Contract Bargaining with a Risk-Averse Agent
-
Sanxi Li
Abstract
This article is the first to study a bargaining model in a moral hazard framework where the principal is risk neutral and the agent is risk averse. We show that the power of incentives increases with the agent’s bargaining power if the contracts induce a high effort. However, under reasonable assumptions about the agent’s utility function, the contracts induce a high effort less often as the agent’s bargaining power increases. As for the social welfare, we are surprised to find that a utilitarian, who cares about the sum of the two parties’ certainty equivalents, is worse off as the agent’s bargaining power increases. These results are in sharp contrast to the literature, which features risk-neutral agents protected by limited liability.
Appendix
Proof of Corollary 2
Denote 
, which is increased as the agent’s bargaining power increases. Hence, we only need to show that both w and b increase with 
. From the Proposition 2, we have
Therefore,
and
where the last inequality holds, because 
is an increasing function.
Proof of Lemma 3
We want to show that 
. From propositions 2 and 3, we know 
From the corollary’s proof, we know that
It follows that
Denote 
. We will show that 
is strictly convex if 
.
Therefore,
Denote 
, it follows that
where the first inequality is due to the fact that 
is strictly convex, and the second inequality is due to the fact that 
is a decreasing function.
Proof of Corollary 5
CARA implies that 
, and therefore,
IARA implies
DARA is equivalent to 
, and, therefore, the condition 
may be violated. Consider a special class of DARA: 
, where 
. One can easily derive that 
and 
. It follows immediately that 
is equivalent to 
.
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Project supported by the National Natural Science Foundation of China (Grant No. 71303245), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (13YJC790077).
- 1
Limited liability means that the transfer from the principal to the agent must be non-negative.
- 2
With limited liability, the principal can only induce effort by rewarding the agent in the case of a good outcome; hence, the agent can get expected payoffs strictly higher than his reserve utility. The difference between the agent’s expected payoff and his reserve utility is called “limited liability rent” in the literature (see Chapter 4 of Laffont and Martimort (2001) for more detailed discussion).
- 3
For ease of exposition, we work in the space of the certainty equivalent rather than in the space of utility. The certainty equivalent of a random variable
for an agent with utility function u is defined as 
. - 4
Conley and Wilkie (1996) have proved that the symmetric Nash extension is the only solution that satisfies Pareto optimality, symmetry, scale invariance, ethical monotonicity and continuity. They also show the uniqueness of the Nash extension. In this article, we use an asymmetric extension of their symmetric Nash extension.
- 5
That is, the agent gets a higher certainty equivalence from the bargaining.
- 6
It means that the range of the parameters of the contract inducing a high effort decreases as the agent’s bargaining power increases.
©2013 by Walter de Gruyter Berlin / Boston
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Artikel in diesem Heft
- Masthead
- Masthead
- Advances
- Dependence and Uniqueness in Bayesian Games
- Monopolistic Signal Provision†
- Multi-task Research and Research Joint Ventures
- Transparent Restrictions on Beliefs and Forward-Induction Reasoning in Games with Asymmetric Information
- A Simple Bargaining Procedure for the Myerson Value
- On the Difference between Social and Private Goods
- Optimal Use of Rewards as Commitment Device When Bidding Is Costly
- Labor Market and Search through Personal Contacts
- Contributions
- Learning, Words and Actions: Experimental Evidence on Coordination-Improving Information
- Are Trust and Reciprocity Related within Individuals?
- Optimal Contracting Model in a Social Environment and Trust-Related Psychological Costs
- Contract Bargaining with a Risk-Averse Agent
- Academia or the Private Sector? Sorting of Agents into Institutions and an Outside Sector
- Topics
- Poverty Orderings with Asymmetric Attributes
- Dictatorial Mechanisms in Constrained Combinatorial Auctions
- When Should a Monopolist Improve Quality in a Network Industry?
- On Partially Honest Nash Implementation in Private Good Economies with Restricted Domains: A Sufficient Condition
- Revenue Comparison in Asymmetric Auctions with Discrete Valuations
