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Document de travail (Working Paper)

"Crowdfunding: tapping the right crowd"

Belleflamme, Paul ; Lambert, Thomas ; Schwienbacher, Armin

Abstract

The basic idea of crowdfunding is to raise external finance from a large audience (the “crowd”), where each individual provides a very small amount, instead of soliciting a small group of sophisticated investors. The paper develops a model that associates crowdfunding with pre-ordering and price discrimination, and studies the conditions under which crowdfunding is preferred to traditional forms of external funding. Compared to traditional funding, crowdfunding has the advantage of offering an enhanced experience to some consumers and, thereby, of allowing the entrepreneur to practice menu pricing and extract a larger share of the consumer surplus; the disadvantage is that the entrepreneur is constrained in his/her choice of prices by the amount of capital that he/she needs to raise: the larger this amount, the more prices have to be twisted so as to attract a large number of “crowdfunders” who pre-order, and the less profitable the menu pricing scheme.

Référence bibliographique

Belleflamme, Paul ; Lambert, Thomas ; Schwienbacher, Armin. Crowdfunding: tapping the right crowd. CORE Discussion Paper; 2011/32. (2011).

2011/32

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Crowdfunding: tapping the right crowd

Paul Belleflamme, Thomas Lambert and Armin Schwienbacher

Center for Operations Research and Econometrics

Voie du Roman Pays, 34

B-1348 Louvain-la-Neuve Belgium

http://www.uclouvain.be/core

D I S C U S S I O N P A P E R

CORE DISCUSSION PAPER 2011/32

Crowdfunding: tapping the right crowd

Paul BELLEFLAMME 1, Thomas LAMBERT2

and Armin SCHWIENBACHER 3

June 2011

Abstract The basic idea of crowdfunding is to raise external finance from a large audience (the “crowd”), where each individual provides a very small amount, instead of soliciting a small group of sophisticated investors. The paper develops a model that associates crowdfunding with pre-ordering and price discrimination, and studies the conditions under which crowdfunding is preferred to traditional forms of external funding. Compared to traditional funding, crowdfunding has the advantage of offering an enhanced experience to some consumers and, thereby, of allowing the entrepreneur to practice menu pricing and extract a larger share of the consumer surplus; the disadvantage is that the entrepreneur is constrained in his/her choice of prices by the amount of capital that he/she needs to raise: the larger this amount, the more prices have to be twisted so as to attract a large number of “crowdfunders” who pre-order, and the less profitable the menu pricing scheme. Keywords: crowdfunding, pre-ordering, menu pricing.

JEL Classification: G32, L11, L13, L15, L21, L31

1 Université catholique de Louvain, CORE and Louvain School of Management, B-1348 Louvain-la-Neuve, Belgium. E-mail: [email protected]. Other affiliation: CESifo. This author is also member of ECORE, the association between CORE and ECARES. 2 Université catholique de Louvain, Louvain School of Management, B-1348 Louvain-la-Neuve, Belgium. E-mail: [email protected]. 3 Université Lille Nord de France – SKEMA Business School (Université Lille 2, Faculté de Finance, Banque et Comptabilité, F-59020 Lille Cedex, France. E-mail: [email protected]. Other affiliation as research fellow: University of Amsterdam Business School. The authors are grateful to seminar and conference participants in Antwerp, Bologna, Copenhagen, Groningen, Lille, Louvain- la-Neuve, Montpellier, Paris, Prague, and Porto for their helpful comments on a previous version.

This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

1 Introduction

It is well recognized that new ventures face difficulties in attracting external finance at their very initial stage, be it through bank loans or equity capital (see, e.g., Berger and Udell, 1995, Cassar, 2004, and Cosh et al., 2009). While business angels and venture capital funds fill gaps for larger amounts, the smallest amounts are provided by entrepreneurs themselves and friends & family. Still, many ventures remain unfunded, partially because of a lack of sufficient value that can be pledged to investors, partially because of unsuccessful attempts to convince investors (Hellmann, 2007; Casamatta and Haritchabalet, 2010). To circumvent these problems, creative founders have recently made use of a new source of finance–so-called crowdfunding–by tapping the “crowd” instead of specialized investors.

The concept of crowdfunding finds its root in the broader concept of crowdsourcing, which refers to using the crowd to obtain ideas, feedback and solutions in order to develop corporate activities (Howe, 2008; Klee- mann et al., 2008). In the case of crowdfunding, the objective is to collect money for investment; this is generally done by using social networks, in particular through the Internet (Twitter, Facebook, LinkedIn and different other specialized blogs). In other words, instead of raising the money from a very small group of sophisticated investors, the idea of crowdfunding is to obtain it from a large audience (the “crowd”), where each individual will pro- vide a very small amount. This can take the form of equity purchase, loan, donation or pre-ordering of the product to be produced. The amounts that have been targeted through crowdfunding have continuously increased, with Trampoline Systems targeting more than £1 million for the financing of the commercialization stage of their new software. Recently, TikTok+LunaTik raised $941,718 from 13,512 individuals in the form of product pre-ordering of its multi-touch watch kit.

In this paper, we develop a model that associates crowdfunding with pre- ordering and price discrimination, and we study the conditions under which crowdfunding is preferred to traditional forms of external funding (bank loan or equity investor). In this framework, the funding is needed to finance up-front fixed costs of production. Since the remaining consumers will pay a different price, crowdfunding that takes the form of pre-ordering gives the opportunity to price discriminate between the first group (those who pre- order and thus constitute the investing crowd) and the second group (the

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other consumers who wait that production takes place before purchasing directly). However, an entrepreneur is generally unable to identify these consumers. The entrepreneur must then use some self-selecting device so as to induce high-paying consumers to reveal themselves. In this sense, crowdfunding appears as a form of menu pricing.

The trade-off we explore in the model is thus the following: compared to traditional funding, crowdfunding has the advantage of offering an enhanced experience to some consumers and, thereby, of allowing the entrepreneur to practice second-degree price discrimination and extract a larger share of the consumer surplus; the disadvantage is that the entrepreneur is constrained in his/her choice of prices by the amount of capital that he/she needs to raise initially to fund production: the larger this amount, the more prices have to be twisted so as to attract a large number of “crowdfunders”, and the less profitable the menu pricing scheme.

The model highlights the importance of community-based experience for crowdfunding to be a viable alternative to traditional funding. If the entrepreneur is not able to create such benefits, no consumer will find it worthwhile to pre-order the good, unless a discount is offered. However, crowdfunding then becomes less profitable compared to traditional funding. Also, the analysis shows that crowdfunding is the most profitable option only for lower levels of finance. Indeed, crowdfunding yields higher profits only for small amounts where the entrepreneur faces no (or limited) constraints in his/her price setting between the two types of consumers while still securing enough up-front financing. As the amount required becomes larger, the entrepreneur is forced to distort more the prices so that more consumers are willing to pre-order and thus the entrepreneur can collect up-front more money. This in turn reduces the gains from price discrimination. Our results are robust to the possibility that the entrepreneur may take the money collected from the crowdfunding initiative and run away with it.

Another important result is that crowdfunders eventually pay more than other, regular consumers. This outcome is consistent with findings from different case studies where community benefits are important. Indeed, our model shows that through price-discrimination, individuals with the highest willingness-to-pay become crowdfunders. As they are willing to pay the most and additionally enjoy non-monetary community benefits, it turns out that they end up paying more than consumers waiting for the product to

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reach the market. The remaining of this paper is structured as follows. The next section

offers a definition of crowdfunding, discusses crowdfunding practices and provides a survey of related literature. Section 3 presents the theoretical model and discusses its results and implications. Finally, Section 4 concludes with suggested topics for future research.1

2 Crowdfunding as a distinct form of entrepreneurial

finance

Our objective in this section is twofold. First, we aim at providing a general definition of crowdfunding. Second, we discuss how crowdfunding differs from other sources of entrepreneurial finance. To this end, we present se- lected crowdfunding initiatives and then provide a review of the related literature.

2.1 A definition

As mentioned, the concept of crowdfunding can be seen as part of the broader concept of crowdsourcing, which refers to using the “crowd” to obtain ideas, feedback and solutions in order to develop corporate activi- ties. The term “crowdsourcing” has been first used by Jeff Howe and Mark Robinson in the June 2006 issue of Wired Magazine, an American magazine for high technology (Howe, 2008). Kleemann et al. (2008) point out that “crowdsourcing takes place when a profit oriented firm outsources specific tasks essential for the making or sale of its product to the general public (the crowd) in the form of an open call over the internet, with the intention of animating individuals to make a [voluntary] contribution to the firm’s production process for free or for significantly less than that contribution is worth to the firm.” Although this definition of crowdsourcing is a useful starting point, several caveats and clarifications need to be made in order to transpose it to crowdfunding.

Raising funds by tapping a general public (the crowd) is the most impor- tant element of crowdfunding. This means that consumers can volunteer to

1Section 5 is an appendix containing the details of the mathematical developments and

the proofs of the propositions.

