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D86  Economics of Contract: Theory C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C65  Miscellaneous Mathematical Tools G  Financial Economics > G2  Financial Institutions and Services > G22  Insurance ; Insurance Companies ; Actuarial Studies C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods 
Item ID:  37630 
Depositing User:  Mario Ghossoub 
Date Deposited:  28. Mar 2012 16:51 
Last Modified:  05. May 2015 19:28 
References:  [1] C.D. Aliprantis and K.C. Border. Infinite Dimensional Analysis  3rd edition. SpringerVerlag, 2006. [2] R.J. Arnott and J.E. Stiglitz. The Basic Analytics of Moral Hazard. The Scandinavian Journal of Economics, 90(3):383–413, 1988. [3] K.J. Arrow. Essays in the Theory of RiskBearing. Chicago: Markham Publishing Company, 1971. [4] R.J. Aumann. Agreeing to Disagree. The Annals of Statistics, 4(6):1236–1239, 1976. [5] R.J. Aumann. Correlated Equilibrium as an Expression of Bayesian Rationality. Econometrica, 55(1):1–18, 1987. [6] R.J. Aumann. Common Priors: A Reply to Gul. Econometrica, 66(4):929–938, 1998. [7] R.J. Aumann and A. Brandenburger. Epistemic Conditions for Nash Equilibrium. Econometrica, 63(5):1161–1180, 1995. [8] D. Bergemann and S. Morris. Robust Mechanism Design. Econometrica, 73(6):1771–1813, 2005. [9] D. Bergemann and S. Morris. Ex Post Implementation. Games and Economic Behavior, 63(2):527–566, 2008. [10] D. Bergemann and S. Morris. Robust Virtual Implementation. Theoretical Economics, 4(1):45–88, 2009. [11] D. Bergemann and S. Morris. Robust Implementation in General Mechanisms. Games and Economic Behavior, 71(2):261–281, 2011. [12] T.F. Bewley. Knightian Decision Theory. Part I. Cowles Foundation Discussion Papers, Paper no. 807 (1986). [13] V.I. Bogachev. Measure Theory  Volume 1. SpringerVerlag, 2007. [14] K.H. Borch. The Mathematical Theory of Insurance: An Annotated Selection of Papers on Insurance Published 19601972. Lexington Books, 1974. [15] R. Bracewell. The Fourier Transform and Its Applications, 3rd ed. New York: McGrawHill, 2000. [16] A. Chateauneuf, F. Maccheroni, M. Marinacci, and J.M. Tallon. Monotone Continuous Multiple Priors. Economic Theory, 26(4):973–982, 2005. [17] H. Chernoff and L.E. Moses. Elementary Decision Theory. New York: Wiley, 1959. [18] K.S. Chung and J.C. Ely. Foundations of DominantStrategy Mechanisms. The Review of Economic Studies, 74(2):447–476, 2007. [19] D.L. Cohn. Measure Theory. Birkhauser, 1980. [20] B. De Finetti. La Pr´evision: Ses Lois Logiques, Ses Sources Subjectives. Annales de l’Institut Henri Poincare, 7(1):1–68, 1937. [21] B. De Finetti. Theory of Probability: A Critical Introductory Treatment (revised edition) – 1st ed. 1974. New York: Wiley, 1990. [22] J. Dieudonne. Foundations of Modern Analysis. Academic press, 1969. [23] L. Eeckhoudt, C. Gollier, and H. Schlesinger. Economic and Financial Decisions under Risk. Princeton University Press, 2005. [24] P.C. Fishburn. Utility Theory for Decision Making. John Wiley & Sons, 1970. [25] M. Ghossoub. Monotone Equimeasurable Rarrangements with NonAdditive Probabilities. mimeo (2011). [26] M. Ghossoub. Contracting under Heterogeneous Beliefs. PhD thesis, University of Waterloo – Dept. of Statistics & Actuarial Science, 2011. [27] I. Gilboa. Theory of Decision under Uncertainty. Cambridge University Press, 2009. [28] I. Gilboa, F.Maccheroni, M. Marinacci, and D. Schmeidler. Objective and Subjective Rationality in a Multiple Prior Model. Econometrica, 78(2):755–770, 2010. [29] I. Gilboa and M. Marinacci. Ambiguity and the Bayesian Paradigm. mimeo (2011). [30] I. Gilboa and D. Schmeidler. Maxmin