Product and Pricing Decisions in Crowdfunding · Hu, Li, and Shi: Product and Pricing Decisions in...

Vol. 34, No. 3, May–June 2015, pp. 331–345ISSN 0732-2399 (print) � ISSN 1526-548X (online) http://dx.doi.org/10.1287/mksc.2014.0900

© 2015 INFORMS

Product and Pricing Decisions in CrowdfundingMing Hu, Xi Li, Mengze Shi

Rotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada{[email protected], [email protected], [email protected]}

This paper studies the optimal product and pricing decisions in a crowdfunding mechanism by which aproject between a creator and many buyers will be realized only if the total funds committed by the buyers

reach a specified goal. When the buyers are sufficiently heterogeneous in their product valuations, the creatorshould offer a line of products with different levels of product quality. Compared to the traditional situationwhere orders are placed and fulfilled individually, with the crowdfunding mechanism, a product line is morelikely than a single product to be optimal and the quality gap between products is smaller. This paper also showsthe effect of the crowdfunding mechanism on pricing dynamics over time. Together, these results underscorethe substantial influence of the emerging crowdfunding mechanisms on common marketing decisions.

Keywords : crowdfunding; product line design; price discriminationHistory : Received: March 6, 2014; accepted: November 10, 2014; Ganesh Iyer served as the senior editor and

Dmitri Kuksov served as associate editor for this article. Published online in Articles in AdvanceJanuary 28, 2015.

Many hands make light work.—English Proverb

1. IntroductionCristian Barnett is a professional photographer liv-ing in Cambridge, England. Mr. Barnett was so fas-cinated with the Arctic Circle that in 2006 he startedvisiting the countries intersected by the circle. Afterseven years and a dozen trips to that area, he decidedto create a book called Life on the Line, which wouldcontain a selection of portraits he had taken overthe years. To raise funds to pay for the design andprinting of the books, Mr. Barnett launched the Lifeon the Line project on Kickstarter on November 26,2013 (see Figure 1). The funding period would expireon December 31, 2013, with a goal of £10,000. Everybuyer who committed £30 or more would receive asigned copy of the book. A buyer who committed£150 or more would receive a collector’s edition ofthe book, hardbound and in a slipcase, and signedwith the author’s personal dedication. Of course, thepledges would not be redeemed and the books wouldnot be delivered unless the project reached the goal.

Kickstarter is one of the leading online nonequitycrowdfunding sites that match people (the “crowd”)with projects. In such crowdfunding sites, creatorsraise funds from potential buyers to start their ven-tures. In return the creators offer products to thebuyers.1 The creators are entrepreneurs like Cristian

1 In equity crowdfunding sites, funders are also investors for thecreators’ financial endeavors (Agrawal et al. 2014). This paperfocuses on nonequity crowdfunding where buyers are also fundersbut not investors.

Barnett who may be designers, musicians, softwaredevelopers or any kind of inventor. Unlike the con-ventional retail setting, here a buyer not only com-mits to purchasing the product but also prepays tofund the project. A project will be successfully fundedonly if the total value of committed purchases exceedsa specified goal within a certain time.2 Recently thecrowdfunding industry has experienced tremendousgrowth. For example, since its inception in 2009 Kick-starter has raised more than $800 million and sup-ported more than 50,000 projects. With the passing inthe United States of the Jumpstart Our Business Start-ups Act in September 2013, the crowdfunding indus-try acquired legitimacy and is expected to lead a newera of entrepreneurship (Wortham 2013).

In this paper we investigate crowdfunding prod-uct and pricing decisions. Creators often provide aline of products for buyers to choose from. Cris-tian Barnett, for example, offered multiple versionsof his Life on the Line book, including a signed copyfor £30 and a collector’s edition for £150. We moni-tored all of the projects posted on Kickstarter betweenNovember 14–18, 2013. Among 457 newly launchedprojects, 448 (98%) offered more than one level ofrewards and prices. A seller may design a menuof offerings (with different qualities or in differentquantities) when buyers have heterogeneous valua-tions (Moorthy 1984, Monahan 1984). A crucial con-sideration in menu design is incentive compatibility;

2 One reason that creators seek the crowdfunding platform isthat private equity investors or venture capitalists seldom backentrepreneurs who are unable to show evidence of substantial sales.See, e.g., Cortese (2013).

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Hu, Li, and Shi: Product and Pricing Decisions in Crowdfunding332 Marketing Science 34(3), pp. 331–345, © 2015 INFORMS

Figure 1 (Color online) “Life on the Line” Project on Kickstarter

Source. Used with permission from Cristian Barnett.

each type of consumer should be better off choosingthe option designed for them rather than any otheroption. Following the same logic, a creator may pro-vide a menu of crowdfunding project products andpricing options to match different buyer preferences.In addition, a creator needs to consider the anticipatedproject success rate. In the menu design for a crowd-funding project, how would the creator’s considera-tion of project success affect the optimal product lineand pricing decisions?

To investigate this question, we propose and ana-lyze a two-period model where a creator may offer aline of products or charge different prices over timeto sequentially arriving buyers. The decisions of theearlier buyers are posted to the later arrivals. Thebuyers decide whether to purchase and which prod-uct option to choose according to their valuations.Because the project will not succeed unless the totalpurchases from all buyers meet the goal, buyers arelinked together by the common goal of project suc-cess. As a result, given two product options of similaror even identical quality, but at different prices, thebuyer with a high product valuation may choose thehigh-price option to ensure the success of the crowd-funding project. This will be the case as long as abuyer perceives that other buyers may have low prod-uct valuations. Interestingly, this “overpay” behav-ior can also improve buyer total surplus. Hence, theintroduction of discriminatory pricing strategies, suchas a menu, can be win-win for the creator and thebuyers. This result is in stark contrast to the tradi-tional situation wherein the lower price option would

be generally preferred, given little or no difference inquality. Moreover, the buyers’ incentive to collaboratefor the project’s success also affects the optimal prod-uct line design when quality decisions are endoge-nous. Compared to the traditional situation, a creatorusing a crowdfunding mechanism is more likely tooffer a menu of product options, and the difference inquality between product options tends to be smaller.

We extend the main model in several directions.First, we consider buyers’ altruistic motivations suchas warm glow, i.e., deriving positive utility fromspending more money to support a project. We showthat when buyers are altruistic, the main insights foroffering a menu of options remain. Moreover, buyersare willing to contribute more to the projects than inthe selfish setting. Second, we extend the base modelto situations wherein cohorts of buyers arriving intwo periods can have different product valuations orthere can be different numbers of buyers. Our mainfindings about menus of options hold in both cases.Surprisingly, a creator’s profit can be lower when thechance of having a high-type buyer in the secondperiod is greater. That is because the first high-typebuyer, being more optimistic about the second buyer’svaluation, is more likely to consider free-riding. Third,we consider demand uncertainty in the number ofbuyers in each period. Given the possibility of no-shows in one period, we find the results on the opti-mal menu prices to be the same. In addition, whentwo cohorts have an uncertain number of buyers, wefind that the pledged amount can exceed the target,

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Hu, Li, and Shi: Product and Pricing Decisions in CrowdfundingMarketing Science 34(3), pp. 331–345, © 2015 INFORMS 333

resulting in the commonly observed overfunding phe-nomenon. Finally, we extend the model by allowingthe buyers to choose their decision time. When thefirst buyer can choose to wait and decide later, wefind that the first buyer would prefer not to wait, andour main results on the menu strategy hold. When thecreator offers the optimal menu strategy in the basemodel and both buyers decide which period to enter,the creator’s profit can be even higher.

Literature ReviewThis paper contributes to the extensive marketing lit-erature of product line design. Moorthy (1984) pro-vides a general framework for a monopoly seller’sproduct line design problem. The marketing literaturehas investigated how product line design decisionsmay depend on market conditions such as the extentof horizontal differentiation (Desai 2001), the presenceof private information (Balachander and Srinivasan1994) and network externality (Jing 2007), the struc-ture of the distribution channel (Villas-Boas 1998),and customization (Syam et al. 2005, Valenzuela et al.2009). Consistent with this body of literature, ourpaper provides new insights into product line designin an emerging business model.