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provide input to the development of the product, in this case in form of fi- nancial help. How the interaction with the crowd takes place may, however, differ from crowdsourcing. For instance, platforms have recently emerged to facilitate the interaction between entrepreneurial initiatives and poten- tial crowdfunders. Entrepreneurs can post their project on the platform and benefit from the platform’s visibility to reach potential investors. A platform of this sort, Kickstarter, has successfully intermediated more than 7,000 projects already.2

Yet, while the use of the Internet to make an “open call” may be very efficient for crowdsourcing in general, it can become more problematic for crowdfunding, especially if it involves the offering of equity to the crowd. Indeed, making a general solicitation for equity offering is limited to publicly listed equity. In many countries, there is also a limit as to how many private investors a company can have (see Larralde and Schwienbacher, 2010, for an extended discussion). This creates important legal limitations to crowd- funding initiatives, given that the input of the crowd is capital and not an idea or time. Therefore, most initiatives do not offer shares but provide other types of rewards such as a product or membership.

Besides, while the Web 2.0 has been a critical ingredient in the devel- opment of crowdfunding practices, it also differs from open-source practices (Brabham, 2008; Fershtman and Gandal, 2011). An important distinction is that in the case of open-source, the resource belongs to the community, which can then exploit it on an individual basis (there is no restriction on who can use it); in the case of crowdfunding (and also crowdsourcing), it ultimately belongs to the firm, which will be the only one to use it. This distinction with open-source practices becomes even more obvious when re- lated to crowdfunding, since capital cannot be shared. Unlike an idea or a software code, capital is not a public good in the economic sense that assumes non-rivalness and non-excludability.

Based on this discussion and in the spirit of Kleemann et al. (2008), we offer the following, refined definition:

Definition 1 Crowdfunding involves an open call, mostly through the In- ternet, for the provision of financial resources either in form of donation or

2These platforms share some similarities with online lending markets (Everett, 2008;

Freedman and Jin, 2010); while the latter more prominently target social entrepreneurship,

crowdfunding platforms have a broader scope of entrepreneurial initiatives.

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in exchange for the future product or some form of reward and/or voting rights.

As mentioned above, the promised reward can be monetary or non- monetary (such as recognition). This definition encompasses many forms of crowdfunding practices and has been discussed as such. However, in this pa- per, we focus on crowdfunding initiatives that take the form of pre-ordering of products. The following sections are restricted to the latter form.

2.2 Examples

Different reasons may explain recent successes of entrepreneurs who have relied on crowdfunding. Also, there exist many ways to “crowdfund” a project. However, crowdfunding initiatives share some common character- istics, which we stress below in the light of selected cases.

In 2005, the South African singer Verity Price launched the “Lucky Packet Project”. To record her album without assistance of a record label, Verity Price needed to advance an up-front investment of ZAR400,000.3 To this end, she set up a website where she asked people to pre-purchase her album at ZAR200 before she recorded it. In return from their contributions, people were compensated with some form of non-monetary rewards, such as their name credited on her website, the possibility to vote on which songs are recorded, and what artwork and photography are used. Also, 10% of sales would be transferred to charities. Verity Price managed to reach the threshold of ZAR400,000 with the contributions of 2061 individuals. Then, she used the money to record her album. Now, the album has been put on the market and is sold to everyone at ZAR100.

In the same vein, the LINCH three Project aims at making a documen- tary film about the artist David Lynch. The filmmakers ask to David Lynch’s fans to donate $50 each to fund the film project. The fans’ community are rewarded by having online access to exclusive content on the filmmaking process and by receiving limited edition of footage created by Lynch himself either into print, T-shirt, or bag. Once the money is raised, the documentary film will be produced and distributed via the regular distribution channels.4

3ZAR (South African Rands) 400,000 is approximately e35,000. 4Buyacredit.com’s initiative or the film “The Age of Stupid ” are other examples of

independent films for which financing is crowdfunded.

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As exemplified by these two cases, crowdfunding seems more successful in the entertainment industry. However, entrepreneurial ventures in other industries have been financed in the same way and share similar characteris- tics. Initiatives have been undertaken in other industries such as journalism (Spot.Us ), beer (BeerBankroll ), software (Blender Foundation, Trampoline Systems ) and tourism (MediaNoMad ). For instance, by investing in the launch of the Lebanese restaurant mybar, people may enjoy financial and social benefits, such as voting rights, special privileges at mybar, and annual dividends. Also, members of MyFootballClub (who own the football club Ebbsfleet United in United Kingdom) are completely involved in the man- agement of the club through their voting right. The contribution of fans (a membership fee of £35) allowed them to complete the takeover of the club and form a community with real decision power.

To sum up, these cases highlight three recurrent characteristics. First, crowdfunding initiatives often rely on advance purchase of a product, which is not yet on the market in its finished form. At the pre-ordering stage, the entrepreneur offers just a description and promise on what the final prod- uct will be, and also commits that the product will indeed be put on the market. Second, in most of the cases, consumers who pre-order the product pay more than the regular consumers, who wait that production takes place before purchasing directly. In the Verity Price’s experience, the regular con- sumers pay ZAR100 whereas the pre-ordering consumers pay ZAR200. The crowdfunders are therefore willing to pay more for the product. Third, the crowd must identify themselves as such. Crowdfunders must feel that they are being part of a community of privileged consumers. This community can take many forms, starting with receiving rewards up to direct involvement in the project. In the case of LINCH three Project, they gain access to ex- clusive footage of David Lynch. Hence, consumers may self-select into this community and entrepreneurs ensure that consumers enjoy such community benefits and trust in the project.

2.3 Related literature

As crowdfunding is a relatively new phenomenon, it is no surprise that the literature specifically devoted to crowdfunding is only nascent. It is however worthwhile making parallels with other sources of entrepreneurial finance. This allows us to better understand the specificities of crowdfunding as a

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distinct form of finance. First, looking at crowdfunding from a pure financial perspective, connec-

tions can be made with bootstrap finance. This form of financing consists of using internal financing ways rather than traditional sources of external finance (e.g., bank loan, angel capital and venture capital). Several studies provide evidence of the different forms of internal sources used by boot- strapping entrepreneurs (see Bhidé, 1992, Winborg and Landstrom, 2001, and Ebben and Johnson, 2006, just to cite a few). Bhidé (1992) shows that even among the Inc. 500 companies in the US, most of them started by bootstrapping the company. Further financing methods for startups com- panies are analyzed, for instance, by Cosh et al. (2009), who examine a broader range of financing alternatives. None of these studies however con- sider the “crowd” as possible alternative (regardless of whether it constitutes potential consumers or simply profit-driven individuals).

A few studies have recently focused on crowdfunding more specifically. One is by Agrawal, Catalini and Goldfarb (2011) that examines the geo- graphic origin of consumers who invest on the SellaBand platform.5 The authors observe that “the average distance between artist-entrepreneurs and investors is about 3,000 miles, suggesting a reduced role for spatial proxim- ity.” However, they establish that distance still plays a role insofar as “local investors are more likely than distant ones to invest in the very early stages of a single round of financing and appear less responsive to decisions by other investors.”

The idea that investors may be responsive to other investors’ decisions is also present in Ward and Ramachandran (2010). The goal of this paper is to estimate the extent to which demand for crowdfunding projects is driven by peer effects. Like in our model, it is assumed that consumption cannot hap- pen until projects successfully complete their funding. What differs is the

5SellaBand is an online platform based in Amsterdam that enables musicians to raise

money to produce their album. SellaBand ’s business model is as follows. Artists can post

a number of songs (demos) on the platform; visitors to the site can then listen to the

music for free and choose the artists they want to invest in; artists seek to raise $50,000 by

selling “Parts” at $10 each; during the fundraising stage, money is held in an escrow until

the threshold of $50,000 is reached. The $50,000 will be used to fund the artist’s recording

project; finally investors (the “Believers”) are compensated by receiving 10% of revenue

from the album. SellaBand has been one of the first website of this kind; followers are,

e;g., MyMajorCompany in France, Akamusic in Belgium, and ArtistShare in the United

States.

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link that the authors make between crowdfunding and information. While we assume that crowdfunding allows the firm to gain information about its consumers, they posit that crowdfunding allows consumers to refine their information about the quality of an experience good. In their model, crowd- funders may update their prior based on information from their investor social network. Analyzing also data from SellaBand, they find that crowd- funders are influenced by the success or failure of related projects and use the actions of other crowdfunders as a source of information in their funding decisions.

Finally, when crowdfunding is associated with pre-ordering and price discrimination, some strand of literature in the realm of industrial orga- nization provides useful insight. Nocke et al. (2011) have recently linked product pre-ordering to price discrimination, however in a context of infor- mation asymmetry.6 There, the true quality of the product is revealed later so that the firm faces consumers with different expected valuations for its forthcoming product. This induces consumers with highest expected valua- tion to pre-order before the quality is known. Advance-purchase then leads to price discounts, in contrast to our setting that abstracts from information asymmetry. Also, their setting does not induce any network effect like in our setting.

