Bounds on the welfare loss from moral hazard with limited liability☆
Introduction
The principal–agent model with moral hazard has been the workhorse paradigm to understand many interesting economic phenomena where incentives play a crucial role, such as the theory of insurance under moral hazard (Spence and Zeckhauser, 1971), the theory of managerial firms (Alchian and Demsetz, 1972, Jensen and Meckling, 1979), optimal sharecropping contracts between landowners and tenants (Stiglitz, 1974), the efficiency wages theory (Shapiro and Stiglitz, 1984), financial contracting (Holmström and Tirole, 1997, Innes, 1990), and job design and multi-tasking (Holmström and Milgrom, 1991). Casual observation also suggests that moral hazard could be of practical importance. In fact, most sales workers are paid according to a fixed wage and either a bonus paid when a certain sales target is achieved or a commission rate over total sales. Franchisees are also motivated by contracts that entail a fixed payment and an agreement about how to share profits or sales. Additionally, managerial contracts usually consist of a combination of fixed wages and payments that are conditioned on performance. In short, incentive contracts are ubiquitous to market economies.
Regardless of the reason for moral hazard, in most cases it entails welfare losses that remain as far as we know quantitatively uncounted for. This paper begins the task of quantifying the welfare losses implied by the existence of moral hazard in a principal–agent relationship with risk neutral individuals and limited liability. The main consequences of moral hazard are by now well understood and deeply rooted in the economics of information literature, thus the moral-hazard paradigm is ripe for a deeper analysis of the quantitative, rather than qualitative, consequences of it.
The setting we consider consists of a risk-neutral principal who hires a risk-neutral agent subject to limited liability to exert costly effort. The effort level and outcome space are discrete and finite, and effort influences the distribution of output. Because the agent's effort is not observable, the principal can only design contracts based on the agent's observable performance. If the principal wishes to induce the agent to choose a given effort level, he should reward the agent when the realization of output is most indicative of the desired effort level having been chosen, and he should punish him when a different outcome is observed. We assume throughout that the probability distribution of output, which is parameterized by the effort level, satisfies the monotone likelihood ratio property (MLRP).1 This implies that the highest output is the most indicative that the highest effort level has been chosen and therefore the principal pays the agent a salary in excess of the limited-liability amount only when the highest outcome is observed, and pays the limited-liability amount otherwise. Because limited liability imposes a lower bound on the size of the punishment, the equilibrium contract leaves a limited-liability rent to the agent. As a result, the equilibrium contract might not maximize social welfare and the first-best outcome might not be attained; instead, the constrained contract will be second-best.2
Before going into the main results it is worthwhile to notice that the contract considered here is quite implausible in the empirically relevant case of a continuous-outcome space. Under the assumptions considered and a continuous-outcome space, the principal pays a bonus only when the highest outcome takes place, which is a measure-zero event. The incentive compatibility will require the principal to pay an infinite bonus in this state, which is unrealistic. This unappealing feature could be avoided, for example, by introducing risk aversion or by assuming that agents will behave perversely to reward systems that are not properly monotonic. The model considered here is still useful as a first step towards a more general study of the welfare loss arising from moral hazard in settings that are more plausible in the real world.
Finally, the model here can be interpreted in a very intuitive way as a situation where the principal is a monopsonist and the agent is a multi-product firm supplying the monopsonist.3 In this context, states of nature correspond to potential markets, probabilities correspond to quantities sold in each market and the principal-monopsonist restrict itself to use linear-pricing contracts. The agent-supplier's outputs in the potential markets are joint outputs determined by his single effort level, and his production function is given by the vector of probability distribution conditional on the effort level chosen. Under this interpretation, the optimal contract follows from the standard intermediate microeconomics analysis of a monopsonist where the agent-supplier is a competitive price-taking firm and the principal-monopsonist pays the supplier a price equal to the marginal cost corresponding to the principal's target effort level. Compared with the case of a competitive input market, inefficiency is introduced because the principal-monopsonist buys a positive quantity from just one market and this is lower than the welfare-maximizing quantity. Therefore our results also quantify the efficiency loss of the competitive input market relative to a monopsonized input market.
