﻿ 协同效应视角下VC与创业企业的最优收益模型研究 Research on the Optimal Income Model of VC and Entrepreneurial Enterprises from the Perspective of Synergy Effect

Finance
Vol. 09  No. 06 ( 2019 ), Article ID: 33097 , 16 pages
10.12677/FIN.2019.96073

Research on the Optimal Income Model of VC and Entrepreneurial Enterprises from the Perspective of Synergy Effect

Xiaoqing Yu

Donghua University, Shanghai

Received: Oct. 30th, 2019; accepted: Nov. 15th, 2019; published: Nov. 22nd, 2019

ABSTRACT

Based on C-D production function and synergy effect between VC and entrepreneur, this article mainly discusses four differential game models in the continuous time, which is namely cooperative game model, non-cooperative one, VC-dominating Stackelberg model, entrepreneur-dominating Stackelberg model. HJB equation is used to find the optimal return strategy. It analyses the effect on different VC shareholding ratios to the optimal return and synergy incentives. Matlab 7.0 is used to draft the relationship between relative parameters and optimal return strategy. Sensitivity analysis of model parameters is performed. The results show that entrepreneurs are more inclined to provide synergy incentives and bring higher total return when VC shareholding ratio is less than 50%; while it is greater than 50%, VC is more inclined to provide synergy incentives and bring higher total return.

Keywords:Synergy Effect, Venture Capital, HJB Equation

1. 引言

2. 模型相关说明

2.1. 模型相关变量

Table 1. The main variables of the model

1. 模型主要变量

2.2. 模型相关假设

${H}^{\prime }=\delta {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}-\theta H,H\left(0\right)={H}_{0}\ge 0$

$\pi \left(t\right)=c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)$

2.3. VC和创业企业协同的目标函数推导

$\pi \left(t\right)=c*{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)$

(1) VC提供协同效应激励的双方目标函数

${\pi }_{V}=\underset{0}{\overset{+\infty }{\int }}{\text{e}}^{-rt}\left\{\eta \left(t\right)\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)\right]-\frac{1}{2}{p}_{V}{e}_{V}^{2}-\tau \frac{1}{2}{p}_{E}{e}_{E}^{2}\right\}\text{d}t$

${\pi }_{E}=\underset{0}{\overset{+\infty }{\int }}{\text{e}}^{-rt}\left\{\left[1-\eta \left(t\right)\right]\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)\right]-\frac{1}{2}\left(1-\tau \right){p}_{E}{e}_{E}^{2}\right\}\text{d}t$

$\pi =\underset{0}{\overset{+\infty }{\int }}{\text{e}}^{-rt}\left\{\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)\right]-\frac{1}{2}{p}_{V}{e}_{V}^{2}-\frac{1}{2}{p}_{E}{e}_{E}^{2}\right\}\text{d}t$

(2) 创业企业提供协同效应激励的双方目标函数

${\pi }_{V}=\underset{0}{\overset{+\infty }{\int }}{\text{e}}^{-rt}\left\{\eta \left(t\right)\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)\right]-\frac{1}{2}\left(1-\rho \right){p}_{V}{e}_{V}^{2}\right\}\text{d}t$

${\pi }_{E}=\underset{0}{\overset{+\infty }{\int }}{\text{e}}^{-rt}\left\{\left[1-\eta \left(t\right)\right]\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)\right]-\frac{1}{2}{p}_{E}{e}_{E}^{2}-\frac{1}{2}\rho {p}_{V}{e}_{V}^{2}\right\}\text{d}t$

$\pi =\underset{0}{\overset{+\infty }{\int }}{\text{e}}^{-rt}\left\{\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)\right]-\frac{1}{2}{p}_{V}{e}_{V}^{2}-\frac{1}{2}{p}_{E}{e}_{E}^{2}\right\}\text{d}t$

3. 最优策略模型构建与求解

3.1. 完全信息条件下最优微分收益模型构建与求解

${V}^{*}\left(H\right)=\frac{d}{r+\theta }H+\frac{{\left[c\left(r+\theta \right)+\delta d\right]}^{2}}{4r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(1)

${\pi }_{E}=\underset{0}{\overset{+\infty }{\int }}{\text{e}}^{-rt}\left\{\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\left(t\right)\right]-\frac{1}{2}{p}_{V}{e}_{V}^{2}-\frac{1}{2}{p}_{E}{e}_{E}^{2}\right\}\text{d}t$

