Linear Contracts Research Articles

Gaussian agency problems with memory and linear contracts

Gaussian agency problems with memory and linear contracts

Relative performance evaluation in spillover networks

In a multi-agent contracting problem, agents are linked in performance through two channels, effort spillover, governed by spillover network, and risk correlation, governed by risk structure. Assigning compensation weights on peers' performances can not only filter out common risks but also alter agent's incentives. We study how the network and risk structure jointly determine the optimal linear contract. First, the relative compensation sensitivity is determined by ratio of the dot product, between spillover vector and pure hedge portfolio, to unhedgeable risk. We then propose an index named informativeness along the spillover direction and argue that this index measures how precisely the principal can infer the agent's effort. By showing that both the implemented effort and induced welfare are increasing in the informativeness index, we argue that this index captures how central each agent is in this economy. Finally, results regarding relative sensitivities still hold under optimal contract with bounded compensation.

Simple contracts with double-sided moral hazard and adverse selection

We examine a linear contract model involving a risk-neutral venture capitalist (VC) hiring a risk-averse manager to run a project. The project output is determined by the joint efforts of both parties, the manager’s private ability and a random shock. We explicitly derive a truth-telling menu of contracts, which explain that the incentives provided by VC increase with the manager’s ability, under the widely-recognized monotone hazard rate condition. The fixed compensation would be negative; thus, the manager provides a portion of the financial investment and shares risks with VC. VC’s contracts can not distinguish the types of managers in a range of low-ability managers, leading to a partially pooling equilibrium. Conversely, within a range of high-ability managers, VC’s contracts identify the types of managers, resulting in convex information rents that increase with abilities. We explicitly provide ability thresholds specifying partially or completely pooling and separating equilibrium.

Managing Industry 4.0 supply chains with innovative and traditional products: Contract cessation points and value of information

In supply chain management, how contract designs relate to the production system with traditional and Industry 4.0 innovative products (I4I products) is under-explored. In this paper, we fill this literature gap by studying a two-period model in which a manufacturer strategically chooses to incorporate an I4I product line into a traditional product line. In this context, we show that there exists a critical market potential threshold above which the sale of the I4I product is higher than that of the traditional product even when the former is sold at a higher price. For both wholesale price and linear two-part tariff contracts, we show that the contract cessation points and critical market potential threshold behave oppositely. We also find that the innovation level for the I4I product would be higher when the manufacturer’s expected valuation towards the retailer’s cost is higher than the actual one. We extend our base model to demonstrate that the innovation level of the I4I product is higher when the manufacturer uses a linear two-part tariff contract instead of a wholesale price contract. Finally, we uncover that the quantity-dependent innovation investment results in increased profitability and innovation level for the manufacturer provided that the innovation investment coefficient is sufficiently high.

Toward a positive compensation policy for rail transport via mechanism design: The case of China Railway Express

This paper addresses one of the key challenges in the China Railway Express (CR Express) service industry, which is the asymmetry of information between the government and a CR Express operator. We formulate the CR Express service delivery problem with asymmetric information and other unique characteristics as the incentive compensation problem via the principal-agent approach. This is the first paper to develop an incentive-compatible (IC) compensation mechanism to improve CR Express service in the presence of adverse selection and moral hazard. Compared to the generic form of compensation, this paper proves that such an IC scheme with a linear compensation contract is optimal. Our optimal compensation mechanism provides a “positive incentive”, where the monetary compensation is higher for the CR Express operator with lower operating costs, as opposed to the current compensation policy, which can create a “negative incentive”. We find both underestimation and underinvestment problems, which become smaller when the CR Express operator's gross margin is relatively high. When the CR Express service level is discrete-valued, it is optimal to adopt a fixed compensation scheme. Finally, in the scenario with multiple CR Express operators, we propose a generalized Vickrey (GV) auction, which implicitly obtains both true gross margin and service level. The compensation offered by the government is indeed negative in the case of GV auction. Surprisingly, in this scenario, we find problems of both underestimation and overestimation.

Vertical Contracts and Downstream Entry*

We study the implications of different contractual forms in a market with an incumbent upstream monopolist and free downstream entry. We show that traditional conclusions regarding the desirability of linear contracts radically change when entry in the downstream market is endogenous rather than exogenous. By triggering more entry than two‐part tariffs, wholesale price contracts can generate higher aggregate output, consumer surplus, and welfare. In light of this, the upstream monopolist may prefer to trade with wholesale price contracts as well as to give up part of its bargaining power when it is high.

