Compensation incentive contract of the subject librarians based on the H-M model

Subject service is the main way of university libraries to serve and integrate the construction of ‘first-class subjects’, and the key to first-class subject service lies in the subject service team. The subject service team mainly provides services for administrative decision-making departments, scientific researchers, key laboratories through subject dynamic analysis, subject resource promotion, information retrieval and document delivery. Subject librarians are proficient not only in a variety of databases and analysis tools but also in using them to sort, analyse, predict and evaluate data, so as to provide comprehensive, timely objective decision support for the subject construction in universities. Therefore, the subject service team is not only closely related to the construction of first-class subjects in universities but also vital to it. However, outstanding problems such as ‘insufficient incentives’ and ‘ineffective incentives’ faced by the subject service team have caused subject librarians to encounter problems such as lack of competitiveness, insufficient ability, insufficient innovation, low work enthusiasm, negative attitudes and serious brain drain. As a result, subject service quality and efficiency are low, which severely restricts the construction of ‘first-class subjects’ in universities. Therefore, the incentive mechanism of the subject service team is a key issue to accelerate, promote and ensure the construction of ‘first-class subjects’ and the sustainable and healthy development of subject service teams. How to construct a scientific and effective incentive mechanism for discipline service teams is worthy to further research.

Subject librarians are very important in the university library and cannot be replaced by experts [1]. Based on the perspective of management, many scholars have summarised the incentive factors and measures to motivate subject librarians through experience judgement and analysis. For examples, they found that salary incentive is one of the most effective incentive methods. Subject librarians with longer work service are more likely to strive for higher pay, and the number of male subject librarians involved in the fight for higher pay tends to be higher than the number of women; satisfied and stable income guarantee can effectively motivate librarians to work hard [2, 3]. Many researchers have found that the compensation level, office conditions, fair chance, justice, professional promotion, emotion, learning training, faculty status and tenure, and organisational socialisation factors have a significant effect on librarian satisfaction. By implementing the above factors, the passion and enthusiasm of subject librarians can be significantly improved. High income has a vigorous effect in promoting the work efficiency of subject librarians and providing quality service to customers [4, 5, 6, 7, 8, 9, 10]. The existing research mainly summarises the incentive factors and measures of subject librarians from the perspective of management, which is not only lack of solid theoretical support but also lack of empirical test based on data.

Principal-agent theory is an important economic incentive theory. ‘Principal-agent theory’, and vigorously advocated the separation of management right and ownership, the owner has the residual claim, and then transferred the management right. The principal-agent relationship begins with the existence of ‘specialisation’. Once ‘specialisation’ exists, the principal-agent relationship may appear. In this principal-agent relationship, the agent acts on behalf of the principal due to his own comparative advantages. Principal-agent theory studies the principal-agent relationship within and among enterprises from the perspective of general microeconomics, which is better than traditional microeconomics while explaining some organisational phenomena. Its core task is how the principal should construct the best contract to incentive the agent under the condition of private information and interest conflict. Holmstrom and Milgrom [11, 12] established the basic traditional principal-agent model. Subsequently, they extended a single task to multiple tasks and established their theoretical analysis framework. In addition, the principal-agent theory has been widely used in various fields.

There exists typical principal-agent relationship between the academic managers and subject librarians. In view of this, we adopt the economic incentive theory, that is, multi-task principal-agent analysis framework, to discuss the incentive problem of subject librarians. The main contributions of this paper are as follows: ① Construct an optimisation model of subject librarian compensation incentives with inter-task correlation. ② Design the best compensation incentive contract. ③ Analyse the characteristics of the best incentive contract. ④ Establish an empirical model based on data and empirically test the theoretical model and relevant theoretical conclusions.

Model

2.1

Hypothesis

Hypothesis 1: Both library managers and subject librarians are rational economic people, whose behaviour strategies are to maximise their own interests, such as the managers aim to maximise performance and the subject librarians pursue vacation, professional title, entertainment, learning and training, promotion, etc. The interests of both sides are in conflict. In particular, the benefits of both parties are closely related to the efforts of the subject librarians. Their effort level vector for the two tasks is (e₁,e₂), where (e₁,e₂) represents the effort intensity, and although the managers cannot fully monitor the effort intensity, they can observe the performance of the output.

Hypothesis 2: Multiple tasks are not independent of each other, and they have certain mutual relations or mutual attributes, and the service cost function is set as $C (e_{1}, e_{2}) = 1 / 2 c_{1} e_{1}^{2} + 1 / 2 c_{2} e_{2}^{2} + δ e_{1} e_{2}$ C({e_1},{\kern 1pt} {e_2}) = 1/2{c_1}e_1^2 + 1/2{c_2}e_2^2 + \delta {e_1}{e_2} , where δ measures the relevance (dependency) between tasks, C(e₁,e₂) is a strictly increasing convex function satisfying ∂C(e₁,e₂)/∂e_i > 0 and ∂C(e₁,e₂)/∂e_i² > 0, and i = 1, 2.

Let C₁₂ = ∂C(e₁,e₂)/∂e₁∂e₂, so the sign of C₁₂ determines the sign of δ.

If C₁₂ = 0, that is, δ = 0, it means that the tasks are independent of each other, which means that an increase in the effort intensity of one task will not cause a change in the effort level of the other task.

If C₁₂ > 0, that is, δ > 0, it means that the tasks are interchangeable, which means that the completion of one task will increase the difficulty of completing the other task.

If C₁₂ < 0, that is, δ < 0, it means that the tasks are mutually complementary, which means that the completion of one task is more conducive to the completion of the other task.

