Abstract
I investigate a simple model of advancepurchase contracts as a mode of financing costly projects. An entrepreneur has to meet a capital requirement in order to start production and sell the related good to a limited number of potential buyers who are privately informed about their willingness to pay. I find that advancepurchase arrangements enable more costly projects to be financed than is true for traditional funding sources. The entrepreneur uses advancepurchase surcharges as a price discrimination device. However, the discriminatory power is limited by the problem of freeriding.
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Notes
 1.
Berndt and Hurvitz (2005) provide a comprehensive discussion of this proposal that focuses on practical issues. Berndt et al. (2007) estimate the costs and effectiveness of such advancepurchase arrangements. Dalberg (2013) evaluates the process and design of the advance market commitment for pneumococcal vaccines, which was organized as a pilot project by Gavi and engaged the manufacturers GlaxoSmithKline and Pfizer.
 2.
Qualitatively, the results also apply to an extension of the model with mixed financing in which the entrepreneur cannot commit to rely exclusively on revenues from advancepurchases for meeting the capital requirement but may supplement them by traditional funds whenever the project is profitable from an ex interim perspective (see Sect. 6.1).
 3.
For smaller fixed costs, the entrepreneur faces a tradeoff between the informational advantage of using APF as a screening device and the associated incentive costs of preventing the agents from freeriding on the advance payments of others.
 4.
See Heckler and Koch Investor Presentation, April 2017 (downloaded from https://www.hecklerkoch.com/de/ir/irmitteilungen.html on 11/22/2018). On the other side of the market, governments explicitly consider advancepurchase contracts as a valuable instrument for incentivizing research, development, and production of modern weapons. According to its own mission statement, for example, the Undersecretariat for Defence Industries (SSM) of the Turkish government is “designated to...provide advance loans and determine longterm orders and other financial and economic incentives” as well as to “enter into contracts covering technical and financial issues taking into account long term procurement decisions...” (accessed at https://www.ssb.gov.tr/WebSite/contentlist.aspx?PageID=39&LangID=2 on 11/22/2018).
 5.
 6.
 7.
 8.
Examples of regularity include uniform, normal, and exponential distributions.
 9.
The assumption of riskneutral customers is particularly appropriate for institutional buyers as in some of the introductory examples.
 10.
Section 3.3 discusses the assumptions on the entrepreneur’s bargaining power in more detail.
 11.
The case without commitment includes mixed financing and is discussed in Sect. 6.1.
 12.
In contrast to this simple contract that specifies a single advancepurchase price, Ellman and Hurkens (2019) consider general crowdfunding mechanisms that consist of an arbitrary funding threshold (which may differ from the capital requirement) and a set of advancepurchase prices.
 13.
Cornelli (1996) specifies the optimal general advancepurchase contract under the assumption that the entrepreneur can commit to deliver the product only to preorderers: The entrepreneur commits to a prohibitively high regular price.
 14.
Symmetry requires that any two individuals with the same valuation \(\theta\) hold the same beliefs and make the same decisions in equilibrium.
 15.
The reason is that, for any such rule, the entrepreneur’s expectation about the willingness to pay is the same for all remaining customers. The pricing decision is thus based only on this expectation, whereas the number of remaining customers is just a factor that scales additional profits.
 16.
The proofs of the following lemma and all subsequent results can be found in “Appendix A”.
 17.
To see this, note that \(v_{\theta _a}(\theta ) := \theta  \frac{F(\theta _a)F(\theta )}{f(\theta )}\) strictly increases for all \(\theta \in [0,\theta _a)\) because \(v^{\prime }_{\theta _a}(\theta ) = 2 + \frac{[F(\theta _a)F(\theta )]f'(\theta )}{[f(\theta )]^2} > 0.\) This is clear for \(f'(\theta )\ge 0\); for \(f'(\theta )<0\) it follows from the fact that v strictly increases in \(\theta\) and thus
$$\begin{aligned} v^{\prime }_{\theta _a}(\theta ) = 2 + \frac{[F(\theta _a)F(\theta )]f'(\theta )}{[f(\theta )]^2}> 2 + \frac{[1F(\theta )]f'(\theta )}{[f(\theta )]^2} = v'(\theta ) > 0.\end{aligned}$$  18.
By contrast, if all potential buyers have to purchase in advance to meet the capital requirement, each agent will be pivotal with certainty, and the entrepreneur can choose the advance purchase price equal to the valuation threshold without leaving any information rent to the marginal preorder buyer.
 19.
