Advance-Purchase Financing of Projects with Few Buyers


I investigate a simple model of advance-purchase 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 advance-purchase arrangements enable more costly projects to be financed than is true for traditional funding sources. The entrepreneur uses advance-purchase surcharges as a price discrimination device. However, the discriminatory power is limited by the problem of free-riding.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 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 advance-purchase 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. 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 advance-purchases 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. 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 free-riding on the advance payments of others.

  4. 4.

    See Heckler and Koch Investor Presentation, April 2017 (downloaded from on 11/22/2018). On the other side of the market, governments explicitly consider advance-purchase 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 long-term 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 on 11/22/2018).

  5. 5.

    There are the few exceptions: Advance-purchase surcharges may be optimal if the entrepreneur is capacity-constrained and consumers differ in their likelihood of regretting pre-orders (Nasiry and Popescu 2012), or if consumers differ in their experience with the good (Zeng 2013; Loginova 2016).

  6. 6.

    Thus their findings conflict with the result that, in a continuum economy, the behavioral assumption of community benefits is critical for crowdfunding to be more profitable than TF (Belleflamme et al. 2014; Sahm et al. 2014).

  7. 7.

    Schmitz (1997) and Norman (2004) show that the monopolist will indeed find it optimal to rely on simple contracts (such as average cost pricing) if the number of potential buyers becomes very large.

  8. 8.

    Examples of regularity include uniform, normal, and exponential distributions.

  9. 9.

    The assumption of risk-neutral customers is particularly appropriate for institutional buyers as in some of the introductory examples.

  10. 10.

    Section 3.3 discusses the assumptions on the entrepreneur’s bargaining power in more detail.

  11. 11.

    The case without commitment includes mixed financing and is discussed in Sect. 6.1.

  12. 12.

    In contrast to this simple contract that specifies a single advance-purchase 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 advance-purchase prices.

  13. 13.

    Cornelli (1996) specifies the optimal general advance-purchase contract under the assumption that the entrepreneur can commit to deliver the product only to pre-orderers: The entrepreneur commits to a prohibitively high regular price.

  14. 14.

    Symmetry requires that any two individuals with the same valuation \(\theta\) hold the same beliefs and make the same decisions in equilibrium.

  15. 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. 16.

    The proofs of the following lemma and all subsequent results can be found in “Appendix A”.

  17. 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

    $$\begin{aligned} v^{\prime }_{\theta _a}(\theta ) = 2 + \frac{[F(\theta _a)-F(\theta )]f'(\theta )}{[f(\theta )]^2}> 2 + \frac{[1-F(\theta )]f'(\theta )}{[f(\theta )]^2} = v'(\theta ) > 0.\end{aligned}$$
  18. 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 pre-order buyer.

  19. 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 advance-purchase price \(p_a\) below the uniform price \(p_0\).

  20. 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 pre-order, 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. 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. 22.

    Recall that the superscript n refers to the case in which the entrepreneur relies on a minimum of n pre-orders (see page 9).

  23. 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 \(i-1\) pre-orders must not be sufficient to cover the fixed costs is relaxed as K increases.

  24. 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. 25.

    E.g. contractual arrangements with third parties such as the use of crowdfunding platforms

  26. 26.

    K may then be reinterpreted as the capital requirement beyond the debt limit.

  27. 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. 28.

    Again, the technical details and explicit formal solution of the respective game can be found in “Appendix B”.

  29. 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. 30.

    Positive network externalities may represent another source of advance purchase discounts (e.g., Bensaid and Lesne 1996).

  31. 31.

    All examples are taken from reward-based crowdfunding campaigns on All regular price information has been collected as of January 9, 2018, from except for Filippo Loreti which has been collected from 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. 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. 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. 34.

    As the pharmaceutical industry exhibits significant elasticities of innovation to expected market size (Dubois et al. 2015), advance-purchase arrangements might provide a valuable tool for screening the profitability of R&D projects.

  35. 35.

    Notice, however, that \(p_a^1<1/2=p_0\) will be optimal if \(\theta _a^1<\frac{\sqrt{5}-1}{2} \quad \Leftrightarrow \quad K<{\check{K}}\approx 0.08\).


  1. Belleflamme, P., Lambert, T., & Schwienbacher, A. (2014). Crowdfunding: Tapping the right crowd. Journal of Business Venturing, 29(5), 585–609.

    Article  Google Scholar 

  2. Bensaid, B., & Lesne, J.-P. (1996). Dynamic monopoly pricing with network externalities. International Journal of Industrial Organization, 14(6), 837–855.

    Article  Google Scholar 

  3. Berndt, E. R., Glennerster, R., Kremer, M. R., Lee, J., Levine, R., Weizsäcker, G., et al. (2007). Advance market commitments for vaccines against neglected diseases: Estimating costs and effectiveness. Health Economics, 16(5), 491–511.