This hypothesis shows that as the level of effort of the subject librarian on the two tasks continues to increase, the cost of the two tasks increases and the cost increases at an increasing rate.

Hypothesis 3: Performance outputs are very important to managers. It directly affects the income of subject librarians. Set the performance outputs of subject librarians in different tasks, respectively, as S₁ = e₁ + θ₁ and S₂ = e₂ + θ₂, where θ₁ ∼ N(0, σ₁²) and θ₂ ∼ N(0, σ₂²).

Hypothesis 4: Subject librarians have private information and interest that conflict with managers. Namely, the subject librarians aim to enjoy the vacation, professional title, relaxing, professional technical learning, promotion, etc. In order to effectively supervise the subject librarians to complete the task and achieve high performance, the managers provide the subject librarians with a linear compensation incentive contract w = α +b₁S₁ +b₂S₂, where α is the fixed compensation, b₁ and b₂, respectively, denote the incentive factors obtained by the subject librarian, and the contract is complete.

Hypothesis 5: For better research, we assume that subject librarians have an absolute risk aversion utility function μ = −e^−ρ[w−C], where ρ = −μ″/μ′ > 0 is the risk preference, and the reservation utility is ū. This hypothesis implies that the expected utility is equivalent to their expected income for both parties.

2.2

Model building

Based on Hypotheses 3 and 4, we can get the expected net utility of the principal as follows:

E (S_{1} + S_{2}) - E (w) = E (e_{1} + θ_{1} + e_{2} + θ_{2}) - E (α + b_{1} S_{1} + b_{2} S_{2}) = (1 - b_{1}) e_{1} + (1 - b_{2}) e_{2} - α

E({S_1} + {S_2}) - E(w) = E({e_1} + {\theta _1} + {e_2} + {\theta _2}) - E(\alpha + {b_1}{S_1} + {b_2}{S_2}) = (1 - {b_1}){e_1} + (1 - {b_2}){e_2} - \alpha

Based on Hypotheses 2 and 4, we can obtain the expected net return of the agent (subject librarian) as follows:

E (ω) - C (e_{1}, e_{2}) = α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} c_{1} e_{1}^{2} - \frac{1}{2} c_{2} e_{2}^{2} - δ e_{1} e_{2}

E(\omega) - C({e_1},{e_2}) = \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{c_1}e_1^2 - {1 \over 2}{c_2}e_2^2 - \delta {e_1}{e_2}

The certainty equivalent income, namely CE, can directly describe the agent's expected utility or income, and CE is equal to the difference between the agent's expected utility and the cost of risk. Then, CE can be obtained as follows:

CE = α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} c_{1} e_{1}^{2} - \frac{1}{2} c_{2} e_{2}^{2} - δ e_{1} e_{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}{c_1}e_1^2 - {1 \over 2}{c_2}e_2^2 - \delta {e_1}{e_2} - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2

Then, under the inter-task dependence, the best incentive model can be expressed as follows: ${\begin{array}{l} max_{{\bar{b}}_{1}, {\bar{b}}_{2}, {\bar{e}}_{1}, {\bar{e}}_{2}, \bar{α}} (1 - b_{1}) e_{1} + (1 - b_{2}) e_{2} - α \\ s . t . (1) α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} c_{1} e_{1}^{2} - \frac{1}{2} c_{2} e_{2}^{2} - δ e_{1} e_{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \geq \bar{u} \\ (2) e_{1}, e_{2} \in \arg max_{e_{1}, e_{2}} α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} c_{1} e_{1}^{2} - \frac{1}{2} c_{2} e_{2}^{2} - δ e_{1} e_{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \end{array}$ \left\{{\matrix{{\mathop {\max}\limits_{{{\bar b}_1}{\rm{,}}{\kern 1pt} {{\bar b}_2}{\rm{,}}{\kern 1pt} {{\bar e}_1}{\rm{,}}{\kern 1pt} {{\bar e}_2}{\rm{,}}{\kern 1pt} \bar \alpha} (1 - {b_1}){e_1} + (1 - {b_2}){e_2} - \alpha} \hfill \cr {s.t.\;(1)\;\alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{c_1}e_1^2 - {1 \over 2}{c_2}e_2^2 - \delta {e_1}{e_2} - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2 \ge \bar u} \hfill \cr {\quad \;\;{\kern 1pt} (2)\;{e_1}{\rm{,}}{\kern 1pt} {e_2} \in \arg \mathop {\max}\limits_{{e_1}{\rm{,}}{\kern 1pt} {e_2}} \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{c_1}e_1^2 - {1 \over 2}{c_2}e_2^2 - \delta {e_1}{e_2} - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2} \hfill \cr}} \right. where (1) denotes individual rationality, referred to as IR, in which participation constraint satisfies the minimum utility of subject librarians to participate in incentive finance contract and (2) represents the incentive compatible constraint, indicated as IC, in which the incentive compatible constraint can satisfy the subject librarian's maximum benefit.

