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CROWDFUNDING: BACKERS REWARDED

Ahmed Sewaid1*, Miguel Garcia Cestona 1, Florina Silaghi 1

1Departament d’Empresa, Universitat Autonoma de Barcelona, 08193,

Bellaterra, Spain

ABSTRACT

Crowdfunding is becoming a significant source of funds for entrepreneurial startups. Recent

literature has theoretically modelled the pre-ordering scheme under crowdfunding in the

context where entrepreneurs price discriminate through charging crowdfunders a premium

above that of retail consumers. However, more than 50% of total funds raised through

Kickstarter, the leading reward-based crowdfunding platform, represent projects that offer a

discount to early purchasers. We contribute to the literature by modelling pre-ordering using

an advance purchase discount as a price discrimination device, while employing future retail

price commitment. Moreover, we derive the entrepreneur’s optimal choice between opting

for crowdfunding and spot selling for two scenarios: unconstrained and financially

constrained entrepreneur. In the latter the entrepreneur is essentially choosing between

consumer vs investor financing. We further develop our analysis by discussing welfare and

public policy implications.

Keywords; Crowdfunding, Pre-ordering, Price Discrimination, Financing Platforms,

Financing Constraints

JEL classification: L11, L12, L21, G23, G32

*Corresponding author. Tel: +34 93 581 1209. Fax +34 93 581 2555

E-mail addresses: ahmed.sewaid@uab.cat, miguel.garcia.cestona@uab.cat, florina.silaghi@uab.cat

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1. INTRODUCTION

The ability of entrepreneurs and innovators to start new ventures is the engine of every

economy, or as U.S. Congress representatives Scott Peters, Ron Kind and Patrick Murphy

put it “the bedrock of America’s economy”.2 Nevertheless, a common issue that most

entrepreneurs are facing in the traditional financing market is their inability to access

sufficient capital for their ventures. Entrepreneurial startups typically lack tangible assets, so

debt financing is usually not an option (Denis, 2004). This is why firms with higher capital

needs have to turn to alternative sources of financing.

Anecdotal evidence shows that crowdfunding has become a major source of

alternative financing. An example is the Pebble smartwatch project. Eric Migicovsky, the

founder of Pebble, envisioned a smartwatch that would be connected to a smartphone and

display messages on the go. After raising money to launch an earlier project under their

former name Allerta, which produced smartwatches for Blackberry devices, Migicovsky was

unable to attract traditional investors under their new brand name Pebble. Migicovsky took

his idea through the Y Combinator, a business incubator program, and was able to generate

revenues during the program. He was also able to raise $375,000 through angel investors, but

was left hanging, unable to raise further funds to undergo production. On April 11, 2012

Migicovsky launched a Kickstarter campaign to raise $100,000 in the form of pre-orders

from the crowd in order to launch his venture. When the campaign ended on May 18, 2012

he had successfully raised $10,266,845 from 68,929 backers.

Since its appearance, crowdfunding has brought to life projects that would not have

seen light otherwise. Eric Migicovsky’s Pebble came to life post its crowdfunding

campaign’s success, since all other sources of funds were exhausted. Crowdfunding served

as his last financing resort and was determinant to the launch of his venture. In this context,

a deep understanding of crowdfunding and its underlying mechanisms is of great importance

not only for academics, but also for market agents and regulators. In this paper, we contribute

to the scarce literature on crowdfunding by developing a framework to analyze an

entrepreneur’s optimal pricing and financing strategy. This will be followed with a welfare

analysis.

The most general definition for crowdfunding is given by Cross (2011): “The term

‘crowdfunding’ is used to describe a form of capital raising whereby groups of people pool

money, typically comprised of very small individual contributions, to support an effort by

others to accomplish a specific goal”. The most striking feature of crowdfunding is the

dispersion of the investors, with a reduced role of spatial proximity, as shown by Agrawal,

Catalini, & Goldfarb’s (2011), since the platform eliminates distance related economic

frictions. Crowdfunding can take different forms and can be categorized into four main

2 The Huffington Post, “Entrepreneurs: Engines of our economic growth”, November 18, 2014.

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categories: peer-to-peer lending, equity crowdfunding, donation-based crowdfunding, and

reward-based crowdfunding. Reward-based crowdfunding is the crowdfunding mechanism

of particular interest to us in this paper. It involves soliciting funds from the public in the

form of pre-orders, where the crowdfunders “backers” fund the entrepreneur by advance

purchasing the good. Depending on the platform, the entrepreneur might or might not need

to publicize the minimum capital that he needs to raise in order for his venture to launch.

Belleflamme, Lambert, & Schwienbacher (2014) present one of the first attempts to

theoretically model crowdfunding. The authors explore and compare two forms of

crowdfunding: reward-based and equity crowdfunding. Due to community benefits that

crowdfunders enjoy, the entrepreneur can discriminate against crowdfunders and charge

them a price premium. The intuition behind the community benefits significance has been

disputed by Cholakova & Clarysse’s (2014) results which conclude that nonfinancial

motives, including community benefits, are not significant factors affecting the decision to

back a project. Kumar, Langberg, & Zvilichovsky (2016) propose an alternative model

where the price premium paid by the crowdfunders is the entrepreneur’s optimal strategy

given the pivotal role of the crowdfunder.3 Even though in the two models the backers’ price

premium is attributed to different aspects, they both show that crowdfunding enables the

entrepreneur to price discriminate against the backers who value the good more than the

regular consumers.

In both models there is no price commitment by the entrepreneur when launching his

campaign on the platform, such that the entrepreneur does not publicize the future retail price.

However, when analyzing campaigns on Kickstarter, one can see that almost all projects in

the product design and consumer product categories do publicize their future retail price.

Furthermore, for these projects publicizing their pre-ordering and future prices, we observe

that the entrepreneurs offer a discount to the backers during the crowdfunding campaign.

Although some consumers voluntarily pay a price higher than that requested by the

entrepreneurs in order to aid the entrepreneurs in achieving their goal, in aggregate the

amount that the crowdfunders pay is lower than the aggregate future retail value of the units

sold in pre-orders. Therefore, the previous theoretical models fail to explain both the

mechanism of crowdfunding as well as the entrepreneur’s optimal pricing strategy for a

sizable part of crowdfunding projects, those in the product design and consumer product

category.4 According to the statistics made available through Kickstarter, the category of

3 The pivotal consumer is a consumer that believes that without his contribution the project would not be

successful in meeting the minimum capital requirement. Therefore, the outside option for the consumer of not

backing the product is not having the product at all, rather than buying the product later when the venture

launches. 4 As argued by Belleflamme et al. (2014), their model is most suitable for art related projects (music, movies,

photography, etc.). They give the example of an artist who raises funds in pre-orders at a price premium over

the regular future retail price due to the fact that backers decide on which songs the artist would include in the

album.

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product design and consumer products represents a significant portion of the projects on the

crowdfunding platform and accounts for more than 50% of the funds raised.

In this paper, we contribute to the literature by developing a theoretical model that

explains the crowdfunding mechanism of projects in the category of product design and

consumer products. The novelty in our model is that we employ price commitment where the

entrepreneur publicizes both the pre-ordering and the future retail price during the

crowdfunding campaign. Given these prices, rational consumers in an intertemporal setting

maximize their utility by choosing whether to pre-order today, or wait until the product is

commercialized and purchase it later in the spot market. We also show that rewarding the

backers with a price discount is the entrepreneur’s optimal pricing policy. Similarly to

Kumar et al. (2016) we compare reward-based crowdfunding and standard debt. Thus, the

entrepreneur is choosing between consumer financing versus investor financing (sale of

securities on future cash flows). Moreover, we provide some interesting results regarding the

welfare implications of our proposed model which depend on the specific characteristics of

the project.

The model proposed is closely related to the literature on advance purchase discounts

and intertemporal price discrimination. Advance purchase discount has been identified as an

optimal profitable strategy in different settings and the optimality of this discount depends

on capacity costs and demand uncertainty (Dana, 1998; Gale & Holmes, 1993; Möller &

Watanabe, 2010). We specifically build on the general model of Nocke, Peitz, & Rosar

(2011) that focuses on advance purchase discount as a price discrimination device, which can

be an optimal pricing strategy when compared to advanced selling and spot selling. We

incorporate elements of this general model into a crowdfunding framework to help us derive

the optimal crowdfunding pricing strategy as well as reach some general implications.

A strand in the nascent but booming crowdfunding literature has emphasized the

crowdfunding ability to aggregate vague information about consumer preferences and to act

as a demand exploration tool (Ellman & Hurkens, 2014; Chemla & Tinn, 2016; Gruener &

Siemroth, 2016). Although we assume perfect symmetric information, our results can be

related to those of this literature. In particular, Hakenes & Schlegel's (2014) reasoning behind

the entrepreneur opting for crowdfunding is to exploit the consumers’ private information

rather than financial sources. They show that good projects are more likely to get funded

through crowdfunding rather than through standard debt and that bad firms strictly prefer

standard debt. In our model we see that entrepreneurs prefer crowdfunding over standard

debt for projects that are less probable to deliver on their claims while for other projects the

entrepreneur’s financing strategy depends on the capital requirement and the prevailing

interest rate in the credit market.

The rest of the paper is organized as follows: Section 2 describes our proposed basic

model. In Section 3 we compare the strategies available for the entrepreneur under spot

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selling and crowdfunding and derive the optimal contract for the entrepreneur in the

unconstrained case. In Section 4 we proceed by analyzing the case where the entrepreneur is

financially constrained and we derive the optimal contract for the entrepreneur in the

constrained case. Section 5 presents a welfare analysis of spot selling and crowdfunding.

Finally, in Section 6 we conclude the paper and discuss the model’s implications and

limitations, as well as propose future areas of research.

2. THE BASIC MODEL

Our basic model considers a monopolist who is starting up a venture. At t = 0 the

monopolist chooses whether to spot sell the good or use crowdfunding since the latter

provides the possibility of generating demand while raising capital pre-production. The

prices will be determined at t = 0 when the monopolist decides which option to pursue. Under

spot selling, the consumer observes the quality of the good at t = 1 and decides whether to

purchase the good or not for a price 𝑃𝑆. Delivery takes place at t = 2. In the case of

crowdfunding the monopolist solicits pre-orders at t = 0 for a price 𝑃𝐶, the delivery of these

pre-orders will take place at t = 1. At t = 1, the quality would be observed by the regular

consumers who would then be able to buy the product in the spot market for a price 𝑃𝑅 and

the delivery will take place at t = 2 (see Figure 1).

Figure 1: Timeline for spot selling and crowdfunding.

Building on Nocke et al. (2011) but assuming a uniform distribution there is a unit

mass of consumers with a unit demand. The consumer´s valuation of the good is represented

by his θ, where θ is uniformly distributed between [0,1]. The higher the value of θ, the higher

is the consumer’s valuation of the good. At t = 1, vz(θ), the consumer’s ex-post valuation, can

take one of two values, where z ∈ {L,H}, vH(θ) = θ + αH with probability λ and vL(θ) = θ + αL

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with probability 1- λ, where αL ∈ (-1

2 ,0) and αH ∈ (0,

1

2).5 αz represents the shock that the

consumer receives to the standardized quality and is independent across consumers. The

standardized quality is the industry benchmark and, without loss of generality, it is assumed

to be zero. By construction the expected value of the shock is zero. To illustrate this, consider

again the Pebble example where Eric Migicovsky was revolutionizing the nascent smart

watch industry. The positive shock in this example is the additional utility that the consumer

receives if the promised claims are fulfilled and the negative shock is the consumer’s

disutility if the claims are not fulfilled.