Such network effects and similar coordination issues among consumers are present in another strand of literature that can be related to our work, namely the literature on threshold (or discrete) public goods. These goods can be provided only after a sufficient amount of contributions is reached so as to cover their cost of provision (think, e.g., of a lighthouse or a bridge). The parallel with crowdfunding is clear as the entrepreneur also needs to collect private contributions before producing her good (which is private rather than public in the present case). When contributing, an individual exerts a positive externality on other individuals by raising the likelihood that the good (public or private) will be produced. Free-riding may then be observed. In our setting, as will be explained below, we treat this problem by requiring that consumers’ expectations (about the contribution behavior of other consumers) be fulfilled at equilibrium and by assuming that the entrepreneur is able to somehow coordinate consumers’ decisions so as to

6Other studies have shown that advance-purchase discounts may arise in environments

where production capacity is limited or the aggregate level of demand is uncertain (Gale

and Holmes, 1992, 1993; Dana, 1998, 1999, 2001).

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avoid free-riding. This simplification allows us to focus on other important issues of crowdfunding. In contrast, the literature on threshold public goods explicitly addresses the possibility of free-riding and examines the design of optimal mechanisms for the provision of those goods when, for instance, each individual has private information about the cost or benefit associated with his or her participation in the provision of the good (Gradstein, 1994), or when permitting continuous rather than binary “all-or-nothing” contribu- tions (Cadby and Maynes, 1999), or under different rules of allocating excess contributions if the total amount collected exceeds the required threshold (Spencer et al., 2009).

3 Crowdfunding, pre-ordering and menu pricing

In this section, we focus on crowdfunding experiences where consumers are invited to pre-order the product. For the entrepreneur to be able to launch production, the amount collected through pre-ordering must cover the fixed cost of production. Since the remaining consumers will pay a different price, crowdfunding that takes the form of pre-ordering gives the opportunity to price discriminate between the first group (those who pre-order and thus constitute the investing “crowd”) and the second group (the other consumers who wait that production takes place before purchasing directly).

Since the consumers who pre-order are those with a high willingness to pay for the product, these will generally constitute the bulk of the “crowd”. However, an entrepreneur is generally unable to identify these consumers. The entrepreneur must then use some self-selecting device so as to induce high-paying consumers to reveal themselves. The sort of ‘community ex- perience’ that web-based crowdfunding offers may be a means by which the entrepreneur enhances the perceived quality of the product for the con- sumers who agree to pre-order it. In this sense, crowdfunding appears as a form of menu pricing (i.e., of second-degree price discrimination).

The trade-off we explore in our model is thus the following: compared to external funding, crowdfunding has the advantage of offering an enhanced experience to some consumers and, thereby, of allowing the entrepreneur to practice second-degree price discrimination and extract a larger share of the consumer surplus; the disadvantage is that the entrepreneur is constrained in the first period by the amount of capital that she needs to raise. The

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larger this amount, the larger the number of consumers that have to be attracted to cover it, which eventually reduces the profitability of the menu pricing scheme.

In what follows, we first present our model; we then derive, in turn, the outcome under traditional sources of financing (such as debt) and under crowdfunding; finally, we derive the optimal funding choice.

3.1 Model

Suppose a unit mass of consumers identified by θ, with θ uniformly dis- tributed on [0, 1]. The parameter θ denotes a consumer’s taste for an in- crease in product’s quality. Consumers have unit demand (they buy one or zero unit of the product). All consumers have a reservation utility r > 0 for the product; any increase from the basic quality is valued in proportion to the taste parameter θ. Normalizing basic quality to zero, we have that if consumer θ buys one unit of product of increased quality s sold at price p, her net utility is r + θs−p.7 To ensure interior solutions at the pricing stage, we assume:

Assumption 1. r < s < 2r.

The product is marketed by a monopolist. In our simple model, we consider the quality of the product, s, as exogenous and known by the consumers before purchase.8 For simplicity, we set to zero the marginal cost of production. There is, however, a fixed cost of production K > 0. The timing of the game is as follows. In period zero, the entrepreneur chooses her funding mechanism—traditional funding or crowdfunding—with the following implications. If the entrepreneur chooses traditional funding, then, in period 1, it incurs the fixed cost K, which is financed through, e.g., a bank loan; in period 2, the entrepreneur sets a price p for her product, and consumers decide to buy or not.

On the other hand, if the entrepreneur chooses crowdfunding, then it is able to set a menu pricing scheme. In period 1, the entrepreneur sets

7This problem was initially examined by Mussa and Rosen (1978). We use here the

results of the extended analysis of Bhargava and Choudary (2001). 8A natural extension of this framework would be to assume that the quality of the

product is unknown to the consumers before purchase. As observed in some crowdfunding

experiences, the entrepreneur may then use web-based crowdfunding to reveal information

about the product and, thereby, alleviate the experience good problem. We discuss this

possibility in the concluding section.

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p1, the price for consumers who pre-order the product; the total revenue collected through pre-orders is meant to cover the fixed cost of production. In period 2, the entrepreneur sets two prices: pc, the price to be paid by those consumers who have contributed to the financing of the venture (the so-called “crowdfunders”), and pr, the price to be paid by those consumers who have not (the so-called “regular consumers”).9 As for consumers, they choose in period 1 whether to pre-order or not; in period 2, they decide whether to purchase the product or not (as long as the product has been put on the market, i.e., if total contributions in period 1 are at least as large as K).

The key feature of crowdfunding from the point of view of the consumers is that the participation to the mechanism may provide consumers with an increase in the product quality. We have indeed observed in Section 2 that entrepreneurs resorting to crowdfunding use the Internet to maintain an in- teraction with their funders so as to provide them with so-called ‘community benefits’. Various forms of rewards are sometimes offered but it appears that these rewards are mostly symbolic and that crowdfunders essentially value the feeling of belonging to a group of ‘special’ or ‘privileged’ consumers. It is therefore important for the entrepreneur to attract a sufficient number of regular (i.e., ‘non-privileged’) consumers to whom crowdfunders can feel somehow ‘superior’. Denoting the number of regular consumers by nr and the minimal number of them by ρ (with ρ ≥ 0), we capture these findings in the model through the following assumption:

Assumption 2. Crowdfunders perceive the quality of the product to be equal to s + σ, with σ > 0 if nr ≥ ρ and σ = 0 otherwise.

Assumption 2 translates the idea that crowdfunders enjoy some addi- tional utility from the product compared to regular consumers as long as the number of those regular consumers is above some threshold. To elim- inate two sub-cases of very little interest, we put an upper bound on the parameters σ and ρ:

Assumption 3. σ < s (r + s) / (2r).

Assumption 4. ρ < r/ (2s). 9The entrepreneur is able to recognize consumers who pre-ordered in period 1 and

therefore to tell them apart from regular consumers; no personal arbitrage is thus possible

in period 2.

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The entrepreneur maximizes the present discounted value of her profits over the two periods. Consumers maximize the present discounted value of their net utility over the two periods. We assume that the entrepreneur and the consumers have the same discount factor and we let 0 < δ ≤ 1 denote it.

Two comments are warranted on the pricing schedule, which is meant to be very general. First, contributors pay p1 + δpc in total, other consumers pay δpr. This framework encompasses several, more restrictive schemes, including full pre-payments (where contributors pay one single amount up- front, equal to p1 + δpc) as well as ex post price discrimination (where each type pays a different price). This means that we do not exogenously impose pc and pr to be identical, although the framework here allows for this. Sec- ond, by enabling the participating crowd to pay up-front only part of their contribution, we avoid any price setting in which the entrepreneur would raise funds well beyond what she really needs, namely K. Here, contribu- tors provide in period 1 merely what is needed for starting production, the rest being paid in period 2. As we will see, this implies that the first-period price charged to crowdfunders (p1) is not directly set by the entrepreneur but is rather indirectly determined by the interplay between the capital require- ment K and the number of crowdfunders (noted nc), which depends itself on the two second-period prices, pc and pr. These two prices are therefore the two actual choice variables of the entrepreneur.

We now consider the choice of prices under the two funding mechanisms. We then compare optimal profits in the two cases and address the choice of funding mechanism.

3.2 Traditional funding

The case of traditional funding is straightforward. In period 1, the en- trepreneur gathers funds and in period 2, she sets a uniform price p. All consumers perceive that the product has quality s. Hence, the indifferent consumer is such that r + θs−p ≥ 0, or θ ≥ (p−r) /s ≡ θ̂. As we assume a unit mass of consumers uniformly distributed on the unit interval, we have that the quantity demanded is equal to q (p) = 1 − θ̂ = 1 − (p−r) /s. From the first-order condition for profit-maximization, we easily find that the optimal price is p∗ = (r + s) /2. It follows that θ̂∗ = (s−r) /2s, which is positive according to Assumption 1. We can then compute the optimal

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gross profit as p∗ (

1 − θ̂∗ )

= (r + s)2 / (4s). The net profit under traditional funding is thus equal to

πT =

{ δ

(r+s)2

4s −K for K < δ (r+s)

2

4s ,

0 otherwise. (1)

3.3 Crowdfunding

The crowdfunding case is more complex to analyze for two reasons. First, the entrepreneur tries to achieve a form of second-degree price discrimina- tion; profit is thus maximized under a set of incentive compatibility and participation constraints. Second, in period 1 consumers who contemplate pre-ordering the product must form expectations regarding the number of consumers who will do likewise: the larger this number, the lower the pre- ordering price as the fixed cost will be spread over more consumers, which generates a form of network effects. We look for a subgame-perfect equilib- rium of the game played by the entrepreneur and the consumers over the two periods.

