in a simple setting, assume that the agents’ profits, π, are increasing and concave in the equilibrium price,
and that the principal’s surplus function, S, is decreasing and linear in the equilibrium price.5 Consider
next what happens if the principal substitutes two independent contracts by a Demsetz auction. Clearly,
this eliminates the price variability described by F . And since π is concave in price and the participation
constraint always binds, the price that results from a Demsetz auction must be lower than the average with
two agents and ex post competition, i.e., competition for the field leads to lower prices than competition in
the field. Thus, a Demsetz auction is better for the principal when her surplus function is linear. It is easily
seen, as well, that if the surplus function S is sufficiently convex, separate contracts may be better, because
then the principal likes price variability. Our main result generalizes this intuition and shows that a Demsetz
auction is unambiguously better when the composition of the principal’s surplus function and the inverse
profit function, S ◦ π-1, is strictly concave. Conversely, when this composite function is strictly convex two
separate contracts are unambiguously better.
As in the theory of expected utility, we find that this general result is equivalent to a simple condition that
compares the curvatures of the surplus and profit functions. This condition is quite similar to the necessary
and sufficient condition for a utility function to be more risk averse than the other (Pratt’s [1964] theorem)
and makes it easier to compare competition in the field with competition for the field. The condition amounts
to checking a relation that involves only the first and second derivatives of S and π. We illustrate the
usefulness of this condition in the applications section, showing that a Demsetz auction is preferred by the
principal in all cases considered—procurement, royalty contracts, dealerships—whenever marginal profits
are decreasing in quantities.
Our paper is related to the literature of monopoly regulation via franchising which was pioneered by
Chadwick (1859) and Demsetz (1968) (see also Stigler [1968], Posner [1972], Williamson [1975], Riordan
and Sappington [1987], Spulber [1989, ch. 9], Laffont and Tirole [1993, chs. 7 and 8], Harstad and Crew
[1999] and Engel, Fischer and Galetovic [2001 a, b]). We extend this literature by studying Demsetz auctions
in contexts where imperfect competition “in the field” is feasible and is an alternative to a standard Demsetz
auction.
The applications we study suggest that our paper is also related to the literature on the “double marginal-
ization” problem in monopoly pricing (see Spengler [1950] for the seminal contribution and Tirole [1988,
ch. 4] for a review of the literature). Our result implies that when marginal revenue is decreasing in the quan-
tity sold and downstream competition is imperfect, auctioning an exclusive contract is better than relying on
ex post imperfect competition.
The rest of the paper proceeds as follows. In Section 2 we describe the general model and prove the main
result of the paper. In section 3 we apply this general result to study four applications. Section 4 concludes
5While we assume that profit and surplus functions are linear in money (that is, they are risk neutral in money terms), neither
the agents’ profit function nor the principal’s surplus function need, in general, be linear in the equilibrium price, i.e., they are risk
averse (or loving) in prices. For example, profit functions are typically quasiconcave in price. By contrast, when the agent is a
planner who wants to maximize consumer surplus, the principal’s objective function is convex in prices.