Note that in order for the expected value of the shock to be zero, we need to impose

some restrictions on the probability of having a positive shock λ. In particular, consider λ >1

1+2αH . In such a case the expected value of the shock to quality is always strictly positive,

and not equal to zero. We therefore make the following assumption:

Assumption 1.

𝜆 ≤1

1 + 2𝛼𝐻

3. THE UNCONSTRAINED MONOPOLIST

We start our analysis by studying the optimal strategies available for a financially

unconstrained monopolist in the previously described framework. The unconstrained

monopolist is defined either as an entrepreneur who requires no specific level of capital in

order to launch his venture, or, alternatively, as an entrepreneur who has sufficient

endowments to launch a venture with no need for external funding sources. Given that the

monopolist is unconstrained, he could pursue his venture either through spot selling or

through crowdfunding. In the following subsections we first analyze each of these strategies

independently, and then determine the optimal one by comparing both.

3.1 UNCONSTRAINED SPOT SELLING

The entrepreneur can decide to choose spot selling as the strategy to launch the

venture. In this case, following Nocke et al. (2011), the entrepreneur sets the price 𝑃𝑆 at t=0.

The entrepreneur is uncertain regarding the consumers’ realization of the shock at t = 1. What

the entrepreneur knows is that the marginal consumer willing to purchase the good in case of

a negative shock has a higher valuation than that of the marginal consumer who would only

purchase the good in case of a positive shock. The marginal consumer who would purchase

5 Since we are working with a uniform distribution between [0,1], the parameters have been set in order to

guarantee interior solutions for the demand.

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the good at t = 1 if a negative shock is observed is defined by 𝜃𝐻 = 𝑃𝑆 − 𝛼𝐿 and will be

referred to as the high type. Whereas the marginal consumer who would only purchase the

good in case of a positive shock is defined by 𝜃𝐿 = 𝑃𝑆 − 𝛼𝐻 and will be referred to as the

low type hereafter, where 𝜃𝐻 ≥ 𝜃𝐿 (see Figure 2).

Figure 2: Demand in case of positive and negative shock.

When setting 𝑃𝑆 the entrepreneur faces a trade-off. On the one hand, if the price is set

low enough, then he will attract demand from consumers with valuation 𝜃 ≥ 𝜃𝐻 in case of a

negative shock. However, in case of a positive shock he foregoes the possibility of charging

a high price. On the other hand, setting a high price in order to extract more rents from

consumers in case of a positive shock would imply foregoing demand in case of a negative

shock. Thus, when setting the price the entrepreneur has two strategies available depending

on the probability of a positive shock being realized. The available strategies are

characterized below.

Strategy A: Attract demand for both the positive and negative shock.

The monopolist’s maximization problem when he wants to attract demand under both

a positive and a negative shock can be written as follows:

𝐦𝐚𝐱𝑷𝑺

𝝅(𝑷𝑺) = 𝝀𝑷𝑺(𝟏 − 𝜽𝑳) + (𝟏 − 𝝀)𝑷𝑺(𝟏 − 𝜽𝑯)

s.t. 𝜃𝐻 = 𝑃𝑆 − 𝛼𝐿 ; 𝜃𝐿 = 𝑃𝑆 − 𝛼𝐻 ; 𝜃𝐻 ∈ [0,1] ; 𝜃𝐿 ∈ [0,1] (1)

Here we have that �̂�𝑆 ≤ 𝜃𝐻 + 𝛼𝐿 . In this case, the low type consumers (with 𝜃𝐿 ≤ 𝜃 < 𝜃𝐻)

participate only when observing a positive shock. The high type consumers (with 𝜃 ≥ 𝜃𝐻)

participate both in case of a positive and of a negative shock. The first order conditions yield

that the profit maximizing spot selling price is characterized by �̂�𝑆 = 1

2 . The expected

quantity demanded at t = 1 is 𝐸[�̂�𝑆] =1

2 . The expected profits in this case are 𝐸[�̂�𝑆] =

1

4 .6

Strategy B: Attract demand for the positive shock only.

6 See the Appendix for a proof of these results, in particular, the first part of the proof of Proposition 1.

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The entrepreneur wants to attract demand from those consumers who demand the

positive shock to the industry standards and who would only buy if the positive shock is

realized. The monopolist’s maximization problem is as follows:

𝐦𝐚𝐱𝑷𝑺

𝝅(𝑷𝑺) = 𝝀𝑷𝑺(𝟏 − 𝜽𝑳)

s.t. 𝜃𝐿 = 𝑃𝑆 − 𝛼𝐻; 𝜃𝐿 ∈ [0,1] (2)

Here we have that �̌�𝑆 > 𝜃𝐻 + 𝛼𝐿 . In this case, we have both types of consumers

participating in case a positive shock is realized and no consumers participating when a

negative shock is observed. The profit maximizing spot selling price is characterized by �̌�𝑆 =

1+𝛼𝐻

2 . The expected quantity demanded at t = 1 is 𝐸[�̌�𝑆] =

𝜆(1+𝛼𝐻)

2 . The expected profits

given this case are 𝐸[�̌�𝑆] =𝜆(1+𝛼𝐻)2

4 .

Comparing the two strategies we note that when 𝜆 ≥1

(1+𝛼𝐻)2 the entrepreneur will

not find it optimal to attract the high type consumers in case a negative shock is observed.

Indeed, the profit generated by charging a price premium for a positive shock would dominate

the profit generated by offering a price discount in order to ensure demand in case of a

negative shock. Thus, ventures with a relatively high probability of delivering a positive

shock would implement Strategy B, only attracting consumers in case a positive shock to

industry standards is realized. For projects with a relatively low probability of a positive

shock, the entrepreneur finds it optimal to attract the high type consumers in case of a

negative shock, by implementing Strategy A. The monopolist’s strategies under spot selling

are illustrated below in Figure 3.

Figure 3: Entrepreneur's optimal spot selling strategies.

According to Mollick’s (2016) empirical study on the impact of Kickstarter funding

“over 50% of the projects were reported as being innovative by both backers and creators,

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and projects produced over 2,601 patent applications”. Therefore, crowdfunding projects

have a relatively high probability of delivering a positive shock to the industry standards

since they are creative and innovative. This suggests that the comparable spot selling strategy

is the one where the entrepreneur believes that the consumers will more probably realize a

positive shock which is characterized by spot selling Strategy B. In order to focus on the

comparable spot selling strategy, where the entrepreneur attracts demand for the positive

shock (Strategy B), we make the following assumption:

Assumption 2.

𝝀 >𝟏

(𝟏 + 𝜶𝑯)𝟐

In the following sections, we will proceed by considering Strategy B as the

comparable benchmark spot selling strategy. 7

3.2 UNCONSTRAINED CROWDFUNDING

Under crowdfunding the entrepreneur now has the ability to solicit pre-orders from

consumers before production takes place at t = 0 and deliver these orders when the products

are produced at t = 1. Moreover, being able to take pre-orders from crowdfunders does not

hinder the entrepreneur´s ability to generate demand in the spot market from regular

consumers. In this section we will derive the optimal crowdfunding contract offered by the

entrepreneur assuming that at t = 0 the entrepreneur commits to the prices for both the

crowdfunding and the spot selling period. Previous literature assumes no price commitment

and maximizes profits in two stages backwards. In our case, following the literature on

advance purchase discount (Nocke et al., 2011) and consistent with anecdotal evidence, the

entrepreneur commits to the pre-ordering and the future retail prices and maximizes his

profits at time 0.

At t = 0, the entrepreneur publicizes the crowdfunding price, PC, and the retail price,

PR. The entrepreneur commits to offering a discount to crowdfunders such that PC < PR.

Since the entrepreneur is committing to a discounted price for crowdfunders, a rational

consumer that intends to purchase the good irrespective of the shock would pre-order, taking

advantage of the backer’s discount (as the expected value of the shock is zero). A rational

consumer would only delay the choice of purchasing the good until t = 1 if he is not planning

to buy the good in case a negative shock is realized. Without loss of generality we assume

no discounting. Thus, we have the following conditions describing the marginal consumer at

the crowdfunding period, 𝜃𝐶 , and the marginal consumer in the spot market, 𝜃𝑅:

7 As will be seen later on, comparing crowdfunding with the spot selling Strategy A will result in crowdfunding

always dominating. We thus focus on the more interesting and comparable spot selling alternative to

crowdfunding, that is, Strategy B.

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t = 0 : 𝜃 − 𝑃𝐶 ≥ 𝜆(𝜃 + 𝛼𝐻 − 𝑃𝑅) → 𝜃𝐶 ≥𝑃𝐶−𝜆(𝑃𝑅−𝛼𝐻)

1−𝜆 (3)

t = 1 : 𝜃 + 𝛼𝐻 − 𝑃𝑅 ≥ 0 → 𝜃𝑅 ≥ 𝑃𝑅 − 𝛼𝐻 (4)

We have that 𝜃𝐶 > 𝜃𝑅, therefore we observe that crowdfunders have a higher

valuation relative to consumers who wait to observe a positive realization of the shock, 𝛼𝐻,

before making their purchase decision. Crowdfunders will pre-order the good at t = 0, while

the regular consumers will wait until t = 1 to see if the positive shock is realized in order to

purchase the good. Demand at t = 0 is (1 - 𝜃𝐶), while expected demand at t = 1 is 𝜆(𝜃𝐶 − 𝜃𝑅).

Thus, at t = 0 the entrepreneur has certain demand from crowdfunders and uncertain future

demand depending on the realization of the positive shock. The monopolist´s maximization

problem is:

𝐦𝐚𝐱𝑷𝑪,𝑷𝑹

𝝅(𝑷𝑪, 𝑷𝑹) = 𝑷𝑪(𝟏 − 𝜽𝑪) + 𝝀𝑷𝑹(𝜽𝑪 − 𝜽𝑹)

s.t. 𝜃𝐶 =𝑃𝐶−𝜆(𝑃𝑅−𝛼𝐻)

1−𝜆 ; 𝜃𝑅 = 𝑃𝑅 − 𝛼𝐻 (5)

From the first order conditions we obtain that the entrepreneur maximizes his profit

by setting �̂�𝐶 =1

2 and �̂�𝑅 =

1+𝛼𝐻

2 ; therefore, indeed we see that the entrepreneur’s optimal

pricing strategy is such that �̂�𝐶 < �̂�𝑅 . The quantity demanded in pre-orders during the

crowdfunding period is �̂�𝐶 =1

2−

𝜆𝛼𝐻

2(1−𝜆), while the expected demand in the retail market is

𝐸[�̂�𝑅] =𝜆𝛼𝐻

2(1−𝜆). The total expected quantity demanded during both periods is 𝐸[�̂�𝐶𝐹] =

1

2 .

Profits during the crowdfunding period are �̂�𝐶 =1

4−

𝜆𝛼𝐻

4(1−𝜆) , while during the retail period

expected profits are 𝐸[�̂�𝑅] =𝜆𝛼𝐻+𝜆𝛼𝐻

2

4(1−𝜆). Total profits during both periods are 𝐸[�̂�𝐶𝐹] =

1

4+

𝜆𝛼𝐻2

4(1−𝜆) , which increase as the probability of fulfilling the promised shock increases.8

3.3 UNCONSTRAINED OPTIMAL STRATEGY

In the previous subsections we have derived the entrepreneur’s optimal strategy under

spot selling and crowdfunding, assuming he is financially unconstrained. The entrepreneur

now compares the two alternatives and chooses the one that maximizes his profits. These

optimal strategies are from the entrepreneur’s perspective and do not imply optimality from

a welfare standpoint.

Proposition 1.

a) The unconstrained entrepreneur’s unique optimal strategy is to use crowdfunding.