3.3.1 Consumer choices

Suppose that each consumer expects that a mass nec of consumers will choose to pre-order and pay the price p1 set by the entrepreneur in period 1. We adopt the fulfilled-expectations approach: consumers base their decision on their expectation on the mass of contributors, and attention is restricted on equilibria in which these expectations turn out to be correct (i.e., are ratio- nal; see Katz and Shapiro, 1985). Two cases have to be distinguished. First, if nec = 0, then it is optimal for each consumer not to contribute.

10 As the initial expectation is realized, we have a fulfilled expectations equilibrium. Naturally, crowdfunding is doomed to failure under such equilibrium. As some successful crowdfunding experiences exist in reality, it seems natural to assume that entrepreneurs can find some ways to coordinate consumers so that this ‘bad’ equilibrium is not selected.11

The second case is the case of interest. For any nec > 0, the entrepreneur can set p1 such as p1nec ≥ K. As there is no need to gather more capital

10This is so because each consumer is infinitesimal and thus cannot on his own make

sure that the product will be put on the market. 11Another reason to select the ‘good’ equilibrium is that it clearly Pareto-dominates the

‘bad’ equilibrium.

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than needed, we have p1 = K/nec. So, if consumers expect a positive mass of contributors, they can be sure that the good will be produced.12 They also realize that the lower their expectation, the larger the value of p1, i.e., the contribution that will be asked by the entrepreneur.

To decide whether to pre-order or not, consumer θ compares his ex- pected utility in the two options. Let us consider for the moment that the entrepreneur attracts a sufficient number of regular consumers at equilib- rium: nr ≥ ρ. (Naturally, we will have to check below if this condition is actually met.) If the consumer pre-orders, he pays p1 = K/nec today and gets tomorrow a product of enhanced quality (s + σ). We can thus express the expected utility of a crowdfunder as

Uec = − K

nec + δ (r + θ (s + σ) −pc) .

If the consumer decides not to pre-order, he does not pay anything today and he gets tomorrow a product of quality s at price pr. Hence, his expected utility as regular consumer is

Uer = δ (r + θs−pr) .

So, for a consumer to contribute, we must have

Uec ≥ U e r ⇔ δ (θσ + pr −pc) ≥

K

nec

⇔ θ ≥ K

δσnec − pr −pc σ

≡ θ̄ (nec) .

All consumers with a value of θ larger than θ̄ (nec) prefer to pre-order. We observe logically that the mass of crowdfunders increases as (i) the expected number of contributors (nec) increases, (ii) the capital requirement (K) de- creases, (iii) the enhancement in quality (σ) resulting from pre-ordering increases, (iv) the difference between the price for regular consumers and for crowdfunders (pr −pc) increases.

To ease the exposition, we define ∆ ≡ pr−pc; that is, ∆ is the difference between the second period prices for regular consumers and crowdfunders. For a given expected mass of crowdfunders nec, the actual mass of crowd- funders is equal to nc = 1 − θ̄ (nec). We require fulfilled expectations at

12Provided that the entrepreneur does not find it profitable to run away with the contri-

butions at the start of period 2. We consider this issue in Subsection 3.5. For the moment,

we assume that the entrepreneur is able to credibly commit that she will not run away.

15

equilibrium: nc = nec. We must thus solve

nc = 1 + ∆ σ −

K

δσnc .

This equation is represented in Figure 1: solutions are the intersection between the 45◦ line (nc) and the function 1 + ∆σ −

K δσnc

, which is increasing and concave in nc. Figure 1 depicts the latter function for different values of ∆. We observe that an intersection exists as long as

∆ ≥ σ (

2 √ K/ (δσ) − 1

) ≡ ∆. (2)

To understand the meaning of this condition, let us describe what happens when it is violated. For ∆ < ∆, the price charged to crowdfunders is not sufficiently smaller than the price charged to regular consumers, so that a large value of nec is needed to convince consumers to pre-order the prod- uct; indeed, the larger the expected number of crowdfunders, the lower the price p1 each crowdfunder has to pay in period 1, which increases the at- tractiveness of pre-ordering, other things being equal. Yet, the number of consumers who actually decide to pre-order always remains smaller than the expected number, meaning that expectations cannot be fulfilled (i.e., there is no solution to the above equation). Note that the threshold ∆ logically increases with K: the higher the capital requirement, the more difficult it becomes for expectations to be fulfilled. Note also that at ∆ = ∆, there is a unique solution, which is easily computed as nc =

√ K/ (δσ). This value

is strictly lower than unity as long as K < δσ, which we assume for the moment.13 For ∆ > ∆, there are two intersections. As we expect the mass of crowdfunders to increase with ∆, we select the largest value of nc, which is computed as

nc = 1

2σ

( σ + ∆ +

√ (σ + ∆)2 − (σ + ∆)2

) . (3)

As shown in Figure 1, this value is strictly smaller than unity for ∆ < K/δ.

3.3.2 Optimal prices

Suppose for now that nc < 1. We have then that nc consumers pre-order the product at price p1 and buy it in period 2 at price pc. As for the

13For K ≥ δσ, the fulfilled-expectations equilibrium is such that all consumers become crowdfunders. This implies that there can be no regular consumers and that σ inevitably

falls to zero. This possibility is examined under Case 2b below.

16

1

1

n

K /(!")

! = !

! = K /"

Figure 1: Fulfilled expectations equilibrium

other consumers, they buy the product as long as r + θs − pr ≥ 0, or θ ≥ (pr −r) /s ≡ θ̂. If 0 < (pr −r) /s < 1 − n, the number of regular consumers is equal to nr = 1 − n − (pr −r) /s. As long as nr > ρ, the entrepreneur’s profit can be written as

π = p1nc −K︸ ︷︷ ︸ =0

+δpcnc + δprnr

= δpr

( 1 −

pr −r s

) −δ∆

1 2σ

( σ + ∆ +

√ (σ + ∆)2 − (σ + ∆)2

) ,

where the second line is obtained by substituting expression (3) for nc, and ∆ for pr −pc. It is equivalent to maximize π over pr and pc, or over pr and ∆; we choose the latter option.

It is easily found that the first-order condition with respect to pr yields the optimal value p∗r = (r + s) /2, which implies that θ̂ = (p

∗ r −r) /s =

(s−r) /2s. The derivative of profit with respect to ∆ is

dπ

d∆ = −

δ

2σ

[ σ + 2∆ +

√ (σ + ∆)2 − (σ + ∆)2 + ∆(σ+∆)√

(σ+∆)2−(σ+∆)2

] . (4)

It is clear that the bracketed term is strictly positive for positive values

17

of ∆. Hence, any interior solution must be such that ∆ < 0 (i.e., that pr < pc, meaning that crowdfunders pay more than other consumers in period 2). Because of the constraint imposed by (2), this is only possible if ∆ is negative, which is equivalent to K < (δσ) /4. We therefore have to distinguish between two cases (we sketch the results here and refer the reader to the appendix for the detailed computations).

Case 1: K < (δσ) /4. In this case, we solve dπ/d∆ = 0 for ∆ and find:

∆∗ = 4K −δσ

2δ .

We verify that K < (δσ) /4 implies that ∆∗ < 0, i.e. that p∗c > p ∗ r: crowd-

funders pay more than other consumers in period 2 (we will return to this below). We also compute that the number of crowdfunders is given by n∗c = 1/2. Hence, at (p

∗ r, ∆

∗), consumers split into three groups: those with θ ∈ [0, (s−r) /2s] do not consume, those with θ ∈ [(s−r) /2s, 1/2] buy in period 2, and those with θ ∈ [1/2, 1] pre-order in period 1. We compute the equilibrium number of regular consumers as n∗r = 1/2 − (s−r) /2s = r/2s. From Assumption 4, we have that n∗r > ρ, which implies that σ > 0 as initially assumed. We then compute the optimal profit as

π = δ (r + s)2

4s + δσ

4 −K. (5)

In the present case, the capital requirement imposes no constraint what- soever on the entrepreneur. To see this, let us first compute the total price paid by crowdfunders. It is equal to p1 + δpc = δ2 (r + s + σ). Next, we observe that this is exactly the price that the entrepreneur would set if it was only selling in period 1 a product of quality (s + σ) to be delivered in period 2. Indeed, the indifferent consumer would be identified by θ0 such that −p + δ (r + θ0 (s + σ)) = 0, which is equivalent to θ0 = 1s+σ

( 1 δ p−r

) .

The entrepreneur would then maximize π = δ (p (1 −θ0)). It is easy to check that the optimal price is indeed p = 1

2 δ (r + s + σ).

Case 2: K ≥ (δσ) /4. Here, ∆ ≥ 0 under condition (2). Then, expres- sion (4) is clearly negative, meaning that the optimal choice is the lowest admissible value of ∆, i.e., ∆ = ∆ > 0. The intuition goes as follows: the higher capital requirement, combined to the fulfilled expectations require- ment, forces the entrepreneur to give a discount to crowdfunders (pc < pr)

18

but the entrepreneur prefers to keep this discount as small as possible. The number of crowdfunders is then given by

nc = 1 2

+ ∆ 2σ

= √ K/ (δσ) ≥

1 2 .

