Market imperfections and crowdfunding

Abstract

This article is the first one that considers the choice between the different types of crowdfunding and traditional financing under different types of market imperfections. In contrast to most existing literature, we focus on financial aspects of crowdfunding rather than on price discrimination between customers using a new approach on the demand side. The model provides several implications, most of which have not yet been tested. For example, we find that when asymmetric information is important, high-quality projects prefer reward-based crowdfunding. A low-quality firm may find it unprofitable to mimick this strategy as it will be taking more risk to achieve a threshold. This result is contradictory to the spirit of the results in Belleflamme et al. (Journal of Business Venturing: Entrepreneurship, Entrepreneurial Finance, Innovation and Regional Development, 29(5), 585–609, 2014), which finds that asymmetric information favours equity-based crowdfunding. In contrast to Belleflamme et al. (Journal of Business Venturing: Entrepreneurship, Entrepreneurial Finance, Innovation and Regional Development, 29(5), 585–609, 2014), in our model, crowdfunding does not have any ad-hoc non-monetary benefits.

Appendix

Proof of Proposition 1

After shares are sold, the firm chooses p (correspondingly q = a − p) to maximize

$$ (1-\alpha )(p(a-p)+M-cq)-e(a-p) $$

(21)

subject to M ≥ cq. Two cases are possible. If $\frac {a+c+e\diagup (1-\alpha )}{2}\geq a-\frac {M}{c}$, we have$p=\frac {a+c+e\diagup (1-\alpha ) }{2}$. Otherwise, we have a corner solution $p=\frac {ca-M}{c}$. In both cases, under theoptimal strategy chosen by the firm M = cq.

The funders anticipate this and therefore M and α will be connected as follows:

$$ M=\alpha p(a-p)=c(a-p) $$

(22)

Then, we have:

$$\alpha =\frac{M}{p(a-p)}=\frac{c(a-p)}{p(a-p)}=\frac{c}{p} $$

Substituting this into Eq. 21, we get that the entrepreneur‘s expected profit is equal to

$$ (p-c-e)(a-p) $$

(23)

In the beginning, the entrepreneur selects α to maximize (23). The case where $\frac {a+c+e\diagup (1-\alpha )}{2}<a- \frac {M}{c}$ is not optimal. The firm should increase M and α because of the following. Equation 23 is concave in p and $p=\frac {a+c+e }{2}$ is an optimal p in Eq. 23. Further $p=\frac {a+c+e\diagup (1-\alpha )}{2}$ is closer to the optimum than $p=\frac {\text {ca}-M}{c}$. So, we have $p=\frac {a+c+e\diagup (1-\alpha )}{2}$.

Using the above formula for p, Eq. 23 can be converted into

$$ \frac{(a-c)^{2}}{4}-\frac{e^{2}}{4(1-\alpha )^{2}}+\frac{e^{2}}{2(1-\alpha )} -\frac{\text{ea}}{2}+\frac{\text{ec}}{2} $$

(24)

If α = 0, Eq. 24 will be equal to $\frac {(a-c-e)^{2}}{4}$. This is the same value that we under a reward-based crowdfunding scenario. When α is positive, the entrepreneur‘s profit under equity crowdfunding will be smaller since the derivative of Eq. 24 in α is negative. □

Proof of Proposition 2

Consider a situation where l selects reward-based crowdfunding and h selects equity-based crowdfunding. We have (all calculations are based on the symmetric information case for each type described in Section 2):

$$ {\Pi}_{h}=\frac{(a-c_{h})^{2}}{4} $$

(25)

$${\Pi}_{l}=\frac{(a-c_{l})^{2}}{4} $$

where π_j is the equilibrium profit of type j. Also, we have (as follows from Lemma 1):

$$\alpha_{h}=\frac{4M_{h}}{(a-c_{h})^{2}+ 4M_{h}};M_{h}\geq \frac{ c_{h}(a-c_{h})}{2} $$

We will assume$M_{h}=\frac {c_{h}(a-c_{h})}{2}$. The proof is similar for anyvalue of M_h. For simplicity weassume that M_h has the lowest value from the range of possible values. It can also be justified if moral hazard is present (as described in Sections 3 or 5) and therefore minimizing M_h mitigates the extent of moral hazard problems. Note that

$$ \alpha_{h}=\frac{2c_{h}}{a+c_{h}} $$

(26)

h does not have an incentive to mimick l since, as mentioned above, in this section, asymmetric information does not concernreward-based crowdfunding. So if h chose reward-based crowdfunding, it would have the same payoff as it would in equilibrium: $\frac {(a-c_{h})^{2}}{4}$. Now suppose that l mimicks h and chooses equity-based crowdfunding instead. l’s profit π_lh then equals

$$ {\Pi}_{\text{lh}}=(1-\alpha_{h})({pq}+M_{h}-c_{l}q) $$

(27)

In this equation, p and q are discussed below. Note that when l mimicks h, it has to sell a larger stake of equity in thefirm compared to the symmetric information case. Indeed, if l sells equity under symmetric information, we have

$$\alpha_{l}=\frac{2c_{l}}{a+c_{l}} $$

This is smaller than Eq. 26 because c_l < c_h. Note that the amount of funds raised will also be different than under symmetric information.