To quantify the inefficiencies introduced by moral hazard and limited liability, we measure the welfare loss introduced by a given contract, and rely on the concept of price of anarchy. The latter refers to the worst-case welfare loss in a non-cooperative game, that is, the worst possible equilibrium welfare versus that of a socially-optimal solution. The idea of using worst-case analysis to study situations under competition was introduced by Koutsoupias and Papadimitriou (1999) and has gained followers over the last decade. The use of the price of anarchy as a metric of the welfare loss has been widely applied in economics to problems such as the study of competition and efficiency in congested markets (Acemoglu and Ozdaglar, 2007), games with serial, average and incremental cost sharing (Moulin, 2008), price and capacity competition (Acemoglu et al., 2009), Vickrey–Clarke–Groves mechanisms (Moulin, 2009), resource allocation problems (Kelly, 1997, Johari and Tsitsiklis, 2004), and congestion games (Roughgarden and Tardos, 2004, Correa et al., 2008). In our setting, we define the welfare-loss ratio as the ratio between the social welfare of a socially-optimal solution—the sum of the principal's and agent's payoffs when the first-best effort level is chosen—and that of the sub-game perfect equilibrium in which the principal offers the agent a performance-pay contract and then the agent chooses the effort level.
The goal of this paper is two-fold. First, we provide simple parametric bounds on the welfare-loss ratio in a given instance of the problem in the presence of limited-liability and moral hazard. These bounds allow one to directly quantify the inefficiency in a given instance of the problem without the need to determine its first-best and second-best effort levels, and additionally, they shed some light on the structure of those problem instances with high welfare loss. Second, we study the worst-case welfare-loss ratio (i.e., the price of anarchy), which is defined as the largest welfare-loss ratio among all instances of the problem that satisfy our assumptions. In our model, an instance of the problem is given by the outcome vector, the vector of agent's costs of effort, and the probability distribution of outcomes for each effort level.
In order to obtain our bounds for the welfare-loss ratio, we assume throughout that the probability distribution of output, which is parameterized by the effort level, satisfies MLRP—under this property, a higher output is a better signal that the agent has chosen a higher effort level—and that the marginal cost per unit increase in the highest outcome probability is non-decreasing in the effort level (IMCP).4 The latter assumption ensures that local incentive compatibility constraints are sufficient to induce an effort level, and is weaker than the well-known convexity of the distribution function condition (CDFC) that is also related to the idea of decreasing marginal returns to effort (see, for instance, Rogerson, 1985 and Mirrlees, 1999). Furthermore, it is assumed that the sequence of prevailing social welfare levels, as effort levels increase, is quasi-concave (QCSW); i.e., social welfare is single-peaked in the effort level.
Under our assumptions, inefficiency is introduced when the socially-optimal effort level is high and the principal finds it optimal to induce a low effort level. This happens when the net social gain of increasing the expected output is smaller than the rent given to the agent when inducing a higher effort level. Inducing a higher effort level is costly for the principal because of the agent's limited-liability constraint; the principal cannot severely punish the agent when a bad outcome is realized. Therefore the principal is restricted to rewarding the agent when a good outcome is realized, which may not be enough to compensate for the opportunity cost incurred when a bad outcome is observed.
An unbounded welfare loss would arise if the social welfare of the high effort level were arbitrarily larger than that of the low effort level and, regardless, the principal prefers to induce the lower effort level. Because social welfare is the sum of the principal's and the agent's utilities, the latter implies that the agent is capturing most of the social welfare at the high effort level. Our results preclude an arbitrarily large welfare loss because the principal's utility at the high level cannot be arbitrarily small when the social inefficiency at the low effort level is arbitrarily large. Under assumptions MLRP, IMCP and QCSW, we establish that for any instance of the problem, the welfare-loss ratio is bounded from above by a simple formula involving the probabilities of the highest possible outcome. The results arise from the fact that MLRP implies that the principal pays a bonus only when the highest outcome is observed.5
Subsequently, we show that the worst-case welfare-loss ratio among all problem instances with a fixed number of effort and outcome levels that satisfy our assumptions is equal to the number of effort levels E. As a consequence, the social welfare of a subgame perfect equilibrium is guaranteed to be at least of that of the social optimum. We prove that the worst-case is attained by a family of problem instances in which the highest outcome probability increases at a geometric rate with the effort level. A disadvantage of the previous bound is that it grows unboundedly with the number of effort levels. Thus, we study the worst-case welfare-loss ratio among problem instances with an arbitrary number of effort levels and the likelihood ratio of the highest outcome bounded from above by . Here, r is defined as the ratio of the highest outcome probability when the highest effort level is exerted to that when the lowest effort level is exerted. We show that the worst-case welfare-loss ratio within the previous family is equal to independently of the number of effort levels. Our results suggest that moral hazard is more problematic in situations where the agent's available actions are more numerous, or when the informational problem is such that the highest outcome is much more likely under the highest than under the lowest effort level.