Results

Proposition 1

Under the inter-task dependence, the best incentive contract is ( $\bar{α}$ \bar \alpha , ${\bar{b}}_{1}$ {\bar b_1} , ${\bar{b}}_{2}$ {\bar b_2} , ${\bar{e}}_{1}$ {\bar e_1} , ${\bar{e}}_{2}$ {\bar e_2} ), where $\begin{array}{l} {\bar{b}}_{1} = \frac{1 + (c_{2} - δ) ρ σ_{2}^{2}}{1 + ρ (c_{1} σ_{1}^{2} + c_{2} σ_{2}^{2}) + ρ^{2} σ_{1}^{2} σ_{2}^{2} (c_{1} c_{2} - δ^{2})} \\ {\bar{b}}_{2} = \frac{1 + (c_{1} - δ) ρ σ_{1}^{2}}{1 + ρ (c_{1} σ_{1}^{2} + c_{2} σ_{2}^{2}) + ρ^{2} σ_{1}^{2} σ_{2}^{2} (c_{1} c_{2} - δ^{2})} \\ \bar{α} = \bar{u} - {\bar{b}}_{1} {\bar{e}}_{1} - {\bar{b}}_{2} {\bar{e}}_{2} + \frac{1}{2} c_{1} {\bar{e}}_{1}^{2} + \frac{1}{2} c_{2} {\bar{e}}_{2}^{2} + δ {\bar{e}}_{1} {\bar{e}}_{2} + \frac{1}{2} ρ {\bar{b}}_{1}^{2} σ_{1}^{2} + \frac{1}{2} ρ {\bar{b}}_{2}^{2} σ_{2}^{2} \end{array}$ \matrix{{{{\bar b}_1} = {{1 + ({c_2} - \delta)\rho \sigma _2^2} \over {1 + \rho ({c_1}\sigma _1^2 + {c_2}\sigma _2^2) + {\rho ^2}\sigma _1^2\sigma _2^2({c_1}{c_2} - {\delta ^2})}}} \hfill \cr {{{\bar b}_2} = {{1 + ({c_1} - \delta)\rho \sigma _1^2} \over {1 + \rho ({c_1}\sigma _1^2 + {c_2}\sigma _2^2) + {\rho ^2}\sigma _1^2\sigma _2^2({c_1}{c_2} - {\delta ^2})}}} \hfill \cr {\bar \alpha = \bar u - {{\bar b}_1}{{\bar e}_1} - {{\bar b}_2}{{\bar e}_2} + {1 \over 2}{c_1}\bar e_1^2 + {1 \over 2}{c_2}\bar e_2^2 + \delta {{\bar e}_1}{{\bar e}_2} + {1 \over 2}\rho \bar b_1^2\sigma _1^2 + {1 \over 2}\rho \bar b_2^2\sigma _2^2} \hfill \cr} and $\begin{array}{l} {\bar{e}}_{1} = \frac{{\bar{b}}_{1} c_{2} - δ {\bar{b}}_{2}}{c_{1} c_{2} - δ^{2}} \\ {\bar{e}}_{2} = \frac{{\bar{b}}_{2} c_{1} - δ {\bar{b}}_{1}}{c_{1} c_{2} - δ^{2}} \end{array}$ \matrix{{{{\bar e}_1} = {{{{\bar b}_1}{c_2} - \delta {{\bar b}_2}} \over {{c_1}{c_2} - {\delta ^2}}}} \hfill \cr {{{\bar e}_2} = {{{{\bar b}_2}{c_1} - \delta {{\bar b}_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \hfill \cr}

Proof

Under the inter-task dependence and information asymmetry, managers cannot fully monitor the efforts made by the agent. Subject librarians will choose rational behaviours that maximise their own interests, instead of satisfying the demand of the academic managers.

Therefore, under the inter-task dependence, the best incentive contract is the solution of the following incentive optimisation model: (1) ${\begin{array}{l} max_{{\bar{b}}_{1}, {\bar{b}}_{2}, {\bar{e}}_{1}, {\bar{e}}_{2}, \bar{α}} (1 - b_{1}) e_{1} + (1 - b_{2}) e_{2} - α \\ s . t . (1) α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} c_{1} e_{1}^{2} - \frac{1}{2} c_{2} e_{2}^{2} - δ e_{1} e_{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \geq \bar{u} \\ (2) e_{1}, e_{2} \in \arg max_{e_{1}, e_{2}} α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} c_{1} e_{1}^{2} - \frac{1}{2} c_{2} e_{2}^{2} - δ e_{1} e_{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \end{array}$ \left\{{\matrix{{\mathop {\max}\limits_{{{\bar b}_1},{\kern 1pt} {{\bar b}_2},{\kern 1pt} {{\bar e}_1},{\kern 1pt} {{\bar e}_2},{\kern 1pt} \bar \alpha} (1 - {b_1}){e_1} + (1 - {b_2}){e_2} - \alpha} \hfill \cr {s.t.\;(1)\;\alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{c_1}e_1^2 - {1 \over 2}{c_2}e_2^2 - \delta {e_1}{e_2} - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2 \ge \bar u} \hfill \cr {\quad \;\;{\kern 1pt} (2)\;{e_1},{\kern 1pt} {e_2} \in \arg \mathop {\max}\limits_{{e_1},{\kern 1pt} {e_2}} \alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{c_1}e_1^2 - {1 \over 2}{c_2}e_2^2 - \delta {e_1}{e_2} - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2} \hfill \cr}} \right.