Two cases are possible. First,

$$ \frac{c_{h}(a-c_{h})}{2}>\frac{c_{l}(a-c_{l})}{2} $$

(28)

The left side of this condition shows the cost of production by type h under symmetric information and the right side shows that for type l. If (28) holds, l will be able to raise enough funds to produce an optimal quantity of goods. Indeed,the choice of p is determined by maximizing Eq. 27 under the condition that

$$ M_{h}\geq c_{l}q $$

(29)

Absolute maximum is $p=\frac {a+c_{l}}{2}$ and if condition (28) holds, the constraint (29) will not be binded. Then,${\Pi }_{\text {lh}}=(1- \frac {2c_{h}}{a+c_{h}})((\frac {a+c_{l}}{2}-c_{l})(a-\frac {a+c_{l}}{2})+c_{h}(\frac {a-c_{h}}{2}))$. After simplifications, we find that this is less than $\frac {(a-c_{l})^{2}}{4}$ when${c_{h}^{2}}-{c_{l}^{2}}<2a(c_{h}-c_{l})$. This holds because c_l < c_h < a.So l will not mimick h.

Second case is when Eq. 28 does not hold. In this case, l is not able to produce the quantity that would be optimal under symmetric information. Two situations are possible. 1.M_h = c_lq.Then

$$q=\frac{M_{h}}{c_{l}}=\frac{c_{h}(a-p_{h})}{c_{l}} $$

Since$p_{h}=\frac {a+c_{h}}{2}$, this equals$\frac {c_{h}(a-c_{h})}{2c_{l}}$. The entrepreneur’s profit then equals ${\Pi }_{\text {lh}}=(1-\frac {2c_{h} }{a+c_{h}})(a-\frac {c_{h}(a-c_{h})}{2c_{l}})\frac {c_{h}(a-c_{h})}{2c_{l}}$. This is less than$\frac {(a-c_{l})^{2}}{4}$ because Eq. 28 does not hold and c_l < c_h. Indeed the inequality ${\Pi }_{\text {lh}}<\frac {(a-c_{l})^{2} }{4}$ canbe written as ${c_{h}^{2}}(a-c_{h})^{2}(2ac_{l}-ac_{h}+{c_{h}^{2}})<(a-c_{l})^{2}{c_{l}^{2}}c_{h}(a+c_{h})$. Here${c_{h}^{2}}(a-c_{h})^{2}<(a-c_{l})^{2}{c_{l}^{2}}$ because Eq. 28 does not hold and $2ac_{l}-ac_{h}+{c_{h}^{2}}<c_{h}(a+c_{h})$ because c_l < c_h.

2. M_h > c_lq. This is only possible if p ≤ c_l. Otherwise, it makes no sense to keep unused cash because the production of extra units brings profit. In this case, the firm’s profit equals$(1-\frac {2c_{h}}{ a+c_{h}})c_{h}(\frac {a-c_{h}}{2})$. This is less than$\frac {(a-c_{l})^{2}}{4 }$ becausec_l < c_h.

Therefore, l will not mimick h.

Now consider a situation where h selects reward-based crowdfunding and l selects equity-based crowdfunding. Suppose that h mimicks l and chooses equity-based crowdfunding instead. Using similar reasoning one can show that h’s profit π_hl equals

$${\Pi}_{hl}=\left( 1-\frac{c_{l}}{a}\right)\left( a-\frac{c_{l}(a-c_{l})}{2c_{h}}\right)\frac{ c_{l}(a-c_{l})}{2c_{h}} $$

This is greater than Eq. 25 because c_l < c_h. Therefore, h will mimick l. This means that such an equilibrium does not exist.   □

Proof of Proposition 3.

Consider a pooling equilibrium where both types select reward-based crowdfunding, which is supported byoff-equilibrium market beliefs that the firm is h if the market participants observe equity-based crowdfunding.First of all, let us verify non-deviation for each type to equity-based crowdfunding. Since under pooling withreward-based crowdfunding l’s payoff is the same as in the separating equilibrium in Proposition 2, it follows from the proof of Proposition 2 that l does not deviate. h does not deviate because it gets the same amount as inequilibrium.

