Equilibrium Contract Research Articles

Delegated recruitment and statistical discrimination

We study how delegated recruitment shapes talent selection. Firms often pay recruiters via refund contracts, which specify a payment upon the hire of a suggested candidate and a refund if a candidate is hired but terminated during an initial period of employment. We develop a model of delegated recruitment and show that refund contracts with strong screening incentives lead to statistical discrimination in favor of candidates with more precise productivity information. This contrasts with a first-best direct-hiring benchmark, where the firm has option value from uncertain candidates. Under tractable parametric assumptions, we provide a closed-form expression for the unique equilibrium contract and show that it features strong screening incentives. As a result, candidates with lower expected productivity but more informative signals (“safe bets”) are hired over candidates with higher expected productivity but less informative signals (“diamonds in the rough”).

Robust contracts in common agency

AbstractBusiness activities often involve a common agent managing a variety of projects on behalf of investors with potentially conflicting interests. The extent of the agent's actions is also often unknown to investors, who have to design contracts that provide incentives to the manager despite this lack of crucial knowledge. We consider a game between several principals and a common agent, where principals know only a subset of the actions available to the agent. Principals demand robustness and evaluate contracts on a worst‐case basis. This robust approach allows for a crisp characterization of the equilibrium contracts and payoffs and provides a novel proof of equilibrium existence in common agency by constructing a pseudo‐potential for the game. Robust contracts make explicit how the efficiency of the equilibrium outcome relative to collusion among principals depends on the principals' ability to extract payments from the agent.

Labour Market Screening and the Design of Social Insurance: An Equilibrium Analysis of the Labour Market for the Disabled

Abstract This article studies how firms’ screening incentives in the labour market affect the optimal design of social insurance programs and quantitatively assesses the U.S. disability policies accounting for firms’ screening of the disabled. We develop an equilibrium search model where workers with different productivities have heterogeneous preferences over non-wage benefits and firms cannot offer an employment contract that explicitly depends on worker types. In this environment, firms may use contracts to screen out a certain type of workers, distorting employment rates and contracts in equilibrium. Therefore, the optimal structure of social insurance policies depends on firms’ screening incentives. We extend and structurally estimate this framework to quantitatively understand the inefficiencies arising from firms’ incentives to screen out disabled workers and examine the optimal joint design of disability insurance (DI) and various forms of firm subsidies. We find that hiring subsidies mitigate screening distortions; at the same time, they interact with DI by reducing the labour supply disincentives it generates. The optimal policy structure leads to a considerable welfare gain by simultaneously making firm subsidies and DI benefits more generous.

Imperfect Private Information in Insurance Markets

Abstract This paper studies imperfectly-perceived private information in insurance markets when contracts endogenously respond. Equilibrium contracts, pooling and welfare depend on the joint distribution of risk and misperception. In the Health and Retirement Study (HRS), I show that misperceptions typically co-vary with (medical, long-term care, disability and mortality) risk type: high types under-perceive their risk, low types over-perceive. I develop a general model and algorithm to estimate the equilibrium contracts, pooling and welfare impact of misperceptions that is applicable in many settings. I offer suggestive evidence from US annuity markets that contracts are distorted due to misperceptions, with welfare likely increasing.

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

In this article, we study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _\infty$</tex-math></inline-formula> norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields, and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide incremental input-to-state stability and finite input-state gain properties for contracting systems, and a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system.

The stability of eigenvalues and eigenvectors and their impact on differential systems

In this article, we apply the stability of eigenvalues and eigenvectors and their impact on differential systems. To achieve this goal, the eigenvalues and eigenvectors are studied and their differential systems, nature in terms of being different real values, compound eigenvalues, or equal eigenvalues.And to identify how to solve linear differential systems with fixed coefficients with the initial condition a complete solution, which depends on the eigenvalues and the corresponding eigenvectors, finding the general solution and the geometry of teigenvectors graphically and the effect of theigenvalues for the three eigenvalues cases, by drawing the paths and the phase plane and clarifying the state of equilibrium contract and stability

