Optimal Mechanism for Selling Substitutes
Gregory Pavlov∗†
Boston University
February 9, 2006
Abstract
We study a problem of a multiproduct monopolist selling substitutable goods to a buyer
with unknown valuations. Under the standard distributional assumptions we find that in
the optimal menu every nontrivial contract delivers some good with certainty. Using
this result we apply control-theoretic tools to the case of two goods and solve a number
of examples. The optimal menus generally have a simple structure and sometimes are
substantially more profitable than the deterministic menus. We also extend the approach
to the case when the buyer desires more than a single unit of the good.
JEL classification: C61; D42; L11
Keywords: Multidimensional screening; Price discrimination; Optimal selling strategies
1
Introduction
We study a problem of a multiproduct monopolist with constant marginal costs selling to
a buyer with unknown valuations. From the buyer’s perspective all goods are substitutes,
i.e. he desires only a single unit. We show that under the standard regularity conditions on
∗
Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA. E-mail:
gpavlov@bu.edu. webpage: http://people.bu.edu/gpavlov/.
†
This paper is based on Chapter 3 of my dissertation at Northwestern University. I am especially grateful to
my dissertation advisor Asher Wolinsky. I also thank Eddie Dekel, Jeff Ely, Maria Goltsman, Preston McAfee,
Jean-Charles Rochet, Charles Zheng and bag lunch seminar participants at Boston University. All mistakes
are mine.
1
the distribution of the buyer’s valuations it never pays for the seller to randomize over the
aggregate probability of selling: in the optimal menu each nontrivial contract delivers some
good with certainty. However this result does not rule out stochastic contracts, because it
may be profitable for the seller to offer lotteries to screen over the differences of the buyer’s
valuations.
On the technical side we contribute an analytical technique for solving a class of multidimensional mechanism design problems and characterize the solution. In particular we show
how to apply control-theoretic tools to study the case of two goods, and solve a number of
examples. The optimal menus are generally quite simple and sometimes are substantially
more profitable than the best deterministic menus.
The most related papers to the problem we study are Thanassoulis (2004), and Manelli
and Vincent (2004a), (2004b). Thanassoulis (2004) considers a model of substitutes restricted
to the case of two goods. He shows that the seller often benefits from offering lotteries in
addition to the individually priced goods.1 There is a negative effect from offering a lottery
at a discount relative to the individually priced goods, because the buyers who are relatively
indifferent between the goods switch to purchasing a lottery. But there is also a positive effect
from attracting additional buyers who would not participate otherwise. Thanassoulis (2004)
solves for the optimal menus in some examples using numerical methods, but does not provide
the analytical characterization for the general case.
The model in Manelli and Vincent (2004a), (2004b) is similar to the problem we study
with the only difference that the buyer may desire more than a single unit of the good.
The superiority of the mixed bundling strategy over the individual pricing of the goods in
this model was established long ago.2 However the characterization of the fully optimal
mechanism, which may include stochastic contracts, was not available.3 Manelli and Vincent
(2004a), (2004b) in the first paper derive the sufficient conditions for the optimality of the
deterministic contracts, and in the second paper give the necessary conditions for a feasible
1
There is an example in Thanassoulis (2004) where the profit gain relative to the best deterministic mechanism is 8%.
2
See McAfee, McMillan and Whinston (1989).
3
McAfee and McMillan (1988) claim that the optimal mechanism in the case of two goods is deterministic,
but Thanassoulis (2004) points out a mistake in their proof.
2
incentive compatible mechanism to be optimal for some distribution of the buyer’s valuations.
It turns out that the class of potentially optimal mechanisms is very large. However the
distributions of the valuations required to rationalize many of the mechanisms may be very
irregular. The characterization of the optimal mechanisms for the case of economically relevant
distributions is still an open question. In Section 6 we extend our approach to this model and
obtain some results for a class of symmetric problems.
In Section 3 we show that under the standard conditions on the distribution of the buyer’s
valuations the optimal mechanism is deterministic with respect to the aggregate probability
of selling.4 An implication of this property is that the buyer who chooses some nontrivial
contract is automatically guaranteed that his gross benefit is no smaller than the valuation
for the least liked of the goods. Hence the seller effectively can discriminate among the
buyers only on the differences in their valuations, i.e. all participating buyers with the same
differences in the valuations are pooled to receive the same allocation. The role of the levels
of the valuations is only to determine the decision of whether to participate or to take the null
contract. For the case of two goods these observations allow to find the optimal mechanism
by control-theoretic techniques.
Incidentally, the mathematical structure of the resulting problem is very similar to the
model in Rochet and Stole (2002), who study the problem of nonlinear pricing when the buyers
have heterogeneous outside options. The main difference is that in their model the costs are
taken to be quadratic, while we consider the case of constant marginal costs. In both cases the
task of finding the optimal mechanism is quite formidable because the problems lack recursive
structure. The solutions to these two models are qualitatively different: in Rochet and Stole
(2002) the optimal allocation is (for the most part) separating, while in our model there is
a significant amount of pooling. Moreover, in order to solve particular examples Rochet and
Stole (2002) ultimately had to resort to numerical computations, while we provide analytical
solutions. For these reasons the analyses performed in Rochet and Stole (2002) and in this
paper are best viewed as complementary.