Let us now verify that off-equilibrium beliefs survive the intuitive criterion of Cho and Kreps (1987). To show this, let us calculate the maximal payoff of type h in the case that it plays equity-based crowdfunding. Its payoff is evidently maximized if the market’s beliefs place the probability 1 on type l observing equity. If off-equilibrium beliefs survive the intuitive criterion, this expression must be not less than the payoff of h inequilibrium.³³It follows from our analysis of the separating equilubrium above that the payoff of h will be higher than itsequilibtum payoff if the market places the probbaility of 1 on type l.

Now let us analyze the mispricing. First, note that there is no mispricing in the case of a pooling equilibrium with reward-based crowdfunding because the profit of both types of firms equals their profit under symmetric information. Consider pooling with equity-based crowdfunding. The payoff of type l is then${\Pi }_{l}=(1-\frac {2c_{m}}{a+c_{m}})((\frac { a+c_{l}}{2}-c_{l})(a-\frac {a+c_{l}}{2})+c_{m}(\frac {a-c_{m}}{2}))$.This is less than $\frac {(a-c_{l})^{2}}{4}$.   □

Proof of Lemma 2

Consider crowdfunding. After the shares are sold, the firm chooses p to maximize(1 − α)(γp(a − p) + M − (c + b)q) subject to

$$ M\geq (c+b)q $$

(30)

It implies

$$ p=\frac{a+(c+b)/\gamma }{2} $$

(31)

The firm’s expected profit is $\gamma p(a-p)=\gamma \frac {(a+(c+b)/\gamma )(a-(c+b)/\gamma )}{4}$. The funders‘ expected earnings should cover their investment cost or

$$ \alpha \left( \gamma \frac{(a+(c+b)/\gamma )(a-(c+b)/\gamma )}{4}\right)\geq M $$

(32)

Under the optimal solution the conditions (30) and (32) will be binded because the firm can always make α as small as necessary to satisfy them. Then we have

$$\alpha =\frac{(c+b)q}{\gamma \frac{(a+(c+b)/\gamma )(a-(c+b)/\gamma )}{4}}= \frac{2(c+b)}{\gamma a+c+b} $$

The entrepreneur’s expected profit equals

$$\begin{array}{@{}rcl@{}} && \left( 1-\frac{2(c+b)}{\gamma a+c+b}\right)\left( \gamma \frac{(a+(c+b)/\gamma )(a-(c+b)/\gamma )}{4}\right)\\ &&=\frac{(\gamma a-c-b)^{2}}{4\gamma } \end{array} $$

(33)

Consider bank loan financing. The firm maximizes γ (p (a − p) − F) − (1 − γ)B − c(a − p) subject to: pq = p(a − p) ≥ cq = c(a − p). F is the face value of debt.

The solution gives us

$$ p=\frac{a+c/\gamma }{2} $$

(34)

The comparison of Eqs. 31 and 34 leads to the first part of Lemma 2.

The banker’s expected payoff equals $\gamma F=c(a-p)=c(a-\frac {a+c/\gamma }{ 2})=c\frac {a-c/\gamma }{2}$. It implies $F=\frac {c(\gamma a-c)}{2\gamma ^{2}}$.

The firm’s profit equals

$$ {\Pi} =\frac{\gamma (a-c/\gamma )^{2}}{4}=\frac{(\gamma a-c)^{2}}{4\gamma } -(1-\gamma )B $$

(35)

The comparison of Eqs. 33 and 35 leads to the second part of Lemma 2.   □

Proof of Proposition 4

Consider crowdfunding. After the shares are sold, the entrepreneur decides whether to run away or start the production. If he decides to start the production, the firm chooses p to maximize(1 − α)(γp(a − p) + M − cq) subject to

$$ M\geq cq $$

(36)

It implies

$$ p=\frac{a+c/\gamma }{2} $$

(37)

The firm’s expected profit is $\gamma p(a-p)=\gamma \frac {(a+c/\gamma )(a-c/\gamma )}{4}$. The funders‘ expected earnings should cover their investment cost or

$$ \alpha \left( \gamma \frac{(a+c/\gamma )(a-c/\gamma )}{4}\right)\geq M $$

(38)

Under the optimal solution, the conditions (36) and (38) will be binded because the firm can always make α as small as necessary to satisfy them. Then we have

$$\alpha =\frac{cq}{\gamma \frac{(a+c/\gamma )(a-c/\gamma )}{4}}=\frac{2c}{ \gamma a+c} $$

Theentrepreneur‘s expected profit equals

$$ \left( 1-\frac{2c}{\gamma a+c}\right)\left( \gamma \frac{(a+c/\gamma )(a-c/\gamma )}{4}\right)=\frac{ (\gamma a-c)^{2}}{4\gamma } $$

(39)

If the entrepreneur decides to run away, his profit is equal to$M=\text {cq}=c(a- \frac {a+c/\gamma }{2})=c\frac {a-c/\gamma }{2}$. This is not greater than (33) if

$$ c\leq \frac{\gamma a}{3} $$

(40)

Consider bank loan financing. The firm maximizes γ (p(a − p) − F) − (1 − γ)B − c(a − p) subject to pq = p(a − p) ≥ cq = c (a − p). F is the face value of debt.