Second, this paper contributes to the public eco-nomics literature. Research in public economics hasextensively examined provision point mechanisms forimproving the private provision of public goods (see,e.g., Bagnoli and Lipman 1989, Varian 1994). In theprovision point mechanism and crowdfunding, goodswill be provided only when the preset contributionlevel is reached. Otherwise, the parties are not boundto carry through and any monetary contributionsare refunded. However, unlike the provision of pub-lic goods, a buyer in crowdfunding cannot benefitfrom a successfully funded project without pledg-ing at a suggested price. Moreover, price decisionsand the threshold (i.e., the provision point) are typ-ically treated as exogenous in the public economicsliterature.

The recent emergence of online crowdfunding hasalso attracted the attention of academic researchers(Agrawal et al. 2014). Several papers have empiricallyinvestigated the herding behavior caused by asym-metric quality information and observational learn-ing from early contributions (see, e.g., Agrawal et al.2011, Freedman and Jin 2011, Zhang and Liu 2012,Mollick 2014). We isolate the effect of asymmetricinformation and study specific marketing decisionsunder the provision point mechanism, as comparedto the conventional selling mechanism. Our paperyields new insights into how the crowdfunding busi-ness model may affect decisions about optimal prod-uct line and pricing.

A mechanism similar to crowdfunding is groupbuying wherein a seller offers promotional discounts

to a sufficiently large number of committed pur-chasers. Companies such as Groupon and LivingSo-cial have used this mechanism. Existing theoreticalresearch has examined many aspects of the groupbuying mechanism, including the ability to respondto market size uncertainty (Anand and Aron 2003),the use of informed consumers to enhance the val-uation of uninformed consumers in their social net-works (Jing and Xie 2011), disclosure of the cumula-tive sign-up information to increase deal success rates(Hu et al. 2013), and to signal a merchant’s high qual-ity (Subramanian and Rao 2014). Empirical researchhas studied consumers’ sign-up behavior in groupbuying with multiple levels of thresholds (Kauffmanand Wang 2001) and revealed two types of threshold-induced effects (Wu et al. 2014). While group buy-ing deals are often offered by established businesses,crowdfunding projects studied in this paper are typi-cally associated with new ventures. However, becauseboth mechanisms involve product sales, the insightsdeveloped in this paper can be applied to group buy-ing for exclusively designed products and services.

Finally, our model is related to the pay-what-you-want (PWYW) selling scheme wherein buyers canchoose to pay any price they wish. Like productdesign in crowdfunding, PWYW can be an effectiveprice discrimination mechanism (Chen et al. 2012).An important factor for the success of PWYW is thebuyer’s altruism and fairness concerns (Kim et al.2009, Chen et al. 2012, Gneezy et al. 2010). We also takeinto account similar behavioral considerations, such asthe altruism and warm glow effect, and examine theirinfluence on the crowdfunding design.

2. The Base Model and AnalysisIn this section, we develop a two-period model tostudy the creator’s decisions and the buyers’ sign-upbehavior. Online crowdfunding firms such as Kick-starter allow creators to set a target for each project;a project is deemed to be realized only when thetotal amount pledged exceeds the target. We startwith a base model, which assumes an exogenouslydetermined quality, to develop novel insights into thecrowdfunding buyer’s behavior. We analyze the cre-ator’s product line decisions in §2.1.

2.1. Model SetupConsider a simple two-period game wherein a risk-neutral creator adopts a sequential crowdfundingmechanism for selling products. The creator posts aproposed project, with specific product quality andprice information, on a crowdfunding platform. Thesign-up process expires after two periods. In eachperiod t, one buyer arrives at the proposed project.We denote the buyer at time t as Bt , with t = 112.For the proposed project to succeed, both buyers must

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sign up. This is necessary to model the coordinationbetween buyers. In practice, all crowdfunding projectsrequire many buyers. There are many reasons that aproject needs a certain number of buyers and a cer-tain amount of funding to start. For instance, on thesupply side, there may be economies of scale due tohigh initial set-up costs. Most digital products fall intothis category. On the demand side, the product mayexhibit positive externality. For the product to be of suf-ficient value, there must be enough users.

Buyers may have different product valuations. Tomodel this heterogeneity, we assume their valuationsare i.i.d. with the following two-point distribution:

Vt =

{

H with probability �1

L with probability 1 −�1

where H >L> 0. On arrival at the project in period 1,buyer B1 realizes private product valuation, makesthe purchase decision, and leaves the site. The creatorobserves and then announces the purchase decisionof B1. Then buyer B2 arrives at the project, realizes aprivate product valuation, observes the purchase deci-sion of buyer B1, and makes her own purchase deci-sion. Both buyers are rational; each makes a purchasedecision to maximize her own expected utility.

As mentioned, at the beginning of the game, thecreator makes product and pricing decisions, andposts them on a crowdfunding website. In addition,the creator decides the funding target, denoted by T .The creator commits to a provision point mechanismsuch that the project succeeds only if the total amountpledged reaches or exceeds T . Otherwise, the projectfails. When making decisions, the creator knows thedistribution of product valuations of two buyers, butdoes not know the exact realized valuations. The cre-ator’s goal is to maximize the expected profit from theproposed project.3 We assume, without loss of gener-ality, that there is no transaction cost associated withpledging or rewarding, and that there is no time dis-counting over the sign-up horizon.

Next we define alternative pricing strategies withthe target endogenized to be consistent with the strat-egy. We then analyze the profitability of the strategy.

2.2. Alternative Pricing Policies

Uniform Pricing. With a uniform pricing strategy,the creator posts a single price p for her product.Because the project succeeds only if both buyers signup and pay p, the creator can effectively set T = 2p.

3 In reality, creators can have different goals. An entrepreneur islikely to concentrate on expected revenue and profit, but an artistmay also want to reach a wide audience. We have chosen the profit-maximization framework so that our results will be directly com-parable to existing work on product-line design.

Although price p can take any positive value, giventhe two-point distribution of product valuations, theoptimal price should be p = H or p = L. Thus, it suf-fices to consider the following two cases.

Margin Strategy 4H5. With this strategy, the creatorsets the price at pH = H and the target at T H = 2H .Any target beyond 2H would consign the project tofailure. Any target in 4H +L12H7 is equivalent to 2H ,which sells only to high-type buyers, consistent withthe term margin strategy. Under this strategy, a high-type buyer will sign up, but a low-type buyer willdecline. This strategy has a success rate of sH = �2 andthe creator has an expected profit of �H = 2�2H .

Volume Strategy 4L5. With this strategy, the creatorsets the price at pL = L and the target at T L = 2L.Any target below 2L is equivalent to 2L, with only thelow price being paid, consistent with the term vol-ume strategy. Under this strategy, both buyers signup, regardless of their types, and the project alwayssucceeds, i.e., sL = 1. The creator’s profit is �L = 2L.

Compared with the margin strategy, the vol-ume strategy gives the creator a higher chanceof project success; however, given that the projectalways succeeds, the volume strategy yields a lowerprofit margin.

Intertemporal Pricing 4D5. With this strategy, a cre-ator sets different prices for different periods, denotedby pDt for period t, t = 112. Following the samelogic as before, given the two-point distribution ofproduct valuations, the optimal price in each periodmust be L or H . That leads to two candidate strate-gies, 4pD1 1 p

D2 5 = 4H1L5 or 4pD1 1 p

D2 5 = 4L1H5. The cre-

ator’s goal is T D = H + L. Any target in 42L1H +

L7 is equivalent to H + L, which leads to differentprices charged in different periods, consistent withthe term intertemporal pricing strategy. The successrate is sD = � and the creator’s expected profit is�D = �4H +L5. Note that since the buyers arrivingat different periods have the same distribution ofproduct valuations, the creator is indifferent betweenthese two intertemporal pricing strategies. However,as shown later in §4.2, the creator may prefer onetype over another when buyers have different prod-uct valuations.