Subsidization of public transit service under double moral hazard

This paper aims to explore incentive issues and characteristics of optimal contract schemes for subsidizing public transit services under double moral hazard, which refers to situations where both the regulator and the operator face incentives to engage in opportunistic behavior due to information asymmetry in the provision of public transit service. This specific aspect of the regulator-operator interrelationship has not been thoroughly investigated in the existing literature. To address this gap, we utilize principal-agent theory and develop a non-cooperative game-theoretic model to analyze the relationship between the regulator (the principal) and the operator (the agent). Through backward induction, we examine the first-best equilibrium (without moral hazard) and second-best equilibrium (under double moral hazard) solutions. Numerical analysis based on a linear contract reveals some key characteristics of the optimal subsidy contract arrangements. Our findings reveal that under double moral hazard, both the regulator and the operator only exert their second-best efforts. The result analysis shows that while higher incentives can reduce the effort required by the regulator, they can also induce the operator to work harder. The marginal effort utilities of the two parties have distinct impacts on determining the optimal sharing rule. Furthermore, we illustrate how the interaction dynamics of the efforts exerted by the two involved parties vary with the sharing rule. Our study extends the current literature by shedding light on the optimal incentive arrangements and contractual requirements for public transport subsidy contracts when considering the double moral hazard issue.

Input material reduction incentives versus scrap recycling for closed‐loop supply chains

Motivated by interactions with a major player in the aerospace industry, we consider the relationship between a supplier of specialty material forgings and a buyer that manufactures airplane components by extensively machining down these forgings as per component design specifications. Due to high material removal costs, the buyer prefers these forgings to be as similar in geometry and size to the component as possible, that is, near‐net‐shape. The supplier, by default, is unable to deliver such near‐net‐shape forgings as per technological constraints, but can utilize costly effort and/or invest in the required technologies to achieve such capabilities. By taking into account uncertainty regarding the correspondence between supplier's effort and resulting forging size, we assess the implications of two innovative approaches for improving supply chain performance: (i) input material reduction incentives via contracting and (ii) scrap material recycling. We characterize the optimal decisions with respect to final component geometry, various costs, and which party in the supply chain controls the strategic recycling decision. We find that the supply chain should utilize both approaches in a complementary way for components with complex geometry, yet deliberately limit recycling and eliminate contracting for components with simple geometry—a strategy the buyer always implements when controlling the recycling decision. Furthermore, we show these contracting and recycling strategies to be robust by considering linear, expected cost sharing, and nonlinear contract alternatives. Finally, we study supply chain inefficiencies that result from decentralizing the recycling and/or contracting decisions, and highlight whether expected cost sharing and nonlinear contracts can outperform linear contracts.

A continuous-time model of self-protection

This paper deals with an optimal linear insurance demand model, where the protection buyer can also exert a time-dynamic costly prevention effort to reduce her risk exposure. This is expressed as a stochastic control problem that consists in maximising an exponential utility of a terminal wealth. We assume that the effort reduces the intensity of the jump arrival process and interpret this as dynamic self-protection. We solve the problem by using a dynamic programming approach, and we provide a representation of the certainty equivalent of the buyer as the solution to a backward stochastic differential equation (BSDE). Using this representation, we prove that an exponential utility maximiser has an incentive to modify her effort dynamically only in the presence of a terminal reimbursement in the contract. Otherwise, the dynamic effort is actually constant, for a class of compound Poisson loss processes. If there is no terminal reimbursement, we solve the problem explicitly and identify the dynamic certainty equivalent of the protection buyer. This shows in particular that the Lévy property of the loss process is preserved under exponential utility maximisation. We also characterise the constant effort as the unique minimiser of an explicit Hamiltonian, from which we can determine the optimal effort in particular cases. Finally, after studying the dependence of the BSDE associated to the insurance buyer on the linear insurance contract parameter, we prove the existence of an optimal linear cover that is not necessarily zero or full insurance.

