The impact of blockchain on optimal incentive contracts for online supply chain finance

Abstract

In China, supply chain finance is still in infancy. However, it is the problems of information sharing, trust transfer, and risk management that have been making it difficult to meet the financing needs of small- and medium-sized enterprises (SMEs) in supply chain. The emerging blockchain technology, with its unique decentralization, traceability, and other characteristics, has found a digital solution for traditional supply chain finance. Although blockchain has attracted widespread attention and there are more general descriptions of blockchain application areas, there are few researches on the impact mechanisms of blockchain in-depth. Especially in the field of supply chain finance, there is little research on optimal incentive contract in online supply chain finance empowered by blockchain technology. Therefore, this paper explores the influence of blockchain technology maturity on participants, and thus finds the optimal incentive contract in online supply chain empowered by blockchain technology. Because of the mastery of blockchain technology, platforms believe they are well protected against risk and may behave irrationally. Therefore, this paper considers the overconfident behavior of blockchain supply chain finance platform in actual operation, and then applies the principal-agent model and incentive theory to design the incentive mechanism between platforms, banks, and central banks. Finally, numerical analyses show that overconfident behavior and the maturity of blockchain technology have an impact on the optimal decision for the whole supply chain.

Appendix 1. Proof of property 1

The optimization of the platform is as follows:

$$\begin{aligned} {E\pi _e{_1}}= & {} {\varphi _B{_1}}+{\varphi _a{_1}}-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\lambda {{S}_a}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\frac{1}{2}\tau \rho {\beta _a}^2{\sigma _a}^2\end{aligned}$$

(A1)

$$\begin{aligned} {\varphi _B{_1}}= & {} {\omega _B}+{\beta _B}{{y}_B{_1}}={\omega _B}+{\beta _B}({{S}_B}{\theta _B}+{{S}_a}{\theta _B{_a}}+{Z_B{_1}}{\eta }+{\varepsilon _B})\end{aligned}$$

(A2)

$$\begin{aligned} {\varphi _a{_1}}= & {} {\beta _a}{{y}_a{_1}}={\beta _a}({{S}_a}{\theta _a}+{Z_a{_1}}{\eta }+{\varepsilon _a}) \end{aligned}$$

(A3)

Formula Eqs. A2 and A3 are substituted into formula Eq. A1 respectively to get formula Eq. A4

$$\begin{aligned} {E\pi _e{_1}}= & {} {\omega _B}+{\beta _B}({{S}_B}{\theta _B}+{{S}_a}{\theta _B}{_a}+({{x}_B}+{k}{\xi _B})\eta ) +{\beta _a}({{S}_a}{\theta _a}+({{x}_a}+{k}{\xi _a})\eta ) \nonumber \\&-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\lambda {{S}_a}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\frac{1}{2}\tau \rho {\beta _a}^2{\sigma _a}^2\end{aligned}$$

(A4)

$$\begin{aligned} \frac{{dE}{\pi _e{_1}}}{{d}{{S}_B}}= & {} {\beta _B}{\theta _B}-\lambda {{S}_B}\nonumber \\ \frac{{dE}{\pi _e{_1}}}{{d}{{S}_a}}= & {} {\beta _B}{\theta _B{_a}}+{\beta _a}{\theta _a}-\lambda {{S}_a}\nonumber \\ \frac{{dE}{\pi _e{_1}}}{{d}{{S}_B}}= & {} 0, \frac{{dE}{\pi _e{_1}}}{{d}{{S}_a}} \nonumber \\ {{S}_B}^*= & {} \frac{{\beta _B}{\theta _B}}{\lambda }\end{aligned}$$

(A5)

$$\begin{aligned} {{S}_a}^*= & {} \frac{{\beta _B}{\theta _B{_a}}+{\beta _a}{\theta _a}}{\lambda } \end{aligned}$$

(A6)

Appendix 2. Proof of property 2

The expected income of the Central Bank is:

$$\begin{aligned} {E\pi _a{_1}}={{y}_a{_1}}-{\varphi _a{_1}}=(1-{\beta _a})({{S}_a}^*{\theta _a}+({{x}_a}+{k}{\xi _a})\eta ) \end{aligned}$$