Menu Pricing 4M5. With this strategy, the creatorposts a menu containing a high price pMh and a lowprice pMl , where pMl ≤ L ≤ pMh ≤ H . Unlike intertem-poral pricing, the optimal prices in the menu maynot be equal to the two valuation points H and L(see Lemma 1 below). The creator sets the targetat the sum of the high and low prices, i.e., T M =

pMh + pMl . Any target in 42pMl , pMh + pMl ] is equivalent topMh + pMl , which requires at least one buyer to pay thehigh price.

The menu pricing strategy proposed here may notappear to be valid because the same quality product

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has different prices. This is done intentionally to teaseout a buyer’s incentive to overpay in crowdfunding.4

With the traditional selling mechanism, such a menucould not work because each buyer would alwayschoose the lower price option. However, in crowd-funding, a buyer (who is also a funder) may choosethe higher price if such a choice could substantiallyincrease the likelihood of project success. In otherwords, one buyer’s behavior affects another buyer’sexpected utility. Positive externality arises through thecommon goal of project success.

To solve the optimal menu strategy, we use thebackward induction method. In period 2, buyer B2observes the contribution of the earlier buyer B1 andmakes her own decision accordingly. Specifically, if B1has signed up at pMh , B2 always signs up at the lowprice pMl ≤ L, regardless of her product valuations. Onthe other hand, if B1 has signed up at pMl , B2 eitherpledges pMh or does not sign up at all: It is meaning-less for B2 to sign up at pMl because the project willfail. Hence, buyer B2 should sign up at pMh if her prod-uct valuation is H . Otherwise, she should not sign up.Recall that the probability is � for a buyer’s productvaluation to be H .

Next we return to the first period and considerbuyer B1. If her product valuation is L, she alwayssigns up at pMl . Otherwise, she can choose fromtwo options. By choosing the low-price option pMl ,B1 expects a larger surplus H − pMl but a lower suc-cess rate at �. Alternatively, by choosing the high-price option pMh , the buyer expects a smaller surplusH − pMh but a higher success rate at 1. A high-type B1would prefer the high-price option pMh over the low-price option pMl if and only if the following incentive-compatibility (IC) condition is satisfied:

�4H − pMl 5≤H − pMh 0 (IC)

The creator decides the optimal menu of prices tomaximize the expected profit, subject to the condi-tion (IC). Analyzing the creator’s problem leads to thefollowing lemma.

Lemma 1 (Optimal Menu Strategy). With thismenu strategy, the creator’s optimal prices are pMh =

41 −�5H + �L, pMl = L, and the optimal target is T M =

41 −�5H + 41 +�5L. The corresponding success rate issM = �42 −�5, and the expected profit is �M = �42 −�5 ·441 −�5H + 41 +�5L5.

Lemma 1 indicates that with the menu pricing strat-egy (given that all others are the same), as longas the high price pMh is low enough to satisfy theIC condition, the high-type buyer B1 prefers to pay

4 The price menu can be close to actual practice. Many projects onKickstarter contain levels with minute quality differences.

the high price, even though a lower price option isavailable. The amount of overpayment is pMh − pMl =

41 −�54H −L5, which increases with product valua-tion gap H −L and decreases with �. Thus, whenproduct valuation is more heterogeneous between dif-ferent types of buyers or when buyer B1 is more pes-simistic about the product valuation of buyer B2, ahigh-type buyer B1 has a greater incentive to overpay.With a larger gap H −L, a high-type buyer B1 derivesmore utility from the project and is thus more will-ing to make a sacrifice to ensure the project’s success.Similarly, with a smaller �, the risk of project failure ishigh, and thus the incentive to overpay is higher. As �decreases from 1 to a small value � > 0, the amount ofoverpayment increases from 0 to 41 − �54H −L5, closeto H −L.

When a high-type B1 chooses price option pMh , sheenjoys a surplus of �4H −L5. Because the project willsucceed, buyer B2 will choose the low price pMl = L.Otherwise, had buyer B1 chosen pMl , a high-type buyerB2 would have to pay pMh and incur a reduction of sur-plus 41 −�54H −L5. Thus, when the high-type buyerB1 pays the higher price, the overpayment has a pos-itive externality effect on the second buyer’s surplus.Moreover, the size of the positive externality effectincreases with the high price option in the menu.Thus, the crowdfunding mechanism not only artifi-cially creates an externality effect among buyers’ deci-sions but also endogenously determines the size ofthat externality effect.

2.3. Optimal Pricing StrategyThe creator determines the optimal pricing strategyby comparing the expected profits from each of thealternative pricing strategies. We summarize our anal-ysis in the proposition below.

Proposition 1 (Optimal Pricing Strategy). Thecreator’s optimal strategy is

(i) volume strategy, if H/L≤ 42 −�25/4�42 −�55;(ii) menu strategy, if 42 − �25/4�42 − �55 ≤ H/L

and � ≤ 43 −√

55/2, or 42 − �25/4�42 − �55 ≤ H/L ≤

41 +�−�25/43�−�2 − 15;(iii) intertemporal strategy, if 1

2 ≤ � and 41 +�−�25/

43�−�2 − 15 ≤ H/L ≤ 1/42�− 15, or 43 −√

55/2 ≤

�≤12 and 41 +�−�25/43�−�2 − 15≤H/L;

(iv) margin strategy, if 12 ≤ � and 1/42�− 15≤H/L.

Proposition 1 indicates that each of the four pric-ing strategies can be optimal within certain parametersubspaces. We illustrate the results in Figure 2. Thevertical axis represents the ratio of high-type to low-type buyer’s product valuations H/L; the horizontalaxis represents the fraction of high-type buyers (�)in the market. The figure shows that uniform pricingstrategies are optimal in two extreme cases. Specif-ically, given the valuation L, the volume strategy is

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Figure 2 Optimal Strategy with Different Values of H/L and �

0.2 0.4

Fraction of high end consumers �

0.6 0.8 1.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

H/L

Volumestrategy

Menustrategy Margin

strategy

Intertemporalstrategy

optimal when buyers are unlikely to have high prod-uct valuation (i.e., small �) or when high- and low-type buyers have a narrow valuation gap (i.e., H/L isclose to 1). Intuitively, if both buyers are very likelyto have a low product valuation, the creator shouldpursue the volume strategy. The expected gain fromtargeting high-type buyers only is not worth the riskof project failure. At the other extreme, if buyers arevery likely to have a high valuation (i.e., large �) andthe valuation gap between high- and low-type buy-ers is large (i.e., large H/L), the creator should choosethe margin strategy. In other words, if both buyers arevery likely to have a product valuation that is muchhigher than the low valuation, the creator should pur-sue the margin strategy. The gain from targeting high-type buyers is large, and the risk is bearable.

Two types of discriminatory pricing strategies, theintertemporal and the menu, can be more profitablethan uniform pricing strategies when the fraction ofhigh-type buyers (�) is not very large and the valua-tion ratio (H/L) is large enough, in other words, whenbuyers believe that others may have low productvaluations that are sufficiently heterogeneous. (Thedifference between these two discriminatory pricingstrategies is timing. While the same menu of twooptions exists in both periods with a menu pricingstrategy, a unique option is available in each periodwith an intertemporal pricing strategy.) First, with ourmodel, note that an optimal discriminatory pricingstrategy degenerates into a uniform pricing strategyin the traditional selling setting. The intertemporal ormenu pricing strategy becomes optimal with a crowd-funding mechanism because two buyers are linked bya common target. As a result, a high-type buyer maychoose the high-price option to compensate for thesmall contribution from a low-type buyer. Second, thebuyers can achieve coordination without any explicit

Table 1 Success Rate and Target

Success rate Target

Volume strategy 1 2LMenu strategy 2�− �2 41− �5H + 41+ �5L

Intertemporal strategy � H + L

Margin strategy �2 2H

interactions. In our model, self-interested buyers donot incorporate other buyers’ utilities into their objec-tives (as in Chen and Li 2013), nor do some buyerscommunicate with others to increase their valuations(as in Jing and Xie 2011).

