Crowdfunding investors, intermediaries and risky entrepreneurs

Abstract

In this paper, we consider a two-period model where individual investors supply funds to entrepreneurs either indirectly, through a financial intermediary, or directly, using equity crowdfunding. The entrepreneurs vary in terms of the quality of their business projects and ex ante, both the investors and intermediaries are imperfectly informed about their types. The main trade off between the two forms of investment is that crowdfunding is assumed to involve lower costs and higher risk relative to how intermediaries invest their funds. Given this framework, we study investor behavior and find that at intermediate levels of risk for the crowdfunding investment, investors elect to utilize both crowdfunding and financial intermediation in equilibrium. Furthermore, we find that when transaction costs of investment are high, as can be the case with opaque types of small business ventures, this increases the incidence of crowdfunding as the optimal form of investment.

Plain English Summary

In a market where risky entrepreneurs are seeking external funds, crowdfunding and financial intermediation can be compatible methods of investment. Using a model where investors can finance entrepreneurs using either crowdfunding or an intermediary, we find that in some cases investors will prefer to utilize both methods. The availability of the two types of funding depends critically on how much risk new investors face when trying crowdfunding for the first time. The implication of this study is that investors will most benefit when they have access to both risky crowdfunding investment opportunities as well as the safer returns provided by financial intermediaries.

Appendix

Proof of Lemma 2.

There are two cases to consider, depending on whether the crowdfunding investor monitors or not in the first period. First consider the case where $\lambda <\overline {\lambda }$, implying that d_j = d_H and m_j = 0. We show that the payoff in Eq. 9 exceeds the payoff in Eq. 10. Namely,

$$ \begin{array}{@{}rcl@{}} \lambda \left[2R -\frac{d_{H} -d_{L}}{1 -p} -m_{L} -f_{L}\right] +(1 -\lambda )\left[\lambda \left( R -\frac{d_{H}}{1 -p}\right) -f\right] > \end{array} $$

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$$ \begin{array}{@{}rcl@{}} R -\frac{d_{L}}{1 -p} -m_{L} -c +\lambda \left( R -\frac{d_{H}}{1 -p}\right) -f \end{array} $$

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With a bit of simplification we then have

$$ \lambda \left[R -\frac{d_{L}}{1 -p} -m_{L} -f_{L}\right] +(1 -\lambda )\left[\lambda \left( R -\frac{d_{H}}{1 -p}\right) -f\right] >R -\frac{d_{L}}{1 -p} -m_{L} -c $$

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Since λ > λ₁, it must be that $\lambda \left (R -\frac {d_{H}}{1 -p}\right ) -f >R -\frac {d_{L}}{1 -p} -c$, so the above inequality holds as long as

$$ -\lambda m_{L} +(1 -\lambda )\lambda \left( R -\frac{d}{1 -p}\right) > -m_{L} $$

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This holds due to Assumption A1.

The second case is where $\lambda \geq \overline {\lambda }$, meaning that d_j = d_L and m_j = m_L. Again, the payoff in Eq. 9 exceeds that in 10 if

$$ \begin{array}{@{}rcl@{}} &&\lambda \left[2\left( R -\frac{d_{L}}{1 -p}\right) -m_{L} -f_{L}\right]\\ &&-m_{L} -f +(1 -\lambda )\left[\lambda \left( R -\frac{d_{L}}{1 -p}\right) -m_{L} -f\right] > \end{array} $$

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$$ \begin{array}{@{}rcl@{}} &&R -\frac{d_{L}}{1 -p} -m_{L} -c +\lambda \left( R -\frac{d_{L}}{1 -p}\right) -m_{L} -f \end{array} $$

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With some simplification we have

$$ \lambda \left[R -\frac{d_{L}}{1 -p} -f_{L}\right] +(1 -\lambda )\left[\lambda \left( R -\frac{d_{L}}{1 -p}\right) -f\right] >R -\frac{d_{L}}{1 -p} -c $$

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Since λ > λ₁, this implies that λ[R − d_L/(1 − p)] − f > R − d_L/(1 − p) − c, which along with assumption A1, is sufficient to guarantee that the above inequality holds. □

Proof of Proposition 1.

In Section 3.1 of the paper we established the optimal behavior of the investors in the second period of their lives. Furthermore, in Lemma 2 we proved that if λ > λ₁, it is optimal for investors to always choose crowdfunding. For the case where λ ≤ λ₁, we must pin down exactly when crowdfunding is a better choice than intermediation for the young investor. Specifically, the payoff in Eq. 11 exceeds the payoff in Eq. 12 if

$$ \begin{array}{@{}rcl@{}} &&\lambda \left [2R -\frac{d_{j} +d_{L}}{1 -p} -m_{L} -f_{L}\right ]\\ &&-m_{j} -f +(1 -\lambda )\left [R -\frac{d_{L}}{1 -p} -m_{L} -c\right ] > \end{array} $$