As an extension to the basic model, we study the welfare loss when contracts are restricted to be linear. This is motivated by the prevalence of simple contracts in real life (see, e.g., Salanié, 2003). Surprisingly, we show that a similar bound on the welfare loss holds when the principal is restricted to choose linear contracts, and that the worst-case welfare-loss ratio is again equal to the number of effort levels E. Our results provide bounds on the welfare loss but do not shed light on whether, for the same instance of the problem, the restriction to linear contracts increases or decreases welfare loss. In another extension, we study the welfare loss when there are multiple identical and independent tasks, and for each task the agent chooses between two effort levels. We give a simple bound on the welfare loss involving the probabilities that all tasks are successful when the agent exerts a high and a low effort level, and find that the worst-case welfare-loss ratio is 2, regardless of how many tasks the agent has to work on.
In related work, Demougin and Fluet (1998) derive the optimality of the bonus contract in the setting considered here; i.e., with a discrete outcome space. Kim (1997) presents conditions that guarantee the existence of a bonus contract that achieves a first-best allocation under limited liability but for a continuous outcome and effort space. Our results extend this work by quantifying the impact on efficiency when the first-best allocation is not achieved. In this context, our work should be viewed as a preliminary step in a broader agenda of how to quantify the welfare loss of moral hazard in different settings. Along those lines, Babaioff et al., 2009, Babaioff et al., 2012 study a principal–agent problem with an approach similar to ours. They introduce a combinatorial agency problem with multiple agents performing two-effort–two-outcome tasks, and study the combinatorial structure of dependencies between agents' actions and the worst-case welfare loss for a number of different classes of action dependencies. They show that this loss may be unbounded for technologies that exhibit complementarities between agents, while it can be bounded from above by a small constant for technologies that exhibit substitutabilities between agents. In contrast, our model deals with a single agent and its complexity lies in handling more sophisticated tasks, rather than the interaction between tasks.
The rest of the paper is organized as follows. In Section 2, we introduce the model with its main assumptions and present some preliminary results that will prove useful in the rest of the paper. Section 3 presents our main results on bounds for the welfare loss. Section 4 extends our results in several directions, while Section 5 concludes with some remarks and future directions of study.
Section snippets
The basic setup
We consider a risk-neutral principal and a risk-neutral agent in a setting with effort levels and outcomes.6 The agent chooses an effort level and incurs a personal nonnegative cost of . Effort levels are sorted in increasing order with respect to costs; that is, if . Thus, a higher effort level demands more
Bounding the welfare loss
The goal of this section is two-fold. First, we aim to provide simple parametric bounds on the welfare-loss ratio of a given instance of the problem in the presence of limited-liability and moral hazard under the previous assumptions. Second, we study the worst-case welfare-loss ratio among all problem instances satisfying our assumptions, which is commonly referred to as the Price of Anarchy in the computer science literature (Nisan et al., 2007).
Extensions
In this section we look at different extensions of our results, which illustrate the flexibility of our approach. First, we establish that the worst-case welfare-loss ratio does not change if we restrict the contracts to be linear. The motivation to look at this class of contracts arises from their prevalence in practice. Then, we show how our bounds can be tightened if the agent has to work on many identical tasks with two effort levels each. In Supplementary Appendix B.2 we show that in the
Conclusions
This paper quantifies the welfare loss that arises from the principal's inability to observe the agent's effort level when there is limited liability. We have provided a simple parametric bound on the welfare-loss ratio involving the probabilities of the highest possible outcome. This bound leverages the structure of the optimal contract that pays a bonus only when the highest outcome is observed. The general structure of the bounds found in this paper suggests that the welfare loss in a
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We greatly appreciate the comments of two anonymous referees that helped us refine the content and significantly improve the presentation of the paper. This work was partially supported by FONDECYT Chile through grants 1140140 and 1130671, by Instituto Milenio para la Investigación de Imperfecciones de Mercado y Políticas Públicas grant ICM IS130002, by the Instituto Milenio Sistemas Complejos de Ingeniería, by the Millenium Nucleus Information and Coordination in Networks ICM/FIC RC130003, and by CONICET Argentina Grant PIP 112-201201-00450CO, Convenio Cooperación CONICET–CONICYT Resolución 1562/14, and FonCyT Argentina Grant PICT 2012-1324.