Table 1 shows the target amount and project suc-cess rate for each type of pricing strategy. The tar-get amount increases in the following order: volumestrategy, menu strategy, intertemporal strategy, andmargin strategy; the project success rate decreases inthe same order. When the fraction of high-type buyers(�) increases, the differences in the success rate amongdifferent strategies diminish, but the differences in thetarget levels remain large. A high target leads to a lowproject success rate, and vice versa.5

The corollary that follows summarizes the socialwelfare implications of discriminatory pricingstrategies.

Corollary 1 (Social Welfare). The introduction ofdiscriminatory strategies can improve creator and buyersurplus.

(i) If 42−�5/�<H/L<max843−√

55/2, 41+�−�25/43�−�2 −159, the introduction of menu strategy improvesboth creator and buyer surplus over the optimal strategy ofthe other three strategies.

(ii) If 12 ≤ � and 42 − �541 + �5/45� − 2 − �25 ≤

H/L≤ 1/42�−15, the introduction of intertemporal strat-egy improves both creator and buyer surplus over the opti-mal strategy of the other three strategies.

Corollary 1 states that discriminatory pricing strate-gies can improve buyer surplus and social welfare.With discriminatory pricing strategies, the creator canserve more buyers. Margin strategy serves high-typebuyers only. As to volume strategy, because of lowprofitability it is often not sustained as the optimalpricing strategy, even though it serves all buyers. Incontrast, with the menu and intertemporal strategies,the creator can reach all high-type buyers and a frac-tion of low-type buyers and, at the same time, achievehigh profitability.

5 Our model can be easily adapted to situations wherein a not-for-profit creator wants to maximize the success rate, subject to rais-ing enough funds to cover set-up costs in advance. The amountof fixed set-up costs can become the exogenous target. Table 1can be used as a guide for the optimal pricing strategy given theexogenous target. For example, if the exogenous target T falls inthe range 42L1 41−�5H+41+�5L7, the menu strategy maximizes thesuccess rate.

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Discussions. In a model without product quality dif-ferentiation, the menu pricing strategy can be optimalin a crowdfunding mechanism. When buyers are suf-ficiently heterogeneous in product valuation, offeringa menu of prices can help achieve a better balancebetween volume (or success rate) and margin. Notethat a self-interested high-type buyer might be willingto pay extra to ensure the project’s success. Thus, byturning buyers into funders, the crowdfunding mech-anism enhances coordination among different buyers.Moreover, the menu pricing strategy, by choosing aproper level of high-price options below H , moder-ates the high-type buyer’s sacrifice and optimizes thecoordination incentive. Crowdfunding creates com-patibility between the purchases of two buyers whoshare the common goal of the project success. As aresult, each buyer’s behavior has an external effect onother buyers’ utilities. The extent of that externalityeffect is regulated by the price difference in the menu.

The high-type buyer’s incentive to pay extra relieson the belief that the creator is committed to the pro-vision point mechanism, i.e., the project will not becarried out unless the target is met. Although it isnot the focus of this paper, we have conducted a for-mal analysis to confirm that when a menu strategyis optimal, it is in the best interest of the creator tocommit to the provision point mechanism. Intuitively,the absence of such a commitment would jeopardizethe ability of the menu strategy to price discriminateamong buyers. If a high-type buyer expects the cre-ator to continue with the project even if the target isnot met, there would be little incentive for the buyerto pay extra. (Please see the online appendix (avail-able as supplemental material at http://dx.doi.org/10.1287/mksc.2014.0900) for a detailed analysis.)

3. Product Line DesignIn this section, we extend the base model to allow thecreator to offer two vertically differentiated products.We explore how the crowdfunding mechanism mayalter the optimal product line design, specifically, theoptimal quality gap between product options. For thisanalysis, we modify the base model as follows: First,buyers’ valuations depend on the quality of the prod-ucts. Specifically, for a quality level Q, we assumeproduct valuation Vh = QH for a high-type buyer,and valuation Vl =QL for a low-type buyer. The basemodel is a special case, with the quality level fixedat Q = 1. Second, in keeping with the existing litera-ture, we assume that the production cost is a functionof quality. In particular, we assume that the unit pro-duction cost of a good with a quality level Q is Q2/2.This quadratic form is widely used in the marketingliterature (see, e.g., Guo and Zhang 2012). Finally, toaccount for the development cost, we assume that the

creator must produce two units of products, whichmay be available in different qualities.

We denote the creator’s menu decision by 4QMl 1PM

l 5and 4QM

h 1PMh 5. (To distinguish the product line model

from the base model, we use capital letters.) The cre-ator’s funding target is T M = PM

h + PMl . Analysis of

this model is similar to that of the base model. Themajor distinction is the IC condition that ensures self-selection by a high-type buyer B1. It now becomes

�4HQMl − PM

l 5≤HQMh − PM

h 1 (1)

where the buyer’s surplus hinges on quality, too. Thecreator’s expected profit is given by

çM= 4�2

+ 2�41 −�55

(

PMh + PM

l −4QM

h 52

2−

4QMl 52

2

)

1

(2)

subject to the IC condition (1) and PMl ≤ LQM

l . Forgiven quality levels 4QM

h 1QMl 5, clearly the optimal

prices are achieved when both inequalities in the con-straints are binding, i.e., PM

h = HQMh − �4H −L5QM

l ,PMl = LQM

l 0 The first order condition of the creator’sproblem yields the optimal quality choices QM

h =H ,QM

l = L−�4H −L5. We summarize the optimal (inte-rior or boundary) solution as follows:

Proposition 2 (Optimal Product Line Design).In the product line model,

(i) if H/L≤ 41+�5/�, the optimal quality levels are4QM

h 1QMl 5= 4H1ã5, where ã≡L−�4H−L5≤L, and the

optimal prices are 4PMh 1PM

l 5= 4H 2 −�4H−L5ã1Lã5. Thecorresponding expected profit of the creator is çM =

4�42−�5/25·441+�25H 2 −2�41+�5HL+41+�52L253(ii) if H/L ≥ 41 + �5/�, the optimal quality lev-

els are 4QMh 1QM

l 5 = 4H105, and the optimal prices are4PM

h 1PMl 5 = 4H 2105. The corresponding expected profit of

the creator is çM = 4�42 −�5/25H 20

To see the effect of the crowdfunding mechanismon product line design, we compare the results of thismodel with the traditional selling situation. In thatsituation, the current model would have a differentself-selection IC condition

HQTl − P T

l ≤HQTh − P T

h 1 (3)

where the superscript T specifies the traditionalmodel. Here the creator’s goal is to maximize herexpected profit

çT= �

(

P Th −

4QTh 5

2

2

)

+ 41 −�5

(

P Tl −

4QTl 5

2

2

)

1 (4)

subject to the IC condition (3) and P Tl ≤ LQT

l . Sim-ilarly, the optimal prices are achieved when bothinequalities in the constraints are binding, i.e., P T

h =

HQTh − 4H −L5QT

l 1PTl = LQT

l . For purpose of compar-ison, we summarize the optimal solution to the tradi-tional problem as follows:

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Lemma 2. In the traditional product line design prob-lem, the optimal strategy is:

(i) if H/L≤ 1/�, QTh =H , QT

l = 4L−�H5/41−�5 andP Th = 4H 2 − 41 +�5HL+L25/41 −�5, P T

l = 44L−�H5L5/41 −�5;

(ii) if H/L ≥ 1/�, QTh = H , QT

l = 0 and P Th = H 2,

P Tl = 0.

Comparing the quality differences between4QT

h−QTl 5 and 4QM

h −QMl 5, we have:

Proposition 3 (Less Differentiation in Crowd-funding). Qualities are less differentiated in the crowd-funding scenario.6

To understand Proposition 3, note first that inthe case of perfect discrimination, the creator knowseach buyer’s true type and can supply products ofquality H to high-type buyers and products of qual-ity L to low-type buyers. When buyers hold privateinformation on quality valuations, to satisfy the self-selection condition the creator must distort qualitylevels downward. According to our analysis, in bothtraditional and crowdfunding settings, QT

h =QMh =H .

This is the familiar “no distortion at-the-top” result(Moorthy 1984). The downward quality distortioncomes from the low-quality product, and is smaller inthe crowdfunding setting, i.e., QT

l <QMl <L.