    Article  Google Scholar 

  4. Berndt, E. R., & Hurvitz, J. A. (2005). Vaccine advance-purchase agreements for low-income countries: Practical issues. Health Affairs, 24(3), 653–665.

    Article  Google Scholar 

  5. Brito, D. L., & Oakland, W. H. (1980). On the monopolistic provision of excludable public goods. American Economic Review, 70(4), 691–704.

    Google Scholar 

  6. Cornelli, F. (1996). Optimal selling procedures with fixed costs. Journal of Economic Theory, 71, 1–30.

    Article  Google Scholar 

  7. Dalberg Global Development Advisors. (2013, February 15). The advance market commitment for pneumococcal vaccines: Process and design evaluation. In Full report. Retrieved November 13, 2019, from

  8. Dana, J. D. (1998). Advance-purchase discounts and price discrimination in competitive markets. Journal of Political Economy, 106(2), 395–422.

    Article  Google Scholar 

  9. Dana, J. D. (1999). Equilibrium price dispersion under demand uncertainty: The roles of costly capacity and market structure. The RAND Journal of Economics, 30(4), 632–660.

    Article  Google Scholar 

  10. Dana, J. D. (2001). Monopoly price dispersion under demand uncertainty. International Economic Review, 42(3), 649–670.

    Article  Google Scholar 

  11. Dubois, P., de Mouzon, O., Scott-Morton, F., & Seabright, P. (2015). Market size and pharmaceutical innovation. The RAND Journal of Economics, 46(4), 844–871.

    Article  Google Scholar 

  12. Ellman, M., & Hurkens, S. (2019). Optimal crowdfunding design. Journal of Economic Theory, 184, 104939.

    Article  Google Scholar 

  13. Gale, I. L., & Holmes, T. J. (1992). The efficiency of advance-purchase discounts in the presence of aggregate demand uncertainty. International Journal of Industrial Organization, 10(3), 413–437.

    Article  Google Scholar 

  14. Gale, I. L., & Holmes, T. J. (1993). Advance-purchase discounts and monopoly allocation of capacity. American Economic Review, 83(1), 135–146.

    Google Scholar 

  15. Kumar, P., Langberg, N., & Zvilichovsky, D. (2019). Crowdfunding, financing constraints and real effects. Management Science,.

    Article  Google Scholar 

  16. Loginova, O. (2016). Pricing strategies in advance selling: Should a retailer offer a pre-order price guarantee? Review of Industrial Organization, 49(3), 465–489.

    Article  Google Scholar 

  17. Möller, M., & Watanabe, M. (2010). Advance purchase discounts versus clearance sales. The Economic Journal, 120(547), 1125–1148.

    Article  Google Scholar 

  18. Myerson, R. B. (1981). Optimal auction design. Mathematics of Operations Research, 6(1), 58–73.

    Article  Google Scholar 

  19. Nasiry, J., & Popescu, I. (2012). Advance selling when consumers regret. Management Science, 58(6), 1160–1177.

    Article  Google Scholar 

  20. Nocke, V., & Peitz, M. (2007). A theory of clearance sales. The Economic Journal, 117(522), 964–990.

    Article  Google Scholar 

  21. Nocke, V., Peitz, M., & Rosar, F. (2011). Advance-purchase discounts as a price discrimination device. Journal of Economic Theory, 146, 141–162.

    Article  Google Scholar 

  22. Norman, P. (2004). Efficient mechanisms for public goods with use exclusion. Review of Economic Studies, 71, 1163–1188.

    Article  Google Scholar 

  23. Sahm, M., Belleflamme, P., Lambert, T., & Schwienbacher, A. (2014). Corrigendum to ‘Crowdfunding: Tapping the right crowd’. Journal of Business Venturing, 29(5), 610–611.

    Article  Google Scholar 

  24. Schmitz, P. (1997). Monopolistic provision of excludable public goods under private information. Public Finance/Finance Publiques, 52, 89–101.

    Google Scholar 

  25. Strausz, R. (2017). A theory of crowdfunding: A mechanism design approach with demand uncertainty and moral hazard. American Economic Review, 107(6), 1430–1476.

    Article  Google Scholar 

  26. Xie, J., & Shugan, S. M. (2001). Electronic tickets, smart cards, and online prepayments: When and how to advance sell. Marketing Science, 20(3), 219–243.

    Article  Google Scholar 

  27. Zeng, C. (2013). Optimal advance selling strategy under price commitment. Pacific Economic Review, 18(2), 233–258.

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Marco Sahm.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


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 pre-orders \(N_a\) regardless of the rule for the advance-purchase decision \(\Omega\).

