for all p ∈ [p, pm). Since, by assumption, π0 > 0 and S0 < 0 in the relevant range, equation (1) is equivalent
to
00 0 0 00
(2) S π < S π .
Moreover, the converse of condition (1) is necessary and sufficient for S ◦ π-1 to be convex.
Proof: See the Appendix. ∣
Corollary 1 Ifπ is strictly concave, then the concavity ofS is sufficient for a joint contract to be better than
two separate contracts. ∣
We can use Proposition 2 and Figures 1 and 2 to examine the intuition underlying our main result.
Suppose that S is linear, π strictly concave, two separate contracts are auctioned, and in equilibrium p can
take only two values, p and pm, with equal probability. In this case each agent makes expected profits equal
to 1 π(p) + 2π(pm) = 2π(pm) and the principal’s surplus equals 2S(p) + 2S(pm) (see Figure 1). Since S
is linear and π concave, Proposition 2 holds, and a joint auction is better than a separate auction. Why?
Condition 1 implies that 2π(pm) = π(pJ). As is straightforward from Figure 1a, concavity of π implies that
pJ < 2p + 2pm. Hence the principal obtains a lower average price with a joint auction.11 Because in this
example S is linear, 2S (p) + 2S(pm) = S ( 1 p + 2pm) < S(pJ) (see Figure 1b). Note that the same reasoning
applies to any probability distribution F with support in the interval [p,pm].
It can now easily be seen why strict concavity of S is sufficient for a joint auction to be better when
π is concave. Eliminating variability in p is an added bonus for the principal, since ES(p) < S(Ep) for
all distributions F . Conversely, when S is convex, a separate auction may (but need not) be better. Figure
2 depicts exactly the same case as Figure 1, except that S is convex, so that now the principal likes price
variability. For the particular distribution depicted in this figure, the principal is indifferent between a joint
and a separate auction. Essentially, the gain of a lower expected price p attained with a joint auction is
exactly offset by the fall in the expected surplus due to lower price variability. With S sufficiently convex
and for a given π, the gains from a lower expected price are outweighed by the utility loss which stems from
losing “high” surpluses.
3 Applications
In this section we use Proposition 2 to study three canonical applications: procurement (the principal buys
the production of the plants), dealerships (agents buy an input from the principal and incur some costs to
transform and resell it) and royalties (the principal receives a fixed fee per unit sold by the agent without
11 This can be put in the more standard terms of principal-agent theory. From Condition 1 it follows that the agent’s participation
constraint is 2π(pm) = π(pJ). Since π is concave, the average price that the agent requires in order to participate is lower with a
joint contract, which eliminates risk.