We explain this result through IC conditions (1)and (3), together with objective functions (2) and (4).The low-quality option is associated with project suc-cess rate � in crowdfunding’s IC condition (1); theassociated success rate is 1 in the IC condition (3)of the traditional setting. The difference in the suc-cess rates is also reflected in the objective functions.Intuitively, we can see that in crowdfunding, high-type buyers are concerned about both the success rateof the project and surplus from purchase. If a high-type buyer had deviated from the high-quality optionand chosen the low-quality product, the buyer wouldexperience a higher surplus from the option, but facea lower success rate. In light of this trade-off, whichitself deters deviation and does not exist in the tradi-tional setting, the creator can afford a less downwarddistortion, i.e., to set the low-option quality higher.

3.1. Strategy ComparisonNext we evaluate the profitability of the menu strat-egy compared to the three other strategies: margin (H ),volume (L), and intertemporal (D). For each strategy,we analyze the optimal quality level or levels, andits corresponding expected profit. We summarize theresults in the next lemma.

6 Unless otherwise specified, any comparison or monotonicity is inits weaker sense. For example, less means no more and increasingmeans non-decreasing.

Lemma 3. When product qualities are endogenized, thecreator’s optimal decisions under each of the three strategiesare as follows:

(i) margin strategy: QH =H1PH =H 2, and çH =�2H 2;(ii) volume strategy: QL = L1P L = L2, and çL = L2;(iii) intertemporal strategy: QD

t = H , PDt = H 2,

QD3−t = L1 PD

3−t = L2, where the solutions are symmetric fort = 112. Moreover, çD = 4�/254H 2 +L25.

In the intertemporal strategy, we can also measurethe optimal quality gap between the two products,only one of which is offered in a period. Clearly, theoptimal quality gap with the intertemporal strategy,H −L, is also smaller than that in the traditional prod-uct design. The driving force is similar to that whichwe explained for the menu strategy. We compare theprofitability of various pricing strategies, leading tothe following proposition:

Proposition 4. When product qualities are endoge-nized, the creator’s optimal strategy is

(i) volume strategy, if H/L ≤ �1, where �1 = 4−2�2 −

�3 + �4 −√

4�− 6�2 +�4 + 2�5 −�65/4−2� + �2 −

2�3 +�45;(ii) menu strategy, if �1 ≤ H/L ≤ �2, or �1 ≤ H/L and

� ≤23 , where �2 = min441 + �5/�1 4−2� − �2 + �3 −

√−4 + 9�2 + 2�3 − 3�45/4−2 + 3�− 2�2 +�355;(iii) volume strategy, if �2 ≤H/L and 2

3 ≤ �.

Proposition 4 confirms that the same insightobtained from Proposition 1 for the base modelapplies to the generalization under which productqualities are endogenized. That is, in crowdfundingthe menu pricing strategy, as a discriminatory choice,can be more profitable than uniform pricing strate-gies. Unlike the base model, when product qualitiesare endogenized the intertemporal strategy can beshown to be dominated by other strategies, thoughit may still be more profitable than uniform pricingstrategies under certain conditions.7 As in Figure 2in the base model, Figure 3 demonstrates the optimalproduct and pricing strategy under different marketconditions. From Figure 3, we see again that the menustrategy becomes optimal when the fraction of high-type buyers (�) is not too large and the valuationdifferentiation between two types of buyers (H/L) islarge enough. Again, this figure implies that no sin-gle marketing strategy dominates in all parameterspaces. Creators should tailor their product and pric-ing strategies to the specific market conditions associ-ated with their projects.

Our main insights about the high-type buyer’sbehavior and implications for menu design have

7 The intertemporal strategy is more profitable than the volumestrategy when H/L≥

√

42 −�5/�, and more profitable than the mar-gin strategy when H/L≤ 1/

√2�− 1 or �≥

12 .

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Figure 3 Optimal Strategy with Different Values of H/L and � inProduct Line Design

0 0.2 0.4 0.6 0.8 1.01

2

3

4

5

H/L

Menustrategy

Volumestrategy

Marginstrategy

Fraction of high end consumers �

strong relevance to crowdfunding practices. Althoughhigh-type buyers may not be the majority, their contri-butions are critical to project success. To collect someempirical evidence, we monitored the progress of allnew projects launched on Kickstarter from 15:15 to23:15 on January 15, 2014. There were a total of 60such projects; 58% succeeded; 95% posted a menuconsisting of multiple levels of reward-price offers.On average, the top 10% of buyers contributed 52.2%of the total pledged amount. There were a total of8,754 buyers, 87.6% of whom paid the exact pricessuggested. Thus, consistent with our results, a major-ity of consumers paid according to the menus.

4. Model ExtensionsIn this section we discuss a number of extensions tothe base model. Analyzing these extensions not onlyestablishes the robustness of our results on crowd-funding product and pricing decisions but also deep-ens our understanding of these issues. To simplify thediscussion, we may concentrate on the menu strategywhen similar insights are applicable to the intertem-poral strategy.

4.1. Noneconomic MotivationsThe main model assumes that all buyers are self-interested and make decisions to maximize theirown surplus. However, for many crowdfunding pro-jects, especially those posted by artists and not-for-profit organizations, the buyers can be influencedby noneconomic or altruistic motives (Mollick 2014).Here we consider two commonly examined forms ofsuch motives, i.e., warm glow and the prestige effect.We also examine the implications for product andpricing decisions. In both cases we add a term fornoneconomic motivation to a buyer’s utility function.

In the warm-glow model, a buyer’s utility func-tion becomes U = 4V − p5+� · p, where 4V − p5 is the

economic surplus in the base model and p representsthe amount the buyer contributes to the project. Theparameter � ∈ 60115 represents the significance of thewarm-glow effect. This formulation follows the previ-ous literature on warm glow and altruistic motivation(see, e.g., Andreoni 1990).

We now examine the influence of the warm-gloweffect on the menu strategy. Suppose the creator setsthe prices at p

Mw

h and pMw

l . To motivate a low-typebuyer to participate, we need to make sure that L −

pMw

l +�pMw

l ≥ 00 To ensure that a high-type buyerselects the high-price option, we need to ensure thefollowing IC condition: H − p

Mw

h + �pMw

h ≥ �4H −

pMw

l +�pMw

l 50 Incorporating these two constraints intothe creator’s problem, we can solve for the optimalpricing strategy, summarized as follows:

Proposition 5 (Warm Glow). When buyers experi-ence the warm-glow effect, the optimal menu strategy is4p

Mw

h 1 pMw

l 5= 4441−�5H +�L5/41−�51L/41−�55, whichamplifies the optimal menu prices in the base model by afactor of 1/41 −�5.

The prestige effect is a discrete form of the warm-glow effect based on relative contributions. When acreator offers several categories of contributions, thosewho choose the higher categories could experiencethe prestige values (Harbaugh 1998). Assume that abuyer’s additional utility from the prestige effect is up.To motivate a high-type buyer to choose the high-price option, it is necessary to ensure that H − p

Mw

h +

up ≥ �4H − pMw

l 50 Given this revised IC condition, thebuyer’s optimal pricing strategy is summarized asfollows.8

Proposition 6 (Prestige). When buyers experiencethe prestige effect, the optimal menu strategy is4p

Mw

h 1pMw

l 5= 441−�5H+�L+up1L5.

The above propositions imply that, with either thewarm-glow or prestige effect, the parameter spacesunder which a creator should adopt the menu strat-egy expand. Moreover, both the high-price option andthe price gap increase. Intuitively, we can see thatboth effects increase the expected value of and therelative preference for the high-price option. Thus,both warm-glow and prestige effects provide addi-tional justifications for the creator to offer more thanone option.

The base model considers one buyer in each periodwith the same distribution of valuation. Next weexamine two alternative models. In the first, two buy-ers have different valuations. In the second, a differentnumber of buyers arrive in two periods.

8 When prestige effect is proportional to the additional amountpaid with coefficient � , the IC condition becomes H − pMw

h +

�4pMwh − pMw

l 5 ≥ �4H − pMwl 5. The optimal menu strategy is then

4pMwh 1 pMw

l 5= 441 −�5H + 4�− �5L5/41 − �51L5.