The solution gives us $p=\frac {a+c/\gamma }{2}$.

The banker’s expected payoff equals $\gamma F=c(a-p)=c(a-\frac {a+c/\gamma }{ 2})=c\frac {a-c/\gamma }{2}$. It implies $F=\frac {c(\gamma a-c)}{2\gamma ^{2}}$.

The firm’s profit equals

$$ {\Pi} =\frac{\gamma (a-c/\gamma )^{2}}{4}=\frac{(\gamma a-c)^{2}}{4\gamma } -(1-\gamma )B $$

(41)

The comparison of Eqs. 39 and 41 leads to the first part of Proposition 4.

If b > 0, condition (40) will be $c+b\leq \frac {\gamma a}{3}$ or$\gamma \leq \frac {3(b+c)}{a}$ and the firm’s profit under crowdfunding is

$$\frac{(\gamma a-c-b)^{2}}{4\gamma } $$

Then, one can compare this with Eq. 35; this leads to the second part of the proposition. If$\gamma \leq \max \{\frac {3(b+c)}{a},\frac {4B-2ab- \sqrt {16B^{2}-16abB+ 4a^{2}b^{2}+ 6Bb^{2}}}{8B}\}$

or $\gamma \geq \frac {4B-2ab+\sqrt {16B^{2}-16abB+ 4a^{2}b^{2}+ 6Bb^{2}}}{8B}$ a bank loan is better than crowdfunding.   □

Proof of Lemma 4

Consider a situation where type l selects KIA and type h selects AON. First, we have

$$ {\Pi}_{h}=\frac{\pi (1+\pi )(a_{h}-c)^{2}}{4} $$

(42)

$$ {\Pi}_{l}=\frac{\pi (a_{l}-c)^{2}}{2} $$

(43)

where π_j is the equilibrium profit of type j (all calculations are based on the symmetric information case for each type described in Section 6.1). Suppose that l mimicks h and chooses AON. We have

$${\Pi}_{\text{lh}}=\pi \left( \frac{(a_{h}-c)^{2}}{4}+\pi \frac{(a_{h}-c)^{2}}{4}\right) $$

Comparing this with Eq. 43, we find that the former is greater if

$$ \frac{1+\pi }{2}<\left( \frac{a_{l}-c}{a_{h}-c}\right)^{2} $$

(44)

Suppose that h mimicks l and chooses KIA. We have

$${\Pi}_{\text{hl}}=\pi \frac{(a_{l}-c)^{2}}{4}+\pi \frac{(a_{l}-c)^{2}}{4} $$

Comparing with Eq. 42, we find that h does not deviate if

$$ \frac{1+\pi }{2}\geq \left( \frac{a_{l}-c}{a_{h}-c}\right)^{2} $$

(45)

Note that conditions (44) and (45) can not hold simultaneously because π ≤ 1 and, therefore, this equilibrium does not exist.

The proof is similar for other cases so it is omitted for brevity.   □

Proof of Proposition 5

Consider a situation where type l selects KIA and type h selects AON. First, we have

$$ {\Pi}_{h}=\frac{\pi (1+\pi )(a_{h}-c)^{2}}{4} $$

(46)

$$ {\Pi}_{l}=\frac{\pi (a_{l}-c)^{2}}{2} $$

(47)

where π_j is the equilibrium profit of type j (all calculations are based on the symmetric information case for each type described in Section 6.1). Suppose that l mimicks h and chooses AON. We have

$${\Pi}_{\text{lh}}=\pi \left( (p_{1h}-c)(a_{h}-p_{1h})+\pi \frac{(a_{l}-c)^{2}}{4}\right) $$

where$p_{1l}=\frac {a_{l}+c}{2}$ and$p_{1h}=\frac {a_{h}+c}{2}$.

We have

$${\Pi}_{\text{lh}}=\frac{\pi (a_{h}-c)^{2}}{4}+\frac{\pi^{2}(a_{l}-c)^{2}}{4} $$

Comparing this with Eq. 47, we find that the former is greater if

$$ \pi <2-(\frac{a_{h}-c}{a_{l}-c})^{2} $$

(48)

and, therefore, type l has no incentive to deviate.