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$$ \begin{array}{@{}rcl@{}} &&2\left[R -\frac{d_{L}}{1 -p} -m_{L} -c\right] \end{array} $$

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With a bit of simplification we then have

$$ \begin{array}{@{}rcl@{}} &&\lambda \left[2R -\frac{d_{j} +d_{L}}{1 -p} -m_{L} -f_{L} -R +\frac{d_{L}}{1 -p} +m_{L} +c\right]\\ &&\quad-m_{j} -f >R -\frac{d_{L}}{1 -p} -m_{L} -c \end{array} $$

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$$ \lambda >\frac{R -\frac{d_{L}}{1 -p} -c +f +m_{j} -m_{L}}{R -\frac{d_{j}}{1 -p} +c -f_{L}} $$

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This proves that crowdfunding is preferred by the young investor when λ₂ < λ ≤ λ₁. It also implies that if λ < λ₂, then intermediation must be the optimal choice in the first period. Just to confirm that λ₂ < λ₁, note that

$$ \frac{R -\frac{d_{L}}{1 -p} -c +f +m_{j} -m_{L}}{R -\frac{d_{j}}{1 -p} +c -f_{L}} <\frac{R -\frac{d_{L}}{1 -p} -c +f +m_{j} -m_{L}}{R -\frac{d_{j}}{1 -p}} $$

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$$ 0 <c -f_{L} $$

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This holds due to assumption A3. □

Proof of Corollary 1.

Taking the partial derivatives we have

$$ \frac{ \partial }{ \partial \gamma }\widetilde{\lambda }_{1} =\frac{ -c +f}{R -\frac{d_{j}}{1 -p}} <0 $$

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$$ \begin{array}{@{}rcl@{}} &&\frac{ \partial }{ \partial \gamma }\widetilde{\lambda }_{2}\\ &&=\frac{(-c + f)\left[R - \frac{d}{1 -p} + \gamma c - \gamma f_{L}\right] - (c - f_{L})\left[R - \frac{d_{L}}{1 -p} - \gamma c + \gamma f + m_{j} - m_{L}\right]}{\left[R -\frac{d_{j}}{1 -p} +\gamma c -\gamma f_{L}\right]^{2}} \end{array} $$

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$$ =\frac{(-2c +f +f_{L})\left[R -\frac{d_{j}}{1 -p}\right] +(m_{j} -m_{L})f_{L}}{\left[R -\frac{d_{j}}{1 -p} +\gamma c -\gamma f_{L}\right]^{2}} $$

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Note that in the numerator, − 2c + f + f_L < 0 and m_j − m_L ≤ 0, meaning $\frac { \partial }{ \partial \gamma }\widetilde {\lambda }_{2} <0$. □

Proof of Corollary 2.

Suppose entrepreneur pays cost b and λ₂ drops to $\widetilde {\lambda }_{2}$. Consider a case where, $\lambda >\widetilde {\lambda }_{2}$. By Proposition 1, the young investor now chooses crowdfunding in the first period. Since $\lambda <\overline {\lambda }$, the investor chooses to not monitor. As a result, the entrepreneur expects a first period payoff of pd_H/(1 − p). This is all worthwhile for the entrepreneur if the increase in the first period payoff exceeds the initial cost, b. That is

$$ \frac{pd_{H}}{1 -p} -\frac{pd_{L}}{1 -p} >b $$

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□

Proof of Corollary 3.

Consider the following partial derivatives.

$$ \begin{array}{@{}rcl@{}} \frac{ \partial }{ \partial x}\lambda_{1} &=&\frac{ -\frac{d_{L}}{1 -p}\left[R -\frac{xd_{L}}{1 -p}\right] +\frac{d_{L}}{1 -p}\left[R -\frac{xd_{L}}{1 -p} -c +f\right]}{\left( R -\frac{xd_{L}}{1 -p}\right)^{2}}\\ &=&\frac{\frac{d_{L}}{1 -p}(-c +f)}{\left( R -\frac{xd_{L}}{1 -p}\right)^{2}} <0 \end{array} $$

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$$ \begin{array}{@{}rcl@{}} \frac{ \partial }{ \partial x}\lambda_{2} &=&\frac{ -\frac{d_{L}}{1 -p}\left[R -\frac{xd_{l}}{1 -p} +c -f_{L}\right] +\frac{d_{L}}{1 -p}\left[R -\frac{xd_{L}}{1 -p} -c +f\right]}{\left( R -\frac{xd_{L}}{1 -p} +c -f_{L}\right)^{2}}\\ &=&\frac{\frac{d_{L}}{1 -p}(-2c +f +f_{L})}{\left( R -\frac{xd_{L}}{1 -p} +c -f_{L}\right)^{2}} <0 \end{array} $$

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Crowdfunding occurs in equilibrium for all λ > λ₂. Since λ₂ decreases as x rises, this means that as entrepreneurs’ bargaining power over their equity share rises, there is a larger set of λ values over which crowdfunding will be selected. □