When plants are awarded jointly, the principal’s benefit is denoted WJ ≡ S(pJ). On the other hand, when
when they are awarded separately, the principal’s expected benefit depends on the distribution of collusive
prices F and equals
pS
WS ≡ S(p)dF(p).
pp_
From the assumptions we made on π, it follows that π 1 : [π(p), π(pm)] → [p,pm] is well defined, increasing
and convex. We then have the following central result of the paper:
Proposition 1 If S ◦ π-1 is strictly concave, then WJ > WS. If S ◦ π-1 is strictly convex, then WS > WJ.
Proof: We consider the case where S ◦ π-1 is concave. The case where it is convex is analogous. We have:
pS
WS ≡ S(p)dF(p)
pP-
pSs 1
= S ◦ π-1 [π(p)]dF(p)
pp
pS
π(p)dF(p)
pP
S ◦ π 1[π( pj )]
S(pJ)
WJ,
where the inequality follows from Jensen’s inequality and our assumption that F is not degenerate, and the
identity after the inequality from Condition 1. ∣
A surprising feature of this result is that we have not imposed any condition on the distribution of
possible collusive outcomes F . Hence, in order to compare joint and separate auctions, it is sufficient to
examine the ‘primitive’ functions π, and S , and one can ignore the exact specification of the ex post game
between the agents.
This result depends crucially on Condition 1, which ensures that softer competition when the partici-
pation constraint becomes more demanding (that is, и +1 increases). In the case of joint production, this
means softer competition for the franchise, while in the case of separate production it means less competition
between both agents after they begin producing.
The following result provides a simple characterization for the concavity of S ◦ π-1.
Proposition 2 A necessary and sufficient condition for S ◦ π 1 be concave is that
(1)
S00 π00
Sf > π7