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4.2. Time-Varying ValuationsTwo buyers arriving in different periods may notfollow the same distribution of product valuations.For example, buyers with a special interest in cer-tain types of projects are more likely to search forsuch projects and arrive at them earlier. A creator mayalso share the proposed project first with family andfriends, who may be willing to pay more to showtheir support. To consider such time-varying valua-tions, we assume that the buyers arriving at two peri-ods follow different value distributions. Specifically,let �t be the probability that buyer Bt , t = 1121 hashigh valuation H . Then the creator’s expected profit is�Hh = 2�1�2H with the margin strategy; the expectedprofit is �Lh = 2L with the volume strategy. With theintertemporal pricing strategy, the creator charges thehigh price during the period with the larger �, andthe expected profit is �Dh = max4�11�254H +L5.

Lemma 4. With time-varying product valuations, thecreator’s optimal menu strategy is p

Mh

h = 41−�25H +�2L,pMh

l = L. The corresponding expected profit is �Mh = 4�1 +

�2 −�1�25441 −�25H + 41 +�25L5.

The proof is analogous to that of Lemma 1. A com-parison of the expected profits with various pricingstrategies leads to the following proposition:

Proposition 7. The menu pricing strategy is moreprofitable

(i) than the volume strategy if 42 − �1 + �2 − �1�25/4�1 +�2 −�1�25≤H/L;

(ii) than the margin strategy if H/L ≤ 441 + �25 ·

4�1 +�2 −�1�255/44�1�2 + �22 − �1 − �2 − �1�

225, or, �1

and �2 satisfy �1 +�2 − 4�1�2 −�22 +�1�

22 ≥ 0;

(iii) than the intertemporal strategy if H/L ≤ −4� +

�22 −�1�

225/4�−2�1�2 −�2

2 +�1�225, or, �1 and �2 satisfy

2�1�2 +�22 −�1�

22 ≤ �, where �≡ min4�11�25.

Proposition 7 is consistent with the results for themenu pricing strategy in the base model. Interestingly,�1 and �2 have very different effects on the creator’sprofit. Taking derivatives of �Mh with respect to �1and �2, respectively, yields the following result:

Corollary 2. With time-varying product valuations,a creator following the menu strategy sees the expectedprofit increasing in �1; increasing in �2 if �2 ≤ �∗

2 ≡

4H − 2�1H +L5/4241−�154H −L55, and decreasing in �2if �2 ≥ �∗

2.

Though it is not difficult to see the first partof Corollary 2, i.e., that the creator’s expectedprofit increases in �1, the second part is intriguing.The latter implies that the buyer may gain morewhen buyer B2’s expected valuation �2H + 41 − �25Ldecreases. To understand this, recall that the creator’sexpected profit hinges on the success rate and thetarget, which equals p

Mh

h + pMh

l . While a larger �2

increases the success rate, it can reduce the price pMh

h

and hence the target: When �2 is large enough, withthe expectation that B2 has a greater chance of being ahigh type, B1 may be reluctant to pay the high price,hoping that B2 pays it instead. In short, B1 is morelikely to free ride when �2 rises. With these forces,the effect of �2 on the creator’s profit is no longermonotonic.

Even after all of the alternative pricing strategiesare considered, the expected profit from the best pric-ing strategy may still decrease in �2. These resultshave implications for how creators should manage thebuyers’ arrival sequence and their beliefs about theproduct valuations of future buyers. Under some con-ditions, persuasive marketing campaigns to increase abuyer’s valuation in period 2 can backfire and reducethe creator’s profit. This implies that a creator maybenefit from certain demarketing efforts in the secondperiod (see, e.g., Miklos-Thal and Zhang 2013, Gerst-ner et al. 1993, Pazgal et al. 2013).

4.3. A Two-Cohort ModelInstead of assuming one buyer in each period, wenow consider a cohort of buyers arriving in eachperiod t, t = 112, but keep the assumption of iden-tical value distributions. We let the size of the twocohorts be n1 and n2, respectively. As in the basemodel, here the valuation distributions Vt for bothperiods are i.i.d., following a two-point distribution,equal to Vt =H with probability � and Vt = L withprobability 41−�5. We further assume that the projectrequires all buyers from both cohorts to participate.

The analysis of the margin and volume strategiesproceeds in the same way as in the base model,except that the targets are now 4n1 + n25H and4n1 +n25L, respectively. The expected profits withthese two strategies are �Hc = �24n1 +n25H and �Lc =

4n1 +n25L, respectively. With the optimal intertem-poral pricing strategy, the creator charges cohort 1price H and cohort 2 price L if n1 ≥ n2 (which isreversed if n1 <n2), and the expected profit is �Dc =

�4n1H +n2L5.With the menu pricing strategy, we generalize the

argument made in the base model that the projectis successful only if one buyer is of the high type.In the extended model, a project succeeds onlywhen one cohort of buyers are of high type. Corre-spondingly, the target will be T Mc = max4n11n25p

Mc

l +

min4n11n25pMc

h , where pMc

l ≤ L1pMc

h ≤ H are the menuprices offered in both periods. Because each buyercan pay between p

Mc

l and pMc

h , the total contributionfrom the first cohort of buyers will be between n1p

Mc

l

and n1pMc

h . Suppose that the first cohort has paid � ∈

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6n1pMc

l 1n1pMc

h 7, and the remaining amount needed isT Mc −�> 0. The conditional success rate is then

sMc =

{

1 if T Mc −�≤ n2pMc

l 1

� if n2pMc

l <T Mc −�≤ n2pMc

h 0

The IC condition for cohort 1 can be written as n1H −

4T Mc −n2pMc

l 5≥ �4n1H − 4T Mc −n2pMc

h 550

Proposition 8. In the two-cohort model, the optimalmenu strategy has a target of T Mc = max4n11n25p

Mc

l +

min4n11n25pMc

h , and an expected profit of �Mc =

�42 −�5T Mc , where(i) p

Mc

h = min4H1 41 − �54H − L54n1/n25 + L5 andpMc

l = L, if n1 ≥ n2;(ii) p

Mc

h = 41 −�5H +�L and pMc

l = L, if n1 ≤ n2.

It can be shown that the optimality of the menupricing strategy, as well as the intertemporal strat-egy, is sustained in parameter spaces similar to thebase model. However, the two-cohort model exten-sion leads to the following new insights:

(i) The menu pricing strategy requires that oneof the two cohorts of buyers have high valuations.The creator should target the higher price towardthe smaller cohort, thus reducing the effectiveness ofthe menu pricing strategy as a price discriminationmechanism.

(ii) If n1 > n2, the price pMc

h is higher, and the sec-ond cohort of buyers finds it harder to free ride onthe contributions from the first cohort. The high pricepMc

h increases in this case because the first cohort’s sur-plus depends not only on individual surplus H − p

Mc

h

but also on the portion of buyers contributing pMc

l .Because the first cohort is less sensitive to p

Mc

h , the cre-ator increases the high price in the menu, which hurtsthe second cohort.

To summarize, in the above two subsections, theoptimality of the menu pricing strategy, as well as theintertemporal strategy, is sustained in similar parame-ter spaces. Additional insights arise from the changedincentives to free ride on another cohort’s contribu-tions. A larger second cohort, or one whose prod-uct valuations are expected to be higher, could creategreater free-riding incentives for the buyers in the firstcohort, thus making it more challenging for the cre-ator to discriminate among buyers through a menustrategy.

4.4. Uncertainty in Number of BuyersWe now examine the implications of demand uncer-tainty. We first consider the possibility of no-show inone period, followed by an uncertain number of buy-ers in two cohorts.

