The incentive contract of subject librarians in university library under the non-linear task importance

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.

1.2

Research status

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 [1] 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 [2] 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 [3] 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 [4] 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. [5] 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 [6] 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 [7] studied the influence of the member size and heterogeneity on CEO incentive factors. Fitoussi and Gurbaxani [8] 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 [9] 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 [10] 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 [11] 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 [12] 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.

Incentive optimisation model

The basic multi-task principal-agent is assumed as follows:

Assumption 1: the agent undertake n ≥ 2 tasks. Let $a = (a_{1}, a_{2}, \cdot \cdot \cdot, a_{n}) \in R_{+}^{n}$ a = \left({{a_1},{a_2}, \cdot \cdot \cdot,{a_n}} \right) \in R_ + ^n represent the effort vector, and a_i ∈ [0,+∞] is the effort level of the agent on the task i = 1,2,…,n.

Assumption 2: C (a) is the private cost function, and it is a strictly increasing convex function.

Assumption 3: B(a) denotes the total expected return obtained by the agent effort vector, where B(a) is a strictly concave function.

Assumption 4: The agent's effort vector produces an observable performance vector q = μ (a) + θ, where $μ (\cdot) : R_{+}^{n} \to R^{k}$ \mu \left(\cdot \right):R_ + ^n \to {R^k} is a concave function.

Assumption 5: The compensation incentive contract designed by the principal is ω (q) = δ + S^T q, where $s = (s_{1}, s_{2}, \dots, s_{k}) \in R_{+}^{k}$ s = \left({{s_1},{s_2}, \cdots,{s_k}} \right) \in R_ + ^k , δ is the fixed salary and $\bar{ω}$ \bar \omega is the reserved salary of the agent.

2.1

Model assumptions

In order to analyse the problem conveniently, the following assumptions are made for the basic model.

Assumption 1: University administrators and subject librarians have a typical principal-agent relationship. Subject librarians are mainly responsible for information retrieval and subject services. The benefits of both sides mainly depend on the efforts (e₁, e₂) of subject librarians. Subject librarians understand their level of effort. Although university administrators do not directly know the degree of their efforts, they can observe the results of their efforts.

Assumption 2: Based on reality, the performance output of the subject librarian in information retrieval and subject service is S₁ = e₁ + θ₁ and S₂ = e₂ + θ₂, respectively, where θ₁, θ₂ are respectively the normally distributed random variables with mean 0 and variance σ₁², σ₂². And the two tasks of information retrieval and subject service are independent of each other.

Assumption 3: Subject service ability has a higher impact on subject service efficiency, in view of which, the effort cost function of the subject librarian considering ability is set as C (e₁,e₂) = φ₁e₁²/2 + φ₂e₂²/2. The ability level is reflected by the variable φ_i, which means that the higher the ability, the greater the variable φ_i. And C(e₁,e₂) has a first-order continuous partial derivative and a second-order differentiable.

Assumption 4: Assuming that the expected utility of university administrators is equal to the expected return, subject librarians are risk aversive, and their aversion to risk is measured by the Arrow–Pratt absolute risk aversion ρ.

Assumption 5: The benefits of university administrators depend on the accomplishment of the two tasks by subject librarians. The benefits are non-linear, that is U = f (S₁,S₂) = S₁^pS₂^(1−p), where p, 1 − p respectively represent the output flexibility of subject librarians in the two tasks, that is, the importance of the two tasks in the income of university administrators.

Assumption 6: University administrators use linear incentive functions to pay for the subject librarians, that is, the salary received by academic librarians is w = α + b₁S₁ + b₂S₂, where α is a fixed salary, which has nothing to do with the output of subject librarians and b₁, b₂ respectively represent the proportion of sharing obtained by subject librarians, that is, the incentive factor.

