Under the information asymmetry, we consider the impact of task importance on the income of university managers, introducing the non-linear income function of university managers. Based on the principal-agent theory, we construct a non-linear incentive optimisation model for subject librarians. We discussed how university managers allocate incentive intensity and how the subject librarians allocate the effort level to achieve the best net expected income for university administrators and subject librarians while satisfying the constraints of participation and incentive compatibility. The results show that the optimal incentive contract for subject librarians is related to task importance, ability level, risk aversion characteristics and the randomness of the external environment in university libraries. The relative incentive intensity increases with the increase of task importance, risk aversion, ability level and randomness of information retrieval tasks, and decreases with the increase of randomness of subject service tasks.
Information retrieval and subject service are the core tasks of subject librarians in university libraries. The information held by university administrators and subject librarians is asymmetric. One party has private information. As long as the behaviour of one party affects the interests of the other party, the relationship between the participants can be regarded as a principal-agent relationship, with private information. One party is called the ‘agent’, and the party that does not own private information is called the ‘principal’. There is an obvious information asymmetry between university administrators and subject librarians, and the degree of effort of subject librarians is not easily observed by university administrators. Due to the asymmetry of information, university administrators seek to maximise the interests of the library, where the focus is on the construction and development of the library. Subject librarians value their income, entertainment time, additional benefits and promotion of professional titles. As a result, subject librarians do not aim at maximising the interests of university administrators, but choose to maximise their own interests by hiding their actions and efforts, resulting in the ‘moral hazard‘ problem.
Principal-agent theory was established in the late 1960s and early 1970s. It is an important development of contract theory. Contract theory believes that the principal-agent relationship is a contractual relationship. The party at a disadvantage for information is the principal, and the party at an advantage for information is the agent. The principal authorises the agent to engage in activities for the benefit of the principal. The principal-agent relationship is a kind of bilateral contractual relationship. The principal and the agent are both economic persons pursuing the maximisation of their own interests, and their interests are related but their respective objective functions are different. The principal and the agent form a mutually acceptable contract through ‘bargaining’, that is, an equilibrium contract. The principal and the agent need to meet two conditions to form an equilibrium contract: One is the participation constraint, the expected utility obtained by the agent by accepting the contract should not be less than the maximum expected utility obtained by the agent without accepting the contract. The other is the incentive compatibility (IC) condition, in which the agent should ensure the maximisation of the expected profit of the principal according to the principle of maximisation of contract performance utility. After the principal and the agent form a contractual relationship, the principal grants the agent considerable discretionary power. The principal hopes that the agent chooses actions according to the principal's interests. However, it is difficult for the principal to directly observe what actions the agent chooses but only some variables, which are determined by the agent's actions and exogenous random factors. Therefore, what the principal observes is only an incomplete information of the agent's actions, which determines that the main problem faced by the principal, that is, to motivate the agent to choose actions that are beneficial to the principal based on the observed variables.
When an agent undertakes multiple tasks, there is a conflict in the agent's energy allocation towards the different tasks. When the tasks are different, the supervisor's ability to supervise is also affected, and it is becomes even more difficult to supervise some jobs. Holmstrom and Milgrom  pointed out that the multi-task analysis logic is roughly the same as the bilateral principal-agent theory, while being much more complicated than the bilateral principal-agent theory. When the agent is engaged in multiple tasks, the traditional principal-agent conclusions are not applicable. In view of this, Holmstrom and Milgrom  believed that when an agent undertakes multiple tasks, the degree of motivation for any one task depends not only on the observability of the task, but also on the observability of other tasks. In particular, when the principal wants the agent to make a certain effort towards a certain task, and the task is unobservable, the principal's incentive compensation should not be used for other tasks. Kirkegaard  proposed a new method for moral hazard, which proved the rationality of the first-order conditional method and played an important role in the analysis of multi-dimensional moral hazard. It proved the rationality of the first-order condition, especially in a specific environment, when the agent undertakes multiple tasks under the multi-dimensional economic behavior. Balmaceda  studied the optimal task allocation problem based on the risk-neutral principal-agent model, and pointed out that multiple tasks would bias the distribution of the agent's effort intensity. Compared with the traditional single task, in addition to the complementary relationship between multiple tasks, that is, when there is an alternative or independent relationship between multiple tasks, the principal wants the agent to undertake multiple tasks. In addition, multi-task principal agent is widely used in many fields. Kossi et al.  discussed how the scientific research environment affects the assignment of scientific researchers’ tasks. Based on