4
One may view this result as an extension of the ”no haggling” result which was established for the one
dimensional case by Riley and Zeckhauser (1983).
3
We characterize the optimal mechanisms, provide a condition for the uniqueness and then
solve a number of examples. These examples suggest that the optimal menus are generally
quite simple, i.e. consist of just a few point contracts. The relative gain in expected revenues
from using the fully optimal mechanisms as opposed to the best deterministic mechanism may
vary substantially depending on the distribution. In Section 5.3 we provide an example where
the gain is 50%.
In Section 6 we extend the results to the model of the multiproduct monopolist studied by
Manelli and Vincent (2004a), (2004b). We establish that in the symmetric case all nontrivial
contracts in the optimal menu must deliver at least one of the goods with probability one. This
fact again allows for the use of the control-theoretic techniques to find the optimal mechanism.
A solved example is provided.
The rest of the paper is organized as follows. A model of substitutes is introduced in
Section 2. The 0-1 property of the optimal mechanisms is established in Section 3. In
Section 4 we apply the control theoretic technique to the case of two goods and establish some
qualitative properties of the solution. Solved examples and their discussion are in Section 5,
and the problem of the multiproduct monopolist when the buyer may desire more than a
single unit is in Section 6. The summary and an outline of other applications of the model is
in Section 7. Long proofs are in Appendices A-D.
2
The Model
There is one seller who owns m ≥ 1 indivisible goods and one buyer who desires at most
a single unit of the good.5 The buyer’s valuations for each of the m goods θ = (θ1 , ..., θ m )
are known only to him. Valuations are distributed on a convex compact support Θ ⊂ <m
+
according to an almost everywhere positive bounded differentiable density f (θ1 , ..., θm ). This
distribution is common knowledge.
Both the buyer and the seller have quasi-linear utilities. The buyer’s utility is θ·p−T where
p = (p1 , ..., pm ) are the probabilities of obtaining each of the goods, and T is his payment.
5
Throughout the paper we use feminine pronouns for the seller and masculine pronouns for the buyer.
4
The seller’s utility is T .6 Since the buyer desires only a single unit of the good, the feasible
©
ª
P
set for the probabilities can be taken to be Σ = σ ∈ <m
+ |
i σi ≤ 1 .
The seller offers a direct mechanism consisting of a set of messages Θ, the probabilities
of assigning each good p : Θ → Σ, and the payment T : Θ → < conditional on the reported
message.7 It is without loss of generality to restrict attention to deterministic payments in
our setup. The seller maximizes her expected payoff E [T (θ)] by choosing a mechanism (p, T )
subject to the following constraints:
• Feasibility: p (θ) ∈ Σ for every θ ∈ Θ;
¡ ¢
¡ ¢
• Incentive Compatibility: θ · p (θ) − T (θ) ≥ θ · p θ0 − T θ0 for every θ, θ0 ∈ Θ;
• Individual Rationality: θ · p (θ) − T (θ) ≥ 0 for every θ ∈ Θ.
We call a mechanism (p, T ) admissible if it satisfies the above constraints.
Denote by ∂Θ the surface of the set Θ, and by n (θ) the outward-pointing unit normal
vector at a point θ on the surface of Θ. Define the following subsets of the surface:
∂Θ+ = {θ ∈ ∂Θ : θ · n (θ) ≥ 0}, ∂Θ++ = {θ ∈ ∂Θ : n (θ) ≥ 0}.
For example, let Θ be a disc on a two dimensional plane with center at (1, 1) and radius
n
o
1. Then the surface of the set Θ is the circle ∂Θ = (θ1 , θ2 ) : (θ1 − 1)2 + (θ2 − 1)2 = 1 , and
the unit normal at a point (θ1 , θ2 ) on the surface is (θ1 − 1, θ2 − 1). It is easy to verify that
∂Θ+ is the subset of the circle such that θ1 ≥ 1 or θ2 ≥ 1, and ∂Θ++ is the subset of the
circle such that θ1 ≥ 1 and θ2 ≥ 1.
Throughout the paper we require the following condition to hold.
Condition 1 ∂Θ++ = ∂Θ+ .
£ ¤
© ª
In a one dimensional case Θ = θ, θ . So we always have ∂Θ++ = ∂Θ+ = θ . In a
multidimensional case only ∂Θ++ ⊆ ∂Θ+ is true in general. We require the reverse inclusion
in order to have a subset of the surface which we could unambiguously call ”the top types”.
6
A more general model would be for the seller to have known constant marginal costs of providing each of
the goods c = (c1 , ..., cm ). All the results of the paper apply to this case if the distribution of θ − c satisfies
the assumptions imposed on the distribution of θ.
7
Using direct mechanisms is without loss of generality by the revelation principle.
5
Condition 1 allows for a variety of supports considered in the literature. For example conµ
¶
¤
¡
¢
m £
m
sider a rectangular support Θ = × θi , θi ⊂ <+ . Then for any ”outer face” × θi , θi ×
i=1
µ
¶ i6=j
¡
¢
© ª
© ª
θj the unit normal is n (θ) = ej , and for any ”inner face” × θi , θi × θj the unit
i6=j
normal is n (θ) = −ej , where ejj = 1 and eji = 0 for i 6= j. Hence ∂Θ++ = ∂Θ+ .8
We also require the distribution to satisfy a version of a ”hazard condition” which is
standard in this literature.9
Condition 2 (m + 1) f (θ) + θ · ∇f (θ) > 0, where ∇f (θ) is the gradient of f evaluated at θ.