The example of two potential buyers with uniformly distributed valuations below illustrates, however, that \(p_a>p_0\) is not always met: In a certain range of fixed costs, the entrepreneur may find it optimal to choose an advancepurchase price \(p_a\) below the uniform price \(p_0\).
 20.
Consider, for example, the case with a uniform distribution of valuations, only one potential buyer (see Sect. 5.1) and a capital requirement of \(\hbox{K}=1/8\). Because the project is then profitable ex ante, it will be realized with certainty under TF, and the product will be offered at a uniform price equal to \(1/2=16/32\). Under APF, the advance purchase price will be identical to the threshold valuation and equal \(9/16=18/32\) (see “Appendix B”). Now suppose that the potential buyer’s realized valuation equals 17/32. Then, he will not preorder, and the project will not be realized under APF. (By contrast, under TF, he will buy and—just like the entrepreneur—realize a positive surplus.)
 21.
The technical details and explicit formal solutions of the respective games can be found below for \(N=1\) and in “Appendix B” for \(N=2\). For \(N=3\), they are provided by the author upon request.
 22.
Recall that the superscript n refers to the case in which the entrepreneur relies on a minimum of n preorders (see page 9).
 23.
Graphically, any two expected profit functions \(E\pi _a^i\), \(E\pi _a^j\) intersect only once and in their downward sloping part. Note that, for \(i>1\), the initial increase of \(E\pi _a^i\) is due to the fact that the constraint that \(i1\) preorders must not be sufficient to cover the fixed costs is relaxed as K increases.
 24.
The graphs of \(E\pi _0\) and the upper envelope of \(E\pi _a^1,\ldots ,E\pi _a^N\) intersect only once.
 25.
E.g. contractual arrangements with third parties such as the use of crowdfunding platforms
 26.
K may then be reinterpreted as the capital requirement beyond the debt limit.
 27.
I use the previous notation and normalize the capital costs of TF to zero. The capital costs of TF may be interpreted as a measure of commitment opportunities ranging from 0 (no commitment) to \(\infty\) (full commitment).
 28.
Again, the technical details and explicit formal solution of the respective game can be found in “Appendix B”.
 29.
The latter case may also be interpreted as uncertainty about the trustworthiness of the entrepreneur under the threat of entrepreneurial fraud (Strausz 2017).
 30.
Positive network externalities may represent another source of advance purchase discounts (e.g., Bensaid and Lesne 1996).
 31.
All examples are taken from rewardbased crowdfunding campaigns on https://www.kickstarter.com. All regular price information has been collected as of January 9, 2018, from https://www.amazon.com except for Filippo Loreti which has been collected from https://shop.filippoloreti.com. Notice, moreover, that many crowdfunding campaigns also allow for donations, which may be interpreted as the surcharge that potential consumers are willing to pay in excess of the regular price in order to increase the probability of realization and availability of the respective product.
 32.
In the example of new drugs and vaccines, their quality is usually approved by regulatory authorities based on clinical tests of their effectiveness and safety.
 33.
In the context of R&D of new drugs—e.g. a vaccine against HIV—the national health agencies usually have more detailed information about the risk exposure and disposable income in their countries as two important determinants of their willingness to pay.
 34.
As the pharmaceutical industry exhibits significant elasticities of innovation to expected market size (Dubois et al. 2015), advancepurchase arrangements might provide a valuable tool for screening the profitability of R&D projects.
 35.
Notice, however, that \(p_a^1<1/2=p_0\) will be optimal if \(\theta _a^1<\frac{\sqrt{5}1}{2} \quad \Leftrightarrow \quad K<{\check{K}}\approx 0.08\).
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Acknowledgements
I am very grateful for insightful comments and helpful suggestions from the editor, two anonymous referees, Paul Belleflamme, Matthew Ellman, Sjaak Hurkens, Thomas Lambert, Martin Peitz, Markus Reisinger, Armin Schwienbacher, Roland Strausz, David Zvilichovsky, and many participants of various economic seminars and conferences.
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Appendices
Appendix A: Proofs of the Basic Results in Sections 3 and 4
Lemma 3
The optimal regular price \(p_r\) does not depend on the number of preorders \(N_a\) regardless of the rule for the advancepurchase decision \(\Omega\).
Proof of Lemma 3
For any given \(p_a\), denote by \(\Omega (p_a) \subset [0,1]\) the set of preordering types: \(\theta _i \in \Omega (p_a)\) implies that individual \(i \in \{1,\ldots ,N\}\) preorders at price \(p_a\) in stage 1. Then \(N_a=\{\theta _i \in \Omega (p_a)  i \in \{1,\ldots ,N\}\}\). Whenever \(N_ap_a \ge K\), the entrepreneur maximizes his additional conditional expected profits
through the choice of \(p_r\). For any conceivable \(\Omega (p_a) \subset [0,1]\), the maximizing regular price \(p_r\) is independent of the realized number of preorders \(N_a\).