We see thus that the entrepreneur has to attract a larger number of crowd- funders than in the previous, unconstrained, case; this number grows with K (and remains smaller than unity as long as K < δσ).

As the number of crowdfunders grows, it is not clear whether the en- trepreneur still finds it optimal to attract a sufficient number of regular consumers in period 2. It does so as long as 1 − nc − (p∗r −r) /s ≥ ρ. As p∗r = (r + s) /2, the latter condition is equivalent to

K ≤ δσ ( r + s

2s −ρ )2 ≡ K1 (σ,ρ) .

It can be checked that Assumptions 1 and 4 ensure that K1 (σ,ρ) is com- prised between (δσ) /4 and δσ; it is also clear that K1 increases with σ and decreases with ρ. There are thus two subcases to consider.

Case 2a: (δσ) /4 ≤ K ≤ K1 (σ,ρ) . Here, as nr ≥ ρ, we have that σ > 0 for crowdfunders and we can compute the equilibrium value of pc as p∗c = p

∗ r − ∆ = (r + s) /2 − 2σ

√ K/ (δσ) + σ (which is positive under

Assumption 3). The equilibrium profit is then equal to

π = δ (r + s)2

4s + √ δσK − 2K. (6)

Case 2b: K > K1 (σ,ρ) . In this case, given nc and p ∗ r, the number of

regular consumers is too low for crowdfunders to enjoy any additional utility through ‘community benefits’. The entrepreneur faces then the following alternative: she either adjusts pr so as to keep nr above ρ and, thereby, σ > 0, or she accepts nr < ρ, σ = 0 and maximizes a different objective function. We show in the appendix that the latter option is equivalent, in terms of equilibrium profits, to traditional funding. As for the former option, the entrepreneur sets the price pr such that nr = ρ, which is equivalent to

pr = r + s−s (√

K/ (δσ) + ρ ) .

The price for crowdfunders is then computed as pc = pr − ∆ = r + s + σ − sρ−(s + 2σ)

√ K/ (δσ). It seems reasonable to exclude negative prices. We

19

have thus that this option is feasible as long as pc ≥ 0, which is equivalent to

K ≤ δσ ( r + s + σ −sρ

s + 2σ

)2 ≡ K2 (σ,ρ) .

We show in the appendix that Assumption 3 implies that K2 (σ,ρ) > K1 (σ,ρ). Hence, for K1 (σ,ρ) < K ≤ K2 (σ,ρ), the option of setting pr so that nr = ρ yields the following profit:

π = δ ( r + s−s

(√ K/ (δσ) + ρ

))(√ K/ (δσ) + ρ

) + √ δσK − 2K. (7)

We will show below that the latter function decreases with K. At K = K2, we have pc = 0, pr > 0 and nr = ρ > 0; it follows that the entrepreneur still earns positive profits in this extreme case. The question remains, however, whether this option is more profitable than traditional funding or not. We will examine this issue in the next section but before, we collect our results and perform some comparative statics exercises.

Summary. Combining expressions (5) to (7), we can express equilibrium profits in the crowdfunding case as

πC =

 

δ (r+s)2

4s + δσ

4 −K for K < δσ

4 ,

δ (r+s)2

4s + √ δσK − 2K for δσ

4 ≤ K ≤ K1,

δ

( r + s−sρ−s

√ K δσ

)(√ K δσ

+ ρ )

+ √ δσK − 2K

for K1 < K ≤ K2. (8)

We show in the appendix that each segment of this profit function is a decreasing function of K and an increasing function of σ; moreover, the third segment is a decreasing function of ρ. We record these results in the following proposition.

Proposition 1 The equilibrium profit under crowdfunding decreases with the capital requirement (K) and with the minimal number of regular con- sumers required to generate community benefits (ρ); it increases with the magnitude of community benefits (σ).

The intuition behind these comparative static results is clear. The capital requirement has a twofold negative impact on profit: on the one hand, it makes production more expensive and on the other hand, it reduces the possibility to implement the optimal menu pricing scheme. An increase in

20

the minimal number of regular consumers required to generate community benefits also constraints price discrimination and thereby negatively affects profits. Indeed, when ρ increases, communuty benefits cannot be shared as broadly as before. Finally, if crowdfunders have a higher valuation for the community benefits, their willingness to pay increases and the entrepreneur is able to increase her margins.

3.4 Choice of funding method

Comparing expressions (8) and (1), we observe first that for small values of K (K ≤ (δσ) /4), crowdfunding clearly yields larger profits than traditional funding. The intuition is obvious: in this region of parameters, crowdfunding allows the entrepreneur to optimally price discriminate between the high- valuation crowdfunders and the remaining consumers. As the ‘enhanced quality’ σ comes at no cost for the entrepreneur, menu pricing performs better than the uniform pricing that prevails under traditional funding.

For larger values of K, however, the entrepreneur is constrained to im- plement corner solutions under crowdfunding. Here, σ is no longer a sort of ‘manna from heaven’ for the entrepreneur: several requirements constrain the prices that the entrepreneur can choose, which inevitably reduces her profits. The first constraining requirements are the network effects among crowdfunders and the imposition of fulfilled expectations. Nevertheless, crowdfunding still dominates for values of K comprised between (δσ) /4 and K1 (σ,ρ): πC −πT =

√ δσK −K, which is positive as K ≤ K1 (σ,ρ) < δσ.

Yet, an additional constraint bites for K1 (σ,ρ) < K ≤ K2 (σ,ρ), namely the minimal number of regular consumers necessary to generate the commu- nity benefits for crowdfunders (combined with the non-negativity of prices). We show in the appendix that the profit under crowdfunding may fall under the profit under traditional funding when the capital requirement becomes large enough. More precisely, we find that for K1 (σ,ρ) < K ≤ K2 (σ,ρ), πC −πT if and only if K ≤ K3 (σ,ρ), with

K3 (σ,ρ) ≡ δσ ( s(r+s+σ−2sρ)+

√ σs(4sρ(r−sρ)+σs+s2−r2) 2s(s+σ)

)2 .

We show in the appendix that K3 (σ,ρ) > K1 (σ,ρ) and that, for given values of the other parameters, there exists a value σ̂ such that K2 (σ,ρ) > K3 (σ,ρ) for σ < σ̂ and K2 (σ,ρ) < K3 (σ,ρ) for σ > σ̂. Moreover, it is intuitive that the three thresholds increase with σ and decrease with ρ.

21

!"

K

! (r +s) 2

4s

!T = 0

!C = !T

K3(!,")

K2(!,")

Crowdfunding

Traditional funding

No activity

s(r +s) 2r

Figure 2: Choice of funding method

We collect our results in the following proposition and we depict them in Figure 2.

Proposition 2 In situations where an entrepreneur can use crowdfunding and pre-sales to induce self-selection of high paying consumers, crowdfund- ing is preferred over traditional funding if the capital requirement is below some threshold value (i.e., K ≤ min{K2 (σ,ρ) ,K3 (σ,ρ)}). This condition becomes less stringent as community benefits become more important, either because their magnitude (σ) increases or because they are generated more easily (ρ decreases).

One important implication of Proposition 1 is that the level of addi- tional benefits accruing to the pre-ordering crowd (i.e., σ) must be suffi- ciently large. If the crowd does not enjoy any of such benefits or utility, crowdfunding does not yield any benefits over traditional funding for the entrepreneur. The parameter σ can be seen as additional utility or bene- fits from a community-based experience. Then, the lack of a community would result in a value of σ equal to zero. An important implication is the need for the entrepreneur to identify and target this community. While con- sumers with a high willingness to pay for the product may self-select into the community, the entrepreneur still needs to ensure that the “crowd” can

22

generate these additional benefits. The following managerial lesson can thus be drawn from our analysis: entrepreneurs who cannot identify or create a community around their products so that this community enjoys additional benefits, will hardly ever opt for crowdfunding. Indeed, we observe on Figure 2 that as σ decreases, the range of values of K for which crowdfunding is preferred narrows down.

The previous finding is consistent with many observed crowdfunding ini- tiatives. Indeed, while some offer monetary rewards, an important other form of reward is recognition or credits offered to crowdfunders. The im- portance of non-monetary benefits for crowdfunders is further stressed by the observation that at equilibrium, crowdfunders always end up paying a larger total price than regular consumers. To see this, we compute the dif- ference (p∗1 + δp

∗ c) −δp∗r in the three regimes. We find that this difference is

equal to δσ/2 for K < δσ/4, and to δσ (

1 − √ K/ (δσ)

) for K ≥ δσ/4. As

crowdfunding is not feasible for K > δσ, we check that (p∗1 + δp ∗ c) > δp

∗ r for

all admissible values of the parameters. This result is not surprising insofar as crowdfunders are high-valuation consumers and that their willingness to pay is further enhanced by the community benefits.