Suppose that h mimicks l and chooses KIA. We have

$${\Pi}_{\text{hl}}=\pi \left( \frac{(a_{l}-c)^{2}}{4}+\pi \frac{(a_{h}-c)^{2}}{4}\right)+(1-\pi )\pi \frac{(a_{l}-c)^{2}}{4} $$

Comparing with Eq. 46, we find that h does not deviate if

$$ \frac{\pi }{2-\pi }>\left( \frac{a_{l}-c}{a_{h}-c}\right)^{2} $$

(49)

Note that conditions (48) and (49) do not contradict each other. It is because the right side of Eq. 49 is smaller than that of Eq. 48. Indeed, let $x=(\frac {a_{l}-c}{a_{h}-c})^{2}$. Then, the following makes the comparison described in the previous sentence

$$x<2-\frac{1}{x}, $$

which always holds. For brevity, we omit the analys is of equilibrium and also the case where type l selects equity-based crowdfunding and type h selects AON.

Consider a situation where type h selects KIA and type l selects AON. First, we have

$$ {\Pi}_{h}=\frac{\pi (a_{h}-c)^{2}}{2} $$

(50)

$$ {\Pi}_{l}=\frac{\pi (1+\pi )(a_{l}-c)^{2}}{4} $$

(51)

Suppose that l mimicks h and chooses KIA. We have

$${\Pi}_{\text{lh}}=\pi \left( \frac{(a_{h}-c)^{2}}{4}+\pi \frac{(a_{l}-c)^{2}}{4}\right)+(1-\pi ) \frac{(a_{h}-c)^{2}}{4} $$

This is greater than Eq. 51because a_h > a_l and π < 1. So, a situation where type h selects KIA and type l selects AON is not an equilibrium.

Finally, consider a situation where type h selects equity-based crowdfunding. We have

$${\Pi}_{h}=\frac{\pi (a_{h}-c)^{2}}{2} $$

$$ {\Pi}_{l}\leq \frac{\pi (a_{l}-c)^{2}}{2} $$

(52)

(if l selects KIA, Eq. 52 holds as an equality). Suppose that l mimicks h and chooses equity-based crowdfunding. l’s profit π_lh then equals

$${\Pi}_{\text{lh}}=(1-\alpha_{h})(\pi p_{1l}(a_{l}-p_{1l})+\pi (p_{2l}-c)(a_{l}-p_{2l})) $$

where

$$\alpha_{h}=\frac{c}{c+\pi (a_{h}-c)} $$

$$p_{1l}=p_{2l}=\frac{a_{l}+c}{2} $$

Itimplies

$${\Pi}_{\text{lh}}=\left( 1-\frac{c}{c+\pi (a_{h}-c)}\right)\frac{\pi a_{l}(a_{l}-c)}{2} $$

This is greater than Eq. 52 because a_l < a_h. Therefore, l will mimick h and such an equilibrium does not exist.   □

Proof of Proposition 6

Consider a pooling equilibrium where both types select KIA, which is supported by off-equilibrium market beliefs that the firm is l if the market participants observe AON or equity-based crowdfunding. First of all, let usverify non-deviation for each type to equity-based crowdfunding. h payoff in equilibrium is

$$ {\Pi}_{h}=\frac{\pi (a_{m}-c)^{2}}{4}+\frac{\pi (a_{h}-c)^{2}}{4} $$

(53)

where a_m = xa_h + (1 − x)a_l.³⁴ If h deviates to equity-based crowdfunding, it gets

$$ {\Pi}_{h}=(1-\alpha_{l})\left( \pi \frac{M_{l}}{c}\left( a_{h}-\frac{M_{l}}{c}\right)+\pi \frac{(a_{h}-c)^{2}}{4}\right) $$

(54)

where

$$\alpha_{l}=\frac{c}{c+\pi (a_{l}-c)} $$

$$M_{l}=\frac{c(a_{l}-c)}{2} $$

Note that it follows from Eq. 17 that $M_{l}=\frac {c(a_{l}-c)}{2} <M_{h}=\frac {c(a_{h}-c)}{2}$ becausea_l < a_h, so h will not be able to produce the optimal quantity if h deviates to equity-based crowdfunding. Equation 53 is greater than Eq. 54 if

$$ (a_{m}-c)^{2}>\frac{\pi (a_{l}-c)^{2}(2a_{h}-a_{l}+c)-c(a_{h}-c)^{2}}{(c+\pi (a_{l}-c))} $$

(55)

So it holds if x and respectively a_m are sufficiently large. If h deviates to AON, it gets

$${\Pi}_{h}=\pi \left( \frac{(a_{l}-c)^{2}}{4}+\pi \frac{(a_{h}-c)^{2}}{4}\right) $$

This is less than Eq. 53 because a_l ≤ a_m so h does not deviate.