8 Comparative statics are provided in Table A in the Appendix.

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b) �̂�𝐶 & �̂�𝑅 set by the unconstrained entrepreneur are indeed such that �̂�𝐶 < �̂�𝑅 , thus

the optimal pricing strategy is such that crowdfunders are rewarded with a price

discount when compared to the retail consumers.

c) Total production under crowdfunding dominates the one under spot selling.

Proof. See Appendix.

Part (a) illustrates a crucial point. An unconstrained entrepreneur is always better off

using crowdfunding than spot selling.9 Therefore, whenever the consumer’s expected value

of the future shock is zero, we have that the market opens for crowdfunders and retail

consumers. Crowdfunders enjoy a price discount as stated by part (b). The reason that an

entrepreneur is better off with crowdfunding than with spot selling, even though the price

paid by crowdfunders is lower than the spot-selling price is clarified in part (c). Indeed, under

crowdfunding the entrepreneur is able to shift uncertain future demand into certain pre-orders

by crowdfunders and therefore expand the market. A financially unconstrained entrepreneur

would thus optimally choose crowdfunding.

So far we have explored the financing options available for the entrepreneur to launch

his venture assuming a financially unconstrained entrepreneur, that is, no minimum capital

requirement had to be met to undergo production. This is compatible with the “Keep it All”

option that entrepreneurs have when listing a project on the second leading crowdfunding

platform, Indiegogo. In this case the entrepreneur gets to keep whatever amount he raised,

even if the posted minimum capital requirement is not met. However, to list a project on the

leading pre-ordering crowdfunding platform “Kickstarter” the entrepreneur has to raise at

least the publicized minimum capital requirement. The capital requirement is in general the

fixed amount of money needed to start production. In case the entrepreneur fails to raise the

minimum capital required, nothing is charged to the consumers who pre-ordered and no

money is transferred to the entrepreneur. The same happens with the “All or Nothing” option

on Indiegogo. According to Cumming, Leboeuf, & Schwienbacher (2014), the “All or

Nothing” option offers a credible signal that the entrepreneur would only launch the venture

if the publicized capital requirement is raised. Moreover, they show that entrepreneurs opting

for this scheme are able to set higher goals and tend to be more successful. Therefore, in the

following section we proceed by analyzing the case of a financially constrained entrepreneur,

assuming that the entrepreneur needs to meet a minimum capital requirement in order to

launch his venture.

9 Note that this holds for the unconstrained monopolist no matter if we compare to strategy A or to strategy B.

Indeed, when analyzing this further, we see that crowdfunding dominates both spot selling strategies, A and B,

for all admissible values of λ given by Assumption 1.

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4. THE CONSTRAINED MONOPOLIST

The case of a financially constrained entrepreneur can be seen either as a situation

where the entrepreneur does not have any personal financial endowments or as a situation in

which the capital requirement is the amount required beyond what the entrepreneur can

personally provide (Chang, 2016). When an entrepreneur opts for spot selling as his

launching strategy he will be constrained for any K > 0 since under spot selling the

entrepreneur receives orders after production has taken place. Whereas, under crowdfunding,

the entrepreneur is able to raise from crowdfunders an amount equal to 𝐾𝐶𝐹 = �̂�𝐶(𝟏 − 𝜽𝑪) =

1

4−

𝜆𝛼𝐻

4(1−𝜆). Thus, the entrepreneur is constrained only for levels of 𝐾 > 𝐾𝐶𝐹 . For lower levels

of K the entrepreneur is able to raise the required capital in the form of pre-orders before

production starts. This is essential to our analysis since crowdfunding not only acts as a

means of securing uncertain demand, but in its essence it is also a financing source for a

constrained entrepreneur.

4.1 CONSTRAINED SPOT SELLING

As previously mentioned, a spot selling entrepreneur is constrained for any required

level of capital greater than zero. In such a case the constrained entrepreneur can attempt to

tap external sources of funds (banks, P2P loans, etc.) which are provided in the form of a

debt obligation. These can be viewed as securities on future cash flows. We are considering

the case where the entrepreneur is able to tap these external sources of funds. Even though

some entrepreneurs might be unable to access the traditional debt market, the development

of the P2P lending market has facilitated access to capital in the form of a debt obligation.

External sources of funds come at a cost and the cost set by the lenders reflects the risk that

they assume, the cost of capital is denoted by r, with R denoting (1+r).

Following Kumar et al. (2016) we assume that lenders have perfect information

regarding the venture’s expected profits. Moreover, they do their due diligence and the

maximum capital that they are willing to lend is such that the entrepreneur is able to pay back

the principal and accrued interest once the venture launches. The information that both the

lender and the entrepreneur have regarding the future profitability of the venture is

symmetric. The entrepreneur has no motive to provide false information regarding the

profitability of the venture or expropriate the loan since these loans take the form of personal

loans that he is liable for regardless of the venture’s success or failure. The entrepreneur is a

rational sophisticated individual who is able to evaluate external sources of funds and opts

for the source charging the lowest interest rate. It would be interesting to consider how the

entrepreneur evaluates different types of loans given the emergence of P2P lending as an

alternative to traditional bank loans, but this is out of the scope of this paper. We will treat

all forms of debt obligations similarly. The entrepreneur opts for the optimal loan and the

13

only cost associated to it is the interest rate payable by the entrepreneur. The constrained

monopolist’s problem is:

𝐦𝐚𝐱𝑷𝑺

𝑪𝝅(𝑷𝑺

𝑪) = 𝝀𝑷𝑺𝑪(𝟏 − 𝜽𝑳) − 𝑹 ∗ 𝑲

s.t. 𝜃𝐿 = 𝑃𝑆𝐶 − 𝛼𝐻 (6)

The optimal price and quantities are the same as that under the unconstrained spot

selling:

�̌�𝑆𝐶 =

(1+𝛼𝐻)

2 , 𝐸[�̌�𝑆

𝐶] =𝜆(1+𝛼𝐻)

2

Since now the entrepreneur is required to take a loan and invest in order to undergo

production he will have to pay back the loan plus interest. Expected profits will be profits

after the payment of the loan and interest. Expected profits are lower than when no capital

was required by the entrepreneur in the unconstrained case. The maximum capital that an

entrepreneur is able to raise under this strategy is 𝐾𝑆 =𝜆(1+𝛼𝐻)2

4𝑅 , which is the maximum

loan the lender would be willing to offer. For levels of capital above this threshold, the

entrepreneur would not be able to launch his venture.

𝑬[�̌�𝑺𝑪] = {

𝜆(1 + 𝛼𝐻)2

4− 𝑅 ∗ 𝐾, 𝑓𝑜𝑟 𝐾 ≤ 𝐾𝑆

0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

A benchmark numerical simulation for the constrained spot selling strategy is

presented in Table 1 along with some comparative statics. The results and dynamics can be

better demonstrated numerically and graphically as presented in Table 1 and Figure 4.

Without loss of generality, we assume that 𝜆 = 0.5 and 𝛼𝐻 = 0.45 for the benchmark case.

We chose these parameter values such that we have a symmetric positive and negative shock.

Our results are robust to these values. The spot selling price and quantities are unaffected by

the loan size. However, the expected profits, which in the constrained case are the profits

after paying back the loan and interest, are affected by the capital requirement and interest

rate as demonstrated by Figure 4.

Spot Selling

Benchmark

λ 𝜶𝑯

0.48 0.52

0.425 0.475

Spot Selling Price (�̂�𝑺𝑪) 0.7250 0.7250 0.7250 0.7125 0.7375

Expected Demand (𝑬[�̂�𝑺𝑪]) 0.3625 0.3480 0.3770 0.3563 0.3688

Maximum Capital (�̅�𝑆):

r = 10.00% 0.2389 0.2293 0.2485 0.2308 0.2472 r = 12.50% 0.2336 0.2242 0.2430 0.2256 0.2417

r = 15.00% 0.2285 0.2193 0.2377 0.2207 0.2365

r = 17.50% 0.2236 0.2147 0.2326 0.2160 0.2314 r = 20.00% 0.2190 0.2103 0.2278 0.2115 0.2266

14

Table 1: Benchmark numerical simulation for constrained spot selling and comparative statics.

Figure 4: Residual profits given capital requirements and different interest rates.

From the analysis of the benchmark case in Table 1 and the graph in Figure 4 we

observe that the maximum capital an entrepreneur could raise through debt obligations

decreases as the interest rate increases, as expected. We also notice that, for any given level

of interest rate, projects with higher probability of delivering the promised positive shock

have access to higher levels of capital. In other words, we could say that more credible

entrepreneurs are able to take loans for comparable projects that less credible entrepreneurs

might not be able to raise funds for. Whereas, when having comparable entrepreneurs with

the same probability of delivering on their promised shocks to the industry standards, we

expect to see that projects with higher promised levels of 𝛼𝐻 will have access to more

generous credit lines, where by credit line we are referring to the maximum capital that the

lender is willing to lend. This follows intuitively as projects promising higher shock levels

are expected to have higher future profits.

4.2 CONSTRAINED CROWDFUNDING

In a prior section we have discussed the ability of crowdfunding to raise capital

through pre-orders prior to production. Given the entrepreneur’s optimal prices and quantities

we have seen that through crowdfunding he is able to raise 𝐾𝐶𝐹 = 1

4−

𝜆𝛼𝐻

4(1−𝜆) without the

need to diverge away from the optimal conditions. In other words, whenever the capital

requirement of the entrepreneur is below 𝐾𝐶𝐹 the entrepreneur is unconstrained and is able

to raise what he needs in order to undergo production. The optimal contract proposed under

the unconstrained case of crowdfunding is the contract which maximizes total profits from

both periods. Now when the entrepreneur is constrained under crowdfunding, he will have a

constrained maximization problem: maximize total profits under the constraint of raising the

necessary capital during the crowdfunding period. Expected profits of crowdfunding would

15

be the sum of both periods profits less the required capital. Therefore, the contract offered by

the constrained entrepreneur would differ from that of the unconstrained case and would

involve diverging away from the optimal pricing strategy. The entrepreneur would

manipulate prices in order to raise more in pre-orders to meet his capital requirement.

Through this strategy the entrepreneur is shifting future demand to pre-orders beyond the

optimal levels suggested by his unconstrained optimality conditions. Thus, for levels of 𝐾 >

𝐾𝐶𝐹 the entrepreneur’s problem is defined below:

𝐦𝐚𝐱𝑷𝑪

𝑪,𝑷𝑹𝑪

𝝅(𝑷𝑪𝑪, 𝑷𝑹

𝑪) = 𝑷𝑪𝑪(𝟏 − 𝜽𝑪) + 𝝀𝑷𝑹

𝑪(𝜽𝑪 − 𝜽𝑹) − 𝑲

s.t. 𝑃𝐶𝐶(1 − 𝜃𝐶) = 𝐾; 𝜃𝐶 =

𝑃𝐶𝐶−𝜆(𝑃𝑅

𝐶−𝛼𝐻)

1−𝜆 ; 𝜃𝑅 = 𝑃𝑅

𝐶 − 𝛼𝐻 (7)

𝑃𝐶𝐶 & 𝑃𝑅

𝐶 denote the constrained entrepreneur´s crowdfunding and future retail price

respectively. Since the constraint is binding, the optimal crowdfunding price is such that the

entrepreneur is able to successfully raise his capital requirement during the crowdfunding

period, 𝑃𝐶𝐶(1 − 𝜃𝐶) = 𝐾. We can thus express the optimal crowdfunding price as a function

of the retail price by solving the polynomial 𝑃𝐶𝐶 (1 −

𝑃𝐶𝐶−𝜆(𝑃𝑅

𝐶−𝛼𝐻)

1−𝜆) = 𝐾. We obtain the

following crowdfunding price as a function of the future retail price that the entrepreneur

commits to and of the level of capital that the entrepreneur requires:

�̂�𝐶𝐶 =

1 − 𝜆(1 + 𝛼𝐻) + √(1 + 𝜆𝑃𝑅𝐶)2 + 𝜆(1 + 𝛼𝐻)[𝜆(1 + 𝛼𝐻 − 2𝑃𝑅

𝐶) − 2] − 4𝐾(1 − 𝜆)

2

We can now plug �̂�𝐶𝐶 back into the entrepreneur’s objective function and solve for

the optimal crowdfunding and future retail price through maximizing the total profits given

the other two constraints. This maximization problem in general has no analytical solution

and has to be solved numerically. Nevertheless, we can derive simple analytical solutions for

the polar case of the maximum capital that the entrepreneur is able to raise under

crowdfunding. The maximum capital that the entrepreneur can raise in pre-orders is achieved

by maximizing first period profits by shifting all future demand to pre-orders such that 𝜃𝑅 =

𝜃𝐶 . For this to hold we have that 𝑃𝑅𝐶 = 𝑃𝐶

𝐶 + 𝛼𝐻 and 𝜃𝐶 = 𝑃𝐶𝐶 . 10 Thus, the entrepreneur

foregoes any profits in the second period in order to undergo production.

