Complete Monotonicity of the Representative Consumer's Discount Factor

A univariate real-valued function is said to be completely monotone if it takes positive values and alternate the signs of its higher order derivatives, starting from everywhere negative first derivatives. We prove that the representative consumer's discount factor of a continuous-time economy under uncertainty is a power function of some completely monotone function of time satisfying certain boundary conditions if and only if it may be derived from a group of consumers having constant and equal relative risk aversion, and constant and yet possibly unequal discount rates.Complete monotonicity, discount factor, discount rate, representative consumer, expected utility, time additivity, relative risk aversion, Bernstein's theorem

Complete Monotonicity of the Representative Consumer's Discount Factor

A univariate real-valued function is said to be completely monotone if it takes positive values and alternate the signs of its higher order derivatives, starting from everywhere negative first derivatives. We prove that the representative consumer's discount factor of a continuous-time economy under uncertainty is a power function of some completely monotone function of time satisfying certain boundary conditions if and only if it may be derived from a group of consumers having constant and equal relative risk aversion, and constant and yet possibly unequal discount rates.Complete monotonicity, discount factor, discount rate, representative consumer, expected utility, time additivity, relative risk aversion, Bernstein's theorem.

Heterogeneous Impatience in a Continuous-Time Model

In a continuous-time economy with complete markets, we study how the heterogeneity in the individual consumers' risk tolerance and impatience affects the representative consumer's risk tolerance and impatience. We derive some formulas, which indicate that the representative consumer's impatience decrease over time, and whether his risk tolerance increases or decreases over time depends on the sign of some weighted covariance between the individual consumers' cautiousness (derivative of risk tolerance with respect to own consumptions) and impatience. These results are then used to show that the short rate tends to decrease over time and the market price of risk is volatile in some special cases of heterogeneous economies.Endogenous Growth; Representative consumer, risk tolerance, impatience, state-price deflator, shortrate process, market price of risk

Heterogeneous Impatience in a Continuous-Time Model

In a continuous-time economy with complete markets, we show how the heterogeneity in the individual consumers' risk attitudes and impatience would affect the representative consumer's counterparts. Specifically, our formulas tell us how his risk tolerance and impatience will change over time, and how his impatience will be affected by the changes in aggregate consumption levels. Under the assumption of equal and constant relative risk aversion across individual consumers, we characterize his discount factor by means of a completely monotone function of time. These results are used to analyze short-rate processes.Representative consumer, expected utility, time additivity, multiplicative separability, impatience, risk tolerance, state-price deflator, short-rate process, complete monotonicity

Heterogeneous Impatience in a Continuous-Time Model

In a continuous-time economy with complete markets, we study how the heterogeneity in the individual consumers' risk tolerance and impatience affects the representative consumer's risk tolerance and impatience. We derive some formulas, which indicate that the representative consumer's impatience decrease over time, and whether his risk tolerance increases or decreases over time depends on the sign of some weighted covariance between the individual consumers' cautiousness (derivative of risk tolerance with respect to own consumptions) and impatience. These results are then used to show that the short rate tends to decrease over time and the market price of risk is volatile in some special cases of heterogeneous economies.Representative consumer, risk tolerance, impatience, state-price deflator, shortrate process, market price of risk

Bargaining Set and Anonymous Core without the Monotonicity Assumption

We give an example of an atomless exchange economy in which consumers'preference relations are not monotone and in which the bargaining set of Mas-Colell (1989) consists of all allocations satisfying resource constraints, although the set of all Walrasian equilibrium allocations, the core, and the anonymous core of Hara (2002) are all empty. We also give an equivalence theorem for the anonymous core when the preference relations may not be monotone.Atomless exchange economies, core, bargaining set, anonymous core, equivalence theorems

Heterogeneous Risk Attitudes in a Continuous-Time Model

We prove that every continuous-time model in which all consumers have time-homogeneous and time-additive utility functions and share a common probabilistic belief and a common discount rate can be reduced to a static model. This result allows us to extend some of the existing results on the representative consumer and risk-sharing rules in static models to continuous-time models. We show that the equilibrium interest rate is lower and more volatile than in the standard representative consumer economy, and that the individual consumption growth rates are more dispersed than is predicted from the first-order conditions.Heterogeneity, risk attitudes, hyperbolic absolute risk aversion, representative consumer, risk-sharing rules, mutual fund theorem, Ito's Lemma, interest rates.

An equilibrium existence theorem for atomless economies without the monotonicity assumption

We consider an exchange economy with an atomless space of consumers whose preference relations need not be monotone. We prove that if for every commodity, there is a group of consumers who regard it as potentially desirable, in a sense to be made precise, then there exists a competitive equilibrium, however small the proportion of such a group may be in the entire economy.

Effectively Complete Asset Markets with Multiple Goods and over Multiple Periods

Following LeRoy and Werner (2001), we propose a definition of effectively complete asset markets in a model with multiple goods and multiple periods, and establish the first and second welfare theorems in such markets. As applications of the first welfare theorem, we derive the sunspot irrelevance theorem of Mas-Colell (1992), and extend the no-retrade theorem of Judd, Kubler, and Schmedders (2003) and Kubler and Schmedders (2003) to the case where the asset prices need not be time-invariant Markov processes.Complete markets, effectively complete markets, welfare theorems, sunspot, Markov environment

Representative Consumer's Risk Aversion and Efficient Risk-Sharing Rules

We study the representative consumer's risk attitude and efficient risk-sharing rules in a single-period, single-good economy in which consumers have homogeneous probabilistic beliefs but heterogeneous risk attitudes. We prove that if all consumers have convex absolute risk tolerance, so must the representative consumer. We also identify a relationship between the curvature of an individual consumer's individual risk sharing rule and his absolute cautiousness, the first derivative of absolute risk-tolerance. Some consequences of these results and refinements of these results for the class of HARA utility functions are discussed