2.2

Model construction

The income of university administrators is U = f (S₁,S₂) = S₁^pS₂^(1−p) where p represents the degree of influence of information retrieval on the income of university administrators and 1 − p represents the degree of influence of subject services on the income of university administrators. So p cannot be negative and 0 < p < 1. The client's expected net income is equal to its expected net income. Therefore, the expected net utility of university administrators is as follows: $E (S_{1}^{p} S_{2}^{(1 - p)} - w) = S_{1}^{p} S_{2}^{(1 - p)} - α - b_{1} S_{1} - b_{2} S_{2}$ E\left({{S_1}^p{S_2}^{(1 - p)} - w} \right) = {S_1}^p{S_2}^{(1 - p)} - \alpha - {b_1}{S_1} - {b_2}{S_2}

The expected income of subject librarian is as follows: $E (w) - C = α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2}$ E(w) - C = \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}{e_1}^2 - {1 \over 2}{\varphi _2}{e_2}^2

Subject librarians have typical characteristics of absolute risk aversion. Furthermore, the deterministic equivalent income of subject librarians is expressed as: $CE = α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2}$ CE = \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}{e_1}^2 - {1 \over 2}{\varphi _2}{e_2}^2 - {1 \over 2}\rho {b_1}^2{\sigma _1}^2 - {1 \over 2}\rho {b_2}^2{\sigma _2}^2 where $\frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} + \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2}$ {1 \over 2}\rho {b_1}^2{\sigma _1}^2 + {1 \over 2}\rho {b_2}^2{\sigma _2}^2 is the risk cost of the subject librarian, which means that the subject librarian is willing to give it up in the random income in exchange for the deterministic equivalent income.

Under the information $α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \geq \bar{u} e_{1}$ \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}e_1^2 - {1 \over 2}{\varphi _2}e_2^2 - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2 \ge \overline u \,{e_1} , $e_{2} \in arg max_{e_{1}, e_{2}} α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \bar{u} \bar{u}$ {e_2} \in \arg \max \limits_{{e_1},{e_2}} \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}e_1^2 - {1 \over 2}{\varphi _2}e_2^2 - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2\bar u\bar u asymmetry, when the incentive contract is given, the subject librarian will choose the optimal level of effort to maximize his deterministic equivalent income, that is, the optimal incentive contract must satisfy the IC constraint. At the same time, the retained income of subject librarian is. When the deterministic equivalent income is greater than or equal to, the subject librarian accepts the contract, otherwise the subject librarian does not accept the contract. The optimal incentive contract must meet the participation constraint (IR). Therefore, considering the importance of the task, when the information is asymmetric, the participation constraint (IR) is expressed as:

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: ${\begin{array}{l} max_{\bar{b_{1}}, \bar{b_{2}}, \bar{e_{1}}, \bar{e_{2}}, α} & S_{1}^{p} S_{2}^{1 - p} - α - b_{1} e_{1} - b_{2} e_{2} \\ s . t . (1) & α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \geq \bar{u} \\ (2) & e_{1}, e_{2} \in arg max_{e_{1}, e_{2}} α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \end{array}$ \left\{{\matrix{{{\max}\limits_{\overline {{b_1}},\overline {{b_2}},\overline {{e_1}},\overline {{e_2}},\alpha}} \hfill & {{S_1}^p{S_2}^{1 - p} - \alpha - {b_1}{e_1} - {b_2}{e_2}} \hfill \cr {s.t.\;(1)} \hfill & {\alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}e_1^2 - {1 \over 2}{\varphi _2}e_2^2 - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2 \ge \overline u} \hfill \cr {(2)} \hfill & {{e_1},{e_2} \in \arg \max \limits_{{e_1},{e_2}} \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}e_1^2 - {1 \over 2}{\varphi _2}e_2^2 - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2} \hfill \cr}} \right.

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.

