×

Arrow’s theorem of the deductible with heterogeneous beliefs. (English) Zbl 1414.91193

Summary: In Arrow’s classical problem of demand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer – or decision maker (DM) – is a deductible contract when the insurer is a risk-neutral expected-utility (EU) maximizer and when the DM is a risk-averse EU maximizer. In Arrow’s framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This article reexamines Arrow’s problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer’s and the DM’s subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow’s classical result is then obtained as a special case.

MSC:

91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alary, D.; Gollier, C.; Treich, N., The Effect of Ambiguity Aversion on Insurance and Self-Protection, The Economic Journal, 123, 573, 1188-1202, (2013)
[2] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis., (2006), New York: Springer-Verlag, New York · Zbl 1156.46001
[3] Amarante, M.; Ghossoub, M., Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer, Risks, 4, 1, 8, (2016)
[4] Amarante, M.; Ghossoub, M.; Phelps, E. S., Contracting on Ambiguous Prospects, Economic Journal
[5] Amarante, M.; Ghossoub, M.; Phelps, E. S., The Entrepreneurial Economy I: Contracting under Knightian Uncertainty, (2011)
[6] Amarante, M.; Ghossoub, M.; Phelps, E. S., Ambiguity on the Insurer’s Side: The Demand for Insurance, Journal of Mathematical Economics, 58, 61-78, (2015) · Zbl 1319.91093
[7] Anwar, S.; Zheng, M., Competitive Insurance Market in the Presence of Ambiguity, Insurance: Mathematics and Economics, 50, 1, 79-84, (2012) · Zbl 1235.91077
[8] Arrow, K. J., Essays in the Theory of Risk-Bearing, (1971), Chicago: Markham, Chicago · Zbl 0215.58602
[9] Aumann, R. J., Agreeing to Disagree, Annals of Statistics, 4, 6, 1236-1239, (1976) · Zbl 0379.62003
[10] Aumann, R. J., Common Priors: A Reply to Gul, Econometrica, 66, 4, 929-938, (1998) · Zbl 1073.91514
[11] Bernard, C.; He, X.; Yan, J. A.; Zhou, X. Y., Oprimal Insurance Design under Rank-Dependent Expected Utility, Mathematical Finance, 25, 1, 154-186, (2015) · Zbl 1314.91134
[12] Biffis, E.; Blake, D., Informed Intermediation of Longevity Exposures, Journal of Risk and Insurance, 80, 3, 559-584, (2013)
[13] Biffis, E.; Millossovich, P., Optimal Insurance with Counterparty Default Risk, (2010)
[14] Borch, K. H., The Mathematical Theory of Insurance: An Annotated Selection of Papers on Insurance Published 1960–1972, (1974), New York: Lexington Books, New York
[15] Bracewell, R., The Fourier Transform and Its Applications., (2000), New York: McGraw-Hill, New York · Zbl 0149.08301
[16] Briys, E.; Viala, P., Optimal Insurance Design under Background Risk, (1995)
[17] Carlier, G.; Dana, R. A., Pareto Efficient Insurance Contracts When the Insurer’s Cost Function is Discontinuous, Economic Theory, 21, 4, 871-893, (2003) · Zbl 1060.91075
[18] Carlier, G.; Dana, R. A.; Shahidi, N., Efficient Insurance Contracts under Epsilon-Contaminated Utilities, GENEVA Papers on Risk and Insurance–Theory, 28, 59-71, (2003)
[19] Carothers, N. L., Real Analysis, (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0997.26003