3
The 0-1 Property of the Allocation
In this section we show that under Conditions 1 and 2 the optimal mechanism must satisfy
the following property.
Definition 1 The mechanism satisfies a 0-1 property if
every θ ∈ Θ.
P
i pi (θ)
is either 0 or 1 for almost
It is convenient to reformulate the problem of the seller in terms of choosing the buyer’s
utility schedule.10 Denote the utility of the buyer of type θ by U (θ) = θ · p (θ) − T (θ). The
problem of the seller is
maxE [θ · ∇U (θ) − U (θ)]
U
subject to
• Feasibility: ∇U (θ) ∈ Σ for every θ ∈ Θ;
• Incentive Compatibility: U is convex;
• Individual Rationality: U (θ) ≥ 0 for every θ ∈ Θ.
Theorem 1 Consider a mechanism that does not satisfy the 0-1 property. Under Conditions
1 and 2 there exists a 0-1 mechanism which brings a strictly higher expected revenue.
8
See Section 5 for other examples of supports which satisfy Condition 1.
See for example McAfee and McMillan (1988), Manelli and Vincent (2004a).
10
See Rochet (1985).
9
6
Proof. See Appendix A.
£ ¤
We show the main logic of the proof for the one dimensional case Θ = θ, θ .11 Take
£ ¤
some admissible mechanism (p, T ) : θ, θ → [0, 1] × < which generates a utility schedule
£ ¤
U : θ, θ → <+ and does not satisfy the 0-1 property. Introduce a new ”take it or leave it”
¡ ¢
¡
¡ ¢¢
mechanism with a price T θ + θ 1 − p θ . Notice that the utility of the highest type under
the new mechanism is left unchanged, and the utilities of all the other types are weakly lower
than before:
¡ ¢
¡ ¡ ¢
¡
¡ ¢¢¢
¡ ¢
¡ ¢
¡ ¢
U θ =θ− T θ +θ 1−p θ
= θp θ − T θ = U θ ,
©
¡ ¡ ¢
¡
¡ ¢¢¢ ª
© ¡ ¢ ¡
¢ ª
U (θ) = max θ − T θ + θ 1 − p θ
, 0 = max U θ − θ − θ , 0 ≤
Zθ ³ ´
¡ ¢
≤U θ − p e
θ de
θ = U (θ).
θ
The inequality follows from the individual rationality, feasibility and the envelope formula
for the buyer’s utility. Notice that the inequality must be strict for a positive measure of
types since the original mechanism does not satisfy the 0-1 property. Using integration by
parts we can represent the expected revenue in terms of the utility schedule as follows.
¡ ¢ ¡ ¢
E [θ · ∇U (θ) − U (θ)] = U θ θf θ − U (θ) θf (θ) −
Zθ
θ
¡
¢
U (θ) 2f (θ) + θf 0 (θ) dθ
(1)
The expected revenue from the ”take it or leave it” mechanism is strictly higher than the
revenue from the original mechanism since 2f (θ) + θf 0 (θ) > 0 by Condition 2. In case of
multiple goods we use the divergence theorem to represent the expected revenue.12
E [θ · ∇U (θ) − U (θ)] =
Z
U (θ) f (θ) [θ · n (θ)] dθ −
Z
U (θ) [(m + 1) f (θ) + θ · ∇f (θ)] dθ (2)
Θ
∂Θ
Similar to the one dimensional case we replace the original mechanism with a 0-1 mechanism
so that the utilities of the ”upper surface” types ∂Θ+ are unchanged and the utilities of all
the other types are lowered. Some extra work is required relative to the one dimensional case
11
Of course the result for the one dimensional case was established long ago under more general conditions
(see for example Myerson (1981), Riley and Zeckhauser (1983)).
12
See for example Schey (1996).
7
because there is a whole set of the ”upper surface” types, and the original mechanism has to
be adjusted in an incentive compatible way individually for all those types.
Hence we found that the seller does not benefit from randomizing over the aggregate
P
allocation i pi (θ). However she may still benefit from offering lotteries over different goods
as a part of the optimal menu. Next we show that Theorem 1 implies that the seller can
optimize over a smaller set of mechanisms in which the screening is performed only on the
differences in the valuations. In Section 4 this result is used to develop a technique for solving
the case of two goods.
Corollary 1 There is no loss for the seller to screen only on the differences in the valuations
conditional on participation, i.e. if θ, θ0 ∈ Θ are such that θ0 = θ + 1 · t for some t ∈ < and
¡ 0¢
P
P ¡ 0¢
i pi (θ) =
i pi θ = 1, then (p1 , ..., pm , T ) (θ) = (p1 , ..., pm , T ) θ .
Proof. Consider types θ, θ0 ∈ Θ such that
(i) θ0 = θ + 1 · t = (θ1 + t, ..., θm + t) for some t ∈ <;
P
P ¡ ¢
(ii) i pi (θ) = i pi θ0 = 1.
Notice that θ0i − θ0j = θi − θj for any i and j, i.e. these types have the same differences in
the valuations across goods.
