Partner Uncertainty and the Dynamic Boundary of the Firm

Long-term Contracts. We now briefly show that the use of a long-term contract cannot improve the outcome. Suppose that Alice and Bob retain individual asset ownership at date 0, but commit to a contract that specifies the price at which they transact at date 4. If the contract is only structured as an option without commitment, then it has no effect at all. However, if the contract is binding, then the two partners face a similar renegotiation game as under joint asset ownership: If they want to switch partners, they cannot do so without the consent of their original trading partner. This in turn implies that a long-term contract cannot prevent retention externalities. Moreover, the division of surplus in equilibrium is determined by the bargaining outcome. The only difference between a long-term contract and joint asset ownership is how the surplus is split between the two partners. Under joint asset ownership, Alice and Bob each get a constant fraction of the profits, as defined by α and β. With a long-term contract the partners agree on a pre-specified price (or pricing formula) that determines the division of surplus. What matters for the model is not the actual distribution at date 4, but the expected distribution at dates 2 and 3. Suppose Bob (upstream) sells the input to Alice (downstream). Let ṽ be the value of the input for the buyer (Alice) at date 4, and c̃ be the cost of the seller (Bob). Their joint surplus is then given by y = ṽ− c̃. We conveniently denote the joint distribution of ṽ and c̃ by Ωvc(ṽ, c̃), so that ∫ ydΩin(y) = ∫ (ṽ− c̃)dΩvc(ṽ, c̃). Let λ be the transfer price specified in the long-term contract. This price can only be made contingent on verifiable information, i.e., on the realizations of ṽ and c̃ at date 4. With a constant inside prospect π, it is easy to see that the two partners agree on a unique transfer price λ∗ that allows them to split the expected surplus according to the bargaining outcome.1 Alternatively, they can define a flexible pricing schedule that implements the bargaining outcome. The flexible pricing schedule must then satisfy α∗y = ṽ− λ∗, which requires λ∗ = (1 − α∗)ṽ + α∗c̃. However, a long-term contract only affects the means through which the total surplus is split, and not the division of surplus itself. Thus, long-term contracts gen1Specifically, we have α∗ ∫ (ṽ − c̃)dΩvc(ṽ, c̃) = ∫ (ṽ − λ)dΩvc(ṽ, c̃), or equivalently, λ∗ = ∫ ṽdΩvc(ṽ, c̃)− α∗ ∫ (ṽ − c̃)dΩvc(ṽ, c̃).