(A7)

Formula Eq. A6 is substituted into formula Eq. A7 to get formula Eq. A8

$$\begin{aligned} {E\pi _a{_1}}= & {} {{y}_a{_1}}-{\varphi _a{_1}}=(1-{\beta _a})(\frac{{\beta _a}{\theta _a}^2+{\beta _B}{\theta {_B}_a}{\theta _a}}{\lambda }+({{x}_a}+{k}{\xi _a})\eta )\end{aligned}$$

(A8)

$$\begin{aligned} \frac{{dE}{\pi _a{_1}}}{{d}{\beta _a}}= & {} -(\frac{{\beta _a}{\theta _a}^2+{\beta _B}{\theta {_B}_a}{\theta _a}}{\lambda }+({{x}_a}+{k}{\xi _a})\eta )+(1-{\beta _a})\frac{{\theta _a}^2}{\lambda }\nonumber \\ \frac{{dE}{\pi _a{_1}}}{{d}{\beta _a}}= & {} 0\nonumber \\ {\beta _a}^*= & {} \frac{1}{2}-\frac{{\beta _B}{\theta _B{_a}}}{2{\theta _a}}-\frac{({{x}_a}+{k}{\xi _a})\eta \lambda }{2{\theta _a}^2} \end{aligned}$$

(A9)

Appendix 3. Proof of property 3

The expected income of the bank is:

$$\begin{aligned} Max{E\pi _B{_1}}={IrT}+(1-{\beta _B})({{S}_B}^*{\theta _B}+{{S}_a}^*{\theta {_B}_a}+({{x}_B}+{k}{\xi _B})\eta )-{\omega _B} \end{aligned}$$

(A10)

Since the game between the bank and the platform exists only when the platform participates in the decision, the bank must simultaneously satisfy constraint condition Eq. A11 in the pursuit of profit maximization:

$$\begin{aligned} {\varphi _B{_1}}-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2\ge \varpi \end{aligned}$$

(A11)

As a rational principal, the bank will not give the platform excessive remuneration, as long as the agent can be guaranteed to participate in the principal-agent relationship, that is, the participation constraint is Eq. A12:

$$\begin{aligned} {\varphi _B{_1}}-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2=\varpi \end{aligned}$$

(A12)

The Lagrange function Eq. A13 is constructed with the constraint of Eqs. A12 and A10

$$\begin{aligned} {L_1}= & {} {IrT}+(1-{\beta _B})({{S}_B}^*{\theta _B}+{{S}_a}^*{\theta {_B}_a}+({{x}_B}+{k}{\xi _B})\eta )-{\omega _B}\nonumber \\&+\alpha ({\omega _B}+{\beta _B}({{S}_B}^*{\theta _B}+{{S}_a}^*{\theta _B{_a}}+({{x}_B}+{k}{\xi _B}){\eta })-\frac{1}{2}\lambda {{S}_B}^*2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\varpi ) \end{aligned}$$

(A13)

Formula Eqs. A5, A6, and A9 are substituted into formula Eq. A13 respectively to get Eq. A14

$$\begin{aligned} {L_1}= & {} {IrT}+(1-{\beta _B})(\frac{{\beta _B}{\theta _B}}{\lambda }{\theta _B}+\frac{{\beta _B}{\theta _B{_a}}+(\frac{1}{2}-\frac{{\beta _B}{\theta _B{_a}}}{2{\theta _a}}-\frac{({{x}_a}+{k}{\xi _a})\eta \lambda }{2{\theta _a}^2}){\theta _a}}{\lambda }{\theta {_B}_a}+({{x}_B}+{k}{\xi _B})\eta )-{\omega _B}\nonumber \\+ & {} \alpha ({\omega _B}+{\beta _B}(\frac{{\beta _B}{\theta _B}}{\lambda }{\theta _B}+\frac{{\beta _B}{\theta _B{_a}}+(\frac{1}{2}-\frac{{\beta _B}{\theta _B{_a}}}{2{\theta _a}}-\frac{({{x}_a}+{k}{\xi _a})\eta \lambda }{2{\theta _a}^2}){\theta _a}}{\lambda }{\theta _B{_a}}+({{x}_B}+{k}{\xi _B}){\eta })-\frac{1}{2}\lambda (\frac{{\beta _B}{\theta _B}}{\lambda })^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\varpi ) \end{aligned}$$