Robust contracting under double moral hazard

We study contracting when both principal and agent have to exert noncontractible effort for production to take place. An analyst is uncertain about what actions are available and evaluates a contract by the expected payoffs it guarantees to each party in spite of the surrounding uncertainty. Both parties are risk‐neutral; there is no limited liability. Linear contracts, which leave the agent with a constant share of output in exchange for a fixed fee, are optimal. This result holds both in a preliminary version of the model, where the principal only chooses to supply or not supply an input, and in several variants of a more general version, where the principal may have multiple choices of input. The model thus generates nontrivial linear sharing rules without relying on either limited liability or risk aversion.

Two-stage stochastic programming with imperfect information update: Value evaluation and information acquisition game

<abstract> <p>We focus on the two-stage stochastic programming (SP) with information update, and study how to evaluate and acquire information, especially when the information is imperfect. The scarce-data setting in which the probabilistic interdependent relationship within the updating process is unavailable, and thus, the classic Bayes' theorem is inapplicable. To address this issue, a robust approach is proposed to identify the worst probabilistic relationship of information update within the two-stage SP, and the robust Expected Value of Imperfect Information (EVII) is evaluated by developing a scenario-based max-min-min model with the bi-level structure. Three ways are developed to find the optimal solution for different settings. Furthermore, we study a costly information acquisition game between a two-stage SP decision-maker and an exogenous information provider. A linear compensation contract is designed to realize the global optimum. Finally, the proposed approach is applied to address a two-stage production and shipment problem to validate the effectiveness of our work. This paper enriches the interactions between uncertain optimization and information management and enables decision-makers to evaluate and manage imperfect information in a scarce-data setting.</p> </abstract>

Stock-Based Compensation and Insider Trading with Liquidity Signal

We study a stock-based executive (CEO) compensation contract for the case where the insider receives a signal about the demand of liquidity traders in the stock market. The singleperiod model of Kyle (1985) is adopted for deriving the stock market equilibrium. Based on the equilibrium stock price, the optimal linear contract for executive compensation is obtained by applying a standard principal-agent model. Surprisingly, it is found that the firm’s liquidation value is not used in the optimal executive compensation contract in both our model and the benchmark model (where there is no signal about liquidity). We also make comparative statics for the equilibrium stock price and the optimal executive compensation contract with respect to exogenous parameters.

Randomization is optimal in the robust principal-agent problem

A principal contracts with an agent, who takes a hidden action. The principal does not know all of the actions the agent can take and evaluates her payoff from any contract according to its worst-case performance. Carroll (2015) showed that there exists a linear contract that is optimal within the class of deterministic contracts. This paper shows that, whenever there is an optimal linear contract with non-zero slope, the principal can strictly increase her payoff by randomizing over deterministic, linear contracts. Hence, if the principal believes that randomization can alleviate her ambiguity aversion, then restricting attention to the study of deterministic contracts is with loss of generality.

Coordination of dual-channel supply chains with uncertain demand information

AbstractUncertain demand information increases the difficulty of decision-making in supply chains, especially in the light of inventory costs. This article researches a two-echelon dual-channel fresh agricultural product (FAP) supply chain, and discusses optimal decision-making and the value of information sharing using the Stackelberg game. To eliminate the double marginalization effect between supply chain parties, a two-part linear tariff contract is introduced for coordination. The results show that information sharing is always profitable for the manufacturer, but not for the retailer. The inventory and shortage costs will reduce the manufacturer's optimal pricing and fresh-keeping effort level. When the manufacturer overestimates the market demand predicted by the retailer, asymmetric information will increase the manufacturer’s inventory. The results provide important managerial implications for effective supply chain management.

Retailer inventory data sharing in a fresh product supply chain

This research explores the value of data sharing in a two-echelon, fresh product supply chain consisting of a retailer and manufacturer. Demand is stochastic and price sensitive. Decision-making is decentralized and the supply chain operates with a simple linear wholesale price contract. We employ a game theoretic model in which the manufacturer is a Stackelberg leader. Although fresh product supply chains are ubiquitous in the marketplace, academic guidance is limited, particularly in the context of data sharing and when price is determined endogenously.Our analysis shows that although data sharing helps to improve product freshness and reduce waste, sufficiently improved product freshness is the key for both suppliers and retailers to benefit financially. Otherwise, either party or the entire supply chain can be worse off. We demonstrate that data sharing is most valuable when product is somewhat perishable and demand is volatile. For highly perishable or non-perishable products, the manufacturer benefits but the retailer is harmed. Our results suggest that the drivers that motivate retailers to share inventory data in a fresh product supply chain are distinct from extant knowledge. The results provide insight into strategies that not only improve both parties’ performance, but also reduce food waste at the same time.