Off-equilibrium beliefs survive the intuitive criterion of Cho and Kreps (1987). The proof is omitted for brevity.

Now let us analyze the mispricing. Consider pooling with equity-based crowdfunding. h’s profit π_h equals

$$ {\Pi}_{h}=(1-\alpha_{m})(\pi \frac{M_{m}}{c}(a_{h}-\frac{M_{m}}{c})+\pi \frac{(a_{h}-c)^{2}}{4}) $$

(56)

where

$$\alpha_{m}=\frac{c}{c+\pi (a_{m}-c)} $$

$$M_{m}=\frac{c(a_{m}-c)}{2} $$

Consider pooling with AON.

$$ {\Pi}_{h}=\pi (\frac{(a_{m}-c)^{2}}{4}+\pi \frac{(a_{h}-c)^{2}}{4}) $$

(57)

From the comparison of Eqs. 53 and 57, it follows that pooling with KIA dominates pooling with AON. Now let uscompare KIA and equity-based crowdfunding. Equation 53 is greater than Eq. 56 if

$$(a_{m}-c)^{2}(2\pi a_{h}-c-2\pi a_{m}+ 2\pi c)<c(a_{h}-c)^{2} $$

If x is sufficiently large, it holds. Indeed, in the extreme case when x = 1 and respectively a_m = a_h, this condition becomes π < 1. This completes the second part of the proposition.

Now compare pooling with KIA with the separating equilibrium where h plays AON

$$\pi (\frac{(a_{h}-c)^{2}}{4}+\pi \frac{(a_{h}-c)^{2}}{4}) $$

This isgreater than Eq. 53 if

$$\pi >\frac{(a_{m}-c)^{2}}{(a_{h}-c)^{2}} $$

So, if x is sufficiently large, pooling with KIA dominates.   □

Proof of Lemma 5

Consider reward-based crowdfunding.

In period 2, the firm chooses p₂ to maximize (p₂ − c)(a − p₂), which gives $p_{2}=\frac {a+c}{2}$.

In period 1, the firm maximizes (p₁ − c)(a − p₁) − I subject to p₁q₁ = p₁(a − p₁) ≥ I + cq₁ = I + c(a − p₁). This condition means that the amount of pre-orders should cover the start-up cost (fixed costs and period 1’svariable costs).

Two cases are possible. If

$$ \frac{(a-c)^{2}}{4}\geq I $$

(58)

then $p_{1}=\frac {a+c}{2}$.

The firm’s profit over the two periods equals

$$ {\Pi} =\frac{(a-c)^{2}}{4}-I+\frac{(a-c)^{2}}{4}=\frac{(a-c)^{2}}{2}-I $$

(59)

If Eq. 58 fails, the firm will not be able to raise the funds needed to launch production. When the required amount ofinitial investment is quite large, reward-based crowdfunding may not be an option.

The analysis of equity-based crowdfunding is omitted for brevity (it is very similar to the analysis in Section 6). □

Proof of Proposition 8

Consider a situation where l selects reward-based crowdfunding and h selects equity-based crowdfunding.

Consider firm h. Calculations are similar to those in Section 4. In period 2, the firm chooses p₂ to maximize the entrepreneur‘s profit (1 − α) (γp₂ − c_h)(a − p₂), which makes $p_{2}=\frac {a+c_{h}/\gamma }{2}$.

In period 1, after the shares are sold, the firm chooses p₁ to maximize (1 − α)(γp₁(a − p₁) + M − c_hq₁) subject to

$$ M\geq cq_{1} $$

(60)

. It implies: $p_{1}=\frac {a+c_{h}/\gamma }{2}$. The firm’sexpected profit in period 1 is $\gamma p_{1}(a-p_{1})=\gamma \frac {(a+c_{h}/\gamma )(a-c_{h}/\gamma )}{4}$. The funders‘ expected earnings should cover their investment cost or:

$$ \alpha (\gamma \frac{(a+c_{h}/\gamma )(a-c_{h}/\gamma )}{4}+\frac{\gamma (a-c_{h}/\gamma )^{2}}{4})\geq M $$

(61)

Under the optimal solution, the conditions (60) and (61) will be binded because the firm can always make α as small as necessary. Then, we have

$$\alpha =\frac{c_{h}q_{1}}{\gamma \frac{(a+c_{h}/\gamma )(a-c_{h}/\gamma )}{4} +\frac{\gamma (a-c_{h}/\gamma )^{2}}{4}}=\frac{c_{h}\frac{a-c_{h}/\gamma }{2} }{\frac{\gamma a(a-c_{h}/\gamma )}{2}}=\frac{c_{h}}{\gamma a} $$