Proof of Lemma 3

For any given \(p_a\), denote by \(\Omega (p_a) \subset [0,1]\) the set of pre-ordering types: \(\theta _i \in \Omega (p_a)\) implies that individual \(i \in \{1,\ldots ,N\}\) pre-orders 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

$$\begin{aligned} E\pi _{r}(N_a,\Omega (p_a))=(N-N_a) \cdot \text{Prob}(\theta \ge p_r \mid \theta \not \in \Omega (p_a)) \cdot p_r \end{aligned}$$

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 pre-orders \(N_a\).

Proof of Lemma 1

Denote by \(m \in \{1,\ldots ,N\}\) the minimum number of pre-orders that are required to finance the project for the given advance-purchase price \(p_a\): \(mp_a \ge K > (m-1)p_a\). Let \(\sigma (k)\) be the probability that the number of pre-orders among \(N-1\) potential buyers will be at least \(k \in {\mathbb{N}}\). Trivially, \(\sigma (k-1) \ge \sigma (k)\). Some customer with willingness to pay \(\theta\) will weakly prefer to pre-order the product if and only if his expected utility from an advance-purchase—\(\sigma (m-1)(\theta -p_a)\)—is at least as high as that from a regular purchase—\(\sigma (m)(\theta -p_r)\)—or, equivalently, if and only if

$$\begin{aligned} (\sigma (m-1) - \sigma (m)) \theta \ge \sigma (m-1) p_a - \sigma (m) p_r. \end{aligned}$$

For \(\sigma (m-1) = \sigma (m)\), nobody (everybody) will pre-order if \(p_a > p_r\) (\(p_a \le p_r\)), and \(\theta _a:=1\) (\(\theta _a:=0\)) has the stated property. For \(\sigma (m-1) > \sigma (m)\), set \(\theta _a := \min \{\max \{\frac{\sigma (m-1) p_a - \sigma (m)p_r}{\sigma (m-1) - \sigma (m)},0\},1\}\).

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 \(\theta _a=\theta _a^{n^*}\).

  1. (a)

    Suppose to the contrary that \(p_a = p_a^{n^*} \le p_r\). Then the equality of (4) and (5) implies

    $$\begin{aligned} A(n^*)&= \left[ \sum _{i=0}^{N-n^*} \left(\begin{array}{c}N-1 \\ n^*-1+i\end{array}\right) [1-F(\theta _a^{n^*})]^{n^*-1+i}[F(\theta _a^{n^*})]^{N-n^*-i} \right] \\ &\quad \cdot (\theta _a^{n^*}-p_a^{n^*}) \\ &\ge \left[ \sum _{i=0}^{N-n^*} \left(\begin{array}{c}N-1 \\ n^*-1+i\end{array}\right) [1-F(\theta _a^{n^*})]^{n^*-1+i}[F(\theta _a^{n^*})]^{N-n^*-i} \right] \\ &\quad \cdot (\theta _a^{n^*}-p_r) \\ &= A(n^*) + P(n^*)(\theta _a^{n^*}-p_r), \end{aligned}$$

    which is a contradiction as both \(P(n^*)\) and \(\theta _a^{n^*}-p_r\) are positive by Eq. (6) and Lemma 2, respectively.

  2. (b)

    If \(n^*=N\), then the equality of (4) and (5) will imply \(0=R(N)=A(N)=[1-F(\theta _a^N)]^{N-1}(\theta _a^N-p_a^N)\) 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 advance-purchase price \(p_a^N\) that makes each potential buyer pivotal for undertaking the project. The corresponding expected profit

$$\begin{aligned} E\pi _a^N = [1-F(p_a^N)]^N(Np_a^N-K) \end{aligned}$$

is a lower bound for the entrepreneur’s optimal profit under APF. The optimal \(p_a^N\) maximizes (11) subject to the constraints

$$\begin{aligned} Np_a^N \ge K > (N-1)p_a^N. \end{aligned}$$

For any regular F, the unconstrained solution to this problem can be derived from the first-order condition and is implicitly given by

$$\begin{aligned} Np_a^N-\frac{1-F(p_a^N)}{f(p_a^N)} = K. \end{aligned}$$

As \(v(p_a^N)\) strictly increases in \(p_a^N\), so does the left-hand 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 \(Np_a^N - K = \frac{1-F(p_a^N)}{f(p_a^N)} >0\). Moreover, it will meet the second constraint \(K-(N-1)p_a^N = p_a^N + K-Np_a^N = p_a^N - \frac{1-F(p_a^N)}{f(p_a^N)} >0\) as well if and only if \(p_a^N>p_0\). In this case, the expected profit equals \(E(\pi _a^N) = [1-F(p_a^N)]^{N+1}>0\). Otherwise the entrepreneur can set \(p_a^N\) arbitrarily close to \(\frac{K}{N-1}<p_0<1\). The corresponding expected profit is then given by \(E\pi _a^N = \left[ 1-F\left( \frac{K}{N-1}\right) \right] ^N\left( \frac{K}{N-1}\right) > 0\).

