Abstract
We investigate competitive equilibria in a special type of incomplete markets, referred to as a comonotone market, where agents can only trade such that their risk allocation is comonotonic. The comonotone market is motivated by the no-sabotage condition. For instance, in a standard insurance market, the allocation of risk among the insured, the insurer and the reinsurers is assumed to be comonotonic a priori to the risk-exchange. Two popular classes of preferences in risk management and behavioral economics, dual utilities (DU) and rank-dependent expected utilities (RDU), are used to formulate agents’ objectives. We present various results on properties and characterization of competitive equilibria in this framework, and in particular their relation to complete markets. For DU-comonotone markets, we find the equilibrium in closed form and for RDU-comonotone markets, we find the equilibrium in closed form in special cases. The fundamental theorems of welfare economics are established in both the DU and RDU markets. We further propose an algorithm to numerically obtain competitive equilibria based on discretization, which works for both the DU-comonotone market and the RDU-comonotone market. Although the comonotone and complete markets are closely related, many of our findings are intriguing and in sharp contrast to results in the literature on complete markets in terms of existence, uniqueness, and closed-form solutions of the equilibria, and comonotonicity of the pricing kernel.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anthropelos, M., Kardaras, C.: Equilibrium in risk-sharing games. Finance Stoch. 21(3), 815–865 (2017)
Araujo, A., Chateauneuf, A., Faro, J.H.: Pricing rules and Arrow–Debreu ambiguous valuation. Econ. Theory 49, 1–35 (2012). https://doi.org/10.1007/s00199-011-0660-4
Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22(3), 265–290 (1954)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)
Barrieu, P., El Karoui, N.: Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 9(2), 269–298 (2005)
Beissner, P., Riedel, F.: Equilibria under Knightian price uncertainty. Econometrica 87(1), 37–64 (2019)
Bernard, C., He, X., Yan, J.A., Zhou, X.Y.: Optimal insurance design under rank-dependent expected utility. Math. Finance 25, 154–186 (2015)
Bewley, T.F.: Existence of equilibria in economics with infinitely many commodities. J. Econ. Theory 4(3), 514–540 (1972)
Boonen, T.J.: Competitive equilibria with distortion risk measures. ASTIN Bull. 45(3), 703–728 (2015)
Boonen, T.J.: Risk redistribution games with dual utilities. ASTIN Bull. 47(1), 303–329 (2017)
Borch, K.: Equilibrium in a reinsurance market. Econometrica 30, 424–444 (1962)
Cai, J., Lemieux, C., Liu, F.: Optimal reinsurance from the perspectives of both an insurer and a reinsurer. ASTIN Bull. 46(3), 815–849 (2016)
Cai, J., Liu, H., Wang, R.: Pareto-optimal reinsurance arrangements under general model settings. Insur. Math. Econ. 77, 24–37 (2017)
Carlier, G., Dana, R.-A.: Pareto efficient insurance contracts when the insurer’s cost function is discontinuous. Econ. Theory 21, 871–893 (2003). https://doi.org/10.1007/s00199-002-0281-z
Chabi-Yo, F.: Pricing kernels with stochastic skewness and volatility risk. Manag. Sci. 58(3), 624–640 (2012)
Chateauneuf, A., Dana, R.-A., Tallon, J.-M.: Optimal risk-sharing rules and equilibria with Choquet-expected-utility. J. Math. Econ. 34(2), 191–214 (2000)
Cheng, H.H.: Asset market equilibrium in infinite dimensional complete markets. J. Math. Econ. 20(1), 137–152 (1991)
Cheung, K.C., Sung, K., Yam, S.C.P.: Risk-minimizing reinsurance protection for multivariate risks. J. Risk Insur. 81(1), 219–236 (2014)
Chew, S.H., Karni, E., Safra, Z.: Risk aversion in the theory of expected utility with rank dependent probabilities. J. Econ. Theory 42, 370–381 (1987)
Dana, R.-A.: Existence and uniqueness of equilibria when preferences are additively separable. Econometrica 61(4), 953–957 (1993)
Dana, R.-A., Le Van, C.: Arbitrage, duality and asset equilibria. J. Math. Econ. 34(3), 397–413 (2000)
Dana, R.-A., Le Van, C.: Overlapping sets of priors and the existence of efficient allocations and equilibria for risk measures. Math. Finance 20(3), 327–339 (2010)
Dana, R.-A., Riedel, F.: Intertemporal equilibria with Knightian uncertainty. J. Econ. Theory 148, 1582–1605 (2013)
De Giorgi, E., Hens, T., Rieger, M.O.: Financial market equilibria with cumulative prospect theory. J. Math. Econ. 46(5), 633–651 (2010)
