Competition In or For the Field: Which is Better

engaging in production). In these cases functions S and π can be derived from standard demand and cost
functions.

In all the cases that follow we assume that the value of the marginal unit atq is P(q), with P⁰ < 0. We also
assume that the inverse function of P, P^-1(p) ≡ D(p) is well defined in the appropriate range. Obviously
D⁰ < 0. Agent i incurs in total cost c(qi) when producing qi units of output at a given plant, with c⁰ > 0 and
c⁰⁰ ≥ 0.¹²

We find that in all three cases a sufficient condition for a joint contract to be better for the principal is
that marginal revenue be decreasing in quantities. In addition, we show that whenever this holds, a social
planner also prefers a joint auction. The following two lemmas, which are proven in the appendix, will be
useful when establishing this result:

Lemma 1 ⁽i) P⁰(q) = DPq)); ⁽ii) P⁰⁰(q) = -{DP^q}3 ; (iii) D⁰(p) = P(D^; ⁽iv) D⁰(p) =   PPD^p}3.

Lemma 2 2P⁰(q) +qP⁰⁰(q) < 0 if and only if DD⁰⁰ - 2(D⁰)² < 0.

3.1 Procurement

We first consider fixed-price procurement.¹³ The principal wants to buy an input as cheaply as possible,
and can choose between one or two suppliers. Clearly, the principal cares (directly) only about the price p
paid per unit, and not about production costs c (of course, as in any principal-agent problem, the principal
cares about the agents’ costs indirectly through the participation constraint). Hence S (p) ≡ _p^∞D(s)ds is the
principal’s surplus, and π(p) = 2pD(p) - c D2p)^) is the surplus of each agent with a separate auction; with
a joint auction the agent’s surplus is 2π(p). In this case pm = argmax 12pD(p) - c D2^p)^) } and p is such
that 2pD(p) - c (D2p)´ = 0. Therefore

S⁰  =  -D < 0,

S⁰⁰  =  -D⁰ >0

(i.e. S is convex and Corollary 1 does not apply). Also,

∏⁰ = 2 [D + (p - c' ) D ],

¹²This condition implies no loss of generality. If c⁰⁰ < 0, marginal and average costs are decreasing and auctioning jointly is
clearly better.

¹³In fixed-price procurement contracts, the buyer and the seller agree on a price, and the seller assumes all cost risk. At the
other extreme, in a cost-plus contract the buyer reimburses the seller’s cost. As argued in Laffont and Tirole (1993, p. 662), only
fixed-price contracts are relevant when it is too costly for the buyer to audit the subcost of the supplier.

<<   7   8   9   10   11   12   13   >>

More intriguing information