Decreasing absolute prudence: Characterization and applications to second-best risk sharing

In this paper, we provide two basic properties of utility functions u which exhibit decreasing absolute prudence i.e. (− u/u″)′ ≤ 0. These properties are used to examine the allocation of risks in an economy when some agents bear non-transferable risks. We show that it is fair Pareto-efficient to let those with a non-transferable risk bear relatively less of the transferable risk in the economy if and only if absolute prudence is decreasing. In another model, there is a complete set of contingent markets, but some agents have no direct access to them. We examine the fair efficient allocation of risk in a pool gathering a trader and a non-trader. Decreasing absolute prudence provides an upper bound to the share of the pool's risk that should be borne by the trader.     

Skewed background risks and higher-order risk preferences: prudent versus temperate behavior

In recent economic experiments, lotteries with left-skewed background risks elicit more prudent choices than lotteries with right-skewed background risks. In this letter, we use an expected utility framework to show that a skewed zero-mean background risk may induce not only prudent but also temperate behaviour. We suggest that the experimental findings could also be due to temperance, rather than merely to prudence.     

Insurance with Frequency Trading: A Dynamic Analysis of Efficient Insurance Markets

This paper extends existing insurance results on the type of insurance contracts needed for insurance market efficiency to a dynamic setting. It introduces continuously open markets that allow for more efficient asset allocation. It also estimates the role of preferences and endowments in the classification of risks, which is done primarily in terms of the actuarial properties of the underlying risk process. The paper further extends insurability to include correlated and catastrophic events. Under these very general conditions the paper defines a condition that determines whether a small number of standard insurance contracts (together with aggregate assets) suffice to complete markets or one needs to introduce such assets as mutual insurance. Journal of Economic Literature Classification Numbers: D81, D99, G11.     

An equilibrium model of a second economy market in a centrally planned economy

We study a CPE in which an identical good is sold on the official market (OM) and a “second economy” market (SEM ). Planners set parameters. Managers divide inputs between markets to maximize expected utility of wealth. Consumers are expected utility maximizers who purchase the good on the OM or SEM. On the OM, excess demand exists at the non-Walrasian price; delivery date is stochastic. The SEM offers immediate availability. Our solution concept involves the rational expectations of managers, the consistency of consumers' decisions, and a market-clearing SEM. We solve for SEM price and supply and investigate various comparative statics.     

On the optimal design of insurance contracts with the restriction of equity risk

This paper focuses on the optimal design of insurance contracts with the restriction of equity risk by Arrow's optimal insurance model [2] and Zhou's optimal insurance model [4]. The insured aims to maximize his/her expected utility of terminal wealth, under the constraint that the insurer wishes to control his/her risk. We establish an optimal insurance model that restrains underwriting risk and investment risk simultaneously. We solve this problem and obtain the results which improve and extend the results given by Arrow [2] and Zhou [4].     

The demand for a risky asset in the presence of a background risk

We examine the demand for a risky asset in the presence of two risks: a financial risk and a background risk which need not be financial. First, we compute the necessary and sufficient condition for a positive demand for a risky asset, showing that it depends on two terms capturing respectively the direct effect of risk premium and the dependence between the two risks. Second, we develop higher order expectation dependence concept and show that the more information about the sign of higher cross derivatives of the utility function we have, the weaker dependence conditions on distribution we achieve.     

Endogenous market structures and contract theory: Delegation, principal-agent contracts, screening, franchising and tying

I study the role of unilateral strategic contracts for firms active in markets with price competition and endogenous entry. Traditional results change substantially when the market structure is endogenous rather than exogenous. They concern (1) contracts of managerial delegation to non-profit maximizers, (2) incentive principal-agent contracts in the presence of moral hazard on cost-reducing activities, (3) screening contracts in case of asymmetric information on the productivity of the managers, (4) vertical contracts of franchising in case of hold-up problems and (5) tying contracts by monopolists competing also in secondary markets. Firms use always these contracts to strengthen price competition and manage to obtain positive profits in spite of free entry.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Parametric characterizations of risk aversion and prudence

Summary. Our first main result says that whether one decision maker is more risk averse than another can be determined from their attitudes toward a given two-parameter family of risks. When all risks belong to this family, risk aversion can be compared even when initial wealth is random. Our second main result solves a long-standing problem in mean-variance analysis: what is the interpretation of the concavity of utility as a function of mean and variance? We show that in the case of normal distributions, this utility function is concave if and only if the agent has decreasing prudence. Received: July 29, 1996; revised: October 2, 1998     

On the theory of risk aversion and the theory of risk

Relationships between the theory of risk aversion and the theory of risk have been noted by Diamond and Stiglitz [12], Kihlstrom and Mirman [24], and Leland [29]. The main result of this paper is a characterization of the Pratt and Kihlstrom-Mirman [24, 46] relation “more risk averse” between utility functions in terms of a stochastic dominance or “riskier” relation between certain probability measures derived from these utility functions. This result is used in the comparison of risk-adjusted portfolio yields to extend to the several risky asset case the intuition provided by the Arrow-Pratt [1, 46] portfolio theorem that “more risk averse” means “will take less risk.”     