3.5 Take the money and run

In the previous analysis, we have abstracted away the possibility that the entrepreneur could “take the money and run” at the start of period 2, i.e., to collect p1 from the crowdfunders without incurring the fixed cost and thus, without producing the product. That is, we implicitly assumed that the en- trepreneur had some form of commitment at her disposal to guarantee her second-period activity. Absent such commitment device, consumers would only be convinced that production will take place if it is indeed in the en- trepreneur’s best interest; otherwise, no consumer would agree to pre-order the product and crowdfunding would fail. The entrepreneur’s net profit when producing must then be at least as large as the total amount that is collected at the end of period 1, i.e., K. We show in the appendix that

πC ≥ K ⇔ K ≤ K4 (σ,ρ)

23

with14

K4 (σ,ρ) ≡

 

δσ

( (r+s(1−2ρ)+σ)+

√ −12sσρ2+4σ(3r+2s)ρ+(r+s+σ)2

2(s+3σ)

)2 for 0 < σ ≤ σ1,

δσ 36

( 1 +

√ 1 + 3 (r+s)

2

σs

)2 for σ1 ≤ σ ≤

s(r+s) 2r

,

σ1 ≡ s(r+s)2

(r+s−2sρ)(3r+s−6sρ).

As depicted on Figure 3, K4 (σ,ρ) lies below the minimum of K2 (σ,ρ) and K3 (σ,ρ) for sufficiently large values of σ and sufficiently large values of K.15 This implies that when consumers may fear that the entrepreneur could take the money and run, crowdfunding becomes harder to implement: there exists indeed a region of parameters (characterized by high values of σ and K) where consumers will not agree to pre-order the product as they rightfully anticipate that the entrepreneur will not put the product on the market. For these parameters, traditional funding appears as the only option (assuming, of course, that banks are better equipped than crowdfunders to prevent the entrepreneur’s default). We record this result in the following proposition.

Proposition 3 When the entrepreneur has no credible way to commit that she will not run away with the money collected in period 1, crowdfunding is less likely to be preferred to traditional funding.

4 Concluding remarks

This paper sheds light on crowdfunding practices of entrepreneurial activ- ities. It stresses the need for building a community that ultimately enjoys additional private benefits from their participation to make crowdfunding a viable alternative to investor- or creditor-based funding such as through banks, business angels or even venture capital. In setting up the initiative, the entrepreneur potentially faces the following trade-off. Crowdfunding al- lows for price discrimination if pre-ordering is used. The capacity to opti- mally implement price-discrimination between pre-ordering consumers (the

14The second branch of K4 only obtains for sufficiently small values of ρ. 15Logically, K4 (σ,ρ) (where πC = K) intersects with K3 (σ,ρ) (where πC = πT ) at

K = δ (r + s) 2 / (8s), which is the value of K such that πT = K. Above this threshold,

we have that for any σ, K4 (σ,ρ) < K3 (σ,ρ).

24

!"

K

! (r +s) 2

4s

!T = 0

!C = !T

K3(!,")

K2(!,")

Crowdfunding

Traditional funding

No activity

s(r +s) 2r

K4(!,")

!C = K

Figure 3: Funding method if the entrepreneur can take the money and run

crowdfunders) and other consumers may however be constrained by the amount of capital that the entrepreneur needs to raise to cover the up-front (fixed) costs. Whenever this amount exceeds some threshold, the distortion in the price discrimination becomes excessive, in which case the profitability of the crowdfunding initiative is reduced and the entrepreneur may be better off approaching a single, larger investor (a bank or a large equity investor) who can cover the full costs on its own.

To our knowledge, this is the very first study offering a theoretical anal- ysis of crowdfunding. It also highlights new follow-up research questions on the topic. For instance, an interesting avenue for future research is to incorporate the fact that the crowdfunders can at times also participate in strategic decisions or even have voting rights. In this case, control rights and voting power become an additional benefit for the participating crowd. Crowdfunding through pre-ordering will have a very different effect on in- formation and voting results than if the crowd purchases equity in the en- trepreneurial firm. Also, outcomes of votes can provide valuable insights into the optimal design of products if the voting community is representative for the overall population of end-consumers.

Future works may further explore information motivations of entrepreneurs. Indeed, while the primary goal of crowdfunding is certainly to raise money, it may also help firms in testing, promoting and marketing their products,

25

in gaining a better knowledge of their consumers’ tastes, or in creating new products or services altogether. In this sense, crowdfunding can be used as a promotion device, as a means to support mass customization or user-based innovation, or as a way for the producer to gain a better knowledge of the preferences of its consumer. Crowdfunding seems thus to have implications that go beyond the financial sphere of an organization: it also affects the flow of information between the organization and its customers.

In any case, a strong advantage of this form of financing is the attention that the entrepreneur may attract on his/her project or company. This can become a vital asset for many of them, especially for artists or entrepreneurs in need to present their talent and product to the crowd (as potential cus- tomers). In other cases, it is a unique way to validate original ideas in front of a specifically targeted audience. This may in turn provide insights into market potential of the product or service offered. From this perspective, crowdfunding may be viewed as a broader concept than purely raising funds: it is a way to develop corporate activities through the process of fundraising.

Also, several platforms have emerged recently, such as IndieGoGo, Kick- starter, Sandawe, SellaBand, MyMajorCompany and Artistshare. These platforms intermediate between entrepreneurs and potential crowdfunders. Therefore, a distinction can be made between direct and indirect fundraising because at times entrepreneurs make use of such crowdfunding platforms in- stead of seeking direct contact with the crowd. These platforms share some similarities with online lending markets (Everett, 2008; Freedman and Jin, 2010); while the latter more prominently target social entrepreneurship, crowdfunding platforms have a broader scope of entrepreneurial initiatives. Our understanding of the role played by platforms is still limited; it is worth investigating the extent to which platforms increase the chances of success of crowdfunding initiatives or solve asymmetric information issues. As an example, for crowdfunders, platforms may facilitate in learning the quality of the product through the possible interaction between crowdfunders (e.g., via other crowdfunder comments on a forum) or by observing the contribu- tions of other crowdfunders. More research could be done along the line of peer effects as it relates to crowdfunding platforms, as suggested by Ward and Ramachandran (2010).

From a more general perspective, crowdfunding practices raise questions with respect to corporate governance and investor protection issues if most

26

individuals only invest tiny amounts. Crowdfunders are most likely to be offered very little investor protection. This may lead to corporate gover- nance issues, which in turn may entail reputation concerns if some cases of fraud or bad governance are uncovered. Crowdfunders have very little scope to intervene to protect their interests as stakeholders. Moreover, the fact that their investment is small is likely to create a lack of incentive to inter- vene. Therefore, trust-building is an essential ingredient for any successful crowdfunding initiative. It is also not a surprise that many of the observed crowdfunded initiatives are either project-based or based on donations. In many cases, the financial return seems to be of secondary concern for those who provide funds. This suggests that crowdfunders care about social rep- utation or enjoy private benefits from participating in the success of the initiative (Glaeser and Shleifer, 2001; Ghatak and Mueller, 2011).

5 Appendix

We give here the details of the mathematical developments that we sketched in the text.

5.1 Optimal prices under crowdfunding

Case 1. If K < (δσ) /4, then there may exist a value of ∆ ≥ ∆ that solves dπ/d∆ = 0, or

σ + 2∆ + √

(σ + ∆)2 − (σ + ∆)2 + ∆(σ+∆)√ (σ+∆)2−(σ+∆)2

= 0 ⇔

√ (σ + ∆)2 − (σ + ∆)2 =

(σ + ∆)2

σ + 2∆ − (σ + ∆) (9)

As long as the RHS is positive, we can take the square of the two sides of the equality:

(σ + ∆)2 − (σ + ∆)2 = (σ + ∆)4

(σ + 2∆)2 + (σ + ∆)2 − 2

(σ + ∆) (σ + ∆)2

σ + 2∆ ,

which, after simplification, yields

∆∗ = 1

2σ

[ (σ + ∆)2 −σ2

] =

1 2δ

(4K −δσ) .

We still need to check whether condition (2) is satisfied:

1 2σ

[ (σ + ∆)2 −σ2

] > ∆ ⇔ 2σ∆ + ∆2 > 2σ∆,

27

which is true. We also need to check that the RHS of expression (9) is positive, as we assumed it. We compute

(σ + ∆)2

σ + 2∆ > (σ + ∆) ⇔

4σK/δ 4K/δ

> 4K + δσ

2δ ⇔ 2δσ > 4K + δσ ⇔ K < δσ/4

which is true. To proceed, we compute

σ + ∆∗ = 1

2σ

[ (σ + ∆)2 + σ2

] ,√

(σ + ∆∗)2 − (σ + ∆)2 = 1

2σ

[ σ2 − (σ + ∆)2

] .

It follows that

n∗ = 1

2σ

( 1

2σ

[ (σ + ∆)2 + σ2

] +

1 2σ

[ σ2 − (σ + ∆)2

]) =

1 2 .

Recall that we need

pr −r s

< 1 −n ⇔ s−r s

< 1 2 ⇔ s < 2r,

which is guaranteed by Assumption 1. We can now compute the optimal profit:

π = δ (r + s)2

4s −δ∆∗

1 2σδ

(δ (σ + ∆∗) −δ∆∗)

= δ (r + s)2

4s + δσ

4 −K.