Optimal Cyber-Insurance Contract Design for Dynamic Risk Management and Mitigation

With the recent growing number of cyberattacks and the constant lack of effective defense methods, cyber risks have become ubiquitous in enterprise networks, manufacturing plants, and government computer systems. Cyber insurance provides a valuable approach to transfer the cyber risks to insurance companies and further improve the security status of the insured. The designation of effective cyber-insurance contracts requires considerations from both the insurance market and the dynamic properties of the cyber risks. To capture the interactions between the users and the insurers, we present a dynamic moral-hazard type of principal–agent model incorporated with Markov decision processes, which are used to capture the dynamics and correlations of the cyber risks as well as the user’s decisions on the protections. We study and fully analyze a case with a two-state two-action user under linear coverage insurance and further show the risk compensation, Peltzman effect, linear insurance contract principle, and zero-operating profit principle in this case. Numerical experiments are provided to verify our conclusions and further extend to cases of a four-state three-action user under linear coverage insurance and threshold coverage insurance.

Public service provision with a limited budget: Service design, privatization and subsidies

While demand for public services has been increasing over time, governments are usually short on capital to invest in public service provision, thereby resulting in low accessibility and heavy congestion in systems. To ease the pressure of public service provision, governments may invite private organizations to invest in and run public service systems, i.e., privatize the systems. Using a queueing framework, this paper studies how capital constraints at both the investment and operation stages affect the social planner’s pricing and capacity decisions for public interest maximization when a limited budget is available. Furthermore, two modes of privatization, complete privatization and privatization with regulation, are discussed. We then derive the optimal design of a family of linear contracts to improve the performance of privatization with a limited budget. Our results show that throughput subsidies are more effective than capacity subsidies. Furthermore, we identify the condition under which the social planner should privatize service provision completely or with regulation. Our findings are useful for improving the utilization efficiency of public service budgets.

Information acquisition, selling effort and pre-order strategy

We study a firm's joint decision on pre-order strategy and information acquisition in the principal-agent framework. The firm may hire a sales agent to promote the pre-order opportunity and/or to boost spot demand. The market signal gleaned from the pre-order outcome is reported to the firm to refine the demand forecast. Considering the compensation contract design problem, we show that the interests of the firm and the sales agent can be aligned by the classical menu of linear contracts (MLC) mechanism. Comparing the performance of the models with one or both of the adverse selection and moral hazard problems, we find that the pre-order strategy dominates the traditional strategy without pre-ordering. The comparison results also show that under the pre-order strategy, a combination of the pre-selling and regular selling efforts may be outperformed by the pre-selling effort only.

Designing a circular contract Template: Insights from the fairphone-as-a-Service project

Recent scholarship has started to recognize the need to understand the role of contracts in processes of circular business model innovation, particularly in Product Service Systems (PSS) business models. Ownership, usage right and risk allocation are key aspects of contract design, all influencing circular business model innovation. However, current knowledge of how contracts support emerging circular business models is limited. In this paper, we address this gap by presenting an in-depth analysis of a case study drawing on data stemming from the Fairphone-as-a-Service project to define key elements for a Circular Service Contract Template. Our findings support the idea that contracts, ownership, and usage right issues strongly influence the core elements of a PSS business model, such as servitization, longevity and modularity. Particularly, evidence from the case study confirmed the presence of tensions between ownership and contractual aspects when attempting to manage the circular business practices.

The role of contract structure for damage quantification: pitfalls and solutions

ABSTRACT Damage quantification in antitrust cases typically revolves around overcharge and pass-on estimation. In vertical industry structures, both measures crucially depend on contract structure, with linear and two-part tariffs being the most common contract types. While most of the damages literature focuses on linear contracts, two-part tariffs are mostly neglected. We discuss implications of tariff structure for damage quantification, highlight common pitfalls and propose practical solutions in damage assessment when two-part tariffs are present.