Model solution

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: ${\begin{array}{l} max_{\bar{b_{1}}, \bar{b_{2}}, \bar{e_{1}}, \bar{e_{2}}, α} & S_{1}^{p} S_{2}^{1 - p} - α - b_{1} e_{1} - b_{2} e_{2} \\ s . t . (1) & α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \geq \bar{u} \\ α = \bar{u} - b_{1} e_{1} - b_{2} e_{2} + \frac{1}{2} φ_{1} e_{1}^{2} + \frac{1}{2} φ_{2} e_{2}^{2} + \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} + \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \\ (2) & e_{1}, e_{2} \in arg max_{e_{1}, e_{2}} α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} φ_{1} e_{1}^{2} - \frac{1}{2} φ_{2} e_{2}^{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \end{array}$ \left\{{\matrix{{{\max}\limits_{\overline {{b_1}},\overline {{b_2}},\overline {{e_1}},\overline {{e_2}},\alpha}} \hfill & {{S_1}^p{S_2}^{1 - p} - \alpha - {b_1}{e_1} - {b_2}{e_2}} \hfill \cr {s.t.\;(1)} \hfill & {\alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}e_1^2 - {1 \over 2}{\varphi _2}e_2^2 - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2 \ge \overline u} \hfill \cr {} \hfill & {\alpha = \overline u - {b_1}{e_1} - {b_2}{e_2} + {1 \over 2}{\varphi _1}e_1^2 + {1 \over 2}{\varphi _2}e_2^2 + {1 \over 2}\rho b_1^2\sigma _1^2 + {1 \over 2}\rho b_2^2\sigma _2^2} \hfill \cr {(2)} \hfill & {{e_1},{e_2} \in \arg \max \limits_{{e_1},{e_2}} \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{\varphi _1}e_1^2 - {1 \over 2}{\varphi _2}e_2^2 - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2} \hfill \cr}} \right.

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: $\begin{array}{l} b_{1} - φ_{1} e_{1} = 0 \\ b_{2} - φ_{2} e_{2} = 0 \end{array}$ \matrix{{{b_1} - {\varphi _1}{e_1} = 0} \hfill \cr {{b_2} - {\varphi _2}{e_2} = 0} \hfill \cr}

And then $\begin{array}{l} e_{1} = \frac{b_{1}}{φ_{1}} \\ e_{2} = \frac{b_{2}}{φ_{2}} \end{array}$ \matrix{{{e_1} = {{{b_1}} \over {{\varphi _1}}}} \hfill \cr {{e_2} = {{{b_2}} \over {{\varphi _2}}}} \hfill \cr}

After substituting the objective function and simplifying: $U_{p} = {(\frac{b_{1}}{φ_{1}})}^{p} {(\frac{b_{2}}{φ_{2}})}^{1 - p} - \bar{u} - \frac{b_{1}^{2}}{2 φ_{1}} - \frac{b_{2}^{2}}{2 φ_{2}} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2}$ {U_p} = {\left({{{{b_1}} \over {{\varphi _1}}}} \right)^p}{\left({{{{b_2}} \over {{\varphi _2}}}} \right)^{1 - p}} - \overline u - {{{b_1}^2} \over {2{\varphi _1}}} - {{{b_2}^2} \over {2{\varphi _2}}} - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2

Find the first order condition for b₁, b₂ respectively: $\begin{array}{r} p {(\frac{b_{1}}{φ_{1}})}^{p - 1} {(\frac{b_{2}}{φ_{2}})}^{1 - p} \frac{1}{φ_{1}} - \frac{b_{1}}{φ_{1}} - ρ σ_{1}^{2} b_{1} = 0 \\ (1 - p) {(\frac{b_{1}}{φ_{1}})}^{p} {(\frac{b_{2}}{φ_{2}})}^{- p} \frac{1}{φ_{2}} - \frac{b_{2}}{φ_{2}} - ρ σ_{2}^{2} b_{2} = 0 \end{array}$ \matrix{\hfill {p{{\left({{{{b_1}} \over {{\varphi _1}}}} \right)}^{p - 1}}{{\left({{{{b_2}} \over {{\varphi _2}}}} \right)}^{1 - p}}{1 \over {{\varphi _1}}} - {{{b_1}} \over {{\varphi _1}}} - \rho \sigma _1^2{b_1} = 0} \cr \hfill {(1 - p){{\left({{{{b_1}} \over {{\varphi _1}}}} \right)}^p}{{\left({{{{b_2}} \over {{\varphi _2}}}} \right)}^{- p}}{1 \over {{\varphi _2}}} - {{{b_2}} \over {{\varphi _2}}} - \rho \sigma _2^2{b_2} = 0} \cr}