𝐦𝐚𝐱𝑷𝑪

𝑪𝝅(𝑷𝑪

𝑪) = 𝑷𝑪𝑪(𝟏 − 𝜽𝑪) − 𝑲

s.t. 𝜃𝐶 = 𝑃𝐶𝐶 (8)

10 𝜃𝑐 & 𝜃𝑅 are defined by our consumer participation constraints. For 𝜃𝑅 = 𝜃𝐶 we have that 𝑃𝑅

𝐶 − 𝛼𝐻 =𝑃𝐶

𝐶−𝜆(𝑃𝑅𝐶−𝛼𝐻)

1−𝜆 solving for 𝑃𝑅

𝐶 we arrive at 𝑃𝑅𝐶 = 𝑃𝐶

𝐶 + 𝛼𝐻. Plugging 𝑃𝑅𝐶 in 𝜃𝐶 we obtaib 𝜃𝐶 = 𝑃𝐶

𝐶 .

16

The maximum capital raised under crowdfunding is always 𝐾𝐶𝐹 =1

4 regardless of

the shock and the probability of the shock being delivered. At this level 𝑃𝐶𝐶 =

1

2 and 𝑃𝑅

𝐶 =

1

2+ 𝛼𝐻. We have no retail demand and the market shrinks to just one period.11

For levels of capital below the maximum, when the entrepreneur is constrained, no

analytical solution is possible due to a polynomial of fourth degree. However, numerical

simulations show that the crowdfunding price and demand exhibit a non-monotonic

relationship with the capital requirement (see Figure 5). We denote the level of capital where

there is an inflection in the behavior of the crowdfunding price and demand by �̃�𝐶𝐹.

a) Prices b) Demands c) Profits

Figure 5: Relationship of Optimal Prices, Demand and Profits with Capital Requirement.

The benchmark results representing the optimal outcomes for different capital

requirements are provided below in Table 2.

Table 2: Benchmark numerical simulation for constrained crowdfunding.

11 Proof in the Appendix.

17

For levels of 𝐾 ≤ 𝐾𝐶𝐹 , the entrepreneur is unconstrained and does not need to diverge

away from the optimal prices and quantities, but profits are lower than in the case where no

capital was required to launch the venture since the entrepreneur now is required to make an

investment in order to undergo production. When this required investment is lower than what

he could raise in pre-orders, the entrepreneur does not find it optimal to change his pricing

strategy.

For levels of K where 𝐾𝐶𝐹 < 𝐾 ≤ �̃�𝐶𝐹, the entrepreneur decides to expand the

crowdfunding market in order to raise more money in pre-orders to finance his capital

requirement. The entrepreneur finds it optimal to increase the backer’s discount through

lowering the crowdfunding price, while increasing the premium that the retail consumers

would have to pay. The future expected demand shrinks, but is offset by the demand gains in

preorders, such that total expected production increases.

For even higher levels of capital, where �̃�𝐶𝐹 < 𝐾 < 𝐾𝐶𝐹, the decrease in the

crowdfunding price does not increase the quantity demanded in the crowdfunding period as

much. The entrepreneur no longer finds it optimal to reduce the price further for

crowdfunders and starts raising the crowdfunding price as the capital requirement increases.

The constrained crowdfunding price remains lower than the unconstrained crowdfunding

price. The premium charged to the future retail consumers increases even more than the

increase in the crowdfunding price. Therefore, the backers’ discount that the crowdfunders

receive relative to the retail price increases as the capital requirement increases for all levels

of 𝐾 > 𝐾𝐶𝐹 . As the entrepreneur raises the crowdfunding and retail price the market size

starts shrinking. The amount raised in pre-orders increases since the loss in demand is offset

by the higher price charged by the entrepreneur. Total production is still larger than the case

where the entrepreneur is unconstrained.

When crowdfunding, an unconstrained entrepreneur can pocket in profits from the

pre-ordering period as well as the retail period. As the entrepreneur becomes constrained, all

profits from the the pre-ordering period are used to meet the investment requirement to

launch the venture. Expected profits become profits from the retail period, which takes place

after the entrepreneur starts production. The higher the capital requirement the more the

entrepreneur needs to shift future demand to the pre-ordering stage. This results in lower

profits in the retail period. Thus, in order to raise higher levels of capital the entrepreneur is

giving up part of the future profits in order to launch his venture.

We presented the benchmark case earlier and we would now like to analyze the effect

of a change in the probability of the shock being delivered and the size of the shock that the

entrepreneur claims to deliver on our previous results. At 𝐾𝐶𝐹 and 𝐾𝐶𝐹, we are able to provide

analytical comparative statics which are summarized in Table B in the Appendix. We notice

that the level of capital starting with which the entrepreneur becomes constrained, 𝐾𝐶𝐹 =1

4−

18

𝜆𝛼𝐻

4(1−𝜆) , decreases in both 𝜆 and 𝛼𝐻. Thus for higher 𝜆 and 𝛼𝐻 the entrepreneur becomes

constrained and needs to diverge away from the optimal unconstrained conditions for lower

capital requirements. At 𝐾𝐶𝐹 and 𝐾𝐶𝐹, as 𝛼𝐻 increases the retail price increases but this does

not have any effect on the crowdfunding price. The probability of the shock being delivered

does not affect prices but does affect demand at 𝐾𝐶𝐹. As 𝜆 or 𝛼𝐻 increase there is a decrease

in the crowdfunding demand but profits increase since demand is shifted to the retail period

at which the price is higher. The maximum amount of capital that the entrepreneur can raise,

𝐾𝐶𝐹 =1

4 , does not depend on 𝜆 or 𝛼𝐻 . In Table 3 and Table 4 we numerically present

comparative statics for the constrained entrepreneur opting for crowdfunding.

λ = 0.48 λ = 0.52

No Capital 𝑲𝑪𝑭 �̃�𝑪𝑭 𝑲𝑪𝑭 No Capital 𝑲𝑪𝑭 �̃�𝑪𝑭 𝑲𝑪𝑭

Capital Requirement (K) 0 0.1462 0.1903 0.2500 0 0.1281 0.1830 0.2500

Crowdfunding Price (�̂�𝑪𝑪) 0.5000 0.5000 0.4830 0.5000 0.5000 0.5000 0.4800 0.5000

Retail Price (�̂�𝑹𝑪) 0.7250 0.7250 0.7997 0.9500 0.7250 0.7250 0.8019 0.9500

Crowdfunders’ Discount 0.2250 0.2250 0.3167 0.4500 0.2250 0.2250 0.3219 0.4500

Crowdfunding Demand (�̂�𝑪𝑪) 0.2923 0.2923 0.3940 0.5000 0.2562 0.2562 0.3812 0.5000

Expected Retail Demand (𝑬[�̂�𝑹𝑪] ) 0.2077 0.2077 0.1230 0 0.2438 0.2438 0.1288 0

Aggregate Demand (𝑬[�̂�𝑪𝑭𝑪 ] ) 0.5000 0.5000 0.5170 0.5000 0.5000 0.5000 0.5200 0.5000

Crowdfunding Profits (�̂�𝑪𝑪) 0.1462 0.1462 0.1905 0.2500 0.1281 0.1667 0.1830 0.2500

Expected Retail Profit ( 𝑬[�̂�𝑹𝑪] ) 0.1506 0.1506 0.0984 0 0.1767 0.1250 0.1113 0

Residual Profit ( 𝑬[𝝅𝑪𝑭𝑪 (𝑲)] ) 0.2967 0.1506 0.0984 0 0.3048 0.1250 0.1113 0

Table 3: Comparative statics for different probabilities λ given the benchmark 𝜶𝑯= 0.45

𝜶𝑯 = 0.425 𝜶𝑯 = 0.475

No Capital 𝑲𝑪𝑭 �̃�𝑪𝑭 𝑲𝑪𝑭 No Capital 𝑲𝑪𝑭 �̃�𝑪𝑭 𝑲𝑪𝑭

Capital Requirement (K) 0 0.1437 0.1916 0.2500 0 0.1312 0.1817 0.2500

Crowdfunding Price (�̂�𝑪𝑪) 0.5000 0.5000 0.4834 0.5000 0.5000 0.5000 0.4798 0.5000

Retail Price (�̂�𝑹𝑪) 0.7125 0.7125 0.7881 0.9250 0.7375 0.7375 0.8133 0.9750

Crowdfunders’ Discount 0.2125 0.2125 0.3047 0.4250 0.2375 0.2375 0.3335 0.4750

Crowdfunding Demand (�̂�𝑪𝑪) 0.2875 0.2875 0.3963 0.5000 0.2625 0.2625 0.3788 0.5000

Expected Retail Demand (𝑬[�̂�𝑹𝑪] ) 0.2125 0.2125 0.1203 0 0.2375 0.2375 0.1415 0

Aggregate Demand (𝑬[�̂�𝑪𝑭𝑪 ] ) 0.5000 0.5000 0.5166 0.5000 0.5000 0.5000 0.5202 0.5000

Crowdfunding Profits (�̂�𝑪𝑪) 0.1437 0.1437 0.1916 0.2500 0.1312 0.1312 0.1817 0.2500

Expected Retail Profit ( 𝑬[�̂�𝑹𝑪] ) 0.1514 0.1514 0.0948 0 0.1752 0.1752 0.1151 0

Residual Profit ( 𝑬[𝝅𝑪𝑭𝑪 (𝑲)] ) 0.2952 0.1514 0.0948 0 0.3064 0.1752 0.1151 0

Table 4: Comparative statics for different shock levels 𝜶𝑯 given the benchmark λ = 0.50

19

4.3 CONSTRAINED OPTIMAL STRATEGY

The analysis of the constrained entrepreneur’s optimal pricing strategy under spot

selling and crowdfunding lays the ground on which we will develop the optimal financing

strategy for the constrained entrepreneur. In this section we compare the options available to

the constrained entrepreneur and check the conditions under which one form of financing

dominates the other. Thus far we know that the maximum capital that an entrepreneur can

raise under crowdfunding is reached by shifting all uncertain future demand into pre-orders

which gave us 𝐾𝐶𝐹 =1

4 . Whereas, the maximum amount that the entrepreneur can take in

loans is 𝐾𝑆 =𝜆(1+𝛼𝐻)2

4𝑅 . In the extreme, for 𝑟 = 0 we have 𝐾𝐶𝐹 < �̿�𝑆 =

𝜆(1+𝛼𝐻)2

4. For a

given capital level to be raised, K, a project is feasibvle through standard debt financing if

the interest rate is below a certain threshold . This interest rate threshold denoted by 𝑟𝐹 is

expressed below:

𝑟𝐹 =𝜆(1+𝛼𝐻)2

4𝐾− 1 (9)

Thus, we have that the entrepreneur faces different financing options depending on

his capital requirement and the interest rate. Below we identify the four possible situations

and the respective strategies available:

i. 𝑟 ≤ 𝑟𝐹 and 0 < 𝐾 ≤ 𝐾𝐶𝐹; Constrained Spot Selling vs Unconstrained Crowdfunding

ii. 𝑟 ≤ 𝑟𝐹 and 𝐾𝐶𝐹 < 𝐾 ≤ 𝐾𝐶𝐹; Constrained Spot Selling vs Constrained Crowdfunding

iii. 𝑟 ≤ 𝑟𝐹 and 𝐾𝐶𝐹 < 𝐾 ≤ 𝐾𝑆; Constrained Spot Selling Only

iv. 𝑟 > 𝑟𝐹 and 0 < 𝐾 ≤ 𝐾𝐶𝐹; Crowdfunding Only

These situations are more intuitively illustrated in Figure 6.

