Core of convex distortions of a probability

This paper characterizes the core of a differentiable convex distortion of a probability measure on a nonatomic space by identifying it with the set of densities which dominate the derivative of the distortion, for second order stochastic dominance. The densities that have the same distribution as the derivative of the distortion are the extreme points of the core. These results are applied to the differentiability of a Yaari's or Rank Dependent Expected utility function. The superdifferential of a Choquet integral at any point is fully characterized. Examples of use of these results in simple models where some agent is a RDEU maximizer are given.     

Context matters

Eliciting the level of risk aversion of experimental subjects is of crucial concern to experimenters. In the literature there are a variety of methods used for such elicitation; the concern of the experiment reported in this paper is to compare them. The methods we investigate are the following: Holt–Laury price lists; pairwise choices, the Becker–DeGroot–Marschak method; allocation questions. Clearly their relative efficiency in measuring risk aversion depends upon the numbers of questions asked; but the method itself may well influence the estimated risk-aversion. While it is impossible to determine a ‘best’ method (as the truth is unknown) we can look at the differences between the different methods. We carried out an experiment in four parts, corresponding to the four different methods, with 96 subjects. In analysing the data our methodology involves fitting preference functionals; we use four, Expected Utility and Rank-Dependent Expected Utility, each combined with either a CRRA or a CARA utility function. Our results show that the inferred level of risk aversion is more sensitive to the elicitation method than to the assumed-true preference functional. Experimenters should worry most about context.     

Stochastically independent randomization and uncertainty aversion

Summary.   This paper proposes a preference-based condition for stochastic independence of a randomizing device in a product state space. This condition is applied to investigate some classes of preferences that allow for both independent randomization and uncertainty or ambiguity aversion (a la Ellsberg). For example, when imposed on Choquet Expected Utility (CEU) preferences in a Savage framework displaying uncertainty aversion in the spirit of Schmeidler [27], it results in a collapse to Expected Utility (EU). This shows that CEU preferences that are uncertainty averse in the sense of Schmeidler should not be used in settings where independent randomization is to be allowed. In contrast, Maxmin EU with multiple priors preferences continue to allow for a very wide variety of uncertainty averse preferences when stochastic independence is imposed. Additionally, these points are used to reexamine some recent arguments against preference for randomization with uncertainty averse preferences. In particular, these arguments are shown to rely on preferences that do not treat randomization as a stochastically independent event. Received: February 10, 2000; revised version: March 30, 2000     

Wage-vacancy contracts and coordination frictions

We consider a directed search model with risk-averse workers and risk-neutral entrepreneurs who can set up firms that post wage-vacancy contracts, i.e., contracts where firms can make payments to more than one applicant, and where the payments can be different for each applicant and be contingent on the number of applicants. We establish that the type of contracts the literature focuses on are not offered if firms can post wage-vacancy contracts. We show that there exists an equilibrium satisfying a Monotonic Expected Utility property which is efficient. Furthermore, we investigate the role of wage-vacancy contracts on welfare and competition.     

Subjective ambiguity, expected utility and Choquet expected utility

Summary. Using the Savage set up, this paper provides a simple axiomatization of the Choquet Expected Utility model where the capacity is an inner measure. Two attractive features of the model are its specificity and the transparency of its axioms. The key axiom states that the decision-maker uses unambiguous acts to approximate ambiguous ones. In addition, the notion of ‘ambiguity’ is subjective and derived from preferences. Received: March 23, 2000; revised version: April 24, 2001     

A simple axiomatization and constructive representation proof for choquet expected utility

>P>Summary. We provide a set of simple and intuitive set of axioms that allow for a direct and constructive proof of the Choquet Expected Utility representation for decision making under uncertainty. Received: October 29, 2002; revised version: November 13, 2002 RID="*" ID="*" We thank Matthew Ryan for very useful comments and suggestions on related work and for encouraging us to write this note. Correspondence to: S. Grant     

Testing the Gaussian and Student's t copulas in a risk management framework

This paper introduces a semiparametric framework for selecting either a Gaussian or a Student's t copula in a d-dimensional setting. We compare the two models using four different approaches: (i) four goodness-of-fit graphical plots, (ii) a bootstrapped correlation matrix generated in each scenario with the empirical correlation matrix used as a benchmark, (iii) Value-at-Risk (VaR) and Expected Shortfall (ES) as risk measures, and (iv) co-Value-at-Risk (CoVaR) and Marginal Expected Shortfall (MES) as co-risk measures. We illustrate this four-step procedure using a portfolio of daily returns of six international stock indices. The VaR results confirm that the t-based copula model is an attractive alternative to the Gaussian. The ES analysis is less conclusive, and indicates that risk managers should jointly use the risk measure as well as the copula model. The results highlight the importance of promoting stress testing rather than ES in the risk management industry, particularly in the aftermath of a financial crisis.     

