and is followed by a brief appendix.
2 General model and main result
A risk neutral principal wants to contract the production of a good at two plants or locations.6 Output from
one plant is a perfect substitute for the output of the other. If the equilibrium price is p, an agent producing
at one plant makes profits π(p), with π0(p) > 0 for p ∈ [p,pm), where π(p) = 0 and pm = argmaxpπ(p).
Furthermore, π0(pm) = 0 and π00 (pm) < 0. On the other hand, the principal’s surplus is S(p) if agents charge
p, with S0 (p) < 0. Hence there is a conflict of interest: while agents would like to increase prices up to pm ,
the principal wants the price to be as low as possible.
The principal may award both plants jointly (J), so that they are run by one agent; or separately (S), so
that two agents run one plant each and compete. The principal auctions both contracts. When both plants
are awarded jointly, the winning bid is denoted bypJ and per-plant profits for the agent are equal to π(pJ).7
On the other hand, when plants are awarded to different agents, the minimum winning bid, common across
plants, is denoted by pS . In this case agents are uncertain both about whether they will be able to collude,8
and, if they do, about the price above p at which they will collude.9 We assume that each agent serves half
the demand at a common equilibrium price p, and denote by F(p) the cdf with support [p,pS] that describes
their common beliefs about the realization of this price.10
We make the essential assumption that ex-ante expected gross profits per plant under a joint or a separate
auction are the same, that is
Condition 1
pS
— u +1,
π(p)dF(p) =π(pJ)
Jp
where u is the agent ,s reservation utility and I stands for any sunk setup cost. There exist many agents that
could produce the good, all of them with the same value of ( u +1 ).
Condition 1 implies that benefits for agents are independent of whether the principal auctions production
at both plants jointly or separately. Or, in the standard guise of principal-agent theory, Condition 1 is the
participation constraint that the principal must obey. Note also that if F (p) is degenerate, Condition 1
implies that under separate auctions the price will be pJ, so that joint and separate auctions are identical. We
rule out this possibility by assumption in what follows.
6All that follows extends trivially to the case of n locations.
7We assume pJ ≤ pm .
8Caillaud and Tirole (2001) consider this possibility in the context of essential facilities.
9That is, we assume that prices are such that agents do not lose money ex post, since π(p) = 0.
10Competition in practice is generally neither static nor symmetric. We avoid complications by concentrating on stationary
equilibria and we use symmetry due to the lack of consensus on how to model collusion in asymmetric games.