Figure 6: Entrepreneur’s available strategies given the capital requirement and interest rate.

20

Having laid the ground for the situations that the entrepreneur might face we proceed

by analyzing these situations in order to determine the entrepreneur’s optimal strategy

depending on the capital requirement and the interest rate faced by the entrepreneur.

i. Constrained Spot Selling vs Unconstrained Crowdfunding

For 0 < 𝐾 ≤ 𝐾𝐶𝐹 and 𝑟 ≤ 𝑟𝐹 , the project is feasible under both launching options.

Given the capital requirement the entrepreneur is constrained when spot selling but

unconstrained when crowdfunding. The entrepreneur would be better off being

unconstrained under crowdfunding than opting for standard debt, which is quite intuitive

since the entrepreneur was always better off under crowdfunding in the unconstrained case.

Thus, we always have that 𝐸[�̂�𝐶𝐹] > 𝐸[�̌�𝑆𝐶], regardless of the size of the shock that the

entrepreneur promises or the probability that the shock is fulfilled. Projects with low capital

requirements, such that the entrepreneur is able to raise the required capital in pre-orders

without the need to diverge away from the optimal prices, would always find that

crowdfunding dominates standard debt.

ii. Constrained Spot Selling vs Constrained Crowdfunding

For 𝐾𝐶𝐹 < 𝐾 ≤ 𝐾𝐶𝐹 and 𝑟 ≤ 𝑟𝐹, the project is feasible under both launching options.

Given the venture’s capital requirement the entrepreneur is now constrained with respect to

both launching strategies. The entrepreneur opts for the strategy with the higher expected

profits. Since now the entrepreneur is diverging away from his optimal prices when opting

for crowdfunding it does not follow that the entrepreneur is always better off under

crowdfunding. The entrepreneur’s optimal choice involves comparing expected profits under

both strategies. The choice will depend on the interest rate that he faces in the credit market

and this brings to the analysis a critical interest rate that we will denote with 𝑟𝑂.

𝑟𝑂 =(

𝜆(1+𝛼𝐻)2

4−𝐸[𝜋𝐶𝐹

𝐶 (𝐾)])

𝐾− 1 (10)

It is when both options are feasible for the entrepreneur where 𝑟𝑂 comes into play.

𝑟𝑂 represents the interest rate threshold which determines the optimality of the financing

option that the entrepreneur opts for. If the interest rate is above 𝑟𝑂 then the entrepreneur

would find it optimal to use crowdfunding as the financing strategy and would prefer to

manipulate prices rather than taking a loan.

iii. Constrained Spot Selling Only

21

For 𝐾 > 𝐾𝐶𝐹 and 𝑟 ≤ 𝑟𝐹, the entrepreneur’s capital requirement can not be met

through crowdfunding and standard debt is the only launching option available to the

entrepreneur.

iv. Crowdfunding Only

For 0 < 𝐾 ≤ 𝐾𝐶𝐹 and 𝑟 > 𝑟𝐹, the entrepreneur’s capital requirement can not be met

through standard debt and crowdfunding is the only launching option available to the

entrepreneur.

We sum up the results from the discussion of the four situations above in Proposition

2 and illustrate in Figure 7 the entrepreneur’s optimal financing strategy given the capital

requirement and the interest rate.

Proposition 2.

a) For 0 < 𝐾 ≤1

4 and 𝑟 > 𝑟𝐹, only crowdfunding is feasible, thus the constrained

entrepreneur’s optimal strategy is to use crowdfunding.

b) For 0 < 𝐾 ≤1

4 and 𝑟 < 𝑟𝑂, both strategies are feasible and the constrained

entrepreneur’s unique optimal strategy is to use standard debt.

c) For 0 < 𝐾 ≤1

4 and 𝑟𝑂 ≤ 𝑟 ≤ 𝑟𝐹, both strategies are feasible and the constrained

entrepreneur’s unique optimal strategy is to use crowdfunding.

d) For 1

4< 𝐾 ≤ �̿�𝑆 and 𝑟 ≤ 𝑟𝐹, only standard debt is feasible, thus the constrained

entrepreneur’s optimal strategy is to use standard debt.

e) For 𝐾 >1

4 and 𝑟 > 𝑟𝐹, the project is not feasible under any strategy.

Figure 7: Entrepreneur’s optimal strategy given the capital requirement and interest rate.

22

From the comparative statics of the key variables, when the entrepreneur is

constrained, we arrive at interesting results that we mention in the corollary below and

proceed by further discussing their implications.

Corollary 1: Projects that promise higher positive shocks or are more probable to deliver

on their promises have a lower 𝐾𝐶𝐹, higher �̿�𝑆 and a higher interest rate thresholds (𝑟𝐹 and

𝑟𝑂).

This poses important implications. From our corollary we can infer that projects

promising higher shock to industry standards are constrained for lower levels of capital

requirement under crowdfunding. Thus, they are more probable to diverge away from their

optimal crowdfunding strategy making standard debt more attractive. This is consistent with

the findings of Chan & Parhankangas (2017) where they investigated the effect of

innovativeness on crowdfunding outcomes. They found that projects with incremental

innovativeness, lower 𝛼𝐻, result in favorable crowdfunding outcomes when compared to

projects with radical innovativeness, higher 𝛼𝐻. Further on we depict that projects with lower

probability of delivering on their promises would more likely opt for crowdfunding which is

in line with the results of an empirical analysis by Xu (2017) that shows that entrepreneurs

tend to launch riskier projects on Kickstarter. Given this we expect to see that projects

launched through the use of debt obligations are more innovative or that the entrepreneurs

opting for standard are more credible and exhibit higher probabilities of delivering on their

promises. Moreover, our model further adds that when both financing options are available

to the entrepreneur, standard debt becomes more attractive as the funding needs increase.

Thus, entrepreneurs with larger funding requirements would prefer to raise their capital in

the form of a debt obligation rather than utilizing the crowdfunding ability of raising funds

in the form of pre-orders.

5. WELFARE ANALYSIS

In order to gain a better understanding from the social planner’s perspective we

compare welfare under crowdfunding with the entrepreneur’s benchmark spot selling

strategy, Strategy A. We have seen that given the capital requirement the entrepreneur could

either be unconstrained for both strategies, or constrained when spot selling but not when

crowdfunding, or constrained in both. We divide our analysis into two subsections

unconstrained entrepreneur and constrained entrepreneur. When the entrepreneur is

constrained in one strategy but not in the other, the analysis will be included in our

constrained entrepreneur’s subsection. In our welfare analysis we follow the convention and

decompose welfare into three components: total production, consumer surplus, and producer

surplus in order to provide insightful results.

5.1 Unconstrained Entrepreneur (K = 0)

23

When the entrepreneur opts for spot selling, total production as we have seen earlier

is 𝐸[�̌�𝑆] =𝜆(1+𝛼𝐻)

2. The maximum price that the consumer is willing to pay is

𝑃𝑆 = (1 + 𝛼𝐻) such that for levels of prices above 𝑃𝑆 there is no demand for the product.

Since now we have �̌�𝑆 and 𝑃𝑆 we easily arrive at 𝐶𝑆𝑆 =𝜆(1+𝛼𝐻)2

8.12 Producer surplus is

basically the entrepreneur’s profits such that 𝑃𝑆𝑆 =𝜆(1+𝛼𝐻)2

4.

For crowdfunding we have that total production is 𝐸[�̂�𝐶𝐹] =1

2. In regards to the

consumer surplus we have that the consumers are deciding whether to crowdfund or to buy

in the retail market. Depending on the consumer’s decision we compute the maximum price

that the consumer is willing to pay. When pre-ordering the maximum price that the consumer

is willing to pay is such that 𝑃𝐶 = �̂�𝑅 . If the crowdfunding price is higher than the future

retail price then a rational consumer would wait for the retail period to realize the shock and

make the purchase decision accordingly. Whereas for the retail consumers the maximum that

they are willing to pay is such that 𝜃𝑅 = 𝜃𝐶 . Therefore, using equation (5) we have that 𝑃𝑅 =

�̂�𝐶 + 𝛼𝐻. The consumer surplus under crowdfunding will be 𝐶𝑆𝐶𝐹 =𝛼𝐻

8. 13 The producer

surplus is basically the entrepreneur’s profit under crowdfunding and we have that 𝑃𝑆𝐶𝐹 =1

4+

𝜆𝛼𝐻2

4(1−𝜆) .

When comparing the welfare components under both strategies we observe that

crowdfunding increases total production, but there is always a consumer surplus loss such

that 𝐶𝑆𝑆 > 𝐶𝑆𝐶𝐹 . Even though production increases, the price discrimination tool that the

entrepreneur employs when crowdfunding extracts some of the consumer’s surplus, such that

there is a consumer surplus loss when the entrepreneur opts for crowdfunding over spot

selling. Consumers in the crowdfunding model have 3 options available to them: pre-

ordering, retail purchasing, or not purchasing the product at all. In the case of spot selling the

only options available to the consumer is purchasing or not. Under crowdfunding consumers

make an intertemporal decision based upon the prices that they face. Due to the discount

offered to the backers in the crowdfunding period the maximum price that the consumers

would be willing to pay in the retail period is lower than that in spot selling although optimal

prices in the retail period and spot selling are the same. The maximum that rational consumers

who wait for the retail period are willing to pay is a premium over the crowdfunding price

equal to the size of the positive shock that they realize; otherwise they will not purchase the

product. Regarding the producer, there is always a producer surplus gain (𝑃𝑆𝐶𝐹 > 𝑃𝑆𝑠) since

crowdfunding allows the entrepreneur to discriminate amongst consumers and take a share

of their consumer surplus as well as expand the market. However, the gain by the producers

12 𝐶𝑆𝑆 =

(𝑃𝑆−�̌�𝑆)∗𝐸[�̌�𝑆]

2

13 𝐶𝑆𝐶𝐹 =(𝑃𝐶−�̂�𝐶)∗�̂�𝐶

2+

(𝑃𝑅−�̂�𝑅)∗𝐸[�̂�𝑅]

2

24

under crowdfunding does not offset the loss by the consumers and we always have a welfare

loss under crowdfunding when both strategies are feasible.