The first-order conditions of effort levels e₁ and e₂ in the incentive compatibility constraint in Eq. (1) are as follows: $\begin{array}{l} b_{1} - c_{1} e_{1} - δ e_{2} = 0 \\ b_{2} - c_{2} e_{2} - δ e_{1} = 0 \end{array}$ \matrix{{{b_1} - {c_1}{e_1} - \delta {e_2} = 0} \hfill \cr {{b_2} - {c_2}{e_2} - \delta {e_1} = 0} \hfill \cr}

Therefore, Eq. (1) can be restated as follows: (2) ${\begin{array}{l} max_{{\bar{b}}_{1}, {\bar{b}}_{2}, {\bar{e}}_{1}, {\bar{e}}_{2}, \bar{α}} (1 - b_{1}) e_{1} + (1 - b_{2}) e_{2} - α \\ s . t . (1) α + b_{1} e_{1} + b_{2} e_{2} - \frac{1}{2} c_{1} e_{1}^{2} - \frac{1}{2} c_{2} e_{2}^{2} - δ e_{1} e_{2} - \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} - \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2} \geq \bar{u} \\ (2) b_{1} - c_{1} e_{1} - δ e_{2} = 0; b_{2} - c_{2} e_{2} - δ e_{1} = 0 \end{array}$ \left\{{\matrix{{\mathop {\max}\limits_{{{\bar b}_1},{\kern 1pt} {{\bar b}_2},{\kern 1pt} {{\bar e}_1},{\kern 1pt} {{\bar e}_2},{\kern 1pt} \bar \alpha} (1 - {b_1}){e_1} + (1 - {b_2}){e_2} - \alpha} \hfill \cr {s.t.\;(1)\;\alpha + {b_1}{e_1} + {b_2}{e_2} - {1 \over 2}{c_1}e_1^2 - {1 \over 2}{c_2}e_2^2 - \delta {e_1}{e_2} - {1 \over 2}\rho b_1^2\sigma _1^2 - {1 \over 2}\rho b_2^2\sigma _2^2 \ge \bar u} \hfill \cr {\quad \;\;{\kern 1pt} (2)\;{b_1} - {c_1}{e_1} - \delta {e_2} = 0;\;{b_2} - {c_2}{e_2} - \delta {e_1} = 0} \hfill \cr}} \right.

From the incentive compatibility constraint in Eq. (2), the following equation can be obtained: (3) $e_{1} = \frac{b_{1} c_{2} - δ b_{2}}{c_{1} c_{2} - δ^{2}}$ {e_1} = {{{b_1}{c_2} - \delta {b_2}} \over {{c_1}{c_2} - {\delta ^2}}} (4) $e_{2} = \frac{b_{2} c_{1} - δ b_{1}}{c_{1} c_{2} - δ^{2}}$ {e_2} = {{{b_2}{c_1} - \delta {b_1}} \over {{c_1}{c_2} - {\delta ^2}}}

When the participation constraint is equal, the subject librarian gets the reserved salary, which is expressed as follows: (5) $α = \bar{u} - b_{1} e_{1} - b_{2} e_{2} + \frac{1}{2} c_{1} e_{1}^{2} + \frac{1}{2} c_{2} e_{2}^{2} + δ e_{1} e_{2} + \frac{1}{2} ρ b_{1}^{2} σ_{1}^{2} + \frac{1}{2} ρ b_{2}^{2} σ_{2}^{2}$ \alpha = \bar u - {b_1}{e_1} - {b_2}{e_2} + {1 \over 2}{c_1}e_1^2 + {1 \over 2}{c_2}e_2^2 + \delta {e_1}{e_2} + {1 \over 2}\rho b_1^2\sigma _1^2 + {1 \over 2}\rho b_2^2\sigma _2^2

Substituting Eqs (3)–(5) into the objective function in Eq. (2), the following equations are obtained: $\begin{array}{l} U_{p} & = (\frac{b_{1} c_{2} - δ b_{2} + b_{2} c_{1} - δ b_{1}}{c_{1} c_{2} - δ^{2}}) - \frac{ρ}{2} [b_{1}^{2} σ_{1}^{2} + b_{2}^{2} σ_{2}^{2}] - \frac{1}{2} c_{1} {(\frac{b_{1} c_{2} - δ b_{2}}{c_{1} c_{2} - δ^{2}})}^{2} - \frac{1}{2} c_{2} {(\frac{b_{2} c_{1} - δ b_{1}}{c_{1} c_{2} - δ^{2}})}^{2} \\ - δ (\frac{b_{1} c_{2} - δ b_{2}}{c_{1} c_{2} - δ^{2}}) (\frac{b_{2} c_{1} - δ b_{1}}{c_{1} c_{2} - δ^{2}}) \end{array}$ \matrix{{{U_p}} \hfill & {= \left({{{{b_1}{c_2} - \delta {b_2} + {b_2}{c_1} - \delta {b_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \right) - {\rho \over 2}\left[ {b_1^2\sigma _1^2 + b_2^2\sigma _2^2} \right] - {1 \over 2}{c_1}{{\left({{{{b_1}{c_2} - \delta {b_2}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)}^2} - {1 \over 2}{c_2}{{\left({{{{b_2}{c_1} - \delta {b_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)}^2}} \hfill \cr {} \hfill & {- \delta \left({{{{b_1}{c_2} - \delta {b_2}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)\left({{{{b_2}{c_1} - \delta {b_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)} \hfill \cr}

And then the simplified optimisation problem is obtained as follows: (6) $\begin{matrix} max_{b_{1}, b_{2}} (\frac{b_{1} c_{2} - δ b_{2} + b_{2} c_{1} - δ b_{1}}{c_{1} c_{2} - δ^{2}}) - \frac{ρ}{2} [b_{1}^{2} σ_{1}^{2} + b_{2}^{2} σ_{2}^{2}] - \frac{1}{2} c_{1} {(\frac{b_{1} c_{2} - δ b_{2}}{c_{1} c_{2} - δ^{2}})}^{2} - \frac{1}{2} c_{2} {(\frac{b_{2} c_{1} - δ b_{1}}{c_{1} c_{2} - δ^{2}})}^{2} \\ - δ (\frac{b_{1} c_{2} - δ b_{2}}{c_{1} c_{2} - δ^{2}}) (\frac{b_{2} c_{1} - δ b_{1}}{c_{1} c_{2} - δ^{2}}) \end{matrix}$ \matrix{{\mathop {\max}\limits_{{b_1},{b_2}} \left({{{{b_1}{c_2} - \delta {b_2} + {b_2}{c_1} - \delta {b_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \right) - {\rho \over 2}\left[ {b_1^2\sigma _1^2 + b_2^2\sigma _2^2} \right] - {1 \over 2}{c_1}{{\left({{{{b_1}{c_2} - \delta {b_2}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)}^2} - {1 \over 2}{c_2}{{\left({{{{b_2}{c_1} - \delta {b_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)}^2}} \cr {- \delta \left({{{{b_1}{c_2} - \delta {b_2}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)\left({{{{b_2}{c_1} - \delta {b_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \right)} \cr}