[20] Chateauneuf, A.; Maccheroni, F.; Marinacci, M.; Tallon, J. M., Monotone Continuous Multiple Priors, Economic Theory, 26, 4, 973-982, (2005) · Zbl 1116.91317
[21] Cohn, D. L., Measure Theory, (1980), Boston: Birkhauser, Boston
[22] Cummins, J. D.; Mahul, O., Optimal Insurance with Divergent Beliefs about Insurer Total Default Risk, Journal of Risk and Uncertainty, 27, 2, 121-138, (2003) · Zbl 1054.91050
[23] Dana, R. A.; Scarsini, M., Optimal Risk Sharing with Background Risk, Journal of Economic Theory, 133, 1, 152-176, (2007) · Zbl 1280.91092
[24] De Finetti, B., La Prévision: ses lois logiques, ses sources subjectives, Annales de l’Institut Henri Poincaré, 7, 1, 1-68, (1937) · JFM 63.1070.02
[25] Dionne, G.; Fombaron, N.; Doherty, N. A.; Dionne, G., Adverse Selection in Insurance Markets, Handbook of Insurance, (2000), Bostan: Kluwer Academic Publishers, Bostan
[26] Doherty, N. A.; Schlesinger, H., Optimal Insurance in Incomplete Markets, Journal of Political Economy, 91, 6, 1045-1054, (1983)
[27] Doherty, N. A.; Eeckhoudt, L., Optimal Insurance without Expected Utility: The Dual Theory and the Linearity of Insurance Contracts, Journal of Risk and Uncertainty, 10, 2, 157-179, (1995) · Zbl 0847.90036
[28] Drèze, J. H., Loss Reduction and Implicit Deductibles in Medical Insurance, (2002)
[29] Drèze, J. H.; Schokkaert, E., Arrow’s Theorem of the Deductible: Moral Hazard and Stop-Loss in Health Insurance, Journal of Risk and Uncertainty, 47, 2, 147-163, (2013)
[30] Fei, W.; Schlesinger, H., Precautionary Insurance Demand with State-Dependent Background Risk, Journal of Risk and Insurance, 75, 1, 1-16, (2008)
[31] Fishburn, P. C., Utility Theory for Decision Making, (1970), New York: John Wiley & Sons, New York · Zbl 0213.46202
[32] Franke, G.; Schlesinger, H.; Stapleton, R. C., Risk Taking with Additive and Multiplicative Background Risks, Journal of Economic Theory, 146, 4, 1547-1568, (2011) · Zbl 1247.91084
[33] Ghossoub, M., Contracting under Heterogeneous Beliefs, (2011)
[34] Ghossoub, M., Equimeasurable Rearrangements with Capacities, Mathematics of Operations Research, 40, 2, 429-445, (2015) · Zbl 1377.91110
[35] Ghossoub, M., Vigilant Measures of Risk and the Demand for Contingent Claims, Insurance: Mathematics and Economics, 61, 27-35, (2015) · Zbl 1403.91195
[36] Gollier, C.; Dionne, G., Economic Theory of Risk Exchanges: A Review, Contributions to Insurance Economics, (1992), Bostan: Kluwer Academic Publishers, Bostan
[37] Gollier, C., Optimal Insurance Design of Ambiguous Risks, (2012) · Zbl 1319.91095
[38] Gollier, C.; Schlesinger, H., Arrow’s Theorem on the Optimality of Deductibles: A Stochastic Dominance Approach, Economic Theory, 7, 359-363, (1996) · Zbl 0852.90047
[39] Gul, F., A Comment on Aumann’s Bayesian View, Econometrica, 66, 4, 923-927, (1998) · Zbl 1032.91560
[40] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, (1988), Cambridge: Cambridge University Press. Reprint of 1952 edition, Cambridge · Zbl 0634.26008
[41] Huang, R.; Tzeng, L. Y.; Wang, C., Revisiting the Optimal Insurance Contract under Deviant Beliefs, (2002)
[42] Huang, R. J.; Snow, A.; Tzeng, L. Y., Ambiguity and Asymmetric Information, (2013)
[43] Huberman, G.; Mayers, D.; Smith, C. W. Jr, Optimal Insurance Policy Indemnity Schedules, Bell Journal of Economics, 14, 2, 415-426, (1983)