The Mnemonomics of Contractual Screening

In many markets, buyers sign advance contracts before actual decisions on transactions or consumptions are made. Therefore, a buyer may have private information on expected payoff at the contracting stage, and as time moves on, new information on other components of payoff may arrive. However, prior information can be losable, forgettable, or unattended. In this paper, we investigate how limited memory may influence the optimal design of contracts for sequential screening. Despite memory loss, the buyer can make ex post inference about her initially informed type from the chosen contract. As ex ante screening facilitates subsequent retrospection, the chosen contract can serve as a self-reminding instrument. This would yield an endogenous demand for separation in ex ante contract choice. In response, distortions in the optimal contract design can be either mitigated or intensified, leading to improved or undermined social welfare, respectively. As a result, the equilibrium buyer surplus can be higher than that under perfect memory. We also show that the buyer can exhibit the so-called flat-rate bias, even though her preference is time consistent and perfectly predicted. In addition, as memory can be perfectly recovered from the equilibrium contract choice, investing on any other memory-improving instrument is redundant. Moreover, the buyer’s demand for screening can induce her to choose dominated refund contract. Nevertheless, when dominance must be obeyed, the seller may offer a menu of refund contracts with two-way distortions. This paper was accepted by Matthew Shum, marketing. Funding: The financial support offered by the Hong Kong RGC General Research Fund [Grant 14500519] is acknowledged.

Optimal equilibrium contracts in the infinite horizon with no commitment across periods

Optimal equilibrium contracts in the infinite horizon with no commitment across periods

Trading off static efficiency and dynamic incentives in apprenticeship relations

In this paper, we study a model of on-the-job training in a monopsonic environment with no enforceability and with imperfect credit markets. A market equilibrium is a relational contract which specifies a sequence of wages, a sequence of productivities for the worker and a termination date which satisfy the worker's and the firm's incentive constraint and maximize profits. We show that no self-enforceable contract can end in a period in which total surplus from employment is negative. If the surplus of employment is always positive, equilibrium contracts are perpetual and involve no wage payment before the worker is fully trained. The worker's expected productivity in an alternative employment grows exactly at the interest rate. For the particular case of a Cobb-Douglas production function for the agent, we show that training is slower for higher productivity coefficient and output elasticity. The effect of the discount rate is ambiguous.

Fair and Square: A Retention Model of Managerial Compensation

We propose a model of how the retention motive shapes managerial compensation contracts. Once employed, a risk-averse manager acquires imperfectly portable skills whose value is stochastic because of industry-wide demand shocks. The manager’s actions are uncontractible, and the perceived fairness of the compensation contract affects the manager’s motivation. If the volatility of profits is sufficiently large and outside offers are sufficiently likely, the equilibrium contract combines a salary with an own-firm stock option. The model’s predictions are consistent with empirical regularities concerning contractual shape, the magnitude of variable pay, the lack of indexation, and the prevalence of discretionary severance pay. This paper was accepted by Axel Ockenfels, behavioral economics and decision analysis.

Goal-oriented agents in a market

Abstract We consider a market where “standard” risk-neutral agents coexist with “goal-oriented” agents who, in addition to the expected income, seek a high-enough monetary payoff (the “trigger”) to fulfill a goal. We analyze a two-sided one-to-one matching model where the matching between principals and agents and incentive contracts are endogenous. In any equilibrium contract, goal-oriented agents are matched with the principals with best projects and receive the trigger with positive probability. Moreover, goal and monetary incentives are complementary: goal-oriented agents receive stronger monetary incentives. Finally, we discuss policy interventions in relevant environments.