By the incentive compatibility
¡ ¢ P ¡
¢ ¡ ¢
¡ ¢
U (θ) ≥ U θ0 + i θi − θ0i pi θ0 = U θ0 − t.
¡ ¢
¡ ¢
Similarly U θ0 ≥ U (θ) + t, and thus U (θ) = U θ0 − t. This implies that
¡ ¢
¡ ¢
¡ ¢
P
P
T (θ) = i θi pi (θ) − U (θ) = i θ0i pi θ0 − U θ0 = T θ0 .
¡ ¢
Hence for all such types θ0 we can replace the allocation and the payment (p1 , ..., pm , T ) θ0
by (p1 , ..., pm , T ) (θ). The new mechanism is incentive compatible and generates the same
utilities for all types of the buyer. Hence it results in the same expected revenue for the seller.
Since by Theorem 1 the optimal mechanism must satisfy the 0-1 property we conclude that
there is no loss for the seller to screen only on the differences in the valuations conditional on
participation.
It is possible that the 0-1 property of the optimal mechanisms holds more generally, and
that the conditions in Theorem 1 are only sufficient and far from being necessary. However
8
we were not able either to extend the proof to a larger class of cases, or to provide a counterexample within the setup of the model with continuous distributions. Below is an example
with a discrete distribution where the 0-1 property does not hold.13
©
© ª
ª
Example 1 Let Θ = θ1 , θ2 , θ3 where θ1 = (0, 1), θ2 = (1, 0), θ3 = (3, 2), and Pr θi = 13 ,
¢
¡
¢ ¡
¢ ¡
¢
¡
i = 1, 2, 3. The optimal menu is p11 , p12 , T 1 = (0, 1, 1), p21 , p22 , T 2 = 12 , 0, 12 , p31 , p32 , T 3 =
(1, 0, 2).
Proof. Let us ignore the incentive constraints for θ1 , θ2 and the individual rationality
constraint for θ3 . We will check them ex post.
Clearly it is optimal to set p11 = 0, p22 = 0, T 1 = p12 and T 2 = p21 . Also it is optimal to
assign the efficient allocation to type θ3 : p31 = 1 and p32 = 0. Thus the incentive constraints
for θ3 are
3 − T 3 ≥ 2p12 − T 1 = p12 ,
3 − T 3 ≥ 3p21 − T 2 = 2p21 .
©
ª
Hence T 3 = 3 − max p12 , 2p21 . The problem of the seller reduces to
¡
ª¢
©
max 13 p12 + p21 + 3 − max p12 , 2p21
p12 ,p21
subject to 0 ≤ p12 , p21 ≤ 1.
Notice that
⎧
⎪
⎪
p12 − p21 < p12 − 12 p12 ≤
⎪
⎪
⎨
ª
©
p12 + p21 − max p12 , 2p21 =
p21 < 12 p12 ≤ 12
⎪
⎪
⎪
⎪ 1 1 1
⎩
2 p2 ≤ 2
Hence p12 = 1, p21 =
1
2
1
2
if
2p21 > p12
if
2p21 < p12 .
if
2p21 = p12
and T 3 = 2.
In the one dimensional case a reduction in the allocation for the lower types always allows
to extract more money from the higher types. This is not necessarily the case when there
are multiple dimensions. In this example both downward incentive constraint for type θ3 are
binding at the optimal contract. A reduction in the allocation for type θ2 does not allow to
extract more money from type θ3 , since he can still mimic type θ1 .
13
This example is similar in spirit to the example in Baron and Myerson (1982).
9
4
The Case of Two Goods
4.1
Reformulation of the Problem
In this section we develop the tools for solving the seller’s problem in the two dimensional
case. Define the difference in the valuations by δ = θ1 − θ2 . Since Θ is convex an compact,
£ ¤
the set of the differences is an interval δ, δ = [minθ∈Θ (θ1 − θ2 ) , maxθ∈Θ (θ1 − θ2 )].
By Corollary 1 to Theorem 1 there is no loss for the seller to offer mechanisms which screen
only on the differences in the valuations conditional on the participation, i.e. (p1 (θ) , p2 (θ) , T (θ))
is either equal to (0, 0, 0) or to (α (δ) , 1 − α (δ) , t (δ)).14
With this reformulation the incentive constraints are
¡ ¢
¡ ¢
δα (δ) − t (δ) + θ2 ≥ δα δ 0 − t δ 0 + θ2 for all participating types,
or alternatively
¡ ¢
£ ¤
¡ ¢
δα (δ) − t (δ) ≥ δα δ 0 − t δ 0 for every δ, δ 0 ∈ δ, δ .
Denote u (δ) = δα (δ) − t (δ). Next we simplify the incentive constraints.
Lemma 1 The incentive compatibility constraints are equivalent to:
(i) α is nondecreasing;
·
(ii) u (δ) = α (δ) at all continuity points of α.
Proof. Standard. See for example Myerson (1981).
The buyer of the type θ participates only if his utility from participation is nonnegative.
Hence U (θ) = max {u (δ) + θ2 , 0}. The revenue from the buyer of the type θ is δα (δ) − u (δ)
whenever u (δ) + θ2 ≥ 0, and 0 otherwise. The measure of the participating types for a fixed
Z
δ is
f (θ) dθ, which we denote by g (u (δ) , δ).