5.2 Constrained Entrepreneur (0 < 𝐾 ≤ 𝐾𝐶𝐹)

We now proceed with the welfare analysis assuming that the entrepreneur requires a

specific amount of capital K in order to start production. For low levels of capital requirement

𝐾 < 𝐾𝐶𝐹, for which the entrepreneur is unconstrained when using crowdfunding, there will

be no divergence from the optimal pricing strategy. When using spot selling, the required

capital is raised in the form of a loan. Neither the interest, nor the capital requirements affect

the entrepreneur’s optimal prices. Thus, here we have that total production and the consumer

surplus remain the same as in the previous scenario such that we have a consumer surplus

loss when the entrepreneur opts for crowdfunding. Whereas, for higher levels of capital

(𝐾𝐶𝐹 < 𝐾 ≤ 𝐾𝐶𝐹) the entrepreneur diverges away from the optimal prices in crowdfunding

and is unable to discriminate and extract as much consumer surplus as before. There still

remains a consumer surplus loss (𝐶𝑆𝑆 > 𝐶𝑆𝐶𝐹) when compared to spot selling, but this

consumer surplus loss decreases as the capital requirement increases. Different projects have

different attributes and from our analysis we observe that the consumer surplus loss is higher

for projects with higher probabilities of delivering the positive shock or projects that promise

higher positive shocks. Regarding the producer surplus, the entrepreneur now needs to invest

in order to undergo production, thus the producer surplus is given by the profits net of the

capital investment under crowdfunding, and by the profits net of the loan payback and interest

under spot selling.

In Proposition 2 we highlighted different optimal launching strategy depending on

the project’s capital requirement and the interest rate faced by the entrepreneur. Below we

will proceed by evaluating the welfare outcomes of the entrepreneur’s optimal choice when

constrained.

a) For 𝟎 < 𝑲 ≤𝟏

𝟒 and 𝒓 < 𝒓𝑶; Standard Debt Optimal

The entrepreneur has both sources of financing available, but finds it optimal to

choose traditional debt rather than crowdfunding. Since the entrepreneur opts for standard

debt to launch his venture it follows that there is a producer surplus gain associated with

standard debt. Regardless of whether the entrepreneur is constrained or not in crowdfunding,

spot selling leads to a consumer surplus gain. Since the entrepreneur’s choice results in both

a consumer and producer surplus gain it is welfare enhancing.

b) For 𝟎 < 𝑲 ≤𝟏

𝟒 and 𝒓𝑶 ≤ 𝒓 ≤ 𝒓𝑭, Crowdfunding Optimal

Here we have that the entrepreneur could either be constrained or not when

crowdfunding. Thus, we divide the analysis here into the two subsections below:

25

i. Unconstrained Crowdfunding vs Constrained Spot Selling (0 < 𝐾 ≤ 𝐾𝐶𝐹)

The entrepreneur always finds it optimal to launch his venture through crowdfunding

for all levels of capital. Thus, there is a producer surplus gain. The welfare under both

strategies is presented below:

𝑊𝑆𝐶 = 𝐶𝑆𝑆

𝐶 + 𝑃𝑆𝑆𝐶 → 𝑊𝑆

𝐶 = (𝜆(1 + 𝛼𝐻)2

8) + (

𝜆(1 + 𝛼𝐻)2

4− 𝑅 ∙ 𝐾)

𝑊𝐶𝐹 = 𝐶𝑆𝐶𝐹 + 𝑃𝑆𝐶𝐹 → 𝑊𝐶𝐹 =𝛼𝐻

8+ (

1

4+

𝜆𝛼𝐻2

4(1 − 𝜆)− 𝐾)

For there to be a welfare gain under crowdfunding (𝑊𝐶𝐹 > 𝑊𝑆𝐶), the producer

surplus gain from crowdfunding should be larger than the consumer surplus loss. The results

depend on the level of capital and the interest rate that the entrepreneur faces in the financial

markets when spot selling. We denote the interest rate threshold above which there is a

welfare gain under unconstrained crowdfunding when compared to constrained spot selling

by 𝑟𝑊:

𝑟𝑊 =3𝜆(1 + 𝛼𝐻)2

8𝐾−

2 + 𝛼𝐻

8𝐾−

𝜆𝛼𝐻2

4(1 − 𝜆)𝐾

Through algebraic manipulations we obtain that 𝑟𝑊 < 𝑟𝐹 , such that there is a welfare

gain and constrained spot selling is feasible. Summing up our findings we have that for low

enough levels of capital (0 < 𝐾 < 𝐾𝐶𝐹) there exists an interest rate, 𝑟𝑊 < 𝑟 ≤ 𝑟𝐹, such that

we have welfare gain under crowdfunding.

ii. Constrained Crowdfunding vs Constrained Spot Selling (𝐾𝐶𝐹 < 𝐾 ≤ 𝐾𝐶𝐹)

The producer surplus now is the amount that the entrepreneur raises in the retail

market since all the funds raised during the pre-ordering are used to meet the venture’s capital

requirement. Under spot selling the producer surplus is profits net of the loan payback.

𝑊𝑆𝐶 = 𝐶𝑆𝑆

𝐶 + 𝑃𝑆𝑆𝐶 ; 𝑊𝐶𝐹

𝐶 = 𝐶𝑆𝐶𝐹𝐶 + 𝑃𝑆𝐶𝐹

𝐶

For there to be a welfare gain under crowdfunding (𝑊𝐶𝐹𝐶 > 𝑊𝑆

𝐶), the producer

surplus gain from crowdfunding should be larger than the consumer surplus loss. The results

depend on the level of capital and the interest rate that the entrepreneur faces in the financial

markets when spot selling. We denote the interest rate threshold above which there is a

welfare gain under crowdfunding when compared to spot selling by 𝑟𝑊𝐶 :

𝑟𝑊𝐶 =

(𝐶𝑆𝑆𝐶 − 𝐶𝑆𝐶𝐹

𝐶) + ( 𝜆(1 + 𝛼𝐻)2

4− 𝑃𝐶

𝐶 ∙ 𝑄𝐶𝐶 − 𝑃𝑅

𝐶 ∙ 𝐸[𝑄𝑅𝐶 ])

𝐾

26

Through an exhaustive numerical simulation using a wide array of admissible

parameter values we verify that there exists an interest rate threshold such that 𝑟𝑊𝐶 < 𝑟𝐹 for

capital levels below 𝐾𝑊, where 𝐾𝐶𝐹 < 𝐾𝑊 < 𝐾𝐶𝐹.

Summing up our findings for the situation where crowdfunding is the entrepreneur’s

optimal strategy we have that for levels of capital 0 < 𝐾 < 𝐾𝑊 there exists an interest rate,

𝑟𝑊𝐶 < 𝑟 ≤ 𝑟𝐹, such that we have welfare gain under crowdfunding.

c) For 𝟎 < 𝑲 ≤𝟏

𝟒 and 𝒓 > 𝒓𝑭, Crowdfunding Only

The outside option of not using crowdfunding as the launching strategy is not

launching the venture at all. Thus, in such a case crowdfunding is bringing to life projects

which would not have been executed otherwise. We will always have a consumer and a

producer surplus gain which translates into a welfare enhancement.

d) For 𝟏

𝟒< 𝑲 ≤ �̿�𝑺 and 𝒓 ≤ 𝒓𝑭, Standard Debt Only

The outside option of not using standard debt as the launching strategy is not

launching the venture at all. Thus, in such a case standard debt is bringing to life projects

which would not have been executed otherwise. We will always have a consumer and a

producer

The welfare outcomes of the entrepreneur’s launching choice discussed earlier are

summarized in Proposition 3 and illustrated in Figure 8.

Proposition 3.

a) For a financially unconstrained there is a welfare loss as a result of crowdfunding.

b) For a financially constrained entrepreneur with both financing options available we

observe the following:

i. Crowdfunding Optimal and 0 < 𝐾 < 𝐾𝑊 – there exists a unique interest rate

threshold, 𝑟𝑊, above which there is a welfare gain.

ii. Crowdfunding Optimal and 𝐾𝑊 ≤ 𝐾 < 𝐾𝐶𝐹 – crowdfunding leads to a welfare

loss.

iii. Standard Debt Optimal – the entrepreneur’s choice is welfare enhancing.

c) For a financially constrained entrepreneur with only one financing option available, the

entrepreneur’s choice is always welfare enhancing.

Proof. See Appendix.

27

Figure 8: Welfare outcomes given the entrepreneur’s optimal financing strategy.

Kumar et al. (2016) note that for an unconstrained entrepreneur there is always a

welfare gain under crowdfunding. Moreover, they argue that when the entrepreneur is

constrained and needs capital to launch his venture policy makers should not ease access to

traditional sources of finance. Policy makers should not lower interest rates to spur

entrepreneurial activity. Their reasoning is due to their prediction that crowdfunding

increases production and welfare. When decomposing their results we observe that in their

model there is both a consumer and a producer surplus gain under crowdfunding. The reason

for the consumer surplus gain with crowdfunding, even though there is a price discrimination

strategy used by the entrepreneur, is due to the pivotal role of the crowdfunder where there

outside option of not pre-ordering the product is not having the product.

In the model that we propose, we observe that there is a welfare loss under

crowdfunding for the unconstrained entrepreneur. For projects with a relatively low capital

requirement (0 < 𝐾 < 𝐾𝑊), there exists an interest rate above which the project is feasible

under both strategies and crowdfunding results in a welfare gain. The policy

recommendations for these types of projects are consistent with Kumar et al. (2016). The

policy maker should raise the interest rate in the debt market for projects with low capital

requirement such that crowdfunding can be welfare enhancing. Whereas, the novelty in our

model and the interesting recommendation is that policy makers should promote

entrepreneurial activity for projects with higher capital requirement (𝐾𝑊 ≤ 𝐾 < 𝐾𝐶𝐹 )

through lowering interest rates such that 𝑟 < 𝑟𝑂. By doing so entrepreneurs would find

standard debt their optimal financing choice and this would result in a welfare enhancing

outcome. For crowdfunding to be welfare enhancing it should be the case that the capital

requirement to launch the venture is low or that entrepreneurs only find their projects feasible

under crowdfunding.

28

6. CONCLUSION

In this paper we develop a theoretical framework that explains the crowdfunding

mechanism for projects in the product design category since previous theoretical models fail

to account for them. Projects in the product design category represent a substantial portion

of the reward-based crowdfunding market (more than 50% of funds raised on Kickstarter).

In the framework that we propose we have that consumers with high expected valuation pre-

order the product while consumers with low expected valuation wait to observe the shock

and make their purchase decision in the retail period. In contrast to previous literature, and

supported by anecdotal evidence, we show that a price discount for backers along with price

commitment by the entrepreneur is his optimal pricing strategy, thus, highlighting a

managerial recommendation for a pricing strategy. We also compare crowdfunding to spot

selling and show that it is optimal for entrepreneurs to choose crowdfunding when the venture

is feasible under both strategies.

From a social planner’s perspective we observe that when projects are feasible under

crowdfunding and standard debt and the capital requirement is low crowdfunding could be

welfare enhancing if the interest faced in the credit market is high. Thus, we propose that

policy makers should not lower interest rates to spur entrepreneurial activity. On the contrary,

policy makers should promote the use of crowdfunding for the launch of these ventures

through increasing the interest rate. Whereas, for projects with a higher capital requirement

we find that the use of standard debt is the welfare enhancing choice. Given the implications

of our model we propose that policy makers should lower the interest rates for ventures with

a large capital requirement. For projects that are only feasible under crowdfunding,

crowdfunding acts as the entrepreneur’s last (and only) resort. For this case we conclude that

crowdfunding is welfare enhancing since without the existence of crowdfunding these

ventures would not have been launched.

In the development of the model we assumed that all projects deliver no shock to the

industry standards ex-ante and at t = 0 the expected value of the shock is zero. What if

entrepreneurs have patents or quality certifications that demonstrate that they are more

probable to deliver a shock to the industry standards and that the expected value of the shock

is no longer zero? This could be an interesting avenue for future research. Moreover, in the

base model that we provided we assume that there are two selling dates when using

crowdfunding: t=0 for pre-orders and t=1 for retail orders. For retail orders the consumers

know their ex post valuation as they observe the shock. For future research, the baseline

model could be extended by assuming that each consumer obtains an informative signal

regarding his ex post valuation at some intermediary date. A mechanism design approach

could be used to analyze such an extension to the base model. Finally, relaxing the

assumption of perfect information could be an interesting extension since consumers’ private

information is an important ingredient of crowdfunding. Crowdfunding does play a role in

29

aggregating this private information and incorporating this in the model can provide us with

further implications. This is left for future research.