the two tasks of scientific research and teaching undertaken by scientific researchers, and under the mutual substitution of scientific research and teaching, a multi-task scientific researcher incentive model was constructed. It was found that the dynamic environment has a significant impact on the scientific research output regardless of the scientific research ability of the researchers. Capponi and Frei  constructed a dynamic multi-task principal-agent model combined with the two tasks of the agent's effort and accident prevention, and designed the optimal incentive contract, the optimal degree of effort, and the behaviour of safety protection measures under the information symmetry. Under the multi-task principal-agent model, Li and Hendrikse  studied the influence of the member size and heterogeneity on CEO incentive factors. Fitoussi and Gurbaxani  considered the moral hazard problem of IT multi-task business outsourcing, and believed that the establishment of multiple incentive index system can effectively reduce the opportunistic behaviour of service providers. Dai  used the mean square utility to build a multi-task business outsourcing incentive model, and pointed out that the incentive intensity is related to the risk aversion of the contracting company and the main parameters of the specific task. The correlation between the motivation intensity and the main variables depends on the degree of relevance between the tasks. From an owner's perspective, Zhang  introduced the same ability, that is, fairness preference and reputation of the contractor, respectively, and constructed the multi-task contractor incentive model in three situations, respectively giving the contractor's second-best salary incentive contract. When both state-owned enterprise executives and government authorities are in a fair preference, Yan  gave the optimal salary incentive mechanism for state-owned enterprise executives, and pointed out that when the tasks are complementary or replaceable, the fairness preference of both parties will promote the return of the optimal salary of state-owned enterprise executives to fair salary through direct and indirect influence channels. Based on the economic and public welfare tasks undertaken by doctors, Guo and Gu  built an independent incentive model for medical staff who were between tasks. He pointed out that under the fixed salary system, doctors chose to undertake tasks with a lower marginal effort cost, and under the sharing system and rent system, doctors chose to undertake economic tasks with higher quantification, higher marginal output and higher marginal effort.
In summary, scholars have improved the traditional principal-agent theory by adding new factors and variables, applying it to all areas of life. But the model assumptions are mainly based on linear returns. This paper considers the impact of task importance on the income of university managers, introduces a non-linear performance income function, and constructs a multi-task incentive optimisation model for subject librarians. Through the model solution, the optimal salary incentive contract is designed and the incentive characteristics are analysed.
The basic multi-task principal-agent is assumed as follows:
In order to analyse the problem conveniently, the following assumptions are made for the basic model.
The income of university administrators is
The expected income of subject librarian is as follows:
Subject librarians have typical characteristics of absolute risk aversion. Furthermore, the deterministic equivalent income of subject librarians is expressed as:
Under the information
And, under asymmetric information, the IC constraint is expressed as:
Therefore, under asymmetric information, after introducing task importance, the incentive optimisation model for the subject librarians is as follows:
IR and IC are the two main constraints that university managers must face to obtain the maximum expected utility. IR indicates that the effectiveness of the salary contract designed by university administrators for subject librarians must be greater than or equal to the maximum opportunity income obtained by subject librarians who refuse the contract. IC indicates that when university administrators are unable to detect subject librarians, subject librarians always choose the best effort to maximise their expected utility.
Considering the task importance, when the information is asymmetric, the university managers can not fully observe the effort level of the subject librarians in information retrieval and subject service. In order to achieve the optimal equivalent income of their own certainty, the subject librarians will not choose the optimal effort level. Therefore, the optimal incentive optimisation model is as follows:
Managers will not pay more income for subject librarians. In the best case, participation constraints are equal, that is
Taking the first order condition for the excitation constraint condition IC, the following results are obtained:
After substituting the objective function and simplifying:
Find the first order condition for
Finally, the best effort levels of subject librarians in the two tasks can be obtained as follows:
So, when Since
Under asymmetric information, this paper considers the impact of task importance on the income of university managers. The non-linear income function and specific task ability of university managers are introduced, the non-linear incentive optimisation model of subject librarians is constructed, the optimal incentive contract is designed through model solution, and the characteristics of relative incentive intensity are analysed. Results show that the optimal incentive contract of subject librarians is related to the task importance, the ability level of subject librarians and the degree of risk aversion, and the randomness of the external environment of the university library. The relative incentive intensity increases with the increase of task importance, risk aversion, ability level and information retrieval task randomness, and decreases with the increase of randomness of the subject service task.
Finally, there are many factors influencing motivation intensity, such as overconfidence level, supervision level, reputation, etc. We don’t consider these factors here, and subsequent research can consider the impact of these factors on the intensity of incentives.