Delbaen, F.: Monetary Utility Functions. Osaka University Press, Osaka (2012)
Denneberg, D.: Non-additive Measure and Integral. Springer, Berlin (1994)
Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R.: Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, New York (2005)
Dybvig, P.H.: Distributional analysis of portfolio choice. J. Bus. 61(3), 369–393 (1988)
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R., Beleraj, A.: An academic response to Basel 3.5. Risks 2(1), 25–48 (2014)
Embrechts, P., Liu, H., Wang, R.: Quantile-based risk sharing. Oper. Res. 66(4), 936–949 (2018)
Embrechts, P., Liu, H., Mao, T., Wang, R.: Quantile-based risk sharing with heterogeneous beliefs. Math. Program. 181(2), 319–347 (2020)
Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 4th edn. Walter de Gruyter, Berlin (2016)
Heath, D., Ku, H.: Pareto equilibria with coherent measures of risk. Math. Finance 14(2), 163–172 (2004)
Heaton, J., Lucas, D.J.: Evaluating the effects of incomplete markets on risk sharing and asset pricing. J. Polit. Econ. 104(3), 443–487 (1996)
Hens, T., Reichlin, C.: Three solutions to the pricing kernel puzzle. Rev. Financ. 17, 1065–1098 (2013)
Huberman, G., Mayers, D., Smith Jr., C.W.: Optimal insurance policy indemnity schedules. ACA Trans. 14(2), 415–426 (1983)
Jarrow, R.: Heterogeneous expectations, restrictions on short sales, and equilibrium asset prices. J. Finance 35(5), 1105–1113 (1980)
Jin, H., Xia, J., Zhou, X.Y.: Arrow–Debreu equilibria for rank-dependent utilities with heterogeneous probability weighting. Math. Finance 29(3), 898–927 (2019)
Jouini, E., Schachermayer, W., Touzi, N.: Optimal risk sharing for law invariant monetary utility functions. Math. Finance 18(2), 269–292 (2008)
Landsberger, M., Meilijson, I.: Co-monotone allocations, Bickel–Lehmann dispersion and the Arrow–Pratt measure of risk aversion. Ann. Oper. Res. 52(2), 97–106 (1994)
Ludkovski, M., Rüschendorf, L.: On comonotonicity of Pareto optimal risk sharing. Stat. Probab. Lett. 78(10), 1181–1188 (2008)
Ludkovski, M., Young, V.R.: Optimal risk sharing under distorted probabilities. Math. Financ. Econ. 2(2), 87–105 (2009)
Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, New York (1995)
Moulin, H.: Egalitarian-equivalent cost sharing of a public good. Econometrica 55(4), 963–976 (1987)
Moulin, H., Shenker, S.: Serial cost sharing. Econometrica 60(5), 1009–1037 (1992)
Müller, A., Scarsini, M., Tsetlin, I., Winkler, R.L.: Between first and second-order stochastic dominance. Manag. Sci. 63(9), 2933–2947 (2017)
Nielsen, L.T.: Asset market equilibrium with short-selling. Rev. Econ. Stud. 56(3), 467–473 (1989)
Quiggin, J.: A theory of anticipated utility. J. Econ. Behav. Org. 3(4), 323–343 (1982)
Quiggin, J.: Generalized Expected Utility Theory: The Rank-Dependent Model. Kluwer, Dohdridge (1993)
Rieger, M.O., Wang, M.: Cumulative prospect theory and the St. Petersburg paradox. Econ. Theory 28, 665–679 (2006). https://doi.org/10.1007/s00199-005-0641-6
Rüschendorf, L.: Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Heidelberg (2013)
Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986)
Schmidt, U., Zank, H.: Risk aversion in cumulative prospect theory. Manag. Sci. 54, 208–216 (2008)
Tsanakas, A., Christofides, N.: Risk exchange with distorted probabilities. ASTIN Bull. 36(1), 219–243 (2006)
Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5(4), 297–323 (1992)
Wang, S.S., Young, V.R., Panjer, H.H.: Axiomatic characterization of insurance prices. Insur. Math. Econ. 21(2), 173–183 (1997)
Werner, J.: Arbitrage and the existence of competitive equilibrium. Econometrica 55(6), 1403–1418 (1987)
Wilson, R.: The theory of syndicates. Econometrica 36, 119–132 (1968)
Xia, J., Zhou, X.Y.: Arrow–Debreu equilibria for rank-dependent utilities. Math. Finance 26(3), 558–588 (2016)
Xu, Z.Q., Zhou, X.Y., Zhuang, S.C.: Optimal insurance under rank-dependent utility and incentive compatibility. Math. Finance 29(2), 659–692 (2019)
Yaari, M.E.: The dual theory of choice under risk. Econometrica 55(1), 95–115 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Fangda Liu is supported by the NNSF of China (No. 11601540) and the Natural Sciences and Engineering Research Council of Canada (RGPIN-2020-04717, DGECR-2020-00340). Ruodu Wang is supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Boonen, T.J., Liu, F. & Wang, R. Competitive equilibria in a comonotone market. Econ Theory 72, 1217–1255 (2021). https://doi.org/10.1007/s00199-020-01319-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-020-01319-4