θ: θ1 −θ2 =δ,
u(δ)+θ2 ≥0
Incidentally, the mathematical structure of the resulting problem is very similar to the
model in Rochet and Stole (2002), who study the problem of nonlinear pricing when the
buyers have heterogeneous outside options. The main difference is that their model has in
One may worry that (α, t) may be not defined over some subset of δ, δ . This happens when there exists
δ such that (p1 (θ) , p2 (θ) , T (θ)) = (0, 0, 0) for all θ ∈ Θ such that θ1 − θ2 = δ. However in what follows it will
be clear that there is no loss for the seller in specifying (α, t) for such types which they would choose if they
were required to participate.
14
10
addition quadratic costs. In both cases the task of finding the optimal mechanism is quite
formidable because the problems lack recursive structure. The solutions to these two models
will turn out to be qualitatively different: in Rochet and Stole (2002) the optimal allocation
is (for the most part) separating, while in our model there will be a significant amount of
pooling.
The seller’s problem is
Zδ
Problem I max g (u (δ) , δ) (δα (δ) − u (δ)) dδ
α,u
δ
subject to
£ ¤
• Feasibility: α (δ) ∈ [0, 1] for every δ ∈ δ, δ ;
·
• Incentive Compatibility: (i) α is nondecreasing; (ii) u (δ) = α (δ) at all continuity points
of α.
Notice that the individual rationality constraint is automatically satisfied since the null
contract is available. In Appendix B we formulate Problem I as an optimal control problem,
provide the necessary conditions, prove their sufficiency and also determine a condition for
uniqueness of the solution. The proof of the sufficiency is required because the standard
Arrow type results for sufficiency of the necessary conditions do not apply.15 The reason is
that the Lagrangian L∗ in the optimal control problem is not jointly concave in (α, u). We
demonstrate uniqueness in the case when the exclusion region of the candidate solution is
Z
P
nondegenerate, i.e. there exists a subset Θ0 ⊂ Θ such that
f (θ) dθ > 0 and i pi (θ) = 0
Θ0
for almost every θ ∈ Θ0 .16
The problem can be considerably simplified when the distribution of the valuations is
symmetric.
Definition 2 The distribution (Θ, f ) is symmetric if
(i) (θ1 , θ2 ) ∈ Θ implies that (θ2 , θ1 ) ∈ Θ;
15
See for example Seierstad and Sydsæter (1987).
Armstrong (1996) showed that the strict convexity of the set of the valuations Θ is a sufficient condition
for an exclusion region to be nondegenerate. Our model allows for supports that are not strictly convex, and
thus the possibility that the optimal mechanism has a degenerate exclusion region cannot be ruled out.
16
11
(ii) f (θ1 , θ2 ) = f (θ2 , θ1 ) for every θ ∈ Θ.
For the rest of the paper we restrict attention to the symmetric case. It is straightforward
to show that in the symmetric case the seller’s problem can be solved by optimizing only over
£ ¤
£
¢
the set 0, δ and then setting α (δ) = 1 − α (−δ) and u (δ) = u (−δ) + δ for δ ∈ −δ, 0 .
Zδ
Problem II max g (u (δ) , δ) (δα (δ) − u (δ)) dδ
α,u(0)
0
subject to
• Feasibility: α (δ) ∈
£1
¤
£ ¤
,
1
for
every
δ
∈
0, δ ;
2
Zδ ³ ´
• Incentive Compatibility: (i) α is nondecreasing; (ii) u (δ) = u (0) + α e
δ de
δ for every
0
£ ¤
δ ∈ 0, δ .
4.2
Properties of the Optimal Mechanism
In this section we discuss the qualitative properties of the solution to the seller’s problem. To
understand the main trade-offs consider the marginal contribution of the allocation α (δ) to
the revenue.
Zδ ³ ³ ´ ´
Zδ
³ ´´
∂ ³ ³e´ e´ ³e ³e´
e
e
e
g u δ , δ δα δ − u e
δ de
δ
V (δ) = g (u (δ) , δ) δ − g u δ , δ dδ +
∂u
δ
{z
} |δ
{z
}
|
Standard Effect
(3)
Participation Effect
The first collection of terms illustrates a standard trade-off between earning an extra dollar
from the type δ and leaving higher informational rents to all the types above δ. The last term
is the revenue effect of the allocation at δ through the participation decisions of the types
above δ. Increasing α (δ) raises the informational rents for all types e
δ ≥ δ, and thus increases
³ ³ ´ ´
∂
the measure of the participating types by ∂u
g u e
δ ,e
δ . Every new participant of the type
³ ´
³ ´
e
δ brings an extra revenue of e
δα e
δ −u e
δ .
The formula for the marginal revenue V reveals the main difficulty: the problem is inher-
ently non-recursive. The choice of the optimal allocation α (δ) for a given δ in general depends
on the whole (optimal) profile of allocations α and rents u.
12
Let us also consider the optimal choice of the buyer’s lowest rent u (0). Notice that −u (0)
is the minimal price the seller charges the participating buyers. The trade-off here is between
attracting marginal consumers and charging all inframarginal consumers a higher price:
Zδ
∂
g (u (δ) , δ) (δα (δ) − u (δ)) dδ −
∂u
0
Zδ
g (u (δ) , δ) dδ = 0.
