Optimal Robust Contract Designs for Moral Hazard Problems in the Presence of Model Uncertainty
離散產出概率分佈函數不確定條件下規避道德風險的魯棒激勵機制設計
Student thesis: Doctoral Thesis
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Award date  14 Apr 2020 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(6dc637ca551e415888408e20a1a37ee6).html 

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Abstract
The agency relationship is pervasive in modern business, and moral hazard is the central issue within agency relationships. By designing incentive contracts, agency theory provides effective tools to resolve moral hazard and receives various applications in practice, e.g., salesforce compensation and franchise contract. The agency theory suggests the optimal contracts should be contingent on the realization of the observable output. Hence, the probability distribution of the random output is crucial to design the optimal incentive contracts.
The classical agency theory commonly assumes the probability distribution of the random output is fully known to the principal. However, in practice the principal only has historical data to estimate the output distribution, hence, the principal is often uncertain about the probability distribution of the random output. Such uncertainty is ubiquitous in practice and inherent in all economic models, which is also referred to as the model uncertainty issue. Because of model uncertainty, the classical agency theory is no longer applicable. If the output distribution is misspecified, the contract designed based on the classical agency theory may fail to incentivize the agent. Hence, this dissertation first investigates how to design the optimal incentive contract that can resolve the canonical moral hazard problem and remain robust to model uncertainty. Furthermore, according to the practice and literature of salesforce compensation and franchise contract, inventory decisions, and the principal’s moral hazard can result in more noised data and aggravate the model uncertainty. Hence, this dissertation also investigates how to design the optimal robust contract with joint considerations of inventory decisions and the principal’s moral hazard, respectively.
The three main works and innovations are summarized as follows.
1. The first study proposes the robust contract design framework to address the canonical moral hazard problem where the principal is uncertain about the actioncontingent distribution of the random output: the parameters of the distribution are in an ellipsoidal uncertainty. In the setting, the riskneutral principal maximizes her minimum expected utility over all possible parameters in the uncertainty set, subject to the agent’s robust incentive compatibility constraint (i.e., the agent does not deviate for all possible distributions in the uncertainty set) and the limited liability constraint. We find that when the agent is riskneutral, a linear contract, which rewards the agent proportional to the realized output plus a fixed payment, is uniquely optimal and robust to model uncertainty, where the optimal rate increases in the degree of model uncertainty. When the agent is riskaverse and has a piecewise linear concave utility, the only optimal robust contract is a piecewise linear contract that consists of progressive fixed payments and linear rewards with progressive commission rates. We further show that when the principal is uncertain about the whole probability distribution function rather than only parameters, a quotalinear contract that only rewards outcomes above some quota proportionally is uniquely optimal. The results hold for other settings, including cases with general lpnorm uncertainty sets, multiple effort levels, the agent with a meanvariance utility, and participation constraints. The robust contract design framework generalizes the classical agency theory, e.g., Grossman and Hart (1983) to the nonBayesian paradigm, and provides a new explanation for the popularity of linear contracts (including piecewise linear contracts and quotalinear contracts) from the perspective of robustness.
2. The second study investigates the principal’s optimal robust contract and optimal inventory decisions when she is uncertain about the demand distribution. In the setting, the sales are limited by the inventory level and the lost sales are unobservable, which leads to poor observations on the demand realization thus aggravates the model uncertainty. The effortcontingent demand distribution falls within ellipsoidal uncertainty sets. The principal combines the robust contract design framework with the newsvendor model to jointly design the optimal robust contract and decide the optimal inventory level under the maxmin preference, subject to the robust incentive compatibility condition and limited liability condition. Contrasting with Dai and Jerath (2013) where a quotabonus contract is optimal and the principal tends to overstock, we show that given the optimal inventory level, the only optimal robust contract is a quotacommission that only pays for outcomes above some quota proportionally and that the firm tends to understock compared with the firstbest solution. Furthermore, the optimal commission rate is decreasing in the inventory level. This study underscores the strategic impacts of model uncertainty on the optimal inventory decision and incentive contract designs compensation.
3. The last study investigates the optimal robust contract for the doubleside moral hazard problem and Paretooptimal effort strategies of the principal and the agent in the presence of model uncertainty. The model considers the franchising setting where the principal (i.e., the franchisor) and the agent (i.e., the franchisee) can both exerts unobservable efforts to affect the probability distribution of the random output of the new outlet. The principal has ambiguity about the parameters of the effortcontingent output distribution, which falls within ellipsoidal uncertainty sets. Based on the robust contract design framework, the franchisor adopts the maxmin preference to design the optimal contract and choose Paretooptimal efforts, subject to the franchisee’s robust incentive compatibility condition and limited liability condition. The franchisor’s own incentive compatibility under the worstcase criterion should also hold. We show that the commonlyused franchisefeeroyalty contract is generally not optimal without model uncertainty, while it is indeed optimal with model uncertainty, where the franchisor demands a higher royalty rate for a lower effort. We also characterize the equilibrium effort strategy and generalize to cases when the output distribution rather than parameters are uncertain, where a quotafranchisefeeroyalty contract is uniquely optimal. We also find that the franchisor prefers to negotiate rather than contract with franchisee when facing higher model uncertainty. This study generalizes the optimality of franchisefeeroyalty in Bhattacharyya and Lafontaine (1995) to general and discrete production functions and highlights the impact of model uncertainty on the boundary of the franchising business.