$r\cdot V\left(H\right)=\underset{\begin{array}{l}{e}_{V}\ge 0\\ {e}_{E}\ge 0\end{array}}{\mathrm{max}}\left\{\left[c{\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}\right]+d\cdot H\left(t\right)-\frac{1}{2}{p}_{V}{e}_{V}^{2}-\frac{1}{2}{p}_{E}{e}_{E}^{2}+{V}^{\prime }\left(H\right)\left[\delta {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}-\theta H\right]\right\}$

${V}^{*}\left(H\right)=\frac{d}{r+\theta }H+\frac{{\left[c\left(r+\theta \right)+\delta d\right]}^{2}}{4r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

3.2. VC和创业企业不合作的最优微分收益模型构建与求解

${V}_{1}^{**}\left(H\right)=\frac{d\eta }{r+\theta }\cdot H+\frac{3{\eta }^{\frac{3}{2}}{\left(1-\eta \right)}^{\frac{1}{2}}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{8r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(2)

${V}_{2}^{**}\left(H\right)=\frac{d\left(1-\eta \right)}{r+\theta }\cdot H+\frac{3{\eta }^{\frac{1}{2}}{\left(1-\eta \right)}^{\frac{3}{2}}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{8r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(3)

${V}^{**}\left(H\right)=\frac{d}{r+\theta }\cdot H+\frac{3{\eta }^{\frac{1}{2}}{\left(1-\eta \right)}^{\frac{1}{2}}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{8r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(4)

$r\cdot {V}_{1}\left(H\right)=\underset{\eta \ge 0}{\mathrm{max}}\left\{\eta \left[c\cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\right]-\frac{1}{2}{p}_{V}{e}_{V}^{2}+{{V}^{\prime }}_{1}\left(H\right)\left[\delta \cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}-\theta \cdot H\right]\right\}$

$S.T.:{e}_{V}\in \mathrm{arg}\mathrm{max}{V}_{1}\left(H\right),{e}_{E}\in \mathrm{arg}\mathrm{max}{V}_{2}\left(H\right)$(5)

$r\cdot {V}_{2}\left(H\right)=\underset{\eta \ge 0}{\mathrm{max}}\left\{\left(1-\eta \right)\left[c\cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\right]-\frac{1}{2}{p}_{E}{e}_{E}^{2}+{{V}^{\prime }}_{2}\left(H\right)\left[\delta \cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}-\theta \cdot H\right]\right\}$

$S.T.:{e}_{V}\in \mathrm{arg}\mathrm{max}{V}_{1}\left(H\right),{e}_{E}\in \mathrm{arg}\mathrm{max}{V}_{2}\left(H\right)$(6)

${e}_{V}=\frac{{\left[c\left(1-\eta \right)+\delta \cdot {{V}^{\prime }}_{2}\left(H\right)\right]}^{\frac{1}{4}}\cdot {\left[c\eta +\delta \cdot {{V}^{\prime }}_{1}\left(H\right)\right]}^{\frac{3}{4}}}{2{p}_{V}^{\frac{3}{4}}{p}_{E}^{\frac{1}{4}}}$(7)

${e}_{E}=\frac{{\left[c\left(1-\eta \right)+\delta \cdot {{V}^{\prime }}_{2}\left(H\right)\right]}^{\frac{3}{4}}\cdot {\left[c\eta +\delta \cdot {{V}^{\prime }}_{1}\left(H\right)\right]}^{\frac{1}{4}}}{2{p}_{V}^{\frac{1}{4}}{p}_{E}^{\frac{3}{4}}}$(8)

$r\cdot {{V}^{\prime }}_{1}\left(H\right)=\left[d\eta -\theta \cdot {{V}^{\prime }}_{1}\left(H\right)\right]\cdot H+\frac{3\cdot {\left[c\eta +\delta \cdot {{V}^{\prime }}_{1}\left(H\right)\right]}^{\frac{3}{2}}\cdot {\left[c\left(1-\eta \right)+\delta \cdot {{V}^{\prime }}_{2}\left(H\right)\right]}^{\frac{1}{2}}}{8{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(9)

$r\cdot {{V}^{\prime }}_{2}\left(H\right)=\left[d\left(1-\eta \right)-\theta \cdot {{V}^{\prime }}_{2}\left(H\right)\right]\cdot H+\frac{3\cdot {\left[c\eta +\delta \cdot {{V}^{\prime }}_{1}\left(H\right)\right]}^{\frac{1}{2}}\cdot {\left[c\left(1-\eta \right)+\delta \cdot {{V}^{\prime }}_{2}\left(H\right)\right]}^{\frac{3}{2}}}{{8}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(10)