Liability Insurance: Equilibrium Contracts under Monopoly and Competition

In liability lawsuits (e.g., patent infringement), a plaintiff demands compensation from a defendant, and the parties often negotiate a settlement to avoid a costly trial. Liability insurance creates bargaining leverage for the defendant in this settlement negotiation. We study the characteristics of monopoly and equilibrium contracts in settings where this leverage effect is a substantial source of value for insurance. Our results show that under adverse selection, a monopolist offers at most two contracts, which underinsure low-risk types and may inefficiently induce high-risk types to litigate. In a competitive market, only a pooling equilibrium with underinsurance may exist. (JEL D41, D42, D82, D86, G22, K13, K41)

Partial Unraveling and Strategic Contract Timing

Renegotiations and last-minute contracting are prevalent in many vertical relationships. Information transmission between firms and buyers can be imperfect as well. In this paper we present a theory to explicate how early/delayed contracting over wholesale prices, and partial unraveling of private information, can sustain each other endogenously in a channel setting with bilateral monopoly. Should the wholesale price be predetermined, the downstream manufacturer would be compelled to fully disclose all private information. By contrast, the to-be-negotiated wholesale price can be potentially affected by manufacturer disclosure or concealment. This can represent a countervailing force for equilibrium revelation to be imperfect, even when disclosure is costless. Thus partial unraveling may emerge if and only if no enforceable contract has been signed (i.e., contracting is delayed). Conversely, partial unraveling can endogenously influence ex ante preferences for contract timing. Therefore, contracting may be deliberately delayed even without learning/cost considerations. Moreover, equilibrium contract timing can be socially too early to alleviate channel distortions, and bilateral bargaining can align private and social preferences for delayed contracting. This paper was accepted by Dmitri Kuksov, marketing.

Contracting for perennial energy crops and the cost-effectiveness of the Biomass Crop Assistance Program

Using a framework that meshes mechanism design theory and real options analysis, we recover the contract terms that a biomass processing plant would offer a farmer to induce conversion of land to perennial energy crops. We consider three contract terms: performance payment (price per dry ton of biomass), establishment payment (one-time payment per acre planted), and acreage payment (annual payment per acre planted). We find that, in equilibrium, the contract signed by farmers and buyers of Miscanthus (the perennial energy crop of choice in our analysis) uses a combination of all three payments. Conditional on the equilibrium contract, we show how changes in the structure of the Biomass Crop Assistance Program affect cost and risk along the vertical supply chain. Our analysis reveals that subsidies to both establishment and acreage payments are equally cost-effective in reducing production cost of cellulosic biofuels. However, establishment subsidies are more effective at reducing risk for the buyer and, therefore, dominate acreage subsidies from a risk-adjusted cost-effectiveness criterion. This suggests that a larger share of the Biomass Crop Assistance Program budget should be allocated to establishment subsidies to the detriment of matching payments (subsidies to performance payments) and maintenance payments (subsidies to acreage payments).

Asymmetric awareness and heterogeneous agents

We consider a principal-agent model with moral-hazard and asymmetric awareness and show how the heterogeneity of agents on their aversion to effort affects contract design. We discuss the optimal contract adopted when a principal is aware of all the impacts of an agent’s action, while agents ignore some of them. When a principal faces two types of agents, where one type is more effort-averse than the other, the equilibrium contract is shaped by agent proportions: it pools the agents, separates them, or excludes the more effort-averse agents from the contract. When efforts are observable, all the agents remain unaware, while when efforts are hidden, a principal increases the awareness of the agents to a level commensurate with the nature of the contract. JEL Codes – D82; D83; D86

On the existence of positive equilibrium profits in competitive screening markets

Frictionless consumer choices and price competition are often associated with competitive markets and vanishing equilibrium profits. We discuss vanishing profits in competitive screening markets like insurance. We assume symmetric firms which exhibit constant returns to scale. Consumer heterogeneity creates the possibility of adverse selection. Firms can offer multiple contracts in equilibrium and (importantly) in any deviation. Nash equilibrium profits vanish if each consumer has a unique optimizing bundle at equilibrium prices or, more generally, if there exists a linear ordering of contracts that dictates the preferences of firms whenever consumers are indifferent between multiple optimal contracts. Of particular interest, equilibrium profits vanish if, for each agent, indifference curves are steeper than iso-profit curves. The results extend to Miyazaki-Wilson-Spence equilibria. We provide examples of economies where there exists an equilibrium with strictly positive profit and show that these examples are robust (hold for an open set of economies).