[44] Inada, K., On a Two-Sector Model of Economic Growth: Comments and a Generalization, Review of Economic Studies, 30, 2, 119-127, (1963)
[45] Jeleva, M., Background Risk, Demand for Insurance, and Choquet Expected Utility Preferences, GENEVA Papers on Risk and Insurance-Theory, 25, 1, 7-28, (2000)
[46] Jeleva, M.; Villeneuve, B., Insurance Contracts with Imprecise Probabilities and Adverse Selection, Economic Theory, 23, 4, 777-794, (2004) · Zbl 1065.91035
[47] Karni, E.; Dionne, G., Optimal Insurance: A Nonexpected Utility Analysis, Contributions to Insurance Economics, (1992), Bostan: Kluwer Academic Publishers, Bostan
[48] Machina, M. J., Non-Expected Utility and the Robustness of the Classical Insurance Paradigm, GENEVA Papers on Risk and Insurance–Theory, 20, 1, 9-50, (1995)
[49] Marshall, J.; Dionne, G., Optimum Insurance with Deviant Beliefs, Contributions to Insurance Economics, (1991), Boston: Kluwer Academic Publishers, Boston
[50] Milgrom, P., Good News and Bad News: Representation Theorems and Applications, Bell Journal of Economics, 12, 2, 380-391, (1981)
[51] Morris, S., The Common Prior Assumption in Economic Theory, Economics and Philosophy, 11, 2, 227-253, (1995)
[52] Ping, X.; Zanjani, G., Optimal Insurance Contracts with Insurer’s Background Risk, (2013)
[53] Promislow, S. D.; Young, V. R., Unifying Framework for Optimal Insurance, Insurance: Mathematics and Economics, 36, 3, 347-364, (2005) · Zbl 1242.91095
[54] Ramsey, F. P.; Braithwaite, R. B., The Foundations of Mathematics and Other Logical Essays, (1950), New York: Humanities Press, New York
[55] Raviv, A., The Design of an Optimal Insurance Policy, American Economic Review, 69, 1, 84-96, (1979)
[56] Rudin, W., Principles of Mathematical Analysis., (1976), New York: McGraw-Hill, New York · Zbl 0148.02903
[57] Savage, L. J., The Foundations of Statistics., (1972), New York: Dover Publications, New York · Zbl 0276.62006
[58] Schlesinger, H., Insurance Demand without the Expected-Utility Paradigm, Journal of Risk and Insurance, 64, 1, 19-39, (1997)
[59] Schlesinger, H.; Dionne, G., The Theory Insurance Demand, Handbook of Insurance, (2000), Bostan: Kluwer Academic Publishers, Bostan
[60] Schlesinger, H.; Doherty, N. A., Incomplete Markets for Insurance: An Overview, Journal of Risk and Insurance, 52, 3, 402-423, (1985)
[61] Schmeidler, D., Subjective Probability and Expected Utility without Additivity, Econometrica, 57, 3, 571-587, (1989) · Zbl 0672.90011
[62] Shaked, M.; Shanthikumar, J. G., Stochastic Orders, (2007), New York: Springer, New York
[63] Stiglitz, J. E., Monopoly, Non-Linear Pricing and Imperfect Information: The Insurance Market, Review of Economic Studies, 44, 3, 407-430, (1977) · Zbl 0381.90021
[64] Neumann, J.; Morgenstern, O., Theory of Games and Economic Behavior, (1947), Princeton: Princeton University Press, Princeton · Zbl 1241.91002
[65] Winter, R. A.; Dionne, G., Moral Hazard and Insurance Contracts, Contributions to Insurance Economics, (1992), Bostan: Kluwer Academic Publishers, Bostan
[66] Young, V. R., Optimal Insurance under Wang’s Premium Principle, Insurance Mathematics and Economics, 25, 2, 109-122, (1999) · Zbl 1156.62364
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.