(A14)

Let $\frac{dL{_1}}{{d}{\beta _B}}=0,\frac{dL{_1}}{{d}{\omega _B}}=0,\frac{dL{_1}}{{d}\alpha }=0$, respectively, the solution is

$$\begin{aligned} {\beta _B{_1}}^*= & {} \frac{{\theta _B}^2+{\theta _B{_a}}^2}{{\theta _B}^2+\tau \lambda \rho {\sigma _B}^2}\nonumber \\ {\omega _B{_1}}^*= & {} \varpi +\frac{({\theta _B}^2+{\theta _B{_a}}^2)^2(\tau \lambda \rho {\sigma _B}^2-{\theta _B}^2-3{\theta _B{_a}}^2)}{2\lambda ({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}+\frac{({\theta _B}^2+{\theta _B{_a}}^2)({\theta _B{_a}}{\theta _a}^2+2\lambda {\theta _a}({{x}_B+{k}{\xi _B}})\eta -\lambda {\theta _B}{_a}({{x}_a+{k}{\xi _a}})\eta )}{2\lambda {\theta _a}({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)} \end{aligned}$$

(A15)

Formula Eq. A15 is substituted into formula Eqs. A5, A6, and A9 respectively to get

$$\begin{aligned} {{S}_B{_1}}^*= & {} \frac{{\theta _B}^3+{\theta _B}{\theta _B{_a}}^2}{2\lambda ({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}\\ {{S}_a{_1}}^*= & {} \frac{3{\theta _B}^2{\theta _B{_a}}+3{\theta _B{_a}}^3}{2\lambda ({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}-\frac{({{x}_a+{k}{\xi _a}})\eta }{2{\theta _a}}+\frac{\theta _a}{2\lambda }\\ {\beta _a{_1}}^*= & {} \frac{1}{2}-\frac{{(\theta _B}^2+{\theta _B{_a}}^2){\theta _B{_a}}}{2{\theta _a}({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}-\frac{({{x}_a+{k}{\xi _a}})\eta \lambda }{2{\theta _a}^2} \end{aligned}$$

Appendix 4. Proof of property 4

The optimization of the platform is as follows:

$$\begin{aligned} {MaxE\pi _e{_2}}= & {} {\varphi _B{_2}}+{\varphi _a{_2}}-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\lambda {{S}_a}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\frac{1}{2}\tau \rho {\beta _a}^2{\sigma _a}^2\end{aligned}$$

(A16)

$$\begin{aligned} {\varphi _B{_2}}=\; & {} {\omega _B}+{\beta _B}{{y}_B{_2}}={\omega _B}+{\beta _B}({{S}_B}{\theta _B}+{{S}_a}{\theta _B{_a}}+({{x}_B}+{a}+{k}({\xi _B}-{a})){\eta }+{\varepsilon _B})\end{aligned}$$

(A17)

$$\begin{aligned} {\varphi _a{_2}}= & {} {\beta _a}{{y}_a{_2}}={\beta _a}({{S}_a}{\theta _a}+({{x}_a}+{a}+{k}({\xi _a}-{a})){\eta }+{\varepsilon _a}) \end{aligned}$$

(A18)

Formula Eqs. A16 and A17 are substituted into formula Eq. A18 respectively to get formula Eq. A19

$$\begin{aligned} {MaxE\pi _e{_2}}={\omega _B}+{\beta _B}({{S}_B}{\theta _B}+{{S}_a}{\theta _B{_a}}+(x{_B}+a+k(\xi {_B}-a)){\eta })\nonumber \\ +{\beta _a}({{S}_a}{\theta _a}+(x{_a}+a+k(\xi {_a}-a)){\eta })-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\lambda {{S}_a}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\frac{1}{2}\tau \rho {\beta _a}^2{\sigma _a}^2 \end{aligned}$$