The classical agency theory commonly assumes the probability distribution of the random output is fully known to the principal. However, in practice the principal only has historical data to estimate the output distribution, hence, the principal is often uncertain about the probability distribution of the random output. Such uncertainty is ubiquitous in practice and inherent in all economic models, which is also referred to as the model uncertainty issue. Because of model uncertainty, the classical agency theory is no longer applicable. If the output distribution is misspecified, the contract designed based on the classical agency theory may fail to incentivize the agent. Hence, this dissertation first investigates how to design the optimal incentive contract that can resolve the canonical moral hazard problem and remain robust to model uncertainty. Furthermore, according to the practice and literature of salesforce compensation and franchise contract, inventory decisions, and the principal’s moral hazard can result in more noised data and aggravate the model uncertainty. Hence, this dissertation also investigates how to design the optimal robust contract with joint considerations of inventory decisions and the principal’s moral hazard, respectively.
The three main works and innovations are summarized as follows.
1. The first study proposes the robust contract design framework to address the canonical moral hazard problem where the principal is uncertain about the actioncontingent distribution of the random output: the parameters of the distribution are in an ellipsoidal uncertainty. In the setting, the riskneutral principal maximizes her minimum expected utility over all possible parameters in the uncertainty set, subject to the agent’s robust incentive compatibility constraint (i.e., the agent does not deviate for all possible distributions in the uncertainty set) and the limited liability constraint. We find that when the agent is riskneutral, a linear contract, which rewards the agent proportional to the realized output plus a fixed payment, is uniquely optimal and robust to model uncertainty, where the optimal rate increases in the degree of model uncertainty. When the agent is riskaverse and has a piecewise linear concave utility, the only optimal robust contract is a piecewise linear contract that consists of progressive fixed payments and linear rewards with progressive commission rates. We further show that when the principal is uncertain about the whole probability distribution function rather than only parameters, a quotalinear contract that only rewards outcomes above some quota proportionally is uniquely optimal. The results hold for other settings, including cases with general lpnorm uncertainty sets, multiple effort levels, the agent with a meanvariance utility, and participation constraints. The robust contract design framework generalizes the classical agency theory, e.g., Grossman and Hart (1983) to the nonBayesian paradigm, and provides a new explanation for the popularity of linear contracts (including piecewise linear contracts and quotalinear contracts) from the perspective of robustness.
2. The second study investigates the principal’s optimal robust contract and optimal inventory decisions when she is uncertain about the demand distribution. In the setting, the sales are limited by the inventory level and the lost sales are unobservable, which leads to poor observations on the demand realization thus aggravates the model uncertainty. The effortcontingent demand distribution falls within ellipsoidal uncertainty sets. The principal combines the robust contract design framework with the newsvendor model to jointly design the optimal robust contract and decide the optimal inventory level under the maxmin preference, subject to the robust incentive compatibility condition and limited liability condition. Contrasting with Dai and Jerath (2013) where a quotabonus contract is optimal and the principal tends to overstock, we show that given the optimal inventory level, the only optimal robust contract is a quotacommission that only pays for outcomes above some quota proportionally and that the firm tends to understock compared with the firstbest solution. Furthermore, the optimal commission rate is decreasing in the inventory level. This study underscores the strategic impacts of model uncertainty on the optimal inventory decision and incentive contract designs compensation.
3. The last study investigates the optimal robust contract for the doubleside moral hazard problem and Paretooptimal effort strategies of the principal and the agent in the presence of model uncertainty. The model considers the franchising setting where the principal (i.e., the franchisor) and the agent (i.e., the franchisee) can both exerts unobservable efforts to affect the probability distribution of the random output of the new outlet. The principal has ambiguity about the parameters of the effortcontingent output distribution, which falls within ellipsoidal uncertainty sets. Based on the robust contract design framework, the franchisor adopts the maxmin preference to design the optimal contract and choose Paretooptimal efforts, subject to the franchisee’s robust incentive compatibility condition and limited liability condition. The franchisor’s own incentive compatibility under the worstcase criterion should also hold. We show that the commonlyused franchisefeeroyalty contract is generally not optimal without model uncertainty, while it is indeed optimal with model uncertainty, where the franchisor demands a higher royalty rate for a lower effort. We also characterize the equilibrium effort strategy and generalize to cases when the output distribution rather than parameters are uncertain, where a quotafranchisefeeroyalty contract is uniquely optimal. We also find that the franchisor prefers to negotiate rather than contract with franchisee when facing higher model uncertainty. This study generalizes the optimality of franchisefeeroyalty in Bhattacharyya and Lafontaine (1995) to general and discrete production functions and highlights the impact of model uncertainty on the boundary of the franchising business.
 Moral Hazard, Robust Optimization, Salesforce Compensation, Inventory, Franchising Contracts