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 Pareto-dominates the allocation under TF for at least all \(K \in (K_0,N)\).

Proof of Proposition 3

  1. (a)

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

  2. (b)

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

  3. (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 \(\theta <\theta _a\).

  4. (d)

    Under APF, consumers with \(\theta \in [\theta _a,1]\) pre-order 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 \(E\pi _0=1/2-K\). Under APF, the entrepreneur can rely on a minimum of either one (\(m=1\)) or two pre-orders (\(m=2\)).

For \(m=1\), the equality of (4) and (5) implies \(\theta _a^1 - p_a^1 = (1-\theta _a^1)(\theta _a^1-p_r)\), or equivalently

$$\begin{aligned} p_a^1=\theta _a^1\left( 1-\frac{1}{2}(1-\theta _a^1)\right) =\frac{1}{2}\theta _a^1(1+\theta _a^1). \end{aligned}$$

Now, the entrepreneur chooses \(\theta _a^1\) to maximize the expected profit

$$\begin{aligned} E\pi _a^1&= (1-\theta _a^1)^2 \cdot (2p_a^1 - K) + 2(1-\theta _a^1)\theta _a^1 \cdot \left( p_a^1 + \frac{\theta _a^1-p_r}{\theta _a^1}p_r - K\right) \\ &= -\frac{3}{2}(\theta _a^1)^3 + \left( \frac{1}{2}+K\right) (\theta _a^1)^2 + \theta _a^1 - K \end{aligned}$$

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

$$\begin{aligned} \theta _a^1=\frac{1}{9}\left( \sqrt{18+(1+2K)^2}+1+2K \right) . \end{aligned}$$

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

$$\begin{aligned} E\pi _a^2 = (1-p_a^2)^2 \cdot (2p_a^2 - K) = 2(p_a^2)^3 - (4+K)(p_a^2)^2 + (2+2K)p_a^2 - K \end{aligned}$$

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 \(E\pi _a^2=\left( \frac{2-K}{3}\right) ^3\). 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 \(E\pi _a^2=(1-K)^2K\).

For \({\bar{K}} \le K \le 1/2\), a reliance on one pre-order is optimal, and customers who pre-order—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 pre-order—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 pre-orders (\(m=2\)).

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 \(E(\pi _0^m)=1/2-K\).

For \(m=1\), the entrepreneur’s problem (15) is, now, constrained by the conditions

$$\begin{aligned} p_1 + \frac{\theta _1}{4} \ge K > 2 \cdot \frac{\theta _1}{4} \end{aligned}$$

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

$$\begin{aligned} 2 p_2 \ge K > p_2 + \frac{\theta _2}{4} \end{aligned}$$

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 \(E(\pi _2^m)=\left( \frac{2-K}{3}\right) ^3\). 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 pre-orders that makes the project ex interim profitable for the given prices \(p_a\) and \(p_r\):

$$\begin{aligned} mp_a + (N-m)\rho (p_a,p_r)p_r \ge K > (m-1)p_a + (N-m+1)\rho (p_a,p_r)p_r, \end{aligned}$$

where \(\rho (p_a,p_r)\) denotes the probability that some customer buys at the regular price \(p_r\) conditional on not having pre-ordered at the advance-purchase 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 pre-orders 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 advance-purchase 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:

$$\begin{aligned} Np_a^N \ge K > (N-1)p_a^N + \frac{F(p_a^N)-F(p_r)}{F(p_a^N)}p_r. \end{aligned}$$

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 right-hand 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 advance-purchase price \(p_a^N\) that makes each potential buyer pivotal is given by \(E(\pi _a^N) = (1-p_a^N)^N(Np_a^N-K)\) and the critical constraint that is given by the right inequality in (18) is \(K > (N-1)p_a^N + \frac{p_a^N}{4}\). The unconstrained solution \(p_a^N=\frac{K+1}{N+1}\) will satisfy this constraint if and only if \(K > \frac{4N-3}{7}\). In this case, the expected profit equals \(E(\pi _a^N) = \left( \frac{N-K}{N+1}\right) ^{N+1}>0\). Otherwise the entrepreneur can set \(p_a^N\) arbitrarily close to \(\frac{4K}{4N-3}\). The corresponding expected profit is then given by \(E(\pi _a^N) = \left( \frac{4(N-K)-3}{4N-3}\right) ^N \cdot \frac{3K}{4N-3} > 0\).

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sahm, M. Advance-Purchase Financing of Projects with Few Buyers. Rev Ind Organ 57, 909–933 (2020).

Download citation


  • Crowdfunding
  • Excludable public goods
  • Innovation and R&D
  • Monopolistic provision
  • Pre-ordering
  • Price discrimination

JEL Classification

  • D42
  • G32
  • H41
  • L12
  • L26
  • O31