(A19)

$$\begin{aligned} \frac{{dE}{\pi _e{_2}}}{{d}{{S}_B}}= & {} {\beta _B}{\theta _B}-\lambda {{S}_B}\\ \frac{{dE}{\pi _e{_2}}}{{d}{{S}_a}}= & {} {\beta _B}{\theta _B{_a}}+{\beta _a}{\theta _a}-\lambda {{S}_a}\\ \end{aligned}$$

Let $\frac{{dE}{\pi _e{_2}}}{{d}{{S}_B}}=0, \frac{{dE}{\pi _e{_2}}}{{d}{{S}_a}}=0$, respectively, the solution is

$$\begin{aligned} {{S}_B}^*= & {} \frac{{\beta _B}{\theta _B}}{\lambda }\end{aligned}$$

(A20)

$$\begin{aligned} {{S}_a}^*= & {} \frac{{\beta _B}{\theta _B{_a}}+{\beta _a}{\theta _a}}{\lambda } \end{aligned}$$

(A21)

Appendix 5. Proof of property 5

The expected income of the Central Bank is:

$$\begin{aligned} {E\pi _a{_2}}={{y}_a{_2}}-{\varphi _a{_2}}=(1-{\beta _a})({{S}_a}^*{\theta _a}+({{x}_a}+a+{k}({\xi _a}-a))\eta ) \end{aligned}$$

(A22)

Formula Eq. A21 is substituted into formula Eq. A22 to get formula Eq. A23

$$\begin{aligned} {E\pi _a{_2}}= & {} (1-{\beta _a})(\frac{{\beta _a}{\theta _a}^2+{\beta _B}{\theta {_B}_a}{\theta _a}}{\lambda }+({{x}_a}+a+{k}({\xi _a}-a))\eta )\end{aligned}$$

(A23)

$$\begin{aligned} \frac{{dE}{\pi _a{_2}}}{{d}{\beta _a}}= & {} -(\frac{{\beta _a}{\theta _a}^2+{\beta _B}{\theta {_B}_a}{\theta _a}}{\lambda }+({{x}_a}+a+{k}({\xi _a}-a))+(1-{\beta _a})\frac{{\theta _a}^2}{\lambda }\nonumber \\ \frac{{dE}{\pi _a{_2}}}{{d}{\beta _a}}= & {} 0\nonumber \\ {\beta _a}^*= & {} \frac{1}{2}-\frac{{\beta _B}{\theta _B{_a}}}{2{\theta _a}}-\frac{({{x}_a}+a+{k}({\xi _a}-a))\eta \lambda }{2{\theta _a}^2} \end{aligned}$$

(A24)

Appendix 6. Proof of property 6

The expected income of the bank is:

$$\begin{aligned} Max{E\pi _B{_2}}={IrT}+(1-{\beta _B})({{S}_B}^*{\theta _B}+{{S}_a}^*{\theta {_B}_a}+({{x}_B}+a+{k}({\xi _B}-a))\eta )-{\omega _B} \end{aligned}$$

(A25)

Since the game between the bank and the platform exists only when the platform participates in the decision, the bank must simultaneously satisfy constraint condition Eq. A26 in the pursuit of profit maximization:

$$\begin{aligned} {\varphi _B{_2}}-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2\ge \varpi \end{aligned}$$

(A26)

As a rational principal, the bank will not give the platform excessive remuneration, as long as the agent can be guaranteed to participate in the principal-agent relationship, that is, the participation constraint is Eq. A27:

$$\begin{aligned} {\varphi _B{_2}}-\frac{1}{2}\lambda {{S}_B}^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2=\varpi \end{aligned}$$

(A27)

The Lagrange function Eq. A28 is constructed with the constraint of Eqs. A27 and A25

$$\begin{aligned} {L_2}= & {} {IrT}+(1-{\beta _B})({{S}_B}^*{\theta _B}+{{S}_a}^*{\theta {_B}_a}+({{x}_B}+a+{k}({\xi _B}-a))\eta )-{\omega _B}\nonumber \\&+\alpha ({\omega _B}+{\beta _B}({{S}_B}^*{\theta _B}+{{S}_a}^*{\theta _B{_a}}+({{x}_B}+a+{k}({\xi _B}-a)){\eta })-\frac{1}{2}\lambda {{S}_B}^*2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\varpi ) \end{aligned}$$