4.4.1. No-Shows. Suppose that in the first periodof the base model, buyer B1 arrives with probabil-ity �. Conditional on the arrival of buyer B1, the ICcondition of the menu strategy remains H − pMh ≥

�4H − pMl 5, then the optimal menu prices are the sameas in Lemma 1. Next, suppose that in the secondperiod, buyer B2 arrives with probability �. In thiscase, the project may still fail even if buyer B1 has con-tributed pMh . The IC condition becomes �4H − pMh 5 ≥

��4H − pMl 5, which is equivalent to the original ICcondition. Hence the optimal menu prices remain thesame as in Lemma 1. In both cases, the project successrate and the creator’s expected profit under the menustrategy shrinks by a factor of �. Because the prof-its under other pricing strategies are also reduced bythe same factor �, the parameter spaces wherein eachpricing strategy is optimal remain the same. There-fore the presence of demand uncertainty in the formof no-show does not change the main results obtainedin the base model.

4.4.2. Uncertain Number of Buyers and Over-funding. To consider the uncertain number of buyersin a two-cohort model, we suppose that the number ofbuyers in the first period is a constant n1 and that thenumber of buyers in the second period N2 is random.Let N2 follow a two-point distribution: With proba-bility �, N2 = nH

2 , and with probability 1 −�, N2 = nL2 .

First consider the case wherein n1 ≤ nL2 <nH

2 . Con-sistent with the base model, to induce a high-typefirst cohort to pay the high price to ensure success,the creator sets the target T Mo = n1p

Mo

h + nL2p

Mo

l fora menu 4p

Mo

l 1 pMo

h 5. Further assume that nH2 is large

enough such that even if the first cohort has low val-uation and chooses to pay p

Mo

l , the project can suc-ceed when N2 = nH

2 , i.e., 4n1 +nH2 5p

Mo

l ≥ T Mo . Then, theIC condition for the first cohort is n14H − p

Mo

h 5≥ 4�+

41 −�5�5n14H − pMo

l 5, which yields the optimal pricingstrategy p

Mo

h = 41−�541−�5H + 4�+�−��5L, pMo

l = L.When the first cohort has high valuation and N2 = nH

2 ,the creator collects n1p

Mo

h + nH2 p

Mo

l , which exceeds thetarget T Mo . Here the target is set ex ante and the firstcohort makes pledge decisions before knowing theactual size of the second cohort. Overfunding occursin equilibrium when the second cohort is large. Thesame phenomenon occurs for the case wherein nL

2 ≤

n1 ≤ nH2 , and the cases wherein the size of the first

cohort is random (see the online appendix). Over-funding is commonly observed in crowdfunding. Forexample, among the 60 projects we tracked on Kick-starter, the creators on average collected 27% morethan the targets.

The base model assumes that each buyer arrivesat the site at an exogenously determined time andmakes the sign-up decision on arrival. In reality, thebuyer who arrives early may choose to postpone the

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decision. If both buyers are fully informed of the post-ing, they may simultaneously choose their purchasetimes. We examine these two possibilities in §§4.5and 4.6.

4.5. Strategic DelayHere we extend the base model by allowing buyerB1 to postpone the pledge decision, and analyze howher decision timing may affect the creator’s optimalproduct and pricing strategy.

Proposition 9. In a sequential crowdfunding modelallowing for strategic delay, the first buyer would makethe sign-up decision in the first period. The optimal prod-uct and pricing decisions are identical to those in the basemodel.

The underlying logic for the above proposition isas follows. First, a low-type buyer B1 is indifferentbetween waiting or making the decision immediatelybecause the creator sets a price that fully extracts thesurplus from such a buyer. Postponing the decisionto the second period will not improve the expectedsurplus. To break the tie, we assume that a low-type buyer B1 always purchases at pMl = L in the firstperiod.

Now consider a high-type buyer B1. If this buyerchooses to postpone the purchase decision, in period 2there will be two buyers in the market, B1 and B2. Inthis simultaneous game of period 2, one may worryabout the possibility that the high-type buyer B1 willchoose the low option in the hope that another (hightype) buyer chooses the high option. However, giventhe uncertainty about buyer B2’s type, buyer B1 stillfaces the same trade-off between paying a high pricefor guaranteed success and paying a low price foran uncertain outcome. Moreover, the optimal menuis designed such that a high-type buyer is indifferentbetween choosing the high and low option. Hence, itis optimal for B1 to pay pMh in the second period. Evenif B1 postpones the sign-up decision, the expected sur-plus is H − pMh , which is identical to the payoff frommaking the sign-up decision immediately in the firstperiod. Therefore, a high-type buyer B1 should haveno incentive to postpone the sign-up decision.

4.6. Endogenous ArrivalsNow we further extend the base model by adding,before the two-period model, a period 0 when twobuyers select an entry period. The sequence of eventsis as follows: (1) Buyers simultaneously decide whichperiod to enter; (2) Buyer(s) entering in period 1choose a price option; (3) Buyer(s) entering in period 2observe decisions made in period 1 and choose a priceoption. Here we assume the creator adopts the opti-mal menu strategy specified in Lemma 1. (The gen-eral game of endogenizing the creator’s pricing strat-egy followed by a three-stage game with endogenous

arrivals can be extremely complicated. As demon-strated below, the subgames may have multiple equi-libria.) We limit our attention to a symmetric equilib-rium wherein a buyer’s strategy depends only on thebuyer’s type.

First, both types of buyers choosing to enter inperiod 2 is an equilibrium outcome. For low-type buy-ers, because their surpluses would be fully extracted,they are indifferent between entering in periods 1and 2. Now consider a high-type buyer. Intuitively,regardless of the entry period chosen, she does notknow the contribution made by the other buyer.Under the menu strategy, because the optimal menuis designed such that a high-type buyer is indiffer-ent between the high and low options, the high-type buyer has no incentive to deviate and enter inperiod 1.

Next we consider other possible pure-strategy sym-metric equilibria.

• Both low- and high-type buyers enter in period 1.This is not an equilibrium because a high-type buyeris always better off waiting until period 2. We fixone buyer who follows the strategy of always enter-ing in period 1 and consider whether the other buyerhas any incentive to deviate. A high-type buyer whoenters in period 1 would optimally pay pMh , earningsurplus H − pMh , because she cannot observe the con-tribution made by the other buyer who also enters inperiod 1. However, if she deviates to period 2, herexpected surplus is higher at H − �pMh − 41 − �5pMl ,because she can free ride when the other buyer inperiod 1 is a high type and has paid pMh .

• Low-type buyers enter in period 2 and high-typebuyers enter in period 1. For the same reason statedabove, this strategy is not an equilibrium.

• Low-type buyers enter in period 1 and high-typebuyers enter in period 2. If no one pays in period 1,both buyers will know that the other buyer is a hightype. Then in period 2, the only symmetric equilib-rium is in mixed-strategies, with each high-type buyerchoosing pMh with probability � such that �4H −pMl 5=

H − pMh . Solving this yields � = �. Therefore, eachhigh-type buyer optimally chooses pMh with probabil-ity � and pMl with probability 1 −�, which results inan expected surplus of H − pMh . If a high-type buyerdeviates and enters in period 1, by the same analysisdescribed above, the expected surplus is also H − pMh .Thus, high-type buyers are indifferent between devi-ating or not. This validates another equilibrium ofendogenous sequencing.

In summary, there are two symmetric equilibriumstrategies for the game with endogenous arrivals.In one equilibrium, both types of buyers enter inperiod 2. Here, the creator expects a higher profit thanin the base model under the same menu strategy asspecified in Lemma 1 (see the online appendix for a

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full analysis of the simultaneous game and its com-parison with the sequential game). In another equilib-rium, low-type buyers enter in period 1 and high-typebuyers enter in period 2. In this equilibrium, althoughthe creator may be in a worse position than in thebase model, he can still achieve higher profits thanthe optimal uniform strategies because a high-typebuyer can be induced to pay the high price option pMhwith some probability.

5. ConclusionCrowdfunding is emerging as a popular platformfrom which thousands of entrepreneurs can raise ini-tial funds and sell innovative products. This paperstudies how this new business model may affect acreator’s product and pricing decisions on the basisof a two-period game where cohorts of buyers arriveat a crowdfunding project and make sign-up deci-sions sequentially. Our results contribute some novelinsights to the literature on product line design. Incrowdfunding, even when product options are virtu-ally the same, high-type buyers may still choose thehigh-price option. This result is unique to the crowd-funding mechanism as tacit coordination among buy-ers is necessary to ensure project success. Moreover,the incentive for a high-type buyer to coordinate withother buyers thusly is shaped by the creator’s pric-ing decisions. When the value of the inferior optiondecreases, the high-type buyer’s coordinating incen-tive diminishes. Our analysis also shows that thecrowdfunding mechanism can change the optimalproduct line design. Compared with the traditionalsetting wherein a creator produces goods before sell-ing to individual consumers, crowdfunding leads toa smaller product line quality gap.