${V}_{1}^{**}\left(H\right)=\frac{d\eta }{r+\theta }\cdot H+\frac{3{\eta }^{\frac{3}{2}}{\left(1-\eta \right)}^{\frac{1}{2}}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{8r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

${V}_{2}^{**}\left(H\right)=\frac{d\left(1-\eta \right)}{r+\theta }\cdot H+\frac{3{\eta }^{\frac{1}{2}}{\left(1-\eta \right)}^{\frac{3}{2}}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{8r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

${V}^{**}\left(H\right)=\frac{d}{r+\theta }\cdot H+\frac{3{\eta }^{\frac{1}{2}}{\left(1-\eta \right)}^{\frac{1}{2}}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{8r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

3.3. 不完全信息条件下VC Stackelberg博弈模型建立与求解

${\tau }^{***}=\frac{7\eta -3}{1+3\eta }$，其中， $\frac{3}{7}<\eta <1$(11)

${V}_{1}^{***}\left(H\right)=\frac{d\eta }{r+\theta }\cdot H+\frac{{\left(1+3\eta \right)}^{2}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(12)

${V}_{2}^{***}\left(H\right)=\frac{d\left(1-\eta \right)}{r+\theta }\cdot H+\frac{3\left(1-\eta \right)\left(1+3\eta \right){\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{32r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(13)

${V}^{***}\left(H\right)=\frac{d}{r+\theta }\cdot H+\frac{\left(7-3\eta \right)\left(1+3\eta \right){\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(14)

$r\cdot {V}_{2}\left(H\right)=\underset{{e}_{E}\ge 0}{\mathrm{max}}\left\{\left(1-\eta \right)\left[c\cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\right]-\frac{1}{2}\left(1-\tau \right){p}_{E}{e}_{E}^{2}+{{V}^{\prime }}_{2}\left(H\right)\left[\delta \cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}-\theta \cdot H\right]\right\}$(15)

(15)式对 ${e}_{E}$ 求一阶导数，并令其等于零，可以求得：

${e}_{E}=\frac{{e}_{V}^{\frac{1}{3}}{\left[\delta \cdot {{V}^{\prime }}_{2}\left(H\right)-c\eta +c\right]}^{\frac{2}{3}}}{{4}^{\frac{1}{3}}{p}_{E}^{\frac{2}{3}}{\left(1-\tau \right)}^{\frac{2}{3}}}$(16)

$r\cdot {V}_{1}\left(H\right)=\underset{\begin{array}{c}{e}_{V}\ge 0\\ 0<\tau <1\end{array}}{\mathrm{max}}\left\{\eta \left[c\cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\right]-\frac{1}{2}{p}_{V}{e}_{V}^{2}-\frac{1}{2}\tau {p}_{E}{e}_{E}^{2}+{V}_{1}{}^{\prime }\left(H\right)\left[\delta \cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}-\theta \cdot H\right]\right\}$(17)

${e}_{V}^{***}=\frac{\left(1+3\eta \right)\left[c\left(r+\theta \right)+d\delta \right]}{8\left(r+\theta \right){p}_{V}^{\frac{3}{4}}{p}_{E}^{\frac{1}{4}}}$

${e}_{E}^{***}=\frac{\left(1+3\eta \right)\left[c\left(r+\theta \right)+d\delta \right]}{8\left(r+\theta \right){p}_{V}^{\frac{1}{4}}{p}_{E}^{\frac{3}{4}}}$

${\tau }^{***}=\frac{7\eta -3}{1+3\eta }$，其中， $\frac{3}{7}<\eta <1$

${V}_{1}^{***}\left(H\right)=\frac{d\eta }{r+\theta }\cdot H+\frac{{\left(1+3\eta \right)}^{2}{\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

${V}_{2}^{***}\left(H\right)=\frac{d\left(1-\eta \right)}{r+\theta }\cdot H+\frac{3\left(1-\eta \right)\left(1+3\eta \right){\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{32r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

${V}^{***}\left(H\right)=\frac{d}{r+\theta }\cdot H+\frac{\left(7-3\eta \right)\left(1+3\eta \right){\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

3.4. 不完全信息条件下创业企业主导型Stackelberg博弈模型建立与求解

${\rho }^{****}=\frac{4-7\eta }{4-3\eta }$，其中， $0<\eta <\frac{4}{7}$(18)

${V}_{1}^{****}\left(H\right)=\frac{d\eta }{r+\theta }\cdot H+\frac{3\eta \left(4-3\eta \right){\left[c\left(r+\theta \right)+\delta d\right]}^{2}}{32r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(19)

