2.1 Unconstrained Newsvendor

If you had no capital constraints, then as long as \(c < p\), you can make a profit. The expected profit from producing quantity \(Q\) is \[\begin{equation}\pi(Q) = \int_0^{Q} (p\fbar(q) - c)dq, \label{eq:newsvendor} \end{equation}\] and the optimal production quantity \(Q^{News}\) solves \[\begin{equation} p\fbar(Q^{News}) -c =0. \label{eq:news-opt} \end{equation}\]

2.2 Constrained

In this world, given the up front payment \(P\), the maximum possible production is \(P/c\). Given contract \(R\), in round one you will choose \[\begin{align} Q^*(R,P) \in & \arg \max_{0 \leq q \leq P/c} \E_{D\sim F}[P - c q + p\min(q,D) - R(q,D)] \nonumber \\ & = \arg \max_{0 \leq q \leq P/c} \pi(q) + P - \E_{D \sim F}[R(q,D)].\label{eq:moralhazard} \end{align}\] To capture the lack of commitment, we assume that if you produce nothing (and therefore sells nothing), then you don't need to pay anything back to investors: \[\begin{equation} R(0,D) = 0. \label{eq:nocommitment}\end{equation}\] In the first stage, investors will only lend at (weakly) favorable terms. That is, \[\begin{equation}P \leq \E_{D\sim F}[R(Q^*(R,P),D)].\label{eq:investorIC} \end{equation}\]

2.3 Special Cases

The paper studies two special cases (possible forms for the function \(R\)). In one ("utility tokens"), investors are entitled to a share \(\alpha\) (in the paper, \(n/m\)) of future revenue. That is, \[R_u^\alpha(Q,D) = \alpha \, p \, \min(Q,D).\] In the other ("equity tokens"), they are entitled to a share \(\alpha\) of future profit: \[R_e^\alpha(Q,D) = \alpha \max ( p \, \min(Q,D) - c Q,0).\] We let \(\alpha^u\) and \(\alpha^e\) denote the optimal choices of \(\alpha\), and \(Q^u\) and \(Q^e\) the corresponding production quantities.

3.1 When can you get a loan?

First, when is it possible to convince investors to lend to you? That is, when is there a choice of \((P,R)\) with \(P > 0\) that satisfies \(\eqref{eq:moralhazard}\), \(\eqref{eq:nocommitment}\), and \(\eqref{eq:investorIC}\)? Clearly, a necessary condition is that production level \(Q^* > 0\) (else, no investor will give you money). By \(\eqref{eq:investorIC}\), the expected payment to investors must be at least \(P\): \[ \E[R(Q^*,D)] \geq P.\] Furthermore, your expected profit from producing \(Q^*\) must exceed your expected profit from producing nothing and pocketing \(P\). By \(\eqref{eq:moralhazard}\) and \(\eqref{eq:nocommitment}\), this implies that \[\pi(Q^*) \geq \E[R(Q^*,D)] .\] Feasibility of \(Q^*\) implies \(P \geq c Q^*\) and \(\eqref{eq:newsvendor}\) implies \(\pi(q) \leq (p - c)Q\). Putting it all together we have \[ (p - c)Q^* \geq \pi(Q^*) \geq P \geq c Q^*.\] For \(Q^* > 0\), this holds only if \[\begin{equation} p > 2 c. \label{eq:necessary} \end{equation}\]

This is a necessary condition to be able to raise money. In fact, under either revenue-sharing or profit-sharing, it is also sufficient: if it is satisfied, there is a choice of \(\alpha\) such that investors will give you \(P > 0\).

To me, this is perhaps the cleanest result in the paper. Due to moral hazard, it is not always possible to get a loan (of any size), even if the product could make profit.

3.2 How profitable will your company be?

Let's consider the case of revenue sharing. In that case, starting from production quantity \(q\), the cost to you of producing an additional item is \(c\), and the benefit to you is \((1-\alpha)p\) times the probability of selling this item, which is \(\fbar(q)\). Thus, you will stop when these are equal: \[ (1 - \alpha^u) p\fbar(Q^u) = c.\] Comparing this to the solution without capital constraints given by \(\eqref{eq:news-opt}\), we see that you always produce less than in the Newsvendor model: \(Q^u < Q^{News}\). Furthermore, in your optimal contract, investors will break even in expectation, implying that your expected profit is \(\pi(Q^{u})\). Because \(\pi\) is increasing on \([0,Q^{News})\), this implies that your profit is also strictly lower. The same idea implies that \(Q^e < Q^{News}\), and in fact the authors show (subject to a technical condition -- I am curious to know whether it is necessary) that \[Q^u < Q^e < Q^{News},\] implying the same ranking in terms of expected profit.

Thus, even projects that are funded are less profitable than they would be without capital constraints. However, raising capital reduces the risk associated with these projects. In particular, under either profit sharing or revenue sharing, you cannot lose money (whereas in the standard Newsvendor model, this happens whenever demand is sufficiently low).

3.3 Optimal contract design

Now let's take things one step beyond what is in the paper, and ask the natural follow-up question: what is the best contract to sign? We start with the following observation.

Observation 1: In an optimal contract, investors will get zero expected profit, so the founder's expected final wealth will simply be \(\pi(Q^*)\).

The problem with revenue and profit sharing is that they diminish your incentive to produce later units, because some of the profit from these units is distributed to investors. This can be fixed if you agree to give all initial revenue to investors (up to some pre-specified maximum), with all remaining revenue going to you.

More formally, if we intend to produce quantity \(Q\), then we sign a contract of the following form: \[\begin{equation} R(Q,D) = p \min(Q,D,M), \label{eq:opt} \end{equation}\] where \(M < Q\) is chosen such that \(p\E[\min(D,M)] = c Q\). This gives investors non-negative expected return. Under this contract, your optimal production quantity is either \(0\) or \(\min(Q,Q^{News})\). The investors will only sign the contract with you if selecting \(Q\) is optimal, which requires that \[\begin{equation} \Delta(Q) = \pi(Q) - c Q \geq 0. \end{equation}\]

Let \(Q^{Max}\) be the (unique) solution to \(\Delta(Q^{Max}) = 0\).

Proposition. The contract in \(\eqref{eq:opt}\) with \(Q = \min(Q^{News},Q^{Max})\) is optimal.

There are two cases to consider.

1. If \(Q^{Max} \geq Q^{News}\), then it is incentive compatible to raise enough money to produce quantity \(Q^{News}\). This contract recovers the full expected profit of the Newsvendor solution (and furthermore guarantees you non-negative profit)! This clearly maximizes expected profit, as the expected sum of investor and your profit is at most \(\pi(Q^{News})\), and investors must expect non-negative profit.¹
2. If \(Q^{Max} < Q^{News}\), then our contract earns expected profit \(\pi(Q^{Max}) < \pi(Q^{News}).\) To see that it is impossible to do better, note that by Observation 1, any better outcome would require a higher production quantity \(Q^* > Q^{Max}\). This implies that you must receive (at least) \(cQ^*\) from investors, which we know is higher than your expected final wealth from production \(\pi(Q^*)\). So the temptation to run would be too great.