The entrepreneur‘s expected profit over the two periods equals

$$ (1-\frac{c_{h}}{\gamma a})(\frac{\gamma a(a-c_{h}/\gamma )}{2})=\frac{ (\gamma a-c_{h})^{2}}{2\gamma } $$

(62)

Consider firm l. In period 2, the firm choosesp₂ to maximize (γp₂ − c_l) (a − p₂), which makes$p_{2}=\frac {a+c_{l}/\gamma }{2}$. The firm’s expected profit in period 2 is $\frac {\gamma (a-c_{l}/\gamma )^{2}}{4}$

In period 1, the firm maximizes $\gamma (p_{1}(a-p_{1})+\frac {\gamma (a-c_{l}/\gamma )^{2}}{4})-c_{l}(a-p_{1})$ subject to p₁q₁ = p₁(a − p₁) ≥ c_lq₁ = c_l(a − p₁). The solution gives us $p_{1}=\frac {a+c_{l}/\gamma }{2}$.

The firm’s profit over the two periods equals

$$ {\Pi}_{l}=\frac{\gamma (a-c_{l}/\gamma )^{2}}{4}+\frac{\gamma^{2}(a-c_{l}/\gamma )^{2}}{4}=\frac{\gamma (1+\gamma )(a-c_{l}/\gamma )^{2}}{ 4} $$

(63)

Suppose that l mimicks h and chooses equity-based crowdfunding instead. l’s profit π_lh then equals

$$\begin{array}{@{}rcl@{}} {\Pi}_{\text{lh}}&=&(1-\alpha_{h})(\gamma p_{1l}(a-p_{1l})-c_{l}(a-p_{1l})\\&&+~\gamma p_{2l}(a-p_{2l})-c_{l}(a-p_{2l})) \end{array} $$

where

$$\alpha_{h}=\frac{c_{h}}{\gamma a} $$

$$p_{1l}=p_{2l}=\frac{a+c_{l}/\gamma }{2} $$

It implies

$$ {\Pi}_{\text{lh}}=(1-\frac{c_{h}}{\gamma a})\frac{\gamma a(a-c_{l}/\gamma )}{2}= \frac{(\gamma a-c_{h})(a-c_{l}/\gamma )}{2} $$

(64)

Equation 64 is smaller than Eq. 63 if the following holds:

$$ \frac{2}{\gamma (1+\gamma )}<\frac{\gamma a-c_{l}}{\gamma a-c_{h}} $$

(65)

The left side of this inequality is decreasing in γ and the right sideis increasing in γ. So, we havetwo cases. If$\frac { a-c_{l}}{a-c_{h}}<2$, the condition (65) does not hold for 0 < γ ≤ 1, and a separating equilibrium does not exist. Otherwise, it holds if γ is sufficiently high.

Secondly, in order to have an equilibrium, h should not have an incentive to switch to reward-based crowdfunding. Inthis case, this is a trade-off between bankruptcy cost and the cost of moral hazard. If h switches to reward-basedcrowdfunding, its payoff equals

$${\Pi}_{\text{hl}}=\frac{\gamma (1+\gamma )(a-c_{h}/\gamma )^{2}}{4} $$

This is less than Eq. 62.

Consider a situation where h selects reward-based crowdfunding and l selects equity-based crowdfunding.

Consider firm l. Similarly to the above analysis, we have$p_{1}=p_{2}= \frac {a+c_{l}/\gamma }{2}$,$\alpha =\frac {c_{l}}{\gamma a}$ and the entrepreneur‘s expected profit over the two periods equals

$$ \frac{(\gamma a-c_{l})(a-c_{l}/\gamma )}{2} $$

(66)

Consider firm h. We have $p_{1}=p_{2}=\frac {a+c_{h}/\gamma }{2}$.

The firm’s profit over the two periods equals

$$ {\Pi}_{h}=\frac{\gamma (1+\gamma )(a-c_{h}/\gamma )^{2}}{4} $$

(67)

Suppose that h mimicks l and chooses equity-based crowdfunding instead. h’s profit π_hl then equals

$$\begin{array}{@{}rcl@{}} {\Pi}_{\text{hl}}=(1&-&\alpha_{l})(\gamma p_{1h}(a-p_{1h})-c_{l}(a-p_{1h})\\&+&~\gamma p_{2h}(a-p_{2h})-c_{l}(a-p_{2h})) \end{array} $$

It equals

$${\Pi}_{\text{hl}}=\frac{(\gamma a-c_{l})(a-c_{h}/\gamma )}{2} $$

This is greater than Eq. 67 because c_l < c_h, and therefore, suchan equilibrium does not exist.   □

Proof of Proposition 9

Consider reward-based crowdfunding. In period 2, the firm chooses p₂ to maximize (p₂ − c − b)(s_ra − p₂) which makes $p_{2}=\frac {s_{r}a+b+c}{2}$ (all calculations are identical to those in Section 2.1. except that the cost equals c + b).