The first-order conditions of the share proportion b₁ and b₂ in unconstrained optimisation problem Eq. (6) are expressed as follows: $\begin{array}{l} b_{1} = \frac{c_{2} - δ + δ b_{2}}{c_{2} + ρ σ_{1}^{2} (c_{1} c_{2} - δ^{2})} \\ b_{2} = \frac{c_{1} - δ + δ b_{1}}{c_{1} + ρ σ_{2}^{2} (c_{1} c_{2} - δ^{2})} \end{array}$ \matrix{{{b_1} = {{{c_2} - \delta + \delta {b_2}} \over {{c_2} + \rho \sigma _1^2({c_1}{c_2} - {\delta ^2})}}} \hfill \cr {{b_2} = {{{c_1} - \delta + \delta {b_1}} \over {{c_1} + \rho \sigma _2^2({c_1}{c_2} - {\delta ^2})}}} \hfill \cr}

Through simultaneous simplification, it can be obtained that under the inter-task dependence, the share proportion for the two tasks is, respectively, given as follows: (7) ${\bar{b}}_{1} = \frac{1 + (c_{2} - δ) ρ σ_{2}^{2}}{1 + ρ (c_{1} σ_{1}^{2} + c_{2} σ_{2}^{2}) + ρ^{2} σ_{1}^{2} σ_{2}^{2} (c_{1} c_{2} - δ^{2})}$ {\bar b_1} = {{1 + ({c_2} - \delta)\rho \sigma _2^2} \over {1 + \rho ({c_1}\sigma _1^2 + {c_2}\sigma _2^2) + {\rho ^2}\sigma _1^2\sigma _2^2({c_1}{c_2} - {\delta ^2})}} (8) ${\bar{b}}_{2} = \frac{1 + (c_{1} - δ) ρ σ_{1}^{2}}{1 + ρ (c_{1} σ_{1}^{2} + c_{2} σ_{2}^{2}) + ρ^{2} σ_{1}^{2} σ_{2}^{2} (c_{1} c_{2} - δ^{2})}$ {\bar b_2} = {{1 + ({c_1} - \delta)\rho \sigma _1^2} \over {1 + \rho ({c_1}\sigma _1^2 + {c_2}\sigma _2^2) + {\rho ^2}\sigma _1^2\sigma _2^2({c_1}{c_2} - {\delta ^2})}}

Substituting Eqs (7) and (8) into Eqs (3) and (4), the efforts made by the subject librarian are denoted as follows: $\begin{array}{l} {\bar{e}}_{1} = \frac{{\bar{b}}_{1} c_{2} - δ {\bar{b}}_{2}}{c_{1} c_{2} - δ^{2}} \\ {\bar{e}}_{2} = \frac{{\bar{b}}_{2} c_{1} - δ {\bar{b}}_{1}}{c_{1} c_{2} - δ^{2}} \end{array}$ \matrix{{{{\bar e}_1} = {{{{\bar b}_1}{c_2} - \delta {{\bar b}_2}} \over {{c_1}{c_2} - {\delta ^2}}}} \hfill \cr {{{\bar e}_2} = {{{{\bar b}_2}{c_1} - \delta {{\bar b}_1}} \over {{c_1}{c_2} - {\delta ^2}}}} \hfill \cr}

Substituting Eqs (7) and (8) into Eq. (5), the fixed compensation can be obtained as follows: $\bar{α} = \bar{u} - {\bar{b}}_{1} {\bar{e}}_{1} - {\bar{b}}_{2} {\bar{e}}_{2} + \frac{1}{2} c_{1} {\bar{e}}_{1}^{2} + \frac{1}{2} c_{2} {\bar{e}}_{2}^{2} + δ {\bar{e}}_{1} {\bar{e}}_{2} + \frac{1}{2} ρ {\bar{b}}_{1}^{2} σ_{1}^{2} + \frac{1}{2} ρ {\bar{b}}_{2}^{2} σ_{2}^{2}$ \bar \alpha = \bar u - {\bar b_1}{\bar e_1} - {\bar b_2}{\bar e_2} + {1 \over 2}{c_1}\bar e_1^2 + {1 \over 2}{c_2}\bar e_2^2 + \delta {\bar e_1}{\bar e_2} + {1 \over 2}\rho \bar b_1^2\sigma _1^2 + {1 \over 2}\rho \bar b_2^2\sigma _2^2

Corollary 1

The best incentive intensity decreases as the subject service selectivity increases, which is $\frac{\partial {\bar{b}}_{1}}{\partial σ_{1}^{2}} < 0, \frac{\partial {\bar{b}}_{2}}{\partial σ_{2}^{2}} < 0$ {{\partial {{\bar b}_1}} \over {\partial \sigma _1^2}} < 0,\;{{\partial {{\bar b}_2}} \over {\partial \sigma _2^2}} < 0