(4)
0
Hence the seller has two rent extraction concerns. One goal is to discriminate on the differences
in the valuations. The second problem is the choice of the optimal minimal price for all
participating types. To see these two concerns more clearly we decompose the seller’s expected
revenue into two parts:
Zδ
|0
⎛
g (u (δ) , δ) ⎝δα (δ) −
{z
Zδ
0
⎞
Zδ
³ ´
e
e
⎠
α δ dδ dδ + g (u (δ) , δ) (−u (0)) dδ
Rent Extraction from Differences in Valuations
The seller earns a revenue of δα (δ) −
Rδ
0
}
|0
{z
Minimal Price Charged
(5)
}
³ ´
α e
δ de
δ from the type δ through discrimination on
the differences in the valuations. In addition the seller charges all participating types with
−u (0). We introduce two benchmark problems to isolate the effects of these rent extraction
concerns.
Lemma 2 (i) Let (α0 , u0 (0)) solves max
Zδ
α,u(0)
0
the required constraints.
⎧
⎪
⎨ 1 if
2
Then α0 (δ) =
⎪
⎩ 1 if
(ii) Let
(α00 , u00 (0))
⎛
⎞
Zδ ³ ´
g (u (δ) , δ) ⎝δα (δ) − α e
δ de
δ ⎠ dδ subject to
0
h ´
δ ∈ 0, b
δ
¡ ¢
δ ∈ 0, δ .
³ i for some b
δ∈ b
δ, δ
solves max
Zδ
α,u(0)
g (u (δ) , δ) (−u (0)) dδ subject to the required constraints.
£0 ¤
Then α00 (δ) = 1 for every δ ∈ 0, δ .
Proof. (i) For the first problem it is clear that the seller will choose u (0) sufficiently high
to ensure full participation. Hence the measure of types with the difference in the valuations
13
δ is g (u (δ) , δ) =
Z
f (θ) dθ. Let us denote it by g (δ).
θ: θ1 −θ2 =δ
Using integration by parts we rewrite the
⎛ objective function⎞
⎛
⎞
Zδ
Zδ ³ ´
Zδ
Zδ ³ ´
⎟
⎜
e
e
g (δ) ⎝δα (δ) − α δ dδ ⎠ dδ = ⎝g (δ) δ − g e
δ de
δ ⎠ α (δ) dδ.
0
0
0
δ
Zδ ³ ´
Denote V (δ) = g (δ) δ − g e
δ de
δ. The solution is of ”bang-bang” form: α0 (δ) =
1
2
for
δ
δ. Since V is continuous and is equal to − 12 at zero, we have
δ <b
δ and α0 (δ) = 1 for δ > b
Zδ
b
δ > 0. Since V (δ) dδ = 0, we have b
δ < δ.
0
(ii) In this problem the seller does not benefit from discriminating on the differences in
the valuations. Hence for any given u (0) it pays to choose the allocation which maximizes
the measure of the participating types. The efficient allocation works best since it provides
each participating type of the buyer with his most preferred good.
It turns out that the optimal allocation in the general problem of the seller is in between
two benchmark allocations.
£ ¤
θ2 = θ for every δ ∈ 0, δ . Then the optimal allocation
θ∈Θ: θ1 −θ2 =δ
£ ¤
0
00
α (δ) ∈ [α (δ) , α (δ)] for every δ ∈ 0, δ .
Proposition 1 Let
min
Proof. See Appendix B.
Because the rent profile also determines the shape and size of the exclusion region, the seller
when choosing the rent profile cares not only about the rent extraction from the differences in
the valuations. Leaving higher rents to the buyers, i.e. making the allocation relatively more
efficient, increases participation. When the exclusion region is large, then the rent extraction
from the differences is less important relative to attracting new buyers, and thus the allocation
is more efficient. When the exclusion region is small, and thus the goal of extracting rents
from the differences is more pronounced, then the solution resembles the optimal allocation in
the first benchmark problem (α0 ). This features can be clearly observed in Section 5.1 where
we discuss an example of a uniform distribution on a square.
A similar feature of the solution arises in the random participation model of Rochet and
14
Stole (2002). They show that the optimal allocation lies in between the efficient schedule and
the classic monopolistic schedule (i.e. when there is no concern about participation).
5
Examples and Discussion
5.1
Uniform Distribution on a Square
Example 2 Let the distribution of the valuations be uniform on Θ = [c, c + 1]2 where c ≥ 0.
Solving the optimal control problem we find the optimal menu of the allocations and prices
(p1 , p2 , T ) depending on the parameter c.
Proposition 2 In Example 2 the optimal menu is
√
(i) {(0, 0, 0) , (1, 0, T ) , (0, 1, T )}, where T = 23 c + 13 c2 + 3
©
¡
¢ ¡
¢ª
(ii) (0, 0, 0) , (α, 1 − α, T ) , (1 − α, α, T ) , 1, 0, T , 0, 1, T ,
¡9
¢√
√
1
where α = 27
16c + 9, T = 13 c + 38 + 18 16c + 9,
32 + 32 − 4 c
¡5
¡9
¢√
¢
T = T + 13 32
− 32
− 14 c 16c + 9
¢ ¡
©
¡
¢ ¡
¢ª
(iii) (0, 0, 0) , 12 , 12 , T , 0, 1, T , 1, 0, T ,
q
2
1
where T = 3 c + 3 c2 + 32 and T = T + 16
if c ∈ [0, 1];
if c ∈ (1, c]
where c ≈ 1.3721;
if c ∈ (c, +∞).