Life Insurance and Life Settlement Markets with Overconfident Policyholders

We analyze how the life settlement market – the secondary market for life insurance – may affect consumer welfare in a dynamic equilibrium model of life insurance with one-sided commitment and overconfident policyholders. As in Daily et al. (2008) and Fang and Kung (2010), policyholders may lapse their life insurance policies when they lose their bequest motives; but in our model the policyholders may underestimate their probability of losing their bequest motive, or be overconfident about their future mortality risks. For the case of overconfidence with respect to bequest motives, we show that in the absence of life settlement overconfident consumers may buy “too much” reclassification risk insurance for later periods in the competitive equilibrium. In contrast, when consumers are overconfident about their future mortality rates in the sense that they put too high a subjective probability on the low-mortality state, the competitive equilibrium contract in the absence of life settlement exploits the consumer bias by offering them very high face amounts only in the low-mortality state. In both cases, life settlement market can impose a discipline on the extent to which overconfident consumers can be exploited by the primary insurers. We show that life settlement may increase the equilibrium consumer welfare of overconfident consumers when they are sufficiently vulnerable in the sense that they have a sufficiently large intertemporal elasticity of substitution of consumption.

Forward Contracts, Market Structure and the Welfare Effects of Mergers

We examine how forward contracts affect economic outcomes under generalized market structures. In the model, forward contracts discipline the exercise of market power by making profit less sensitive to changes in output. This impact is greatest in markets with intermediate levels of concentration. Mergers reduce the use of forward contracts in equilibrium and, in markets that are sufficiently concentrated, this amplifies the adverse effects on consumer surplus. Additional analyses of merger profitability and collusion are provided. Throughout, we illustrate and extend the theoretical results using Monte Carlo simulations. We discuss the practical relevance for antitrust enforcement.

Quality incentive contracts considering asymmetric product manufacturability information: Piece rate vs. Tournament

Product quality depends on not only the quality effort spent during the manufacturing process but also the product manufacturability determined at the product design stage. The manufacturer (he) may keep the product design and a part of production in house and outsource some production when he encounters a large unexpected demand. In this case, he may privately own the product manufacturability information (low- or high-type) and should propose an incentive-compatible contract to signal this information to the uninformed supplier. We first consider that the manufacturer outsources to one supplier and proposes a piece rate contract, with which the supplier is charged a quality penalty proportional to the amount of defective products supplied. In the separating equilibrium, the high-type manufacturer employs a smaller quality penalty rate and thus achieves a lower product quality under asymmetric product manufacturability information than under symmetric information. When the manufacturer outsources to multiple suppliers, he can employ a tournament contract, with which the quality penalty is based on the suppliers’ relative performance. In this situation, the equilibrium tournament contract under symmetric product manufacturability information is also the separating equilibrium contract under asymmetric information, hence resulting in the first-best quality and profit. We further find that the separating equilibrium exists and is stable while no pooling equilibrium is stable.

Optimal Contracts for Agents with Adverse Selection

Due to information asymmetry, adverse selection exists largely in the multiagent market. Aiming at these problems, we develop two models: pure adverse selection model and mixed adverse selection and moral hazard model. We make the assumption that a type of agent is discrete and effort level is continuous in the models. With these models, we investigate the characters that make an optimal contract as well as the conditions under which the utility of a principal and agents can be optimized. As a result, we show that, in the pure adverse selection model, the conditions to reach the optimal utility of a principal and individual agents are that a principal needs to design different contracts for different types of agents, and an individual agent chooses the corresponding type of contracts. For the mixed model, we show that incentive constraint for agents plays a very important role. In fact, we find that whether a principal provides high-type contract or a separating equilibrium contract depends on the probability of existence of low-type agents in the market. In general, if a separating equilibrium contract is issued, then information asymmetry will cause the utility of the high-type agents to be less than that of the case in full information.