Proof

In fact, in the expression of ${\bar{b}}_{1}$ {\bar b_1} , the numerator is a normal number independent of $σ_{1}^{2}$ \sigma _1^2 , and the term with denominator related to $σ_{1}^{2}$ \sigma _1^2 is

$[c_{1} + ρ σ_{2}^{2} (c_{1} c_{2} - δ^{2})] ρ σ_{1}^{2}$ [{c_1} + \rho \sigma _2^2({c_1}{c_2} - {\delta ^2})]\rho \sigma _1^2

And because c₁ > 0, ρ > 0, c₁c₂ − δ² > 0, so

$[c_{1} + ρ σ_{2}^{2} (c_{1} c_{2} - δ^{2})] ρ > 0$ [{c_1} + \rho \sigma _2^2({c_1}{c_2} - {\delta ^2})]\rho > 0

Then,

$\frac{\partial {\bar{b}}_{1}}{\partial σ_{1}^{2}} < 0$ {{\partial {{\bar b}_1}} \over {\partial \sigma _1^2}} < 0

Similarly, it can be proved that

$\frac{\partial {\bar{b}}_{2}}{\partial σ_{2}^{2}} < 0$ {{\partial {{\bar b}_2}} \over {\partial \sigma _2^2}} < 0

Corollary 2

The profit-sharing ratios ${\bar{b}}_{i}$ {\bar b_i} corresponding to different tasks are complementary, which is $\frac{\partial {\bar{b}}_{1}}{\partial σ_{2}^{2}} < 0, \frac{\partial {\bar{b}}_{2}}{\partial σ_{1}^{2}} < 0$ {{\partial {{\bar b}_1}} \over {\partial \sigma _2^2}} < 0,\;{{\partial {{\bar b}_2}} \over {\partial \sigma _1^2}} < 0

Proof

First, by Eq. (7) and the chain guide rule, the following equation is obtained: $\begin{array}{l} \frac{\partial {\bar{b}}_{1}}{\partial σ_{2}^{2}} & = \frac{\partial {\bar{b}}_{1}}{\partial {\bar{b}}_{2}} \times \frac{\partial {\bar{b}}_{2}}{\partial σ_{2}^{2}} \\ = {[c_{2} + ρ σ_{1}^{2} (c_{1} c_{2} - δ^{2})]}^{- 1} δ \times \frac{\partial {\bar{b}}_{2}}{\partial σ_{2}^{2}} \\ < 0 \end{array}$ \matrix{{{{\partial {{\bar b}_1}} \over {\partial \sigma _2^2}}} \hfill & {= {{\partial {{\bar b}_1}} \over {\partial {{\bar b}_2}}} \times {{\partial {{\bar b}_2}} \over {\partial \sigma _2^2}}} \hfill \cr {} \hfill & {= {{[{c_2} + \rho \sigma _1^2({c_1}{c_2} - {\delta ^2})]}^{- 1}}\delta \times {{\partial {{\bar b}_2}} \over {\partial \sigma _2^2}}} \hfill \cr {} \hfill & {< 0} \hfill \cr}

Second, by Eq. (8) and the chain guide rule, the following equation is obtained: $\begin{array}{l} \frac{\partial {\bar{b}}_{2}}{\partial σ_{1}^{2}} & = \frac{\partial {\bar{b}}_{2}}{\partial {\bar{b}}_{1}} \times \frac{\partial {\bar{b}}_{1}}{\partial σ_{1}^{2}} \\ = {[c_{1} + ρ σ_{2}^{2} (c_{1} c_{2} - δ^{2})]}^{- 1} δ \times \frac{\partial {\bar{b}}_{1}}{\partial σ_{1}^{2}} \\ < 0 \end{array}$ \matrix{{{{\partial {{\bar b}_2}} \over {\partial \sigma _1^2}}} \hfill & {= {{\partial {{\bar b}_2}} \over {\partial {{\bar b}_1}}} \times {{\partial {{\bar b}_1}} \over {\partial \sigma _1^2}}} \hfill \cr {} \hfill & {= {{[{c_1} + \rho \sigma _2^2({c_1}{c_2} - {\delta ^2})]}^{- 1}}\delta \times {{\partial {{\bar b}_1}} \over {\partial \sigma _1^2}}} \hfill \cr {} \hfill & {< 0} \hfill \cr}

Corollary 3

The higher the incentive intensity of academic managers for specific tasks, the higher the effort intensity of subject librarians is, which is $\frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{1}} > 0, \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{2}} > 0$ {{\partial {{\bar e}_1}} \over {\partial {{\bar b}_1}}} > 0,\;{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_2}}} > 0

Proof

From Proposition 1, the best effort level is differentiated with respect to the best incentive intensities ${\bar{b}}_{1}$ {\bar b_1} and ${\bar{b}}_{2}$ {\bar b_2} , as shown below: $\begin{array}{l} \frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{1}} = \frac{c_{2} (c_{1} c_{2} - δ^{2})}{{(c_{1} c_{2} - δ^{2})}^{2}} \\ \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{2}} = \frac{c_{1} (c_{1} c_{2} - δ^{2})}{{(c_{1} c_{2} - δ^{2})}^{2}} \end{array}$ \matrix{{{{\partial {{\bar e}_1}} \over {\partial {{\bar b}_1}}} = {{{c_2}({c_1}{c_2} - {\delta ^2})} \over {{{({c_1}{c_2} - {\delta ^2})}^2}}}} \hfill \cr {{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_2}}} = {{{c_1}({c_1}{c_2} - {\delta ^2})} \over {{{({c_1}{c_2} - {\delta ^2})}^2}}}} \hfill \cr}