30

7. APPENDIX

Proof. [Proposition 1]

- We start by deriving the optimal pricing strategy under spot selling. Then we will derive

the optimal strategy under crowdfunding. Finally, we will compare both scenarios to

complete the proof.

- Under spot selling there are two strategies available to the entrepreneur.

Strategy A

The entrepreneur is trying to attract demand in case either shock is realized.

max𝑃𝑆

𝜋(𝑃𝑆) = 𝜆𝑃𝑆(1 − 𝜃𝐿) + (1 − 𝜆)𝑃𝑆(1 − 𝜃𝐻)

s.t. 𝜃𝐻 ≥ 𝑃𝑆 − 𝛼𝐿 ; 𝜃𝐿 ≥ 𝑃𝑆 − 𝛼𝐻 ; 𝜃𝐻 ∈ (0,1) ; 𝜃𝐿 ∈ (0,1)

Since the objective function is increasing in the price �̂�𝑆, the first two constraints will be

binding at the optimum. Replacing the expressions for 𝜃𝐻 and 𝜃𝐿 from these constraints into

the objective function we obtain 𝜆𝑃𝑆(1 − 𝑃𝑆 + 𝛼𝐻) + (1 − 𝜆)𝑃𝑆(1 − 𝑃𝑆 + 𝛼𝐿). Taking the

first order condition we get �̂�𝑆 =1

2. Since αL ∈ (-

1

2 ,0) and αH ∈ (0,

1

2), we always have an

interior solution such that 0 < 𝜃𝐿 < 𝜃𝐻 < 1. The expected quantity is 𝐸[𝑄𝑆] = 𝜆(1 − 𝜃𝐿) +

(1 − 𝜆)(1 − 𝜃𝐻) =1

2. The entrepreneur’s expected profits are 𝐸[�̂�𝑆] =

1

4 .

Strategy B

The entrepreneur is only concerned with attracting demand when the consumers realize a

positive shock.

max𝑃𝑆

𝜋(𝑃𝑆) = 𝜆𝑃𝑆(1 − 𝜃𝐿)

s.t. 𝜃𝐿 ≥ 𝑃𝑆 − 𝛼𝐻 ; 𝜃𝐿 ∈ (0,1)

Similarly to the previous case, the constraint is binding at the optimum. We can thus replace

𝜃𝐿 into the objective function. From the first order condition we have that �̆�𝑆 =1+𝛼𝐻

2 and that

𝐸[�̆�𝑆] =𝜆(1+𝛼𝐻)

2 . It follows that the expected profits are 𝐸[�̆�𝑆] =

𝜆(1+𝛼𝐻)2

4.

Comparing both strategies we can note that 𝐸[�̆�𝑆] > 𝐸[�̂�𝑆] for 𝜆 >1

(1+𝛼𝐻)2, such that

Strategy B dominates Strategy A for projects with a relatively high probability of delivering

a positive shock. We have established in the main text that the comparable benchmark

31

strategy to crowdfunding projects is Strategy B. This is ensured by Assumption 2 which

restricts our analysis to domains of 𝜆 >1

(1+𝛼𝐻)2.

- The optimal crowdfunding pricing strategy is the one that solves the following

maximization problem:

max𝑃𝐶,𝑃𝑅

𝜋(𝑃𝐶 , 𝑃𝑅) = 𝑃𝐶(1 − 𝜃𝐶) + 𝜆𝑃𝑅(𝜃𝐶 − 𝜃𝑅)

s.t. 𝜃𝐶 =𝑃𝐶−𝜆(𝑃𝑅−𝛼𝐻)

1−𝜆 ; 𝜃𝑅 = 𝑃𝑅 − 𝛼𝐻

We replace the constraints, that are binding at the optimum, into our objective function.

From the first order conditions we obtain the following:

Prices:

Quantities:

Profits:

�̂�𝐶 =1

2 ; �̂�𝑅 =

1+𝛼𝐻

2

�̂�𝐶 =1

2−

𝜆𝛼𝐻

2(1−𝜆) ; 𝐸[�̂�𝑅] =

𝜆𝛼𝐻

2(1−𝜆) ; 𝐸[�̂�𝐶𝐹] =

1

2

�̂�𝐶 =1

4−

𝜆𝛼𝐻

4(1−𝜆) ; 𝐸[�̂�𝑅] =

𝜆𝛼𝐻+𝜆𝛼𝐻2

4(1−𝜆) ; 𝐸[�̂�𝐶𝐹] =

1

4+

𝜆𝛼𝐻2

4(1−𝜆)

We can now compare spot selling with crowdfunding and prove each of the two claims of

Proposition 1.

a) The unconstrained entrepreneur’s unique optimal strategy is to use crowdfunding.

Comparing crowdfunding to the benchmark spot selling Strategy A we can see that

crowdfunding dominates if 𝐸[�̂�𝐶𝐹] =1

4+

𝜆𝛼𝐻2

4(1−𝜆)> 𝐸[�̆�𝑆] =

𝜆(1+𝛼𝐻)2

4, which holds for all

admissible values of 𝜆.

b) �̂�𝐶 & �̂�𝑅 set by the unconstrained entrepreneur are indeed such that �̂�𝐶 < �̂�𝑅 which

supports our claim that the optimal pricing strategy is such that crowdfunders receive a

price discount when compared to the retail consumers.

This follows directly as �̂�𝐶 =1

2 < �̂�𝑅 =

1+𝛼𝐻

2, since 𝛼𝐻 > 0.

c) Total production under crowdfunding dominates the one under spot selling.

𝐸[�̂�𝐶𝐹] =1

2> 𝐸[�̆�𝑆] =

𝜆(1+𝛼𝐻)

2, which holds for all admissible values of 𝜆.

32

Table A: Comparative Statics Unconstrained Crowdfunding

𝝏(. )

𝝏𝜶𝑯⁄ 𝝏(. )

𝝏𝝀⁄

Crowdfunding Price (�̂�𝑪 ) 0 0

Retail Price (�̂�𝑹 ) + 0

Crowdfunding Demand (�̂�𝑪 ) - -

Expected Retail Demand (𝑬[�̂�𝑹 ] ) + +

Crowdfunding Profits (�̂�𝑪 ) - -

Expected Retail Profit ( 𝑬[�̂�𝑹 ] ) + +

Proof. [Proposition 2]

- We start by determining the constrained spot selling pricing strategy.

max𝑃𝑆

𝐶𝜋(𝑃𝑆

𝐶) = 𝜆𝑃𝑆𝐶(1 − 𝜃𝐿) − 𝑅 ∗ 𝐾

s.t. 𝜃𝐿 = 𝑃𝑆𝐶 − 𝛼𝐻

We substitute the constraints into our objective function. The optimal price for the

constrained spot selling is unaffected by the capital requirement and we have that �̌�𝑆𝐶 =

1+𝛼𝐻

2

and 𝐸[�̌�𝑆𝐶] =

𝜆(1+𝛼𝐻)

2 . This is for 𝐾 ≤ 𝐾𝑆 where 𝐾𝑆 is the maximum capital that the

entrepreneur can raise when constrained under spot selling. We have that 𝐾𝑆 =𝜆(1+𝛼𝐻)2

4𝑅. For

levels of 𝐾 > 𝐾𝑆 the project is not feasible under spot selling. Rearranging we determine the

interest rate threshold, 𝑟𝐹, with 𝑟𝐹 =𝜆(1+𝛼𝐻)2

4𝐾− 1. 𝑟𝐹 is the interest rate threshold above

which the project is not feasible.

- For 𝟎 < 𝑲 <𝟏

𝟒 and 𝒓 > 𝒓𝑭, the project is only feasible under crowdfunding.

- For 𝟎 < 𝑲 <𝟏

𝟒 and 𝒓 ≤ 𝒓𝑭, the project is feasible using standard debt and crowdfunding.

When comparing standard debt with crowdfunding, the entrepreneur could be unconstrained

or constrained when crowdfunding.

For 𝐾 ≤ 𝐾𝐶𝐹 = 1

4−

𝜆𝛼𝐻

4(1−𝜆);

33

The entrepreneur is unconstrained and his optimal prices and quantities are the same as those

that he exhibits in the unconstrained section.

For 𝐾𝐶𝐹 < 𝐾 < 𝐾𝐶𝐹;

We now have that the entrepreneur is constrained under crowdfunding. Thus, the

entrepreneur would now need to maximize his objective function but constrained by the need

to raise the required capital in the pre-ordering stage.

𝑚𝑎𝑥𝑃𝐶

𝐶,𝑃𝑅𝐶

𝜋(𝑃𝐶𝐶 , 𝑃𝑅

𝐶) = 𝑃𝐶𝐶(1 − 𝜃𝐶) + 𝜆𝑃𝑅

𝐶(𝜃𝐶 − 𝜃𝑅) − 𝐾

s.t. 𝑃𝐶𝐶(1 − 𝜃𝐶) = 𝐾; 𝜃𝐶 =

𝑃𝐶𝐶−𝜆(𝑃𝐶

𝑅−𝛼𝐻)

1−𝜆 ; 𝜃𝑅 = 𝑃𝑅

𝐶 − 𝛼𝐻

We replace 𝜃𝐶 into 𝑃𝐶𝐶(1 − 𝜃𝐶) = 𝐾 . We will have that 𝑃𝐶

𝐶 (1 −𝑃𝐶

𝐶−𝜆(𝑃𝐶𝑅−𝛼𝐻)

1−𝜆) = 𝐾 . Solving

for 𝑃𝐶𝐶 we have that:

�̂�𝐶𝐶 =

1 − 𝜆(1 + 𝛼𝐻) + √(1 + 𝜆𝑃𝑅𝐶)2 + 𝜆(1 + 𝛼𝐻)[𝜆(1 + 𝛼𝐻 − 2𝑃𝑅

𝐶) − 2] − 4𝐾(1 − 𝜆)

2

Plugging back �̂�𝐶𝐶 into the objective function we solve for the optimal retail price. There

exists no analytical solution due to a polynomial of forth degree. But numerical simulations

have been presented within the text.

For 𝐾 = 𝐾𝐶𝐹;

The maximum capital (𝐾𝐶𝐹

) to be raised when crowdfunding is such that all future demand

is shifted to the pre-ordering period such that 𝑃𝐶

𝐶−𝜆(𝑃𝐶𝑅−𝛼𝐻)

1−𝜆= 𝑃𝑅

𝐶 − 𝛼𝐻 . Solving for 𝑃𝑅𝐶 we

have that the publicized future retail price that closes the retail market and switches demand

to the pre-ordering stage is 𝑃𝑅𝐶 = 𝑃𝐶

𝐶 + 𝛼𝐻. Plugging 𝑃𝑅𝐶 into the entrepreneur’s objective

function we now have that:

𝑚𝑎𝑥𝑃𝐶

𝐶,𝑃𝑅𝐶

𝜋(𝑃𝐶𝐶) = 𝑃𝐶

𝐶(1 − 𝜃𝐶) − 𝐾

s.t. 𝜃𝐶 = 𝑃𝐶𝐶

We plug the constraint into the objective function that is binding at the optimum. We yield

from the first order conditions that 𝑃𝐶𝐶 =

1

2 and 𝑃𝑅

𝐶 = 1

2+ 𝛼𝐻. The demand in the pre-

ordering stage is 𝑄𝐶𝐶 =

1

2. The maximum capital is 𝐾𝐶𝐹 =

1

4 .