Foundations of neo-Bayesian statistics

We study an axiomatic model of preferences, which contains as special cases Subjective Expected Utility, Choquet Expected Utility, Maxmin and Maxmax Expected Utility and many other models. First, we give a complete characterization of the class of functionals representing these preferences. Then, we show that any such functional can be represented as a Choquet integral

where is the canonical mapping from the space of bounded Σ-measurable functions into the space of weak^*-continuous affine functions on a weak*-compact, convex set of probability measures on Σ. Conversely, any preference relation defined by means of such functionals satisfies the axioms of the model we study. Different properties of the capacity give rise to different models. Our result shows that the idea of Choquet integration is general enough to embrace all the models mentioned above. In doing so, it widens the range of applicability of well-known procedures in robust statistics theory such as the Neyman–Pearson lemma for capacities [P.J. Huber, V. Strassen, Minimax tests and the Neyman–Pearson lemma for capacities, Ann. Statist. 1 (1973) 251–263], Bayes' theorem for capacities [J.B. Kadane, L. Wasserman, Bayes' theorem for Choquet capacities, Ann. Statist. 18 (1990) 1328–1339] or of results like the Law of Large numbers for capacities [F. Maccheroni, M. Marinacci, A strong law of large numbers for capacities, Ann. Probab. 33 (2005) 1171–1178].     

Option price without expected utility

We consider the meaning of the option price, commonly acknowledged as the preferred ex ante welfare measure, in the rank-dependent expected utility (RDEU) framework. The importance of this pertains to performing benefit-cost analysis when RDEU maximizers are prevalent in society.     

Ambiguous events and maxmin expected utility

We study the properties associated to various definitions of ambiguity [L.G. Epstein, J. Zhang, Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001) 265-306; P. Ghirardato et al., Differentiating ambiguity and ambiguity attitude, J. Econ. Theory 118 (2004) 133-173; K. Nehring, Capacities and probabilistic beliefs: a precarious coexistence, Math. Soc. Sci. 38 (1999) 197-213; J. Zhang, Subjective, ambiguity, expected utility and Choquet expected utility, Econ. Theory 20 (2002) 159-181] in the context of Maximin Expected Utility (MEU). We show that each definition of unambiguous events produces certain restrictions on the set of priors, and completely characterize each definition in terms of the properties it imposes on the MEU functional. We apply our results to two open problems. First, in the context of MEU, we show the existence of a fundamental incompatibility between the axiom of “Small unambiguous event continuity” (Epstein and Zhang, 2001) and the notions of unambiguous event due to Zhang (2002) and Epstein and Zhang (2001). Second, we show that, in the context of MEU, the classes of unambiguous events according to either Zhang (2002) or Epstein and Zhang (2001) are always λ-systems. Finally, we reconsider the various definitions in light of our findings, and identify some new objects (Z-filters and EZ-filters) corresponding to properties which, while neglected in the current literature, seem relevant to us.     

More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model

Summary. This paper studies monotone risk aversion, the aversion to monotone, mean-preserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f. Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel and Lehmann [3,4] in Non-parametric Statistics. We present a characterization of the pairs (u,f) of monotone risk averse decision makers, based on an index of greediness G _u of the utility function u and an index of pessimism P _f of the probability perception function f: the decision maker is monotone risk averse if and only if . The index of greediness (non-concavity) of u is the supremum of taken over . The index of pessimism of f is the infimum of taken over 0 < v < 1. Thus, , with G _u = 1 iff u is concave. If then , i.e., f is majorized by the identity function. Since P _f = 1 for Expected Utility maximizers, forces u to be concave in this case; thus, the characterization of risk aversion as is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if P _f = 1.Received: 10 April 2001, Revised: 18 November 2003, JEL Classification Numbers: D81. Correspondence to: Michéle CohenAlain Chateauneuf, Michéle Cohen, Isaac Meilijson: We are most grateful to Mark Machina, Peter Wakker and two anonymous referees for very helpful suggestions and comments.     

Fair Income Tax

In a model where agents have unequal skills and heterogeneous preferences over consumption and leisure, we look for the optimal tax on the basis of efficiency and fairness principles and under incentive-compatibility constraints. The fairness principles considered here are: (1) a weak version of the Pigou–Dalton transfer principle; (2) a condition precluding redistribution when all agents have the same skills. With such principles we construct and justify specific social preferences and derive a simple criterion for the evaluation of income tax schedules. Namely, the lower the greatest average tax rate over the range of low incomes, the better. We show that, as a consequence, the optimal tax should give the greatest subsidies to the working poor (the agents having the lowest skill and choosing the largest labour time).     