When $δ \in [0, {(c_{1} c_{2})}^{\frac{1}{2}}]$ \delta \in [0,{({c_1}{c_2})^{{1 \over 2}}}] , that is, when the tasks are interchangeable, the following equation is obtained: $\frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{1}} > 0, \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{2}} > 0$ {{\partial {{\bar e}_1}} \over {\partial {{\bar b}_1}}} > 0,\;{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_2}}} > 0

And when $δ \in [- {(c_{1} c_{2})}^{\frac{1}{2}}, 0]$ \delta \in [ - {({c_1}{c_2})^{{1 \over 2}}},0] , when the tasks complement each other, the following equation is obtained: $\frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{1}} > 0, \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{2}} > 0$ {{\partial {{\bar e}_1}} \over {\partial {{\bar b}_1}}} > 0,\;{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_2}}} > 0

Corollary 4

The higher the incentive intensity of university managers for specific tasks, the lower the effort level of subject librarians for another specific task is, which is $\frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{2}} < 0, \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{1}} < 0$ {{\partial {{\bar e}_1}} \over {\partial {{\bar b}_2}}} < 0,\;{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_1}}} < 0

Proof

The best effort level is differentiated with respect to the optimal incentive intensities ${\bar{b}}_{1}$ {\bar b_1} and ${\bar{b}}_{2}$ {\bar b_2} , as shown below: $\begin{array}{l} \frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{2}} = \frac{- δ (c_{1} c_{2} - δ^{2})}{{(c_{1} c_{2} - δ^{2})}^{2}} \\ \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{1}} = \frac{- δ (c_{1} c_{2} - δ^{2})}{{(c_{1} c_{2} - δ^{2})}^{2}} \end{array}$ \matrix{{{{\partial {{\bar e}_1}} \over {\partial {{\bar b}_2}}} = {{- \delta ({c_1}{c_2} - {\delta ^2})} \over {{{({c_1}{c_2} - {\delta ^2})}^2}}}} \hfill \cr {{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_1}}} = {{- \delta ({c_1}{c_2} - {\delta ^2})} \over {{{({c_1}{c_2} - {\delta ^2})}^2}}}} \hfill \cr}

When $δ \in [0, {(c_{1} c_{2})}^{\frac{1}{2}}]$ \delta \in \left[ {0,{{({c_1}{c_2})}^{{1 \over 2}}}} \right] , that is, when the tasks are interchangeable, the following equation is obtained: $\frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{2}} < 0, \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{1}} < 0$ {{\partial {{\bar e}_1}} \over {\partial {{\bar b}_2}}} < 0,\;{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_1}}} < 0

And when $δ \in [- {(c_{1} c_{2})}^{\frac{1}{2}}, 0]$ \delta \in [ - {({c_1}{c_2})^{{1 \over 2}}},0] , when the tasks complement each other, the following equation is obtained: $\frac{\partial {\bar{e}}_{1}}{\partial {\bar{b}}_{2}} < 0, \frac{\partial {\bar{e}}_{2}}{\partial {\bar{b}}_{1}} < 0$ {{\partial {{\bar e}_1}} \over {\partial {{\bar b}_2}}} < 0,\;{{\partial {{\bar e}_2}} \over {\partial {{\bar b}_1}}} < 0

Empirical test

Based on the Proposition 1 and Corollaries 1–4, the following empirical model is established. And the correlation analysis between incentive intensity and main variables is as follows in Table 1.

(model 1)

Ln ({Inc}_{it}) = α + β_{1} Ln {Cos}_{it} + β_{2} {({Ara}_{it})}^{2} + β_{3} {Ara}_{it} + ε_{it}

Ln({Inc}_{it}) = \alpha + {\beta _1}Ln{\rm{Cos}_{it}} + {\beta _2}{({Ara}_{it})^2} + {\beta _3}{Ara}_{it} + {\varepsilon _{it}}

(model 2)

Ln ({Inc}_{it}) = α + β_{1} {({Cos}_{it})}^{2} + β_{2} {Cos}_{it} + β_{3} {({Ara}_{it})}^{2} + β_{4} {Ara}_{it} + ε_{it}

Ln({Inc}_{it}) = \alpha + {\beta _1}{({\rm{Cos}_{it}})^2} + {\beta _2}{\rm{Cos}_{it}} + {\beta _3}{({Ara}_{it})^2} + {\beta _4}{Ara}_{it} + {\varepsilon _{it}}

(model 3)

Ln ({Inc}_{it}) = α + β_{1} {(Ln {Cos}_{it})}^{3} + β_{2} {(Ln {Cos}_{it})}^{2} + β_{3} Ln {Cos}_{it} + β_{4} {({Ara}_{it})}^{2} + β_{5} {Ara}_{it} + ε_{it}

Ln({Inc}_{it}) = \alpha + {\beta _1}{(Ln{\rm{Co}}{{\rm{s}}_{it}})^3} + {\beta _2}{(Ln{\rm{Co}}{{\rm{s}}_{it}})^2} + {\beta _3}Ln{\rm{Cos}_{it}} + {\beta _4}{({Ara}_{it})^2} + {\beta _5}{Ara}_{it} + {\varepsilon _{it}}

(model 4)

Ln ({Inc}_{it}) = α + β_{1} {(Ln {Cos}_{it})}^{3} + β_{2} {(Ln {Cos}_{it})}^{2} + β_{3} Ln {Cos}_{it} + β_{4} {({Ara}_{it})}^{2} + β_{5} {Ara}_{it} + β_{6} {Unc}_{it} + ε_{it}