Expected Utility with a NonUnique Prior. Journal of Mathematical Economics, 18(2):141–153, 1989. [31] F. Gul. A Comment on Aumann’s Bayesian View. Econometrica, 66(4):923–927, 1998. [32] J.Y. Halpern and Y. Moses. Knowledge and Common Knowledge in a Distributed Environment. Journal of the ACM (JACM), 37(3):549–587, 1990. [33] G.H. Hardy, J.E. Littlewood, and G. Polya. Inequalities. Cambridge University Press, Cambridge. Reprint of the 1952 edition, 1988. [34] J.C. Harsanyi. Games with Incomplete Information Played by ’Bayesian’ Players, IIII. Part I. The Basic Model. Management Science, 14(3):159–182, 1967. [35] J.C. Harsanyi. Games with Incomplete Information Played by ’Bayesian’ Players, IIII. Part II. Bayesian Equilibrium Points. Management Science, 14(5):320–334, 1968. [36] J.C. Harsanyi. Games with Incomplete Information Played by ’Bayesian’ Players, IIII. Part III. The Basic Probability Distribution of the Game. Management Science, 14(7):486–502, 1968. [37] G. Huberman, D. Mayers, and C.W. Smith Jr. Optimal Insurance Policy Indemnity Schedules. The Bell Journal of Economics, 14(2):415–426, 1983. [38] K. Inada. On a TwoSector Model of Economic Growth: Comments and a Generalization. The Review of Economic Studies, 30(2):119–127, 1963. [39] M. Jeleva and B. Villeneuve. Insurance Contracts with Imprecise Probabilities and Adverse Selection. Economic Theory, 23(4):777–794, 2004. [40] E. Lehrer and D. Samet. Agreeing to Agree. Theoretical Economics, 6(2):269–287, 2011. [41] J. Marshall. Optimum Insurance with Deviant Beliefs. in G. Dionne (ed.), Contributions to Insurance Economics. Boston: Kluwer Academic Publishers, pages 255–274, 1991. [42] P. Milgrom and N. Stokey. Information, Trade and Common Knowledge. Journal of Economic Theory, 26(1):17–27, 1982. [43] S. Morris. The Common Prior Assumption in Economic Theory. Economics and Philosophy, 11(2):227–253, 1995. [44] F.P. Ramsey. The Foundations of Mathematics and other Logical Essays. The Humanities Press: New York (Ed. R.B. Braithwaite), 1950. [45] A. Raviv. The Design of an Optimal Insurance Policy. The American Economic Review, 69(1):84–96, 1979. [46] M. Rothschild and J.E. Stiglitz. Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information. The Quarterly Journal of Economics, 90(4):629–649, 1976. [47] W. Rudin. Principles of Mathematical Analysis – 3rd ed. New York: McGrawHill Book Company, 1976. [48] S. Saks. Theory of the Integral. 2nd ed. Monografie Matematyczne, Warsaw; English transl. L.C. Young, 1937. [49] L.J. Savage. The Foundations of Statistics (2nd revised edition) – 1st ed. 1954. New York: Dover Publications, 1972. [50] D. Schmeidler. Subjective Probability and Expected Utility without Additivity. Econometrica, 57(3):571–587, 1989. [51] J.K. Sebenius and J. Geanakoplos. Don’t Bet on It: Contingent Agreements with Asymmetric Information. Journal of the American Statistical Association, pages 424–426, 1983. [52] S. Shavell. On Moral Hazard and Insurance. The Quarterly Journal of Economics, 93(4):541–562, 1979. [53] J.E. Stiglitz. Monopoly, NonLinear Pricing and Imperfect Information: The Insurance Market. The Review of Economic Studies, 44(3):407–430, 1977. [54] J.E. Stiglitz. Risk, Incentives and Insurance: The Pure Theory of Moral Hazard. The Geneva Papers on Risk and Insurance, 8(1):4–33, 1983. [55] J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton University Press, 1947. [56] R. Wilson. GameTheoretic Analyses of Trading Processes. In T. Bewley (ed.), Advances in Economic Theory: Fifth World Congress, pages 33–70. Cambridge: Cambridge University Press, 1987. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/37630 