${V}_{2}^{****}\left(H\right)=\frac{d\left(1-\eta \right)}{r+\theta }\cdot H+\frac{{\left(4-3\eta \right)}^{2}\cdot {\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(20)

${V}^{****}\left(H\right)=\frac{d}{r+\theta }\cdot H+\frac{\left(4-3\eta \right)\left(4+3\eta \right)\cdot {\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$(21)

$r\cdot {V}_{1}\left(H\right)=\underset{\begin{array}{l}{e}_{V}\ge 0\\ 0<\tau <1\end{array}}{\mathrm{max}}\left\{\eta \left[c\cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}+d\cdot H\right]-\frac{1}{2}\left(1-\rho \right){p}_{V}{e}_{V}^{2}+{{V}^{\prime }}_{1}\left(H\right)\left[\delta \cdot {\left({e}_{V}{e}_{E}\right)}^{\frac{1}{2}}-\theta \cdot H\right]\right\}$(22)

${e}_{V}^{****}=\frac{\left(4-3\eta \right)\left[c\left(r+\theta \right)+d\delta \right]}{8\left(r+\theta \right){p}_{V}^{\frac{3}{4}}{p}_{E}^{\frac{1}{4}}}$

${e}_{E}^{****}=\frac{\left(4-3\eta \right)\left[c\left(r+\theta \right)+d\delta \right]}{8\left(r+\theta \right){p}_{V}^{\frac{1}{4}}{p}_{E}^{\frac{3}{4}}}$

${\rho }_{1}^{****}=\frac{4-7\eta }{4-3\eta }$，其中， $0<\eta <\frac{4}{7}$

${V}_{1}^{****}\left(H\right)=\frac{d\eta }{r+\theta }\cdot H+\frac{3\eta \left(4-3\eta \right){\left[c\left(r+\theta \right)+\delta d\right]}^{2}}{32r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

${V}_{2}^{****}\left(H\right)=\frac{d\left(1-\eta \right)}{r+\theta }\cdot H+\frac{{\left(4-3\eta \right)}^{2}\cdot {\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

${V}^{****}\left(H\right)=\frac{d}{r+\theta }\cdot H+\frac{\left(4-3\eta \right)\left(4+3\eta \right)\cdot {\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{64r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}$

4. 博弈均衡结果比较分析

(1)时，

① 创业投资企业的最优利润排序为：

${V}_{1}^{**}\left(H\right)<{V}_{1}^{****}\left(H\right)$

② 创业企业的最优利润排序如下：

${V}_{2}^{**}\left(H\right)<{V}_{2}^{****}\left(H\right)$

③ 整个创业链的最优利润排序如下：

${V}^{**}\left(H\right)<{V}^{****}\left(H\right)<{V}^{*}\left(H\right)$

(2) $\frac{3}{7}<\eta <\frac{4}{7}$ 时，

① 创业投资企业的最优利润排序为：

${V}_{1}^{**}\left(H\right)<{V}_{1}^{***}\left(H\right)<{V}_{1}^{****}\left(H\right)$

② 创业企业的最优利润排序如下：

${V}_{2}^{**}\left(H\right)<{V}_{2}^{****}\left(H\right)<{V}_{2}^{***}\left(H\right)$

③ 整个创业链的最优利润排序如下：

$\eta \in \left(\frac{3}{7},\frac{1}{2}\right)$ 时， ${V}^{**}\left(H\right)<{V}^{***}\left(H\right)<{V}^{****}\left(H\right)<{V}^{*}\left(H\right)$

$\eta \in \left(\frac{1}{2},\frac{4}{7}\right)$ 时， ${V}^{**}\left(H\right)<{V}^{****}\left(H\right)<{V}^{***}\left(H\right)<{V}^{*}\left(H\right)$

(3) $\frac{4}{7}<\eta <1$ 时，

① 创业投资企业的最优利润排序为：

${V}_{1}^{**}\left(H\right)<{V}_{1}^{***}\left(H\right)$

② 创业企业的最优利润排序如下：

${V}_{2}^{**}\left(H\right)<{V}_{2}^{***}\left(H\right)$

③ 整个创业链的最优利润排序如下：

${V}^{**}\left(H\right)<{V}^{***}\left(H\right)<{V}^{*}\left(H\right)$

$0<\eta <\frac{3}{7}$ 时，

${V}_{1}^{**}\left(H\right)-{V}_{1}^{****}\left(H\right)=\frac{3\eta \left[4{\eta }^{\frac{1}{2}}{\left(1-\eta \right)}^{\frac{1}{2}}-\left(4-3\eta \right)\right]\cdot {\left[c\left(r+\theta \right)+d\delta \right]}^{2}}{32r{\left(r+\theta \right)}^{2}{p}_{V}^{\frac{1}{2}}{p}_{E}^{\frac{1}{2}}}<0$

