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Representative consumer’s risk aversion and efficient risk sharing rules, mimeo
, 2001
"... We investigate the representative consumer’s risk attitude and efficient risksharing rules in a singleperiod, singlegood economy in which consumers have homogeneous probabilistic belief but heterogeneous risk attitudes. Of our two formulas regarding the representative consumer’s risk attitude, th ..."
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Cited by 17 (3 self)
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We investigate the representative consumer’s risk attitude and efficient risksharing rules in a singleperiod, singlegood economy in which consumers have homogeneous probabilistic belief but heterogeneous risk attitudes. Of our two formulas regarding the representative consumer’s risk attitude, the first one implies that if all consumers ’ relative risk aversions are decreasing, then so is the representative consumer’s counterpart; and if, moreover, the former are unequal, then the latter is strictly decreasing. The second formula implies the analogous result on the convexity of absolute risk tolerance (the reciprocal of absolute risk aversion). We also provide a formula relating the curvature of an individual consumer’s risk sharing rule to his cautiousness (the derivative of the absolute risk tolerance). We analyze the limits of the representative consumer’s risk attitude and individual consumers ’ risksharing rules as the aggregate consumption levels are taken to the upper and lower bounds (which may be infinite and negative infinite). Refinements of these results for the case of linear risk tolerances, extensions to multiperiod economies, and implications on asset pricing and portfolio insurance are also given. Chiaki Hara is at the Faculty of Economics and Politics, University of Cambridge. His email addresss is
Chiaci: “Necessary and Sufficient Conditions for the Efficient RiskSharing Rules and the Representative Consumer’s Utility Function
, 2006
"... We show that for every collection of increasing risksharing rules, one for each consumer, and for every increasing and concave expected utility function, there exists a collection of increasing and concave expected utility functions for which the given risksharing rules are efficient and the given ..."
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Cited by 3 (1 self)
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We show that for every collection of increasing risksharing rules, one for each consumer, and for every increasing and concave expected utility function, there exists a collection of increasing and concave expected utility functions for which the given risksharing rules are efficient and the given utility function coincides with the corresponding representative consumer’s utility function. This result shows that the efficiency property imposes no restriction on the risksharing rules beyond comonotonicity, or on the representative consumer’s utility function beyond monotonicity and risk aversion. We also obtain contrasting results when the individual consumers are assumed to exhibit constant relative risk aversion.
Complete Monotonicity of the Representative Consumer’s Discount Factor
, 2007
"... A univariate realvalued 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 continuoustime economy under ..."
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Cited by 2 (2 self)
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A univariate realvalued 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 continuoustime 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.
Heterogeneous impatience in a continuoustime model, working paper
, 2007
"... In a continuoustime 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 de ..."
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Cited by 1 (1 self)
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In a continuoustime 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.
4 We are grateful for their helpful comments to Kazunori Araki, Jeremy Edwards, Günter Franke,
, 2006
"... the editors and referees for their extremely valuable suggestions on both the contents and exposition of the paper. 5 The genesis of this paper is as follows. Huang (2000a, 2000b) had first obtained most of the results in this paper. Working independently, Hara and Kuzmics established similar result ..."
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the editors and referees for their extremely valuable suggestions on both the contents and exposition of the paper. 5 The genesis of this paper is as follows. Huang (2000a, 2000b) had first obtained most of the results in this paper. Working independently, Hara and Kuzmics established similar results in a series of manuscripts since December 2001. These outputs are now merged into this paper. Its exposition We study the representative consumer’s risk attitude and efficient risksharing rules in a singleperiod, singlegood 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 risktolerance. Furthermore, we discuss some consequences of these results and refinements of these results for the class of HARA utility functions.
The Dynamics of Optimal Risk Sharing∗
, 2006
"... In this paper we study a dynamic contracting problem of optimal risksharing between a principal and an agent who invest in a common constantreturnstoscale risky venture. Investment flowreturns follow a geometric Brownian motion and the two agents ’ riskpreferences are represented by additivel ..."
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In this paper we study a dynamic contracting problem of optimal risksharing between a principal and an agent who invest in a common constantreturnstoscale risky venture. Investment flowreturns follow a geometric Brownian motion and the two agents ’ riskpreferences are represented by additively separable utility functions exhibiting constant relative riskaversion (CRRA). Principal and agent have different coefficients of relative riskaversion. In any time period they invest their wealth in the risky venture and optimally share the underlying return risk and termination risk. When the project ends the two individuals divide the accumulated proceeds as specified in the risksharing contract and consume their final accumulated wealth. The paper characterizes risksharing formulae that approximate the optimal risksharing rules. 1
Aggregation of StateDependent Utilities
, 2008
"... In an exchange economy under uncertainty populated by consumers having statecontingent utility functions, we analyze the nature of the efficient risksharing rules and the representative consumer’s statecontingent utility function. We show that the representative consumer’s responsiveness to sta ..."
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In an exchange economy under uncertainty populated by consumers having statecontingent utility functions, we analyze the nature of the efficient risksharing rules and the representative consumer’s statecontingent utility function. We show that the representative consumer’s responsiveness to state variables will typically depend on aggregate consumption levels even when the individual consumers ’ responsiveness do not depend on own consumptions. We also find that the heterogeneity in the individual consumers ’ responsiveness to state variables gives rise to a “convexifying effect ” on the representative consumer’s utility function, in a sense to be made precise. We also present applications of this result to the cases of heterogeneous beliefs and heterogeneous impatience.