Contracts for venture capital financing with double-sided moral hazard

Abstract

Using a Nash bargaining approach, we analyze the financing contract between the entrepreneur and the venture capitalist with double-sided moral hazard in a start-up enterprise. Our results show that there exists an optimal contract set between the entrepreneur and the venture capitalist, in which all contracts achieve an identical second-best social state. Within the optimal contract set, there exists continuum of joint debt-equity financing. The pure equity financing contract exists in the optimal contract set when the ratio of total social surplus to the amount of investment is greater than a threshold.

Appendix

1.1 Proof of Proposition 2

According to the basic form for the financing contract in Proposition 1, the expected effort levels in contract are $\left(\widehat{L},\widehat{K}\right)$ for the entrepreneur and the VC. At arbitrary effort levels L and K, the entrepreneur receives:

$$ {R}_{\mathrm{en}}=\alpha S\left[\left(L,K\right)\left|\left(\widehat{L},\widehat{K}\right)\right.\right]+\frac{{\widehat{L}}^2}{2\eta }, $$

and the entrepreneur’s cost for effort is:

$$ {g}_{\mathrm{en}}(L)=\frac{L^2}{2\eta }. $$

Thus, the social surplus received by the entrepreneur is:

$$ {R}_{\mathrm{en}}-{g}_{\mathrm{en}}(L)=\alpha S\left[\left(L,K\right)|\left(\hat{L},\hat{K}\operatorname{}\right)\right]+\frac{{\hat{L}}^2}{2\eta }-\frac{L^2}{2\eta }. $$

Similarly, the social surplus received by the VC is:

${R}_{\mathrm{vc}}-F-{g}_{\mathrm{vc}}(K)=\left(1-\alpha \right)S\left[\left(L,K\right)|\left(\hat{L},\hat{K}\right)\right]+\frac{{\hat{K}}^2}{2\theta }-\frac{K^2}{2\theta}\cdotp$ Since both the entrepreneur and the VC are rational, they would choose the optimal effort levels $\tilde{L}$ and $\tilde{K}$ to maximize their respective social surplus:

$$ \Big\{{\displaystyle \begin{array}{l}\overset{\sim }{L}=\arg \underset{L}{\max}\kern0.5em \alpha S\left[\left(L,K\right)|\left(\hat{L},\hat{K}\right)\operatorname{}\right]+\frac{{\hat{L}}^2}{2\eta }-\frac{L^2}{2\eta },\forall K\\ {}\overset{\sim }{K}=\arg \underset{K}{\max}\kern0.5em \left(1-\alpha \right)S\left[\left(L,K\right)|\left(\hat{L},\hat{K}\right)\operatorname{}\right]+\frac{{\hat{K}}^2}{2\theta }-\frac{K^2}{2\theta },\forall L\end{array}}\operatorname{} $$

According to the first-order conditions, the equilibrium solution is:

$$ \left\{\begin{array}{l}\tilde{L}=\frac{c}{2}{\left[\left(1-\alpha \right)\theta \right]}^{\frac{1}{4}}{\left(\alpha \eta \right)}^{\frac{3}{4}}={\left(1-\alpha \right)}^{\frac{1}{4}}{\left(\alpha \right)}^{\frac{3}{4}}{L}^{\ast}\\ {}\tilde{K}=\frac{c}{2}{\left[\left(1-\alpha \right)\theta \right]}^{\frac{3}{4}}{\left(\alpha \eta \right)}^{\frac{1}{4}}={\left(1-\alpha \right)}^{\frac{3}{4}}{\left(\alpha \right)}^{\frac{1}{4}}{K}^{\ast}\end{array}\right.. $$

1.2 Proof of Proposition 3

As per the basic form of the financing contract as in Proposition 1 and the assumption of $S\left(\tilde{L},\tilde{K}\right)>0$, the transfer payment $T\left(\hat{L},\hat{K}\right)$ must make both the entrepreneur and the VC receive a positive social surplus. According to Corollary 2, we have the following:

$$ \left\{\begin{array}{l}{S}_{\mathrm{en}}\left(R,\widehat{L},\widehat{K}\right)=\alpha S\left(\tilde{L},\tilde{K}\right)-T\left(\widehat{L},\widehat{K}\right)+T\left(\tilde{L},\tilde{K}\right)>0\\ {}{S}_{\mathrm{vc}}\left(R,\widehat{L},\widehat{K}\right)=\left(1-\alpha \right)S\left(\tilde{L},\tilde{K}\right)+T\left(\widehat{L},\widehat{K}\right)-T\left(\tilde{L},\tilde{K}\right)>0\end{array}\right. $$

which gives:

$$ \left\{\begin{array}{l}T\left(\widehat{L},\widehat{K}\right)>\alpha F-\left(1-\alpha \right)S\left(\tilde{L},\tilde{K}\right)\\ {}T\left(\widehat{L},\widehat{K}\right)<\alpha F+\alpha S\left(\tilde{L},\tilde{K}\right)\end{array}\right.. $$