After reorganizing: $\begin{array}{l} {(\frac{b_{1}}{φ_{1}})}^{p} {(\frac{b_{2}}{φ_{2}})}^{1 - p} = \frac{b_{1}^{2} (\frac{1}{φ_{1}} + ρ σ_{1}^{2})}{p} \\ {(\frac{b_{1}}{φ_{1}})}^{p} {(\frac{b_{2}}{φ_{2}})}^{1 - p} = \frac{b_{2}^{2} (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{1 - p} \end{array}$ \matrix{{{{\left({{{{b_1}} \over {{\varphi _1}}}} \right)}^p}{{\left({{{{b_2}} \over {{\varphi _2}}}} \right)}^{1 - p}} = {{{b_1}^2({1 \over {{\varphi _1}}} + \rho \sigma _1^2)} \over p}} \hfill \cr {{{\left({{{{b_1}} \over {{\varphi _1}}}} \right)}^p}{{\left({{{{b_2}} \over {{\varphi _2}}}} \right)}^{1 - p}} = {{{b_2}^2({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {1 - p}}} \hfill \cr}

And then $\frac{b_{1}^{2} (\frac{1}{φ_{1}} + ρ σ_{1}^{2})}{p} = \frac{b_{2}^{2} (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{1 - p}$ {{{b_1}^2({1 \over {{\varphi _1}}} + \rho \sigma _1^2)} \over p} = {{{b_2}^2({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {1 - p}}

So $\frac{b_{1}}{b_{2}} = {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{\frac{1}{2}}$ {{{b_1}} \over {{b_2}}} = {\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)^{{1 \over 2}}}

Furthermore, $\begin{matrix} b_{1} = {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{\frac{p}{4}} \cdot {(\frac{p {\frac{1}{φ_{1}}}^{p} {\frac{1}{φ_{2}}}^{1 - p}}{\frac{1}{φ_{1}} + ρ σ_{1}^{2}})}^{\frac{1}{2}} \\ b_{2} = {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{\frac{p}{4}} \cdot {(\frac{(1 - p) {\frac{1}{φ_{1}}}^{p} {\frac{1}{φ_{2}}}^{1 - p}}{\frac{1}{φ_{2}} + ρ σ_{2}^{2}})}^{\frac{1}{2}} \end{matrix}$ \matrix{{{b_1} = {{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)}^{{p \over 4}}} \cdot {{\left({{{p{{{1 \over {{\varphi _1}}}}^p}{{{1 \over {{\varphi _2}}}}^{1 - p}}} \over {{1 \over {{\varphi _1}}} + \rho \sigma _1^2}}} \right)}^{{1 \over 2}}}} \cr {{b_2} = {{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)}^{{p \over 4}}} \cdot {{\left({{{(1 - p){{{1 \over {{\varphi _1}}}}^p}{{{1 \over {{\varphi _2}}}}^{1 - p}}} \over {{1 \over {{\varphi _2}}} + \rho \sigma _2^2}}} \right)}^{{1 \over 2}}}} \cr}