Case 2. We first have to establish under which condition there is a suffi- cient number of regular consumers. We need 1 − nc − (p∗r −r) /s ≥ ρ. As nc =

√ K/ (δσ) and p∗r = (r + s) /2, the condition can be rewritten as

1 − √ K

δσ − s−r

2s ≥ ρ ⇔ K ≤ δσ

( r + s

2s −ρ )2 ≡ K1 (σ,ρ) .

We compute

K1 (σ,ρ) − δσ

4 =

σδ (r + 2s− 2sρ) (r − 2sρ) 4s2

> 0,

δσ −K1 (σ,ρ) = σδ (r + 3s− 2sρ) (s + 2sρ−r)

4s2 > 0,

which are both positive because of Assumptions 1 (s > r) and 4 (ρ < r/ (2s)).

28

Case 2a: (δσ) /4 ≤ K ≤ K1 (σ,ρ) . We need to show that p∗c = p∗r − ∆ = (r + s) /2 − 2σ

√ K/ (δσ) + σ ≥ 0. This condition is equivalent to

K ≤ δσ ( r + s + 2σ

4σ

)2 ≡ K5 (σ,ρ) .

We have K5 (σ,ρ) > K1 (σ,ρ) if and only if

r + s + 2σ 4σ

> r + s

2s −ρ ⇔ ρ > −

s (r + s) − 2rσ 4sσ

≡ ρ̃.

Assumption 3, i.e., σ < s (r + s) / (2r), implies that ρ̃ < 0 and hence, that the inequality is satisfied, meaning that p∗c > 0 when K ≤ K1 (σ,ρ).

Case 2b: K > K1 (σ,ρ) . We showed in the text that pc ≥ 0 as long as

K ≤ δσ ( r + s + σ −sρ

s + 2σ

)2 ≡ K2 (σ,ρ) .

We have that K2 (σ,ρ) > K1 (σ,ρ) is equivalent to

r + s + σ −sρ s + 2σ

> r + s

2s −ρ ⇔

s (r + s) − 2rσ + 4sσρ 2s (s + 2σ)

> 0,

which is true under Assumption 3 as 2rσ < s (r + s). We also need to show that when the entrepreneur accepts nr < ρ (which

implies σ = 0), she achieves the same profit as under traditional funding. Suppose first that the entrepreneur decides to attract no regular consumer whatsoever. She can do so by setting p > r + s. In that case, a consumer decides to pre-order the product (which is the only option) if

− K

nec + δ (r + θs−pc) ≥ 0 ⇔ θ ≥

K

δsnec − r −pc s

.

As we require fulfilled expectations, we must have

nc = 1 − K

δsnc + r −pc s

.

Solving the latter equation and keeping the largest root, we have

n = δ (r + s−pc) +

√ δ2 (r + s−pc)2 − 4δsK 2δs

,

which is valid as long as pc ≤ r + s− 2 √ sK/δ.

29

The entrepreneur choose pc to maximize

π = pc δ (r + s−pc) +

√ δ2 (r + s−pc)2 − 4δsK 2δs

.

Solving for the first-oder condition, we find the optimal price as

pc = r + s

2 −

2s δ (r + s)

K

and we check that it is indeed no larger than r + s− 2 √ sK/δ as required.

We can then compute the number of consumers as n = (r + s) / (2s) and the optimal profit as:

π = δ ( r + s

2 −

2s δ (r + s)

K

) r + s

2s = δ

(r + s)2

4s −K = πT .

Alternatively, the entrepreneur could still attract some regular consumers (but less than ρ). To do so, she must set pr < r + s. A consumer would then choose to pre-order the product if

− K

nec + δ (r + θs−pc) ≥ δ (r + θs−pr) ⇔

K

nec ≤ δ (pr −pc) ,

which is clearly impossible if pc ≥ pr. Suppose then pc < pr. The latter condition can we rewritten as

nec ≥ K

δ (pr −pc) .

Expectations are no longer an issue in the present case. Taking nc = K/(δ (pr −pc)), we express the entrepreneur’s profit as

π = δpc K

δ (pr −pc) +δpr

( 1 −

K

δ (pr −pc) − pr −r s

) = δpr

( 1 −

pr −r s

) −K,

which is exactly the same profit function as under traditional funding; this completes our proof.

5.2 Proof of Proposition 1

We have that profits under crowdfunding are equal to πC1 for K < δσ4 , to πC2 for δσ4 ≤ K ≤ K1, and to πC3 for K1 < K ≤ K2, with

πC1 = δ (r+s)2

4s + δσ

4 −K,

πC2 = δ (r+s)2

4s + √ δσK − 2K,

πC3 = δ ( r + s−sρ−s

√ K δσ

)(√ K δσ

+ ρ )

+ √ δσK − 2K.

30

It is obvious that πC1 increases with σ and decreases with K. As for πC2, it clearly increases with σ while its derivative with respect to K is equal to

∂πC2 ∂K

= √ δσ

1 2 √ K − 2.

The latter expression is negative if K > δσ/16, which is the case here. Regarding πC3, we first compute:

∂πC3 ∂K

= (r + s + σ − 2sρ)

√ σδK − 2 (s + 2σ) K

2σK .

The latter expression is negative if and only if

K > δσ

( r + s + σ − 2sρ

2 (s + 2σ)

)2 ≡ K6 (σ,ρ) .

We consider here values of K > K1 (σ,ρ). Let us show that Assumption 4, i.e., ρ < r/ (2s), makes sure that K1 (σ,ρ) > K6 (σ,ρ). The latter condition is equivalent to

r + s 2s −ρ−

r + s + σ − 2sρ 2 (s + 2σ)

= 1 2 σ

2 (r − 2sρ) + s s (s + 2σ)

> 0.

Second, we have

∂πC3 ∂σ

= 2sK − (r − 2sρ + s−σ)

√ σδK

2σ2 .

If σ > r − 2sρ + s, then the latter expression is clearly positive. Otherwise, it is positive as long as

K > δσ

( r − 2sρ + s−σ

2s

)2 ≡ K7 (σ,ρ) .

We show again that K1 (σ,ρ) > K7 (σ,ρ), which is equivalent to

r + s 2s −ρ−

r − 2sρ + s−σ 2s

= σ

2s > 0.

Finally, we compute

∂πC3 ∂ρ

= δ

( r + s− 2sρ− 2s

√ K

σδ

) .

This expression is negative for

K > δσ

( r + s− 2sρ

2s

)2 = K1 (σ,ρ) ,

which completes the proof.

31

5.3 Proof of Proposition 2

We first derive the value of K such that π3C = πT .

δ

( r + s−sρ−s

√ K

δσ

)(√ K

δσ + ρ

) + √ δσK − 2K = δ

(r + s)2

4s −K ⇔

−(s + σ) K+(r + s + σ − 2sρ) √ δσ √ K+δσ (r + s−sρ) ρ−δσ

(r + s)2

4s = 0.

This second-degree polynomial in √ K has real roots as long as

(r + s + σ − 2sρ)2 δσ + 4 (s + σ) ( δσ (r + s−sρ) ρ−δσ (r+s)

2

4s

) =

σ2δ s

( 4sρ (r −sρ) + sσ + s2 −r2

) > 0,

which is satisfied under Assumptions 1 and 4. The roots are then given by

√ K =

√ δσ s (r + s + σ − 2sρ) ±

√ σs (4sρ (r −sρ) + σs + s2 −r2)

2s (s + σ) .

Some lines of computations establish that both roots are positive and that δσ (r + s−sρ) ρ < δσ (r+s)

2

4s under our assumptions. It follows that πC3 > πT

if and only if

δσ

( s(r+s+σ−2sρ)−

√ σs(4sρ(r−sρ)+σs+s2−r2) 2s(s+σ)

)2 < K

< δσ

( s(r+s+σ−2sρ)+

√ σs(4sρ(r−sρ)+σs+s2−r2) 2s(s+σ)

)2 = K3 (σ,ρ) .

We establish now that K1 (σ,ρ) lies between these two bounds. First, the upper bound (i.e., K3) being larger than K1 (σ,ρ) is equivalent to

s(r+s(1−2ρ)+σ)+ √ σs(4sρ(r−sρ)+σs+s2−r2)

2s(s+σ) >

r + s 2s −ρ ⇔√

σs (4sρ (r −sρ) + σs + s2 −r2) > σ (r − 2sρ) ⇔

σ (s−r + 2sρ) (r − 2sρ + s) (s + σ) > 0,

which is satisfied under Assumptions 1 (s > r) and 4 (ρ < r/ (2s)). As for the lower bound being smaller than K1 (σ,ρ), this is so if

s(r+s(1−2ρ)+σ)− √ σs(4sρ(r−sρ)+σs+s2−r2)

2s(s+σ) <

r + s 2s −ρ ⇔√

σs (4sρ (r −sρ) + σs + s2 −r2) > −σ (r − 2sρ) ,

32

which is clearly satisfied under Assumption 4. We can thus state that when K > K1 (σ,ρ),

πC > πT ⇔ K ≤ δσ ( s(r+s+σ−2sρ)+

√ σs(4sρ(r−sρ)+σs+s2−r2) 2s(s+σ)

)2 = K3 (σ,δ) .