34

- For 𝟏

𝟒< 𝑲 ≤ �̿�𝑺 and 𝒓 ≤ 𝒓𝑭, The project is not feasible under crowdfunding. The

entrepreneur’s only launching choice is through the use of standard debt.

Table B: Comparative Statics at 𝑲𝑪𝑭 and 𝑲𝑪𝑭

𝑲𝑪𝑭 𝑲𝑪𝑭

𝜕(. )

𝜕𝛼𝐻⁄ 𝜕(. )

𝜕𝜆⁄

𝜕(. )𝜕𝛼𝐻

⁄ 𝜕(. )𝜕𝜆

⁄

Crowdfunding Price (�̂�𝑪𝑪) 0 0 0 0

Retail Price (�̂�𝑹𝑪 ) + 0 + 0

Crowdfunding Demand (�̂�𝑪𝑪) - - 0 0

Expected Retail Demand (𝑬[�̂�𝑹𝑪] ) + + 0 0

Crowdfunding Profits (�̂�𝑪𝑪) - - 0 0

Expected Retail Profit ( 𝑬[�̂�𝑹𝑪] ) + + 0 0

Expected Profits ( 𝝅𝑪𝑭𝒄 (𝑲) ) + + 0 0

Maximum Unconstrained Capital ( 𝑲𝑪𝑭 ) - - 0 0

Maximum Constrained Capital ( 𝑲𝑪𝑭

) 0 0 0 0

Having established the optimal strategies under the constrained case we can now proceed

with proving Proposition 2.

a) For 0 < 𝐾 ≤1

4 and 𝑟 > 𝑟𝐹, only crowdfunding is feasible, thus the constrained

entrepreneur’s optimal strategy is to use crowdfunding.

b) For 0 < 𝐾 ≤1

4 and 𝑟 < 𝑟𝑂 , both strategies are feasible and the constrained

entrepreneur’s unique optimal strategy is to use standard debt.

rO is the interest rate below which the entrepreneur finds standard debt optimal. When

the entrepreneur is unconstrained when crowdfunding this interest rate is negative so we

can say it does not exist. For higher levels of capital where the entrepreneur is constrained

35

when crowdfunding there exists an interest rate threshold below which the entrepreneur

finds standard debt optimal. It is such that 𝐸[𝜋𝐶𝐹𝐶 (𝐾)] <

𝜆(1+𝛼𝐻)2

4− 𝑅 ∗ 𝐾. Through

rearranging we arrive at this interest rate threshold below.

𝑟𝑂 =(

𝜆(1 + 𝛼𝐻)2

4− 𝐸[𝜋𝐶𝐹

𝐶 (𝐾)])

𝐾− 1

c) For 0 < 𝐾 ≤1

4 and 𝑟𝑂 ≤ 𝑟 ≤ 𝑟𝐹, both strategies are feasible and the constrained

entrepreneur’s unique optimal strategy is to use crowdfunding.

For 𝑟𝑂 ≤ 𝑟 we never have that profits under spot selling dominate crowdfunding.

Therefore, it holds that 𝐸[𝜋𝐶𝐹𝐶 (𝐾)] ≥

𝜆(1+𝛼𝐻)2

4− 𝑅 ∗ 𝐾 such that the entrepreneur prefers

to use crowdfunding as the financing option for his venture.

d) For 1

4< 𝐾 ≤ �̿�𝑆 and 𝑟 ≤ 𝑟𝐹, only standard debt is feasible, thus the constrained

entrepreneur’s optimal strategy is to use standard debt.

e) For 𝐾 >1

4 and 𝑟 > 𝑟𝐹, the project is not feasible under any strategy.

Proof. [Proposition 3]

In order to start our analysis we need to compute the welfare for the spot selling and the

crowdfunding strategy.

Unconstrained Entrepreneur

Welfare under Spot Selling:

As we have mentioned here the entrepreneur is soliciting demand for both cases of

the world, when there is a positive shock and when there is a negative shock. Thus, the

consumer’s surplus is the expected value of the consumer surplus when faced with either

shock (𝐶𝑆𝑆 =(𝑃𝑆

+−�̌�𝑆)∗𝐸[�̌�𝑆]

2.

𝑃𝑆

+is the maximum that the consumer is willing to pay when faced with a positive

shock. For any price higher than this there exists no demand such that 𝑃𝑆

+is the price that

leads us to have 𝜃 = 1 . Solving for 𝑃𝑆

+ we have that 𝑃𝑆

+= 1 + 𝛼𝐻 .

36

𝐸[�̌�𝑆] is the quantity demanded when the consumer realizes a positive shock. From the

optimal spot selling pricing strategy we know that �̌�𝑆 =1

2 . 𝜃 = �̌�𝑆 − 𝛼𝐻 =

1+𝛼𝐻

2− 𝛼𝐻.

𝐸[�̌�𝑆] is equal to 1- 𝜃, thus, solving for 𝐸 [�̌�𝑆+

] we have that quantity demanded in case of

a positive shock is 𝜆(1+𝛼𝐻)

2. We arrive at 𝐶𝑆𝑆 =

𝜆(1+𝛼𝐻)2

8. The producer surplus is simply the

entrepreneur’s profits when spot selling 𝑃𝑆𝑆 =𝜆(1+𝛼𝐻)2

4 .

Welfare under Crowdfunding:

When crowdfunding the consumer can either pre-order the product or wait to realize

the shock before making his purchase decision. The maximum price that the consumer is

willing to pay when pre-ordering the product is its publicized retail price. For a higher price

the consumer is better off waiting to realize the shock then making his purchase decision.

When pre-ordering the maximum price that the consumer is willing to pay is such that 𝑃𝐶 =

�̂�𝑅. Whereas for the retail consumers the maximum that they are willing to pay is such that

𝜃𝑅 = 𝜃𝐶 . Thus, it is the price that closes the retail market. For 𝜃𝑅 = 𝜃𝐶 we have that 𝑃𝑅 −

𝛼𝐻 =�̂�𝐶−𝜆(𝑃𝑅−𝛼𝐻)

1−𝜆 solving for 𝑃𝑅 we arrive at 𝑃𝑅 = �̂�𝐶 + 𝛼𝐻 . Consumer surplus is defined

by 𝐶𝑆𝐶𝐹 =(𝑃𝐶−�̂�𝐶)∗�̂�𝐶

2+

(𝑃𝑅−�̂�𝑅)∗𝐸[�̂�𝑅]

2. We yield that 𝐶𝑆𝐶𝐹 =

𝛼𝐻

8. Producer surplus is the

entrepreneur’s profits under crowdfunding 𝑃𝑆𝐶𝐹 =1

4+

𝜆𝛼𝐻2

4(1−𝜆) .

Welfare Comparison:

Unconstrained Entrepreneur

a) For a financially unconstrained there is a welfare loss as a result of crowdfunding.

𝐶𝑆𝑆 + 𝑃𝑆𝑆 > 𝐶𝑆𝐶𝐹 + 𝑃𝑆𝐶𝐹

3𝜆(1 + 𝛼𝐻)2

8>

𝛼𝐻

8+

1

4+

𝜆𝛼𝐻2

4(1 − 𝜆)

Thus, when the entrepreneur is unconstrained by opting for crowdfunding there is a

welfare loss even though total production is higher.

Constrained Entrepreneur

b) For a financially constrained entrepreneur with both financing options available we

observe the following:

37

i. Crowdfunding Optimal and 0 < 𝐾 < 𝐾𝑊 – there exists a unique interest rate

threshold, 𝑟𝑊, above which there is a welfare gain.

ii. Crowdfunding Optimal and 𝐾𝑊 ≤ 𝐾 < 𝐾𝐶𝐹 – crowdfunding leads to a welfare

loss.

iii. Standard Debt Optimal – the entrepreneur’s choice is welfare enhancing.

For the constrained entrepreneur with both financing options available there are the three

cases mentioned above.

Crowdfunding Optimal and 𝟎 < 𝑲 < 𝑲𝑾:

1) Unconstrained Crowdfunding vs Constrained Spot Selling

𝑊𝑆𝐶 = 𝐶𝑆𝑆

𝐶 + 𝑃𝑆𝑆𝐶 → 𝑊𝑆

𝐶 = (1

8+

𝜆𝛼𝐻2

2(1 − 𝜆)) + (

1

4− 𝑅 ∙ 𝐾)

𝑊𝐶𝐹 = 𝐶𝑆𝐶𝐹 + 𝑃𝑆𝐶𝐹 → 𝑊𝐶𝐹 =𝛼𝐻

8+ (

1

4+

𝜆𝛼𝐻2

4(1 − 𝜆)− 𝐾)

For 𝑊𝐶𝐹 > 𝑊𝑆𝐶 the interest rate that the entrepreneur faces in the credit market

should be larger than 𝑟𝑊, where:

𝑟𝑊 =3𝜆(1 + 𝛼𝐻)2

8𝐾−

2 + 𝛼𝐻

8𝐾−

𝜆𝛼𝐻2

4(1 − 𝜆)𝐾

2) Constrained Crowdfunding vs Constrained Spot Selling

The results can not be expressed analytically due to the polynomial expression for the

optimal prices but through a numerical simulation we were able to arrive at the results

mentioned below.

We are looking for the interest rate below which crowdfunding results in a welfare

gain. This interest rate is denoted by 𝑟𝑊. As expressed earlier in the text 𝑅 = 1 + 𝑟 .

𝐶𝑆𝐶𝐹𝐶 + 𝑃𝑆𝐶𝐹

𝐶 = 𝐶𝑆𝑆𝐶 + 𝑃𝑆𝑆

𝐶

𝐶𝑆𝐶𝐹 + 𝑃𝐶𝐶 ∙ 𝑄𝐶

𝐶 + 𝑃𝑅𝐶 ∙ 𝐸[𝑄𝑅

𝐶] − 𝐾 = 𝐶𝑆𝑆 +1

4− 𝑅 ∙ 𝐾

Solving for 𝑟𝑊 we arrive at 𝑟𝑊𝐶 =

(𝐶𝑆𝑆𝐶−𝐶𝑆𝐶𝐹

𝐶)+( 𝜆(1+𝛼𝐻)2

4−𝑃𝐶

𝐶∙𝑄𝐶𝐶−𝑃𝑅

𝐶∙𝐸[𝑄𝑅𝐶])

𝐾. Through an

exhaustive numerical simulation using a wide array of admissible parameter values we obtain

38

that 𝑟𝑊 < 𝑟𝐹 for 𝐾 < 𝐾𝑊 where 𝐾𝐶𝐹 < 𝐾𝑊 <𝐾𝐶𝐹. Thus, there exists an interest rate above

which crowdfunding is welfare enhancing.

Crowdfunding Optimal and 𝑲𝑾 ≤ 𝑲 < 𝑲𝑪𝑭:

For 𝐾 > 𝐾𝑊 and when both financing options are feasible there exists no interest rate

that makes crowdfunding welfare enhancing such that crowdfunding leads to a welfare loss.

Standard Debt Optimal:

We have established earlier that crowdfunding leads to a consumer surplus loss in the

unconstrained and the constrained case. Since the entrepreneur finds standard debt optimal

so it follows that there is a producer surplus gain when using spot selling as the launching

strategy. Since there is both a producer and consumer surplus gain associated with the use of

standard debt it follows that the entrepreneur’s choice is welfare enhancing.

c) For a financially constrained entrepreneur with only one financing option available, the

entrepreneur’s choice is always welfare enhancing.

Either financing options brings to life projects which would not have launched otherwise and

the entrepreneur’s outside option of not opting for the available source of funds is not

launching the venture at all. Thus, by opting for the available financing option the launch of

the venture is welfare enhancing.

39

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