Finally, the best effort levels of subject librarians in the two tasks can be obtained as follows: $\begin{array}{l} e_{1} = \frac{1}{φ_{1}} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{\frac{p}{4}} \cdot {(\frac{p {\frac{1}{φ_{1}}}^{p} {\frac{1}{φ_{2}}}^{1 - p}}{\frac{1}{φ_{1}} + ρ σ_{1}^{2}})}^{\frac{1}{2}} \\ e_{2} = \frac{1}{φ_{2}} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{\frac{p}{4}} \cdot {(\frac{(1 - p) {\frac{1}{φ_{1}}}^{p} {\frac{1}{φ_{2}}}^{1 - p}}{\frac{1}{φ_{2}} + ρ σ_{2}^{2}})}^{\frac{1}{2}} \end{array}$ \matrix{{{e_1} = {1 \over {{\varphi _1}}}{{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)}^{{p \over 4}}} \cdot {{\left({{{p{{{1 \over {{\varphi _1}}}}^p}{{{1 \over {{\varphi _2}}}}^{1 - p}}} \over {{1 \over {{\varphi _1}}} + \rho \sigma _1^2}}} \right)}^{{1 \over 2}}}} \hfill \cr {{e_2} = {1 \over {{\varphi _2}}}{{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)}^{{p \over 4}}} \cdot {{\left({{{(1 - p){{{1 \over {{\varphi _1}}}}^p}{{{1 \over {{\varphi _2}}}}^{1 - p}}} \over {{1 \over {{\varphi _2}}} + \rho \sigma _2^2}}} \right)}^{{1 \over 2}}}} \hfill \cr}

Model analysis

Let L = b₁/b₂, where L represents the relative incentive intensity of information retrieval to subject services. University administrators can effectively motivate subject librarians through the relative incentive intensity when they cannot accurately observe the situation. That is $L = \frac{b_{1}}{b_{2}} = {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{\frac{1}{2}}$ L = {{{b_1}} \over {{b_2}}} = {\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)^{{1 \over 2}}}

Since $\frac{\partial L}{\partial p} = \frac{1}{2} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{- \frac{1}{2}} \frac{(\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{{(1 - p)}^{2} (\frac{1}{φ_{1}} + ρ σ_{1}^{2})} > 0$ {{\partial L} \over {\partial {\rm{p}}}} = {1 \over 2}{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)^{- {1 \over 2}}}{{({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {{{(1 - p)}^2}({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}} > 0

Therefore, L is an increasing function of p, and the relative excitation intensity increases as the importance of the task increases. Under other similar conditions, the manager's motivation for tasks with high importance should be greater than his motivation for tasks with low importance.

Since $\frac{\partial L}{\partial ρ} = \frac{1}{2} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{- \frac{1}{2}} \frac{(1 - p) p (σ_{2}^{2} (\frac{1}{φ_{1}} + ρ σ_{1}^{2}) - σ_{1}^{2} (\frac{1}{φ_{2}} + ρ σ_{2}^{2}))}{{((1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2}))}^{2}}$ {{\partial L} \over {\partial \rho}} = {1 \over 2}{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)^{- {1 \over 2}}}{{(1 - p)p(\sigma _2^2({1 \over {{\varphi _1}}} + \rho \sigma _1^2) - \sigma _1^2({1 \over {{\varphi _2}}} + \rho \sigma _2^2))} \over {{{\left({(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)} \right)}^2}}}

Therefore, when $(\frac{1}{φ_{1}} + ρ σ_{1}^{2}) - σ_{1}^{2} (\frac{1}{φ_{2}} + ρ σ_{2}^{2}) > 0$ \left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right) - \sigma _1^2\left({{1 \over {{\varphi _2}}} + \rho \sigma _2^2} \right) > 0 , ∂ L/∂ ρ > 0, which means that the relative incentive intensity increases as the degree of risk aversion increases. Under the other similar conditions, the higher the degree of risk aversion for subject librarians, the lower the relative incentive intensity.