This paper provides a number of considerations thatwill be useful to entrepreneurs interested in adopt-ing the crowdfunding model and to crowdfundingplatform managers. First, optimal product and pric-ing decisions should depend on market characteristics.Specifically, depending on the distribution of buyers’product valuations, a creator may choose a volumestrategy, a margin strategy, an intertemporal strategyor a menu strategy. Second, when the market consistsof a low or moderate fraction of high-type buyers anda moderate level of valuation heterogeneity, it is opti-mal to offer a menu of options. Product-line struc-ture tends to be tighter with crowdfunding. Therefore,when an entrepreneur moves from the crowdfund-ing stage to large scale production and retailing, evenif the market condition remains the same, the prod-uct line design should evolve. Third, creators shouldbe cautious about any marketing activities that maychange the mix of buyers arriving at the projects overtime. The belief about the valuations of buyers who

arrive later has substantial implications for the sign-upbehavior of early arrivals, as well as for the creator’sproduct and pricing decisions. Under some condi-tions, the later arrival of a group of high-type buyerscan encourage free riding.

Crowdfunding as an emerging area deserves moreattention in future research. Theoretical analysis mayinvestigate alternative mechanisms. For example, vir-tually all crowdfunding sites adopt the sequentialmechanism. Alternatively, a creator may arrange forthe buyers to make decisions simultaneously underwhich circumstance buyers do not know others’ sign-up decisions. Our analysis indicates that the mainresults are similar in sequential and simultaneous sit-uations. However, offering a menu of product optionstends to be more profitable in the simultaneousthan in the sequential setting. (Detailed analysis isavailable in the online appendix.) Another importantcrowdfunding feature is buyers’ uncertainty aboutproduct quality, as well as about the creator, whenpledge decisions are being made. Future researchmay investigate possible mechanisms to alleviate suchinformation asymmetry. Empirical researchers shouldalso find crowdfunding a fruitful area for future work.Many hypotheses can be generated from the exist-ing literature in public economics and the currentresearch on project designs and the dynamics of sign-up decisions. These hypotheses can be tested using alarge number of projects that are posted online andpublicly observed.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mksc.2014.0900.

AcknowledgmentsThe authors thank the senior editor, the associate editor, twoanonymous referees, Avi Goldfarb, and Juanjuan Zhang fortheir insightful comments, which helped greatly in improv-ing the quality of the paper. The authors are also grateful toseminar participants at the University of Toronto, McMasterUniversity, and the 2014 Marketing Science conference.

Appendix. Proofs.

Proof of Lemma 1. Similar to the traditional productline design problem, the creator maximizes her profit whenthe IC condition is binding, and pMl = L. This gives thatpMh = 41 − �5H + �L. Furthermore, the deal succeeds whenat least one consumer has the high valuation, so the successrate is sM = 1 − 41 − �52 = �42 − �5. The part on the profitfunction follows immediately. �

Proof of Proposition 1. The results follow directly fromcomparing the profit functions. �

Proof of Corollary 1. Because all costs are zero in thebase model, it suffices to consider the valuations of buyersserved. With the margin strategy, the social welfare (W) is

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WH = 2�2H ; with the volume strategy, the welfare is W L =

24�H+41−�5L5; with the intertemporal strategy the welfareis WD = �441 +�5H + 41 −�5L5; and under the menu strat-egy the welfare is WM = �4H + 41 −�5L5+ 41 −�5�4L+H5.A simple calculation yields that WH < WD < WM < W L0As for buyer surplus, denoted by CS, we have CSL =

2�4H −L51 CSH = 01 CSM = �4H −L5, and CSD = �4H −L+

H − pMh 5+�41 −�54H − pMh 5 where pMh <H . We have CSH <CSD <CSM <CSL.

Without introducing the menu strategy, the volume strat-egy dominates whenever H/L≤ 42−�5/�. Hence in the restpart H/L ≥ 42 − �5/�, the menu strategy improves socialwelfare over the local optimal strategy. A similar argumentholds for buyer surplus as well. The claim that the menustrategy is the dominating strategy when 42−�5/�<H/L<43 −

√55/2 or 42 − �5/� < H/L < 41 + � − �25/43� − �2 − 15

follows from Proposition 1. �

Proof of Proposition 3. If H/L ≤ 1/�, 4QTh − QT

l 5 −

4QMh − QM

l 5 = 4�24H − L55/41 − �5 > 0. If 1/� ≤ H/L <41 +�5/�, 4QT

h − QTl 5 − 4QM

h − QMl 5 = ã > 0. If H/L ≥

41 +�5/�, 4QTh −QT

l 5− 4QMh −QM

l 5= 0. �

Proof of Lemma 3. With the margin strategy, the creatorcharges high-type buyers their valuations, i.e., PH =HQH .His expected profit is çH = 2�24HQH −4QH 52/250 Taking thefirst order derivative with respect to QM solves the problem.

With the volume strategy, the creator charges P L = LQL,so that both buyers make purchases. Then çL =

24LQL − 4QL52/251 whose maximum can be solved by thefirst order condition.

Last, let us consider the intertemporal strategy. The cre-ator sells to high-type buyers in one period (and withoutloss of generality, let us suppose it is period 1), and to buy-ers of any type in the other period (period 2). In period 1,the creator prices at HQD

1 , and in period 2 at LQD2 . Then his

expected profit is çD = �(

HQD1 +LQD

2 − 4QD1 5

2/2− 4QD2 5

2/2)

0Again, the first order conditions give us the solution. �

Proof of Proposition 4. When product qualities areendogenized, the profits of various strategies are given inProposition 2 and Lemma 3. The range in which a spe-cific strategy dominates follows by comparing the profitfunctions. �

Proof of Lemma 4. The IC condition is now �2 ·

4H−pMhl 5≤H−p

Mhh , together with the individual rational-

ity pMhl ≤L. When both inequalities are binding, the cre-

ator maximizes her profit. The success rate is 1−41−�15·41−�25=�1 +�2 −�1�2, and this proves the lemma. �

Proof of Corollary 2. Taking derivatives of �Mh withrespect to �1 and �2 yields that ¡�Mh/¡�1 = 41 −�25

2H +

41 −�225L ≥ 0 and ¡�Mh/¡�2 = 41 − 2�1 − 2�2 + 2�1�25H +

L + 241 −�15�2L1 which is no less than zero when �2 ≤

4H − 2�1H +L5/4241 −�154H − L55, and is less than zerootherwise. �

Proof of Proposition 8. When n1 ≥ n2, T Mc = n1pMch +

n2pMcl and the IC condition is n1H − n1p

Mcl − n2p

Mch +

n2pMcl ≥ �n14H − p

Mcl 50 The creator maximizes her profit

when pMcl = L and the IC condition is binding, which gives

pMcl = L and p

Mch = 41 − �54H − L54H/L5 + L. However, we

need the additional constraint pMch ≤ H here. Moreover, we

have the success rate being �42 − �5, and this proves thefirst part.

When n1 ≤ n2, T Mc = n1pMcl +n2p

Mch . The first cohort pays

either n1pMch or n1p

Mcl , but never anything between them.

If the first cohort pays n1pMch , the conditional success rate

is 1; if the first cohort pays within 6n1pMcl 1n1p

Mch 5, the suc-

cess rate is �. The first cohort cannot pay less than n1pMcl

as long as pMcl ≤ L. Now the IC condition becomes n1H −

n1pMch ≥ �4n1H −n1L5, and this immediately verifies the sec-

ond part. �

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Product and Pricing Decisions in Crowdfunding · Hu, Li, and Shi: Product and Pricing Decisions in...

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