Stochastic Choice and the Allocation of Cognitive Effort

Data from a risky choice experiment are used to estimate a fully parametric stochastic model of risky choice. As is usual with such analyses, Expected Utility Theory is rejected in favour of a form of Rank Dependent Theory. Then an estimate of the risk aversion parameter is deduced for each subject, and this is used to construct a measure of the “closeness to indifference' of each subject in each choice problem. This measure is then used as an explanatory variable in a random effects model of decision time, with other explanatory variables being the complexity of the problem, the financial incentives, and the amount of experience accumulated at the time of performing the task. The most interesting finding is that significantly more effort is allocated to problems in which subjects are close to indifference. This presents us with another reason (in addition to statistical information considerations) why such tasks should play a prominent role in experiments. JEL Classification: C23, C91, D81     

Category index

We compare the R&;D decision of a second best social surplus maximizer to that of a profit maximizing monopolist in a second best world, namely one in which a monopolist makes production decisions. We show that a monopolist underinvests in R&;D relative to a second best social surplus maximizer. We show that the expected present value social surplus under a monopoly situation increases with the production technology except at the technology level where a monopolist stops doing R&;D.     

Foreign Direct Investment, inequality, and growth

This paper examines the interactions between Foreign Direct Investment (FDI), inequality, and growth, both from an empirical and a theoretical point of view. Using a panel of 119 developing countries, we observe that FDI promotes both inequality and growth, and tends to reduce the share of agriculture to GDP in the recipient country. We then set up a growth model of a dual economy in which the traditional (agricultural) sector uses a diminishing returns technology, while FDI is the engine of growth in the modern (industrial) sector. The main predictions of the model are consistent with the stylized facts observed in the data.     

Transfer principles in income redistribution

Dalton's principle of transfer for income redistribution says that society will be better off when a unit of income in transferred from a richer to a poorer individual. This paper investigates the conjunction of a natural extension of Dalton's principle — which says that any combination of a socially desirable transfer with its inverse at uniformly higher levels of income will have positive social benefit — and a traditional utilitarian-egalitarian formulation of social welfare. The transfer principles involved in the extension of Dalton's rule correspond to stochastic dominance relations between income distributions with a common mean, and lead to the conclusion that, when ? and g are income distributions, the social welfare of ? is larger than that of g if $\text{Σe}{}^{\mathrm{-ax}}\text{?(x)<Σe}{}^{\mathrm{-ax}}\text{g(x)}$ for all positive a. This conclusion is intimately connected with a measure of income inequality studied by Kolm.     

个体特征对人才流动风险偏好的影响研究

在研究影响组织人才流动因素的基础上,运用预期效用理论分析人才流动过程中存在的风险,认为个体风险偏好的存在影响人才的流动决策。通过对杭州高新产业集群企业人才流动情况的实证分析,指出了影响人才流动风险偏好的因素,并提出了企业留住人才的对策。     

A simple dynamic general equilibrium model

We analyse a single sector economy with H > 1 infinitely-lived agents that operate in a continuous-time framework. Utility functions are recursive but not additive. Both efficient and perfect foresight competitive equilibrium allocations are considered. The existence and stability of such allocations are investigated locally, i.e., in a neighbourhood of steady-state allocations. The model is shown to be useful for explaining the distribution of wealth and consumption across agents, and for analysing the way in which wealth redistribution can affect the dynamics of aggregate economic variables.     

Staying ahead and getting even: Risk attitudes of experienced poker players

Frequent online poker players with extensive experience calculating probabilities and expected values might be expected to behave as Expected Utility maximizers, in that small shocks to their wealth would not affect risk preferences (Rabin, 2000). By contrast, reference-dependent loss aversion (as in Prospect Theory) (Koszegi and Rabin, 2006, Kahneman and Tversky, 1979) predicts that risk aversion decreases as wealth moves away from the reference point in either direction. In terms of continuing to play, as well as a more aggressive playing style, we find strong evidence for the break-even effect, the increased pursuit of risk as a player is losing within a session. Players' behavior also appears consistent with existing evidence on reference-dependent labor supply, in their tendency to reduce effort and risk-taking in response to being ahead. Our findings provide evidence for reference-dependent behavior in a flexible, high-skilled setting, under conditions of well-understood monetary risk.