Entrepreneurial finance and economic growth

Abstract

This paper incorporates the process of entrepreneurial finance into an endogenous growth model with horizontal innovation (Romer J Polit Econ 98:S71–S102, 1990; Jones J Polit Econ 103(4):759–784, 1995b). To capture the market frictions existing in the financing of innovation, entrepreneurial finance is described as a process of “search and matching” between entrepreneurs proposing their innovative ventures and capitalists selecting and financing the most valuable projects. We determine the amount of resources devoted to innovation along the balanced growth path. The welfare analysis highlights the sub-optimality of the equilibrium innovative efforts due to search and bargaining frictions. We analyze the role of the policy maker to restore the optimality of investments in innovation.

Appendix: welfare analysis

This problem of dynamic optimization defined at the beginning of Sect. 5 presents three control variables, $c(t), l_{F},l_{E}$, and two state variables, $k$ and $A$. The Hamiltonian writes as

$$\begin{aligned} H\left( c,l_{E},l_{F},k,A\right)&= \frac{c\left( t\right) ^{1-\phi }-1}{ 1-\phi }+\lambda _{k}\left( k^{\gamma }\left( Al_{Y}\right) ^{1-\gamma }-c-nk\right) \\&+\,\lambda _{A}\left( \beta L\left( l_{E}\right) ^{a}\cdot \left( l_{F}\right) ^{1-a}\right) , \end{aligned}$$

from which we obtain the following 5 conditions:

$$\begin{aligned} \left\{ \begin{array}{l} H_{c}=c\left( t\right) ^{-\phi }-\lambda _{k}=0 \\ H_{l_{E}}=\lambda _{A}aL\beta \left( \frac{l_{F}}{l_{E}}\right) ^{1-a}-\lambda _{k}\left( 1-\gamma \right) \frac{y}{l_{y}}=0 \\ H_{l_{F}}=\lambda _{A}\left( 1-a\right) L\beta \left( \frac{l_{E}}{l_{F}} \right) ^{a}-\lambda _{k}\left( 1-\gamma \right) \frac{y}{l_{y}}=0 \\ H_{k}=\lambda _{k}\left( \gamma \frac{y}{k}-n\right) =\rho \lambda _{k}-\dot{ \lambda }_{k} \\ H_{A}=\lambda _{A}\left( 1-\gamma \right) \frac{y}{A}=\rho \lambda _{A}-\dot{ \lambda }_{A} \end{array} \right. \end{aligned}$$

(27)

Working on $H_{l_{E}}$ and $H_{l_{F}}$ we immediately obtain that the optimal ratio entrepreneurs/financiers must be $l_{E}/l_{F}=a/\left( 1-a\right)$.

To find the absolute values of $l_{E}$ and $l_{F}$ we now proceed as follows. We know that in steady state it must be $\dot{c}/c=n$. Taking logs and derivative from the first equation in (27), we then obtain $\dot{\lambda }_{k}/\lambda _{k}=-\phi n$. On the other hand, dividing $H_{k}$ by $\lambda _{k}$ and $H_{A}$ by $\lambda _{A}$ we obtain

$$\begin{aligned} \frac{\dot{\lambda }_{k}}{\lambda _{k}}=\rho -\left( \gamma \left( \frac{A}{k} \right) ^{1-\gamma }\left( l_{Y}\right) ^{1-\gamma }-n\right) \text { and \ \ }\frac{\dot{\lambda }_{A}}{\lambda _{A}}=\rho -\left( 1-\gamma \right) \left( \frac{A}{k}\right) ^{-\gamma }\left( l_{Y}\right) ^{1-\gamma } \end{aligned}$$

(28)

where we have also exploited the expression for the production function $y$ stated in one of the problem’s constraints. Solving the first of the two equations in (28) by $A/k$ and plugging it into the second, we obtain

$$\begin{aligned} \frac{\dot{\lambda }_{A}}{\lambda _{A}}=\rho -\left( 1-\gamma \right) \left( l_{Y}\right) ^{1-\gamma }\left[ \frac{1}{\gamma \left( l_{Y}\right) ^{1-\gamma }}\left( \rho +n-\frac{\dot{\lambda }_{k}}{\lambda _{k}}\right) \right] ^{-\frac{\gamma }{1-\gamma }}. \end{aligned}$$

Knowing that in steady state it is $\dot{\lambda }_{k}/\lambda _{k}=\dot{ \lambda }_{A}/\lambda _{A}=-\phi n$, we can finally solve the last equation for $l_{Y}$ in function of all parameters of the model and obtain (22) in the main text. Given that $1=l_{E}+l_{F}+l_{Y}$, and that $l_{E}/l_{F}=a/\left( 1-a\right)$, it is easy to characterize the optimal number of entrepreneurs and financiers as given in, respectively, () and (21).