Proof of Lemma 1
Denote by \(m \in \{1,\ldots ,N\}\) the minimum number of preorders that are required to finance the project for the given advancepurchase price \(p_a\): \(mp_a \ge K > (m1)p_a\). Let \(\sigma (k)\) be the probability that the number of preorders among \(N1\) potential buyers will be at least \(k \in {\mathbb{N}}\). Trivially, \(\sigma (k1) \ge \sigma (k)\). Some customer with willingness to pay \(\theta\) will weakly prefer to preorder the product if and only if his expected utility from an advancepurchase—\(\sigma (m1)(\theta p_a)\)—is at least as high as that from a regular purchase—\(\sigma (m)(\theta p_r)\)—or, equivalently, if and only if
For \(\sigma (m1) = \sigma (m)\), nobody (everybody) will preorder if \(p_a > p_r\) (\(p_a \le p_r\)), and \(\theta _a:=1\) (\(\theta _a:=0\)) has the stated property. For \(\sigma (m1) > \sigma (m)\), set \(\theta _a := \min \{\max \{\frac{\sigma (m1) p_a  \sigma (m)p_r}{\sigma (m1)  \sigma (m)},0\},1\}\).
Proof of Proposition 1
There is some \(n^* \in \arg \max _{n \in \{1,\ldots ,N\}}E\pi _a^n\) such that \(p_a=p_a^{n^*}\) and \(\theta _a=\theta _a^{n^*}\).

(a)
Suppose to the contrary that \(p_a = p_a^{n^*} \le p_r\). Then the equality of (4) and (5) implies
$$\begin{aligned} A(n^*)&= \left[ \sum _{i=0}^{Nn^*} \left(\begin{array}{c}N1 \\ n^*1+i\end{array}\right) [1F(\theta _a^{n^*})]^{n^*1+i}[F(\theta _a^{n^*})]^{Nn^*i} \right] \\ &\quad \cdot (\theta _a^{n^*}p_a^{n^*}) \\ &\ge \left[ \sum _{i=0}^{Nn^*} \left(\begin{array}{c}N1 \\ n^*1+i\end{array}\right) [1F(\theta _a^{n^*})]^{n^*1+i}[F(\theta _a^{n^*})]^{Nn^*i} \right] \\ &\quad \cdot (\theta _a^{n^*}p_r) \\ &= A(n^*) + P(n^*)(\theta _a^{n^*}p_r), \end{aligned}$$which is a contradiction as both \(P(n^*)\) and \(\theta _a^{n^*}p_r\) are positive by Eq. (6) and Lemma 2, respectively.

(b)
If \(n^*=N\), then the equality of (4) and (5) will imply \(0=R(N)=A(N)=[1F(\theta _a^N)]^{N1}(\theta _a^Np_a^N)\) and thus \(p_a^N=\theta _a^N\); else, by Lemma 2, it will imply \(0<R(n^*)=A(n^*)\) and thus \(p_a^{n^*}<\theta _a^{n^*}\).
Proof of Proposition 2
Consider the entrepreneur’s strategy to choose an advancepurchase price \(p_a^N\) that makes each potential buyer pivotal for undertaking the project. The corresponding expected profit
is a lower bound for the entrepreneur’s optimal profit under APF. The optimal \(p_a^N\) maximizes (11) subject to the constraints
For any regular F, the unconstrained solution to this problem can be derived from the firstorder condition and is implicitly given by
As \(v(p_a^N)\) strictly increases in \(p_a^N\), so does the lefthand side of this equation, which rises from a negative value \(1/f(0)\) to N as \(p_a^N\) increases from 0 to 1. The equation thus has an interior solution for all \(K<N\). This solution will satisfy the first constraint \(Np_a^N  K = \frac{1F(p_a^N)}{f(p_a^N)} >0\). Moreover, it will meet the second constraint \(K(N1)p_a^N = p_a^N + KNp_a^N = p_a^N  \frac{1F(p_a^N)}{f(p_a^N)} >0\) as well if and only if \(p_a^N>p_0\). In this case, the expected profit equals \(E(\pi _a^N) = [1F(p_a^N)]^{N+1}>0\). Otherwise the entrepreneur can set \(p_a^N\) arbitrarily close to \(\frac{K}{N1}<p_0<1\). The corresponding expected profit is then given by \(E\pi _a^N = \left[ 1F\left( \frac{K}{N1}\right) \right] ^N\left( \frac{K}{N1}\right) > 0\).
