Advance-Purchase Financing of Projects with Few Buyers

Abstract

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.

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 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

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

(10)

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 $\theta _a=\theta _a^{n^*}$.

• (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.

• (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

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

(11)

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

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

(12)

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

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

(13)

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

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

• (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 $\theta \in [\theta _a,1]$ 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

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

(14)

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

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

(15)

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

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

(16)

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

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

(17)

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.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:

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

(18)

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.