5. 数值仿真分析

${V}^{*}\left(H\right)=5H+86.62$ ；VC与创业企业非合作情形下，协同效应 ${H}_{t=10}^{**}=15.64*{\eta }^{\frac{1}{2}}{\left(\text{1}-\eta \right)}^{\frac{1}{2}}$，创业链最优总收益为 ${V}^{**}\left(H\right)=5H+129.93*{\eta }^{\frac{1}{2}}{\left(\text{1}-\eta \right)}^{\frac{1}{2}}$，创业投资企业的最优收益为 ${V}_{1}^{**}\left(H\right)=5\eta H+129.93*{\eta }^{\frac{3}{2}}{\left(1-\eta \right)}^{\frac{1}{2}}$，创业企业的最优收益为 ${V}_{2}^{**}\left(H\right)=5\left(1-\eta \right)H+129.93*{\eta }^{\frac{1}{2}}{\left(1-\eta \right)}^{\frac{3}{2}}$ ；VC主导型主从博弈情形下，协同效应

${H}_{t=10}^{***}=3.91*\left(1+3\eta \right)$，创业链最优总收益为 ${V}^{***}\left(H\right)=5H+5.41*\left(7-3\eta \right)\left(1+3\eta \right)$，创业投资企业的最优收益为 ${V}_{1}^{***}\left(H\right)=5\eta H+5.41*{\left(1+3\eta \right)}^{2}$，创业企业的最优收益为 ${V}_{2}^{***}\left(H\right)=5\left(1-\eta \right)H+32.48*\left(1-\eta \right)\left(1+3\eta \right)$ ；创业企业主导型主从博弈情形下，协同效应 ${H}_{t=10}^{****}=0.39*\left(4-3\eta \right)$，创业链最优总收益为 ${V}^{***}\left(H\right)=5H+$ $5.41*\left(4-3\eta \right)\left(4+3\eta \right)$，创业投资企业的最优收益为 ${V}_{1}^{***}\left(H\right)=5\eta H+32.48*\eta \left(4-3\eta \right)$，创业企业的最优收益为 ${V}_{2}^{***}\left(H\right)=5\left(1-\eta \right)H+5.41*{\left(4-3\eta \right)}^{2}$

5.1. 不同博弈情形下VC持股比例与最优收益的对比分析

Figure 1. The comparative analysis of VC shareholding ratio and optimal returns in different game situations when t = 10

1. t = 10时，不同博弈情形下VC持股比例与最优收益的对比分析

$\eta \in \left[\frac{3}{7},\frac{1}{2}\right]$，由创业企业为创业投资企业提供协同效应“激励因子”；而VC持股比例 $\eta \in \left[\frac{1}{2},\frac{4}{7}\right]$，由创

5.2. 持股比例对最优激励因子的影响分析

(1) 在不完全信息条件下由创业投资主导型的Stackelberg博弈均衡时，创业投资企业提供给创业企业的最优激励因子为：

${\tau }^{***}=\frac{7\eta -3}{1+3\eta }$，其中， $\frac{3}{7}<\eta <1$

Figure 2. The relationship between the optimal incentive factor and VC shareholding ratio in VC-dominated master-slave game

2. VC主导型主从博弈下最优激励因子与VC持股比例的关系

(2) 在不完全信息条件下由创业企业主导型的Stackelberg博弈均衡时，创业企业提供给创业投资企业的最优激励因子为：

${\rho }^{****}=\frac{4-7\eta }{4-3\eta }$，其中， $0<\eta <\frac{4}{7}$

Figure 3. The relationship between the optimal incentive factor and VC shareholding ratio in the entrepreneurial enterprise-dominated master-slave game

3. 创业企业主导型主从博弈下最优激励因子与VC持股比例的关系

(3) $\frac{3}{7}<\eta <\frac{4}{7}$ 时，VC与创业企业激励因子的对比分析

Figure 4. The relationship between the optimal incentive factor and VC shareholding ratio when $\frac{3}{7}<\eta <\frac{4}{7}$

4. $\frac{3}{7}<\eta <\frac{4}{7}$ 时最优激励因子与VC持股比例的关系

6. 结论

Research on the Optimal Income Model of VC and Entrepreneurial Enterprises from the Perspective of Synergy Effect[J]. 金融, 2019, 09(06): 657-672. https://doi.org/10.12677/FIN.2019.96073

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