This proves part (a) of the Proposition and implies part (b) as \(E\pi _a^N>0=E\pi _0\) at \(K=K_0\). Moreover, it also implies part (c) as the allocation under APF Paretodominates the allocation under TF for at least all \(K \in (K_0,N)\).
Proof of Proposition 3

(a)
The firstorder condition (2) implies \(v(p_r)<0\). Because v strictly increases and \(v(p_0)=0\), this implies \(p_r<p_0\).

(b)
Because \(p_r<p_0\), consumers with \(\theta \in [0,p_r]\) will not buy in either case.

(c)
Because \(p_r<p_0\), consumers with \(\theta \in (p_r,p_0)\) will definitely not buy under TF but may possibly buy under APF and thus derive a positive expected utility unless \(n^*=N\) and \(\theta <\theta _a\).

(d)
Under APF, consumers with \(\theta \in [\theta _a,1]\) preorder and will thus possibly buy at a higher price if \(p_a>p_0\). Moreover, they face a positive probability that the product will not be available.
Appendix B: Technical Details of the Example with \({\mathbf{N}}=2\) in Sections 5 and 6
Case \(N=2\) with TF and APF
Under TF: \(K_0 = 1/2\); \(p_0=1/2\); and \(E\pi _0=1/2K\). Under APF, the entrepreneur can rely on a minimum of either one (\(m=1\)) or two preorders (\(m=2\)).
For \(m=1\), the equality of (4) and (5) implies \(\theta _a^1  p_a^1 = (1\theta _a^1)(\theta _a^1p_r)\), or equivalently
Now, the entrepreneur chooses \(\theta _a^1\) to maximize the expected profit
subject to \(p_a^1 = \frac{1}{2}\theta _a^1(1+\theta _a^1) \ge K\). The unconstrained solution is derived from the necessary condition \(\partial E\pi _a^1/\partial \theta _a^1=0\), which yields
It satisfies \(p_a^1 \ge K\) if and only if \(K < {{\tilde{K}}} \approx 0.76\). Otherwise, the solution is given by \(p_a^1=K\) and \(\theta _a^1 = \sqrt{2K + 1/4}1/2\), which is feasible for all \(K \le 1\).
For \(m=2\), the entrepreneur chooses \(p_a^2\) in order to maximize
subject to \(2p_a^2 \ge K > p_a^2\). The unconstrained solution is derived from the necessary condition \(\partial E\pi _a^2/\partial p_a^2=0\), which yields \(p_a^2=(K+1)/3\). It will satisfy \(2p_a^2 \ge K > p_a^2\) if and only if \(1/2 < K \le 2\). It then yields the expected profit \(E\pi _a^2=\left( \frac{2K}{3}\right) ^3\). For \(K \le 1/2\), no solution exists unless there is a smallest monetary unit \(\mu\). As \(\mu \rightarrow 0\), the optimal price \(p_a^2\) converges to K, yielding the asymptotic expected profit \(E\pi _a^2=(1K)^2K\).
For \({\bar{K}} \le K \le 1/2\), a reliance on one preorder is optimal, and customers who preorder—those with valuations \(\theta \ge \theta _a^1\) as given by (16)—would actually prefer TF over APF because \(p_a^1 > 1/2 = p_0\) in this range.^{Footnote 35} Customers who do not preorder—those with valuations \(\theta < \theta _a^1\)—prefer APF over TF if and only if \((1\theta _a^1)(\theta p_r) \ge \theta  p_0 \quad \Leftrightarrow \quad \theta \le \frac{1\theta _a^1(1\theta _a^1)}{2\theta _a^1} =: {\bar{\theta }}\). As \({\bar{\theta }}\) decreases in \(\theta _a^1\) and \(\theta _a^1\) increases in K, the threshold \({\bar{\theta }}\) decreases in K.
Case \(N=2\) with MF
Under MF, the entrepreneur can rely on a minimum of zero (\(m=0\)), one (\(m=1\)), or two preorders (\(m=2\)).
The choice \(m=0\) is equivalent to TF. As shown above, it is feasible for all \(K \le 1/2\) and implies \(p_0^m=1/2\) as well as \(E(\pi _0^m)=1/2K\).