Ln({Inc}_{it}) = \alpha + {\beta _1}{(Ln{\rm{Cos}_{it}})^3} + {\beta _2}{(Ln{\rm{Cos}_{it}})^2} + {\beta _3}Ln{\rm{Cos}_{it}} + {\beta _4}{({Ara}_{it})^2} + {\beta _5}{Ara}_{it} + {\beta _6}{Unc}_{it} + {\varepsilon _{it}}

(model 5)

\begin{array}{l} Ln ({Inc}_{it}) & = α + β_{1} {(Ln {Cos}_{it})}^{3} + β_{2} {(Ln {Cos}_{it})}^{2} + β_{3} Ln {Cos}_{it} + β_{4} {({Ara}_{it})}^{2} + β_{5} {Ara}_{it} + β_{6} {Unc}_{it} \\ + β_{7} Ln (f p_{it}) + β_{8} Ln (d p_{it}) + ε_{it} \end{array}

\matrix{{Ln({Inc}_{it})} \hfill & {= \alpha + {\beta _1}{{(Ln{\rm{Cos}_{it}})}^3} + {\beta _2}{{(Ln{\rm{Cos}_{it}})}^2} + {\beta _3}Ln{\rm{Cos}_{it}} + {\beta _4}{{({Ara}_{it})}^2} + {\beta _5}{Ara}_{it} + {\beta _6}{Unc}_{it}} \hfill \cr {} \hfill & {+ {\beta _7}Ln(f{p_{it}}) + {\beta _8}Ln(d{p_{it}}) + {\varepsilon _{it}}} \hfill \cr}

(model 6)

\begin{array}{l} Ln ({Inc}_{it}) & = α + β_{1} {(Ln {Cos}_{it})}^{3} + β_{2} {(Ln {Cos}_{it})}^{2} + β_{3} Ln {Cos}_{it} + β_{4} {({Ara}_{it})}^{2} + β_{5} {Ara}_{it} + β_{6} {Unc}_{it} \\ + β_{7} Ln (f p_{it}) + β_{8} Ln (d p_{it}) + β_{9} {Age}_{it} * {Dum}_{it} * {Unc}_{it} + ε_{it} \end{array}

Table 1

Correlation regression analysis of main core variables

Variables	M1	M2	M3	M4	M5	M6

LnCos	0.1073 (0.54)	−1.0254 (−0.51)	2.5067 (0.20)	3.5464 (0.28)	11.1197 (0.98)	9.7586 (0.86)
(LnCos)²		0.2858 (0.56)	−1.5639 (−0.24)	−2.0848 (−0.32)	−5.9094 (−1.00)	−5.3936 (−0.91)
(LnCos)³			0.3174 (0.29)	0.4044 (0.36)	1.0522 (1.04)	0.9937 (0.98)
Ara	0.3899*** (4.92)	0.3919*** (4.93)	0.3912*** (4.92)	0.3922*** (4.93)	0.4475*** (6.00)	0.4468*** (6.00)
Ara²	−0.253*** (−4.33)	−0.2565*** (−4.35)	−0.2552*** (−4.32)	−0.2569*** (−4.34)	−0.2725*** (−4.91)	−0.2715*** (−4.90)
Unc				0.0137 (1.17)	−0.0136 (−1.26)	−0.0267** (−2.03)
Lnfp					0.6614*** (2.95)	0.5993*** (2.65)
Lndp					0.7104*** (2.97)	0.7746*** (3.21)
Age * Dum * Unc						0.0018* (1.69)
Within R²	0.0805	0.0804	0.0821	0.0829	0.1930	0.1912
Between R²	0.1324	0.1448	0.1355	0.1341	0.3361	0.3584
Overall R²	0.0905	0.0845	0.0803	0.0913	0.2597	0.3059
N	315	315	315	315	315	315

***, ** and *

indicate that the correlation coefficient is significant at the level of 1%, 5% and 10%, respectively.

Conclusions

In reality, tasks are not independent of each other but interdependent. When the tasks are interchangeable, it means that the completion of one task will increase the difficulty of the other task. When the tasks are mutually complementary, it means that the completion of one task is more conducive to the other task. Based on the Holmstrom and Milgrom [11, 12] traditional model, we introduce the correlation coefficient between tasks, and the best compensation incentive model is constructed under the inter-task cost function. The best compensation incentive contract is constructed by solving the incentive model, and the incentive characteristics are analysed. The results show that the best incentive intensity decreases as the subject service selectivity increases. The higher incentive intensity of university managers for specific tasks, the lower efforts of subject librarians for another specific task. Moreover, when the tasks are substituted for each other, the profit-sharing ratios corresponding to different tasks are complementary. Finally, we establish the econometric empirical models to test these results.

eISSN:: 2444-8656
Sprache:: Englisch

Zeitrahmen der Veröffentlichung:: Volume Open
Fachgebiete der Zeitschrift:: Biologie, andere, Mathematik, Angewandte Mathematik, Allgemeines, Physik

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Compensation incentive contract of the subject librarians based on the H-M model

Online veröffentlicht: 29. Nov. 2021

Seitenbereich: 553 - 562

Eingereicht: 22. Apr. 2021

Akzeptiert: 29. Mai 2021

DOI: https://doi.org/10.2478/amns.2021.2.00106

Schlüsselwörtersubject librarian, compensation, incentive contract

© 2021 Chunping Wang et al., published by Sciendo

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

Schlüsselwörter
subject librarian, compensation, incentive contract