In period 1, the firm maximizes (p₁ − b − c) (a − p₁) subject to p₁q₁ = (p₁ − c) (a − p₁) ≥ cq₁ = c(a − p₁).

Two cases are possible. If

$$ (a-c-b)^{2}<4I $$

(68)

then the firm will not be able to raise enough funds to launch the production. Otherwise, we have$p_{1}=\frac {a+c+b}{2}$.

The firm’s profit over the two periods equals

$$ {\Pi}_{r}=\frac{(a-b-c)^{2}}{4}+\frac{(s_{r}a-b-c)^{2}}{4} $$

(69)

Consider equity-based crowdfunding. In period 2, the firm chooses p₂ to maximize (1 − α)(p₂ − c − b)(s_ea − p₂), which makes $p_{2}=\frac { s_{e}a+b+c}{2}$.

In period 1, the firm maximizes (1 − α) (p₁ − b − c) (a − p₁), which makes $p_{1}=\frac {a+b+c}{2}$.

The firm’s profit equals

$${\Pi}_{e}=(1-\alpha )(\frac{(a-b-c)^{2}}{4}+\frac{(s_{e}a-b-c)^{2}}{4}) $$

Since

$$\alpha (\frac{(a-b-c)^{2}}{4}+\frac{(s_{e}a-b-c)^{2}}{4})=\text{cq} $$

wehave

$$ {\Pi}_{e}=\frac{(a-b-c)^{2}}{4}+\frac{(s_{e}a-b-c)^{2}}{4} $$

(70)

In the case of bank loan financing, we have $p_{1}=p_{2}=\frac {a+c}{ 2}$.

The firm’s profit is

$$ {\Pi}_{b}=\frac{(a-c)^{2}}{2} $$

(71)

Since Eq. 69 is greater than Eq. 70, we have two cases. Resulting from the comparison of Eqs. 69 and 70, the firm prefers reward-based crowdfunding to equity-based crowdfunding because s_r > s_e. As follows from the comparison of Eqs. 69 and 71, the firm selects reward-based crowdfunding if s_r is sufficiently largeor b is sufficiently small. This is not surprising given that b reflects the degree of the moral hazard cost under crowdfunding and s_r reflects the efficiency of market feedback. Otherwise, the firm takes a bank loan.

Let us now analyze the role of demand (a) on a firm’s decision-making. The firm is indifferent between reward-basedcrowdfunding and a bank loan if

$$\frac{(a-b-c)^{2}}{4}+\frac{(s_{r}a-b-c)^{2}}{4}=\frac{(a-c)^{2}}{2} $$

This equation can be rewritten as

$$ \frac{(a-b-c)^{2}}{2}+\frac{(s_{r}a-b-c)^{2}}{2}=(a-c)^{2} $$

(72)

Since this is a quadratic equation, it implies that for any given value of I, the firm selects equity-based crowdfunding if a is either very small or very large. Otherwise it takes a bankloan.

It was shown previously: the entrepreneur’s profits under the different strategies are equal to the following.

$${\Pi}_{r}=\frac{(a-b-c)^{2}}{4}+\frac{(s_{r}a-b-c)^{2}}{4}-I $$

$${\Pi}_{e}=\frac{(a-b-c)^{2}}{4}+\frac{(s_{e}a-b-c)^{2}}{4}-I $$

$${\Pi}_{b}=\frac{(a-c)^{2}}{2}-I $$

where subscript r stands for reward-based crowdfunding, e means equity-based crowdfunding and b means bank loan.

The firm is indifferent between reward-based crowdfunding and a bank loan if

$$ \frac{(a-b-c)^{2}}{2}+\frac{(s_{r}a-b-c)^{2}}{2}=(a-c)^{2} $$

(73)

The firm is indifferent between equity-based crowdfunding and a bank loan if

$$ \frac{(a-b-c)^{2}}{2}+\frac{(s_{e}a-b-c)^{2}}{2}=(a-c)^{2} $$

(74)

Also if

$$ (a-c-b)^{2}<4I $$

(75)

the firm will not be able to use reward-based crowdfunding. And if (a − c − b)² ≥ 4I, the firm prefers reward-based crowdfunding over equity-based crowdfunding. □