We show next that for given values of the other parameters, there exists a value σ̂ such that K2 (σ,ρ) > K3 (σ,ρ) for σ < σ̂ and K2 (σ,ρ) < K3 (σ,ρ) for σ > σ̂. For δσ > 0, we have

K2 (σ,ρ)−K3 (σ,ρ) ∝ r+s+σ−sρs+2σ − s(r+s(1−2ρ)+σ)+

√ σs(−4s2ρ2+4rsρ+s2−r2+σs) 2s(s+σ)

.

The latter expression tends to (r + s) / (2s) when σ tends to zero, and to −1/2 when σ tends to infinity. As both K2 and K3 are increasing functions of σ, there exists a value of σ below which K2 > K3 and above which K3 > K2. Numerical simulations suggest that this cutoff value of σ is smaller than s (r + s) / (2r), i.e., the upper bound we have assumed for σ.

5.4 Proof of Proposition 3

Recall that profits under crowdfunding are equal to πC1 for K < δσ4 , to πC2 for δσ

4 ≤ K ≤ K1, and to πC3 for K1 < K ≤ K2, with

πC1 = δ (r+s)2

4s + δσ

4 −K,

πC2 = δ (r+s)2

4s + √ δσK − 2K,

πC3 = δ ( r + s−sρ−s

√ K δσ

)(√ K δσ

+ ρ )

+ √ δσK − 2K.

We need to check under which conditions πC ≥ K. We first have that πC1 ≥ K if and only if

δ (r+s)2

4s + δσ

4 > 2K ⇐⇒ K < 1

2

( δ

(r+s)2

4s + δσ

4

) .

The latter threshold is larger than δσ 4

provided that

1 2

( δ

(r+s)2

4s + δσ

4

) − δσ

4 = δ

8s

( (r + s)2 −σs

) > 0.

The latter condition is satisfied because of Assumptions 1 and 3; we have indeed that s (r + s) / (2r) < (r + s)2 /s. It follows that in the space of parameters that we consider, πC1 > K.

Next, we find that πC2 ≥ K if and only if

−3K + √ δσ √ K + δ (r+s)

2

4s > 0.

33

This polynomial has one negative and one positive real root; as the polyno- mial is positive for K = 0, we have that the condition for πC2 ≥ K is

√ K ≤ 1

6

(√ δσ +

√ δσ + 3δ (r+s)

2

s

) or equivalently

K ≤ δσ 36

( 1 +

√ 1 + 3 (r+s)

2

σs

)2 ≡ K8 (σ,ρ) .

Is the latter condition more or less stringent than K ≤ K1 (σ,ρ)? We compute

K1 > K8 ⇔ r+s2s −ρ > 1 6

( 1 +

√ 1 + 3 (r+s)

2

σs

) ⇔ σ > s(r+s)

2

(3r+s−6sρ)(r+s−2sρ) = σ1 (ρ) .

Whether σ1 (ρ) is above or below s (r + s) / (2r) depends on the value of ρ. It can be checked that σ1 (ρ) increases with ρ and σ1 (0) =

s(r+s) 3r+s

< s(r+s)

2r

while σ1 ( r 2s

) = (r+s)

2

s >

s(r+s) 2r

. The exact cutoff value is 0 < ρ̂ < r 2s

, with

ρ̂ = 3r + 2s−

√ 6r2 + 6rs + s2

6s .

Finally, we find that πC3 ≥ K if and only if

δ

( r + s−sρ−s

√ K

δσ

)(√ K

δσ + ρ

) + √ δσK ≥ 3K.

Developing this inequality, we find that it is equivalent to

K < δσ

( (r+s(1−2ρ)+σ)+

√ −12sσρ2+4σ(3r+2s)ρ+(r+s+σ)2

2(s+3σ)

)2 = K9 (σ,ρ) .

Unsurprisingly, we find that K9 > K1 for σ < σ1 (ρ). We have thus that K8 (σ,ρ) and K9 (σ,ρ) constitute the two branches of K4 (σ,ρ) as described in the text.

References

[1] Agrawal, A., C. Catalini, and A. Goldfarb, 2011. The Geography of Crowdfunding. NBER Working Paper, No. 16820.

34

[2] Berger, A., and G. Udell, 1995. Relationship Lending and Lines of Credit in Small Firm Finance. Journal of Business 68, 351-381.

[3] Bhargava, H.K. and Choudhary, V., 2001. Second-degree price discrim- ination for information goods under nonlinear utility functions. In Pro- ceedings of the 34th Hawaii International Conference on System Sci- ences.

[4] Bhidé, A., 1992. Bootstrap Finance: The Art of Start-ups. Harvard Business Review 70 (66), 109-117.

[5] Brabham D.C., 2008. Crowdsourcing as a Model for Problem Solving: An Introduction and Cases. Convergence: The International Journal of Research into New Media Technologies 14, 75- 90.

[6] Cadsby, C.B. and E. Maynes, 1999. Voluntary Provision of Threshold Public Goods with Continuous Contributions: Experimental Evidence. Journal of Public Economics 71, 53–73.

[7] Casamatta, C., and C. Haritchabalet, 2010. Dealing with Venture Cap- italists: Shopping Around or Exclusive Negotiation. Working paper.

[8] Cassar, G., 2004. The Financing of Business Start-ups. Journal of Busi- ness Venturing 19, 261-283.

[9] Cosh, A., D. Cumming and A. Hughes, 2009. Outside Entrepreneurial Capital. Economic Journal 119, 1494-1533.

[10] Dana, J., 1998. Advance-Purchase Discounts and Price Discrimination in Competitive Markets, Journal of Political Economy 106, 395-422.

[11] Dana, J., 1999. Equilibrium Price Dispersion under Demand Uncer- tainty: The Roles of Costly Capacity and Market Structure, Rand Journal of Economics 30, 632-660.

[12] Dana, J., 2001. Monopoly Price Dispersion Under Demand Uncertainty, International Economic Review 42, 649-670.

[13] Ebben, J., and A. Johnson, 2006. Bootstrapping in Small Firms: An Empirical Analysis of Change over Time. Journal of Business Venturing 21, 851-865.

35

[14] Everett, C.R., 2008. Group Membership, Relationship Banking and Loan Default Risk: The Case of Online Social Lending. Mimeo.

[15] Fershtman, C., and N. Gandal, 2011. A Brief Survey of the Economics of Open Source Software. CEPR Discussion Paper No. DP8434.

[16] Freedman, S., and G.Z. Jin, 2011. Learning by Doing with Asymmetric Information: Evidence from Prosper.com. NBER Working Paper, No. 16855.

[17] Gale, I., and T. Holmes, 1992. The Efficiency of Advance-Purchase Dis- counts in the Presence of Aggregate Demand Uncertainty, International Journal of Industrial Organization 10, 413-37.

[18] Gale, I., and T. Holmes, 1993. Advance-Purchase Discounts and Monopoly Allocation of Capacity, American Economic Review 83, 135- 146.

[19] Ghatak, M. and H. Mueller, 2011. Thanks for nothing? Not-for-profits and motivated agents, Journal of Public Economics, 95, pp. 94-105.

[20] Glaeser, E.L., and A. Shleifer, 2001. Not-for-Profit Entrepreneurs. Jour- nal of Public Economics 81, 99-115.

[21] Gradstein, M., 1994. Efficient Provision of a Discrete Public Good. International Economic Review 35, 877-897.

[22] Hellmann, T., 2007. Entrepreneurs and the Process of Obtaining Re- sources. Journal of Economics and Management Strategy 16, 81-109.

[23] Howe, J., 2008. Crowdsourcing. Why the Power of the Crowd is Driving Future of Business. New York: Three Rivers Press.

[24] Katz, M. and Shapiro, C., 1985. Network Externalities, Competition and Compatibility. American Economic Review 75, 424-440.

[25] Kleemann, F., G.G. Voß and K. Rieder, 2008. Un(der)paid Innovators: The Commercial Utilization of Consumer Work through Crowdsourc- ing. Science, Technology & Innovation Studies 4, 5-26.

[26] Larralde, B., and A. Schwienbacher, 2010. Crowdfunding of Small En- trepreneurial Ventures. Book chapter for “Entrepreneurial Finance” (Ed. D.J. Cumming), forthcoming at Oxford University Press.

36

[27] Mussa, M., and Rosen, S., 1978. Monopoly and Product Quality. Jour- nal of Economic Theory 18, 301-317.

[28] Nocke, V., M. Peitz and F. Rosar, 2011. Advance-Purchase Discounts as a Price Discrimination Device. Journal of Economic Theory 146, 141-162.

[29] Spencer, M.A., S.K. Swallow, J.F. Shogren and J.A. List, 2009. Rebate Rules in Threshold Public Good Provision. Journal of Public Economics 93, 798–806.

[30] Ward, C., and Ramachandran, V., 2010. Crowdfunding the Next Hit: Microfunding Online Experience Goods. Mimeo.

[31] Winborg, J., and H. Landstrom, 2001. Financial Bootstrapping in Small Businesses: Examining Small Business Managers’ Resource Acquisition Behaviors. Journal of Business Venturing 16, 235-254.

37

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