Stochastic dominance and behavior towards risk: The market for Internet stocks

Internet stocks registered large gains in the late 1990s, followed by large losses from early 2000. Using stochastic dominance theory, we infer how investor risk preferences have changed over this cycle, and relate our findings to utility theory and behavioral finance. Our major findings are as follows. First, risk averters and risk seekers show a distinct difference in preference for Internet versus “old economy” stocks. This difference is most evident during the bull market period (1998–2000) where Internet stocks stochastically dominate old economy stocks for risk seekers but not risk averters. In the bear market, risk averters show an increased preference for old economy stocks, while risk seekers show a reduced preference for Internet stocks. These results are inconsistent with prospect theory and indicate that investors exhibit reverse S-shaped utility functions.     

Unemployment Risk, Labour Force Participation and Savings

This paper analyses labour force participation and precautionary savings in the presence of risks of being fired or failing to secure a job offer when out of the labour market. We use a finite horizon framework with two employment states and a stage utility function which is CARA in consumption but non-separable in leisure. The results are that there is precautionary labour force participation: employment risk lowers the reservation wage; generally it also reduces consumption. However due to the non-separability assumption, precautionary savings, as usually defined, may not be positive. We characterize the reservation wage and contrast the results with those in which the stage utility is additive in leisure and consumption. We extend the analysis to study the effects of cyclical variation in employment risk, of stochastic future wage rates, and of adding a third employment state of search.     

On relative and partial risk attitudes: theory and implications

This paper develops context-free interpretations for the relative and partial Nth degree risk attitude measures and show that various conditions on theses measures are utility characterizations of the effects of scaling general stochastic changes in different settings. It is then shown that these characterizations can be applied to generalize comparative statics results in a number of important problems, including precautionary savings, optimal portfolio choice, and competitive firms under price uncertainty.     

Background risk in generalized expected utility theory

Summary. In this paper, it is shown that, for a wide range of risk-averse generalized expected utility preferences, independent risks are complementary, contrary to the results for expected utility preferences satisfying conditions such as proper and standard risk aversion. Received: August 10, 2001; revised version: June 18, 2002 RID="*" ID="*"I thank Simon Grant and an anonymous referee for helpful comments and criticism. This research was supported by an Australian Research Council Senior Fellowship and Australian Research Council Large Grant A79800678.     

Precautionary saving in the presence of other risks: a comment

Courbage and Rey (Econ Theory 32:417–424 2007) analyse precautionary saving in the presence of a background risk under specific sets of assumptions on the form of income risk and background risk. Three cases are examined: the case of independent risks, the case of Bernoulli-distributed random variables and the case of risk first-degree stochastic. For each of these cases Courbage and Rey compute the specific sets of conditions related to precautionary saving. This comment shows that some of their conclusions are partially incorrect.      

Restrictions and identification in a multidimensional risk-sharing problem

We consider $H$ expected utility maximizers that have to share a risky aggregate multivariate endowment $X\in {\mathbb {R}}^{N}$ and address the following two questions: does efficient risk-sharing imply restrictions on the form of individual consumptions as a function of $X$ ? Can one identify the individual utility functions from the observation of the risk-sharing? We show that when $H\ge \frac{2N}{N-1}$ efficient risk sharings have to satisfy a system of nonlinear PDEs. Under an additional rank condition, we prove an identification theorem.     

Strategy-proof contract auctions and the role of ties

A contract auction establishes a contract between a center and one of the bidders. As contracts may describe many terms, preferences over contracts typically display indifferences. The Qualitative Vickrey Auction (QVA) selects the best contract for the winner that is at least as good for the center as any of the contracts offered by the non-winning players. When each bidder can always offer a contract with higher utility for the center at an arbitrarily small loss of her own utility, the QVA is the only mechanism that is individually rational, strategy-proof, selects stable outcomes, and is Pareto efficient. For general continuous utility functions, a variant of the QVA involving fixed tie-breaking is strategy-proof and also selects stable outcomes. However, there is no mechanism in this setting that in addition also selects Pareto efficient outcomes.     

Banks' risk race: A signaling explanation

Many observers argue that one of the major causes of the 2007–2009 recession was the abnormal accumulation of risk by banks. This paper provides a signaling explanation for this race for risk. If banks' returns can be observed while risk cannot, the less efficient banks can hide their type by taking more risks and paying the same returns as the more efficient banks. The latter can signal themselves by taking even higher risks and delivering bigger returns. The game presents several equilibria that are all characterized by excessive risk taking as compared to the perfect information case.     

Efficient risk sharing,non-marketable labor income and fixed-wage contracts

This paper seeks to explain fixed-wage labor contracts. The traditional rationale that fixed wages represent an implicit sale of ‘wage insurance’ by risk-neutral firms to risk-averse workers is rejected as being incompatible with the fact that firms are owned by risk-averse investors. Instead, it is shown that fixed-wage contracts might arise from the non-marketability of labor income. When human capital is not marketable, it becomes optimal to shift all the risk in production onto the firm, since trading in equity markets enables efficient allocation of the uncertainty. The fixed-wage contract shifts the risk to equity owners and in fact replicates the first-best equilibrium that would emerge if individuals were paid their realized marginal product and allowed to trade shares in human capital.     

Non-discriminating renegotiation in a competitive insurance market

A variant of the Rothschild-Stiglitz model of a competitive insurance market is considered, where each uninformed firm is allowed to renegotiate the contracts that its customers initially sign, subject to the restriction that renegotiated contracts be offered to all the firm's customers. Such non-discriminating renegotiation is shown to weaken the profitability of cream skimming to the extent that there exists a unique equilibrium outcome. This outcome is that of Miyazaki and Spence i.e., the incentive-compatible pair of zero-profit contracts, if efficient; and the incentive-compatible, zero-profit pair of contracts maximizing low-risk utility, otherwise.