Since $\begin{array}{l} \frac{\partial L}{\partial φ_{1}} = \frac{1}{2} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{- \frac{1}{2}} \frac{(1 - p) p \frac{1}{φ_{1}^{2}} (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{{((1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2}))}^{2}} > 0 \\ \frac{\partial L}{\partial φ_{2}} = \frac{1}{2} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{- \frac{1}{2}} \frac{- (1 - p) p \frac{1}{φ_{2}^{2}} (\frac{1}{φ_{1}} + ρ σ_{1}^{2})}{{((1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2}))}^{2}} < 0 \end{array}$ \matrix{{{{\partial L} \over {\partial {\varphi _1}}} = {1 \over 2}{{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)({1 \over {{\varphi _1}}} + \rho \sigma _1^2)}}} \right)}^{- {1 \over 2}}}{{(1 - p)p{1 \over {{\varphi _1}^2}}\left({{1 \over {{\varphi _2}}} + \rho \sigma _2^2} \right)} \over {{{\left({(1 - p)\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)} \right)}^2}}} > 0} \hfill \cr {{{\partial L} \over {\partial {\varphi _2}}} = {1 \over 2}{{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)}}} \right)}^{- {1 \over 2}}}{{- (1 - p)p{1 \over {{\varphi _2}^2}}\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)} \over {{{\left({(1 - p)\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)} \right)}^2}}} < 0} \hfill \cr}

So, when ∂ L/∂ φ₁ > 0, ∂ L/∂ φ₂ < 0, the relative incentive intensity is positively correlated with the subject librarian ability. The stronger the comprehension ability of subject librarians in information retrieval and subject service, in order to maximise the interests for both sides, the higher the comprehension ability of the subject librarian under other unchanged conditions, university managers should strengthen the incentive factors and improve the sharing portion of subject librarians.

Since $\begin{array}{r} \frac{\partial L}{\partial σ_{1}^{2}} = \frac{1}{2} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{- \frac{1}{2}} \frac{- (1 - p) p ρ (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{{((1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2}))}^{2}} < 0 \\ \frac{\partial L}{\partial σ_{2}^{2}} = \frac{1}{2} {(\frac{p (\frac{1}{φ_{2}} + ρ σ_{2}^{2})}{(1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2})})}^{- \frac{1}{2}} \frac{(1 - p) p ρ (\frac{1}{φ_{1}} + ρ σ_{1}^{2})}{{((1 - p) (\frac{1}{φ_{1}} + ρ σ_{1}^{2}))}^{2}} > 0 \end{array}$ \matrix{\hfill {{{\partial L} \over {\partial \sigma _1^2}} = {1 \over 2}{{\left({{{p({1 \over {{\varphi _2}}} + \rho \sigma _2^2)} \over {(1 - p)\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)}}} \right)}^{- {1 \over 2}}}{{- (1 - p)p\rho \left({{1 \over {{\varphi _2}}} + \rho \sigma _2^2} \right)} \over {{{\left({(1 - p)\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)} \right)}^2}}} < 0} \cr \hfill {{{\partial L} \over {\partial \sigma _2^2}} = {1 \over 2}{{\left({{{p\left({{1 \over {{\varphi _2}}} + \rho \sigma _2^2} \right)} \over {(1 - p)\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)}}} \right)}^{- {1 \over 2}}}{{(1 - p)p\rho \left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)} \over {{{\left({(1 - p)\left({{1 \over {{\varphi _1}}} + \rho \sigma _1^2} \right)} \right)}^2}}} > 0} \cr}

So, when ∂ L/∂ σ₁² > 0, ∂ L/∂ σ₂² > 0, the relative incentive intensity increases with the randomness of information retrieval task, and decreases with the randomness of subject service. In order to achieve the best interests for both sides, university managers should weaken the incentive factors and reduce the sharing portion of subject librarians.

Conclusion

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.

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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The incentive contract of subject librarians in university library under the non-linear task importance

Published Online: Mar 31, 2022

Page range: 721 - 730

Received: May 21, 2021

Accepted: Dec 06, 2021

DOI: https://doi.org/10.2478/amns.2021.2.00161

Keywordsprincipal-agent, incentive contract, subject librarian

© 2021 Chunping Wang et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
principal-agent, incentive contract, subject librarian