For \(m=1\), the entrepreneur’s problem (15) is, now, constrained by the conditions
with \(0 \le \theta _1 \le 1\) (recall that \(p_r=\theta _1/2\)). Using (14), the unconstrained solution (16) will satisfy these conditions if and only if \(\frac{1+\sqrt{15}}{14} < K \le \frac{5}{4}\). For \(K>\frac{5}{4}\), \(m=1\) is not feasible. For \(K \le \frac{1+\sqrt{15}}{14}\), no solution exists unless there is a smallest monetary unit \(\mu\). As \(\mu \rightarrow 0\), the optimal threshold valuation \(\theta _1\) converges to 2K.
For \(m=2\), the entrepreneur’s problem (17) is now constrained by the conditions
with \(p_2=\theta _2\). The unconstrained solution \(p_2=(K+1)/3\) will satisfy these constraints if and only if \(5/7 < K \le 2\), which yields \(E(\pi _2^m)=\left( \frac{2K}{3}\right) ^3\). For \(K \le 5/7\), no solution exists unless there is a smallest monetary unit \(\mu\). As \(\mu \rightarrow 0\), the optimal price \(p_2\) converges to 4K / 5, which asymptotically yields \(E(\pi _2^m)=(1\frac{4}{5}K)^2 \cdot \frac{3}{5}K\).
Appendix C: Technical Details of the Extension to MF in Section 6.1
I review the analysis of Sects. 3 and 4 for the case of MF. I use the same notation as before and focus on the necessary changes to the main Lemmas and Propositions as well as their proofs.
To begin, note that Lemma 1 and its proof hold without any change if one reinterprets m as the minimum number of preorders that makes the project ex interim profitable for the given prices \(p_a\) and \(p_r\):
where \(\rho (p_a,p_r)\) denotes the probability that some customer buys at the regular price \(p_r\) conditional on not having preordered at the advancepurchase price \(p_a\).
As argued in Sect. 6, the lack of commitment does not affect the entrepreneur’s decision on \(p_r\) as given by (2) nor his objective function (8) nor the relation between \(\theta _a\) and \(p_a\) in (7). It affects only the minimum number m of preorders that is necessary for the project to be realized as given in (3). Therefore, Lemma 2 and its proof hold without any change. Similarly, as the proof of Proposition 1 is not based on any considerations of the (modified) constraint (3), it remains valid under MF, too.
Proposition 2 indicates that APF outperforms TF for sufficiently large capital requirements. To see that this is also true for MF, reconsider the entrepreneur’s strategy to choose an advancepurchase price \(p_N\) that makes each potential buyer pivotal for running the project. The corresponding expected profit (11) is a lower bound for the entrepreneur’s optimal profit under MF as well. The relevant constraints (12), however, now become tighter:
Again, the unconstrained solution to the problem of profit maximization is implicitly given by (13) and satisfies the left inequality in (18) for all \(K<N\). Moreover, as the righthand side of the second inequality in (18) is strictly smaller than N due to \(p_r<p_a^N\), the unconstrained solution will satisfy the right inequality in (18) as well if the capital requirement is sufficiently large. Though it is not trivial to specify a universal threshold for general distributions, an exact analog of Proposition 2 holds for a uniform distribution under MF.
To see this, notice that—for a uniform distribution of valuations—the expected profit (11) from choosing an advancepurchase price \(p_a^N\) that makes each potential buyer pivotal is given by \(E(\pi _a^N) = (1p_a^N)^N(Np_a^NK)\) and the critical constraint that is given by the right inequality in (18) is \(K > (N1)p_a^N + \frac{p_a^N}{4}\). The unconstrained solution \(p_a^N=\frac{K+1}{N+1}\) will satisfy this constraint if and only if \(K > \frac{4N3}{7}\). In this case, the expected profit equals \(E(\pi _a^N) = \left( \frac{NK}{N+1}\right) ^{N+1}>0\). Otherwise the entrepreneur can set \(p_a^N\) arbitrarily close to \(\frac{4K}{4N3}\). The corresponding expected profit is then given by \(E(\pi _a^N) = \left( \frac{4(NK)3}{4N3}\right) ^N \cdot \frac{3K}{4N3} > 0\).
Finally, if we replace APF by MF, Proposition 3 and its proof remain valid unless the entrepreneur finds it optimal to rely exclusively on TF.
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Sahm, M. AdvancePurchase Financing of Projects with Few Buyers. Rev Ind Organ 57, 909–933 (2020). https://doi.org/10.1007/s11151019097421
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Keywords
 Crowdfunding
 Excludable public goods
 Innovation and R&D
 Monopolistic provision
 Preordering
 Price discrimination
JEL Classification
 D42
 G32
 H41
 L12
 L26
 O31