Proof of Lemma 1. I start by showing how Seller 1’s investment affects the equilibrium allocation. Without loss of generality, I take b = 0, and substitute U(X∗∣b=0) for U(X^*) in the analysis that follows. Differentiating eq. (5) for xj∗ and j≠i with respect to σ gives:

Because the left-hand side is independent of j, all dxj∗/dσ have the same sign. Now, suppose also that dx1∗/dσ has that same sign. Then, the sum also has that sign, and because U_xx( · ) < 0 and C_xx( · ) > 0, this leads to a contradiction. Now, suppose dx1∗/dσ<0. The other signs therefore have to be positive. By eq. (22), I find that ∑h=1Ndxh∗/dσ<0. However, differentiating x1∗ in (5) with respect to σ

would then have a positive left-hand side and negative right-hand side because of C_xσ( · ) < 0, which is a contradiction. Thus, I have shown (1) and (2) of point (i) of the lemma. Regarding eq. (22), point (3) follows from ∂X∗/∂σ=∑h=1Ndxh∗/dσ. For the effect of the buyer’s investment, I make use of (5), to obtain:

The strict inequality comes from the assumptions of the model and the convexity of the production cost function.

Proof of Lemma 2. Eliminating (b, σ) to simplify, point (i) of the lemma states that X∗>X−{Ji,i}∗+∑j∈Jix~j(Ji). Then, because ∑h≠Ji,ixh∗=X−{Ji,i}∗ the previous condition is equivalent to ∑j∈Jixj∗+xi∗>∑j∈Jix~j(Ji). Because xi∗>0, and if ∑j∈Ji(xj∗−x~j(Ji))>0, the result is shown. If the previous statement is true, it has to be true for any seller j∈J_i. If xj∗>x~j(Ji); thus, the claim is proved. If the converse occurs, xj∗<x~j(Ji), then from the efficient allocation, it has to be that:

and by the concavity of U( · ), the claim is true. Equation (24) also implies that x~j(Ji)>xj∗ for any j ∈ J_i.

Proof of Lemma 3. For any investment profile (b, σ), Ji⊆Ji′, x~i(b,σ∣Ji)≥x~i(b,σ∣Ji′). Eliminating (b, σ), and using a similar argument as given in the proof of Lemma 2 shows that if X−{Ji,i}∗+∑j∈Jix~j(Ji)≥X−{Ji′,i}∗+∑j∈Ji′x~j(Ji′), then x~i(Ji)≤x~i(Ji′). Thus, I obtain:

From Lemma 2, xj∗<x~j(Ji′), and for (25) to be true, I need x~j(Ji)≥x~j(Ji′), which is a contradiction. Then, the only possibility is that X−{Ji,i}∗+∑j∈Jix~j(Ji)≤X−{Ji′,i}∗+∑j∈Ji′x~j(Ji′), giving x~j(Ji)≥x~j(Ji′). The equilibrium transfer is non-increasing with the number of sellers offering latent contracts Ti∗(Ji)≥Ti∗(Ji′). In Ti∗(Ji)=TS∗−TS~−i(Ji)+Ci(xi∗), TS^* and Ci(xi∗) do not depend on J_i. Then, a necessary and sufficient condition for Ti∗(Ji)≥Ti∗(Ji′) is that TS~−i(Ji′)≥TS~−i(Ji). Therefore,

The first strict inequality comes from the convex production cost. The last line results from the fundamental theorem of calculus. The inequality comes from X−{Ji,i}∗+∑j∈Jix~j(Ji)≤X−{Ji′,i}∗+∑j∈Ji′x~j(Ji′) and U_x( · ) > 0.

Proof of Proposition 1. Eliminating (b, σ), the payoff for each seller i is equal to πi(Ji)=Ti∗(Ji)−Ci(xi∗). Introducing the equilibrium transfers in (10) gives πi(Ji)=TS∗−TS~−i(Ji). The buyer obtains:

The proof of point (ii) is immediate from the equilibrium condition considered in Chiesa and Denicolò (2009) in their Proposition 1; a vector (π0,π1,π2,⋅⋅⋅,πN) is a vector of equilibrium payoffs if and only if it satisfies the condition that π0+π1+π2+⋅⋅⋅+πN=TS∗. Then, decreasing the equilibrium transfer reduces the payoff for suppliers and increases the payoff for the buyer.

To show that there is no profitable deviation, consider an equilibrium in which each seller i’s payoff is constrained by a number n of sellers who offer latent contracts. Let mi∗={xi∗,Ti∗} and mi0={0,0} for i ∈ N be the efficient contract and the null contract respectively. Consider that sellers i = n + 2,..., N only offer the efficient and the null contract, Mi={mi∗,mi0} for i = n + 2,..., N. The other sellers also offer latent contracts. Seller 1, offers a latent contract m~1={x~1,T~1}, designed to compete for the equilibrium allocation of any seller i≠1 given that a number n – 1 sellers are also offering latent contracts. Then, M1={m1∗,m10,m~1}. Sellers i = 2, 3,..., n, n + 1 offer two latent contracts, one m~i1={x~i1,T~i1} designed to replace Seller 1, and the other m~ii∈N∖{1}={x~ii∈N∖{1},T~ii∈N∖{1}} for the exclusion of any other seller i≠1. Then, Mi={mi∗,mi0,m~i1,m~ii∈N∖{1}} for i = 2, 3,..., n, n + 1. To show that the payoffs presented in Proposition 1 constitute an equilibrium, I use:

and

The buyer cannot earn more from accepting the latent contracts after excluding seller i than by choosing the efficient contract mj∗ instead of the latent contract m~j for some j ∈ J_i.

The first line stands for the equilibrium payoffs of the buyer. The first equality comes from introducing (26), and the second from (27). The last line represents the payoffs of the buyer by accepting contract mh∗ instead of m~h. Also, the buyer does not obtain larger profits by selecting another latent contract, i. e. instead of choosing m~ji, the buyer chooses m~hp≠i for j, h ∈ J_i.

The buyer does not obtain a larger payoff by choosing mi0∈Ω⊂N for ∣Ω∣>1. Take ∣Ω∣=2 and, because Ti′∗<U(X∗)−U(X−i′∗), and the concavity of U(X) implies that:

If the buyer accepts the null contract for seller i it obtains larger payoffs by choosing the equilibrium contract for seller i^'. This is true for any ∣Ω∣>1.

Proof of Lemma 4. To show point (iii) of the lemma, I use a continuous approximation of γ(J₁). Then, differentiating expression (17) in the lemma with respect to J₁, and applying the Leibniz rule, I obtain:

The sign comes from Lemma 2 and the regularity conditions.

Proof of Lemma 5. To demonstrate that the buyer does not over-invest, I compare the investing threshold in (18) against the efficient investment rule in (7). For a given investment of the seller, I obtain:

A sufficient and necessary condition for K^(σ,J)<K∗ is that for any seller i , TS∗(1,σ)−TS~−i(1,σ∣Ji)>TS∗(0,σ)−TS~−i(0,σ∣Ji). Then,

The inequality results from the inefficient allocation generated when the trading allocation is substituted by the allocation when the buyer does not invest. The fundamental theorem of calculus gives:

To show that K^(σ,J) does not decrease with the increasing intensity of competition, I make use of the following claim.

Claim 1.For a fixed seller’s investment and Ji⊆Ji′,