Arrow–Debreu Equilibria for Rank‐Dependent Utilities with Heterogeneous Probability Weighting

30 Pages Posted: 29 May 2020

See all articles by Hanqing Jin

Hanqing Jin

Mathematical Institute; Oxford-Nie Financial Big Data Laboratory; St. Peter's College

Jianming Xia

Chinese Academy of Sciences (CAS) - Academy of Mathematics and Systems Sciences

Xun Yu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

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Date Written: July 2019

Abstract

We study Arrow–Debreu equilibria for a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents having heterogeneous probability weighting and outcome utility functions. In particular, we allow the economy to have a mix of expected utility agents and rank‐dependent utility ones, with nonconvex probability weighting functions. The standard approach for convex economy equilibria fails due to the incompatibility with second‐order stochastic dominance. The representative agent approach devised in Xia and Zhou (2016) does not work either due to the heterogeneity of the weighting functions. We overcome these difficulties by considering the comonotone allocations, on which the rank‐dependent utilities become concave. Accordingly, we introduce the notion of comonotone Pareto optima, and derive their characterizing conditions. With the aid of the auxiliary problem of price equilibria with transfers, we provide a sufficient condition in terms of the model primitives under which an Arrow–Debreu equilibrium exists, along with the explicit expression of the state‐price density in equilibrium. This new, general sufficient condition distinguishes the paper from previous related studies with homogeneous and/or convex probability weightings.

Keywords: Arrow–Debreu equilibrium, comonotone Pareto optimum, price equilibrium with transfers, probability weighting, rank‐dependent utility, state‐price density

JEL Classification: G11

Suggested Citation

Jin, Hanqing and Xia, Jianming and Zhou, Xunyu, Arrow–Debreu Equilibria for Rank‐Dependent Utilities with Heterogeneous Probability Weighting (July 2019). Mathematical Finance, Vol. 29, Issue 3, pp. 898-927, 2019, Available at SSRN: https://ssrn.com/abstract=3613890 or http://dx.doi.org/10.1111/mafi.12200

Hanqing Jin (Contact Author)

Mathematical Institute ( email )

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Oxford, oxfordshire OX2 6GG
United Kingdom

HOME PAGE: http://www.maths.ox.ac.uk

Oxford-Nie Financial Big Data Laboratory ( email )

Andrew Wiles Building
Woodstock Road
Oxford, Oxfordshire OX2 6GG
United Kingdom

St. Peter's College ( email )

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Oxford, Oxfordshire OX1 2DL
United Kingdom

HOME PAGE: http://www.spc.ox.ac.uk

Jianming Xia

Chinese Academy of Sciences (CAS) - Academy of Mathematics and Systems Sciences ( email )

Zhong-Guan-Cun-Dong-Lu 55, Haidian District
Beijing 100190
China

Xunyu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

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