Proof. See Appendix C.
In this example the marginal revenue in the benchmark problem of rent extraction from the
differences in the valuations is the same for every c, and is equal to V (δ) =
1
2
(1 − δ) (3δ − 1).
According to⎧Lemma 2 the two benchmark allocations are as follows:
¢
£
⎪
⎨ 1 if δ ∈ 0, 1
2
3
0
00
α (δ) =
¡ 1 ¤ and α (δ) = 1 on [0, 1].
⎪
⎩ 1 if δ ∈ , 1
3
By Proposition 1 the optimal allocation is in between α0 and α00 . When c is small then
it is not too costly to exclude types, and thus the exclusion region is relatively large and the
allocation (conditional on participation) is more efficient. However, when c is large then it
is costly to exclude types, and thus the exclusion region is relatively small and the optimal
allocation resembles the α0 allocation.
15
Notice that the optimal menus have a simple structure. The technical reason for this is
roughly as follows. The optimal control problem is of a ”bang-bang” nature in α. A number
of pooling regions for α emerge due to the presence of the monotonicity constraint, however
there are only very few such regions. An important question is whether we can expect the
same to happen in general. We discuss it in Section 5.2.
It is interesting to compare the expected revenues from the fully optimal mechanism and
the optimal deterministic mechanism which makes no use of the lotteries.17 Using the formulas
for the expected revenues it is possible to show that the relative gain from using a fully optimal
mechanism is at most about 1.2%.18 In Section 5.3 we discuss whether we can expect the
optimal deterministic mechanisms to perform that well in general.
5.2
Simplicity of Menus
In Example 2 the optimal menus have only a few point contracts. There is a legitimate
concern that the simplicity of the optimal menu may be special to the two dimensions. The
reasoning can be as follows. The 0-1 property of the optimal mechanisms effectively reduces
the dimensionality of the problem by one. Hence the two dimensional problem reduces to a
one dimensional problem, which we observed to have a ”bang-bang” structure leading to a
small number of the point contracts in the optimal menu. It is not clear what to expect when
there are more than two dimensions.
Next we present an example which suggests that the simplicity of the optimal menus is
not special to two dimensions.
Example 3 Let Θ = [0, 1]m and the valuations θi be independently identically distributed
according to the distribution functions Fi (θi ) = (θi )c where c ≥ 1.
We solve this example using ”the ray technique” developed by Armstrong (1996). Fix
some type profile θ from the upper surface ∂Θ+ , i.e. such that θi = 1 for some i. Consider
now all types e
θ lying on the ray passing through θ and the origin, i.e. there exists s ∈ [0, 1]
17
18
The optimal menu for a deterministic mechanism is given in Proposition 2 in Appendix C.
See Appendix C.
16
such that e
θ = sθ. We show that the conditional distributions of s are the same on every ray.
Then we optimize over each ray independently and verify that there exists a menu of prices
which implements the optimal mechanism.
Proposition 3 In Example 3 the optimal menu of allocations and prices (p, T ) is
n
³
´
o
1
(0, 0) , ei , (mc + 1)− mc for i = 1, ..., m , where eii = 1 and eij = 0 for j 6= i.
Proof. See Appendix C.
Another question suggested by Example 2 is whether in the two dimensional problems
the optimal menus are always simple. If the allocation α is strictly increasing on an interval,
then the marginal revenue function V (δ) evaluated at the candidate solution must be equal
to zero throughout this interval.19 Differentiating the marginal revenue with respect to δ we
get the condition:
·
V (δ) = 2g (u (δ) , δ) +
∂
∂
g (u (δ) , δ) u (δ) + g (u (δ) , δ) δ = 0.
∂u
∂δ
(6)
This expression depends on the rent schedule from the differences in the valuations u and
the exogenous distribution of the valuations. Intuitively, it takes a quite special distribution
to make this condition hold on a nondegenerate interval. Although in the next example we
managed to construct a distribution with a rich optimal menu, we believe that the optimality
simple menus is a quite generic phenomenon.
©
¤ª
£
2 : 21 ≤ θ ≤ z (δ) where δ ∈ − 3 , 3 ,
Example 4 Let the
distribution
be
uniform
on
Θ
=
θ
∈
<
2
16
2
2
⎧
£
¢
⎪
21
⎪
if δ ∈ − 32 , − 12
− 18 δ
⎪
8
⎪
⎨
¢
¡
where z (δ) =
if
− 38 δ 2 − 12 δ + 81
δ ∈ − 12 , 12 .
32
⎪
⎪
⎪
¡
¤
⎪
7
21
⎩
if
δ ∈ 12 , 32
8 − 8δ
The optimal menu of allocations and prices (p1 , p2 , T ) is
´
o
n
³
¡
¢2
where
α
∈
[0,
1]
.20
(0, 0, 0) , α, 1 − α, 12 α − 12 + 27
16
·
¢
£
In this example we constructed a distribution such that V (δ) = 0 on 0, 12 and the optimal
£
¢
allocation is separating on 0, 12 :
19
20
See Lemma 3 in Appendix B.