(A28)

Formula Eqs. A20, A21, and A24 are substituted into formula Eq. A28 respectively to get formula Eq. A29

$$\begin{aligned} {L_2}= & {} {IrT}+(1-{\beta _B})(\frac{{\beta _B}{\theta _B}}{\lambda }{\theta _B}+\frac{{\beta _B}{\theta _B{_a}}+(\frac{1}{2}-\frac{{\beta _B}{\theta _B{_a}}}{2{\theta _a}}-\frac{({{x}_a}+a+{k}({\xi _a}-a))\eta \lambda }{2{\theta _a}^2}){\theta _a}}{\lambda }{\theta {_B}_a}\nonumber \\&+\alpha ({\omega _B}+{\beta _B}(\frac{{\beta _B}{\theta _B}}{\lambda }{\theta _B}+\frac{{\beta _B}{\theta _B{_a}}+(\frac{1}{2}-\frac{{\beta _B}{\theta _B{_a}}}{2{\theta _a}}-\frac{({{x}_a}+a+{k}({\xi _a}-a))\eta \lambda }{2{\theta _a}^2}){\theta _a}}{\lambda }{\theta _B{_a}}\nonumber \\&+({{x}_B}+a+{k}({\xi _B}-a)){\eta })-\frac{1}{2}\lambda (\frac{{\beta _B}{\theta _B}}{\lambda })^2-\frac{1}{2}\tau \rho {\beta _B}^2{\sigma _B}^2-\varpi )+({{x}_B}+a+{k}({\xi _B}-a))\eta )-{\omega _B} \end{aligned}$$

(A29)

Let $\frac{dL{_2}}{{d}{\beta _B}}=0,\frac{dL{_2}}{{d}{\omega _B}}=0,\frac{dL{_2}}{{d}\alpha }=0$, respectively, the solution is

$$\begin{aligned} {\beta _B{_2}}^*=\frac{{\theta _B}^2+{\theta _B{_a}}^2}{{\theta _B}^2+\tau \lambda \rho {\sigma _B}^2} \end{aligned}$$

(A30)

$$\begin{aligned} {\omega _B{_2}}^*= & {} \varpi +\frac{({\theta _B}^2+{\theta _B{_a}}^2)^2(\tau \lambda \rho {\sigma _B}^2-{\theta _B}^2-3{\theta _B{_a}}^2)}{2\lambda ({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}\\&+\frac{({\theta _B}^2+{\theta _B{_a}}^2)({\theta _B{_a}}{\theta _a}^2+2\lambda {\theta _a}({{x}_B}+a+{k}({\xi _B}-a))\eta -\lambda {\theta _B}{_a}({{x}_a}+a+{k}({\xi _a}-a))\eta )}{2\lambda {\theta _a}({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)} \end{aligned}$$

Formula Eq. A30 is substituted into formula Eq. A20, A21, and A24 respectively to get

$$\begin{aligned} {{S}_B{_2}}^*= & {} \frac{{\theta _B}^3+{\theta _B}{\theta _B{_a}}^2}{2\lambda ({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}\\ {{S}_a{_2}}^*= & {} \frac{3{\theta _B}^2{\theta _B{_a}}+3{\theta _B{_a}}^3}{2\lambda ({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}-\frac{({{x}_a}+a+{k}({\xi _a}-a))\eta }{2{\theta _a}}+\frac{\theta _a}{2\lambda }\\ {\beta _a{_2}}^*= & {} \frac{1}{2}-\frac{{(\theta _B}^2+{\theta _B{_a}}^2){\theta _B{_a}}}{2{\theta _a}({\theta _B}^2+\tau \lambda \rho {\sigma _B}^2)}-\frac{({{x}_a}+a+{k}({\xi _a}-a))\eta \lambda }{2{\theta _a}^2} \end{aligned}$$