The proof is available upon request.
17
α (δ) =
5.3
⎧
⎪
⎨
⎪
⎩
1
2
+δ
if
1
if
¢
£
δ ∈ 0, 12
£
¤ .
δ ∈ 12 , 32
Profitability of Stochastic Contracts
In Example 2 the relative gain from using the fully optimal mechanism as opposed to the
optimal deterministic mechanisms is rather small: at most about 1.2%. In this section we
argue that this need not be true in general.21 In the next example we show that the gains
from employing stochastic contracts can be as high as 50%.22
©
¡
ª
¢
© ª
Example 5 Let Θ = θ1 , θ2 , θ3 where θ1 = (1, 0), θ2 = (0, 1), θ3 = 12 , 12 , and let Pr θ1 =
© ª
© ª
Pr θ2 = 14 and Pr θ3 = 12 . Then the relative gain in the expected revenue from using the
fully optimal mechanism rather than the optimal deterministic mechanism is 50%.
Proof. Assign allocations (1, 0), (0, 1) and
1, 1 and
1
2
¡1
1
2, 2
¢
to the the types θ1 , θ2 and θ3 at prices
respectively. This allocation is incentive compatible and efficient. The payoff of
each type of the buyer is zero, which means that the seller captures the whole efficient surplus
( 34 ) and thus cannot do any better.
The deterministic menu of allocations and prices (p1 , p2 , T ) is {(0, 0, 0) , (1, 0, T ) , (0, 1, T )}.
Obviously any price other than
1
2
or 1 is dominated. Both T =
1
2
and T = 1 result in the
revenue of 12 .
Example 5 does not fit the model studied in this paper. Next we present an example
which satisfies the conditions of the model and where the relative gain from using stochastic
contracts is as high as 25%. The underlying logic is the same as in Example 5.
©
ª
Example 6 Let the distribution be uniform on Θ = θ ∈ <2 : 1 ≤ θ1 + θ2 ≤ 1 + ε; θ1 , θ2 ≤ 1 ,
¢
¡
where ε ∈ 0, 12 . The relative gain in the expected revenue from using the fully optimal mechanism rather than the optimal deterministic mechanism tends to 25% as ε goes to zero.23
21
Thanassoulis (2004) also argues in favor of using stochastic contracts, but he only provides an example
where the revenue gain is 8%.
22
This example is similar in spirit to the examples in Adams and Yellen (1976) which are used to demonstrate
the superiority of mixed bundling menus over the individual pricing of the goods.
23
The proof is available upon request.
18
6
The Case When the Buyer Desires More than a Single Unit
of the Good
Consider a modification of the model where the goods offered by the seller are no longer
substitutes, and the buyer desires more than a single unit.24 The only change required in the
formulation of the problem of the seller is that the feasible set for the allocations Σ is now
[0, 1]m .
£ ¤m
Condition 3 Valuations are distributed on θ, θ ⊂ <m
+ according to an almost everywhere
positive bounded differentiable symmetric density f (θ1 , ..., θm ).
We may restrict attention to symmetric mechanisms because the distribution of the valuations is symmetric.25 Next we define nice mechanisms which are counterparts of 0-1 mechanisms in the model of substitutes.
Definition 3 The mechanism is nice if pi∗ (θ) is either 0 or 1 for almost every θ ∈ Θ, where
i∗ = arg max θi .
Theorem 2 Consider a symmetric mechanism that is not nice. Then under Conditions 2
and 3 there exists a nice mechanism which brings a strictly higher expected revenue.
The proof of this result for the case θ = 0 is in McAfee and McMillan (1988). Alternatively
this result can be proved along the lines of the proof of Theorem 1. An important corollary
which allows to solve the case of two goods is a counterpart of Corollary 1 to Theorem 1.
Corollary 2 There is no loss for the seller in not screening over the maximal valuations
conditional on participation, i.e. if θm > max {θ1 , ..., θm−1 } then either (p1 , ..., pm , T ) (θ) =
(0, ..., 0, 0) or pm (θ) = 1.
Next we develop a technique for solving the seller’s problem in the two dimensional case.
The approach is very similar to the one in Section 4, and thus we give only a brief sketch.
24
25
This problem was studied by McAfee and McMillan (1988), Manelli and Vincent (2004a), (2004b).
See Maskin and Riley (1984).
19
Since the optimal mechanism is symmetric we can solve just for this case θ2 ≥ θ1 . By
Corollary 2 to Theorem 2 there is no loss for the seller to offer mechanisms which screen only
on the type θ1 conditional on the participation, i.e. (p1 (θ) , p2 (θ) , T (θ)) is either equal to
(0, 0, 0) or to (α (θ1 ) , 1, t (θ1 )). An analog of Lemma 1 says that α (θ1 ) is nondecreasing and
Zθ1 ³ ´
θ1 .
θ1 de
θ1 α (θ1 ) − t (θ1 ) = u (θ1 ) = u (θ) + α e
θ
The buyer of the type θ participates only if the utility from participation is nonnegative.
Hence U (θ) = max {u (θ1 ) + θ2 , 0}. The revenue from the buyer of the type θ is θ1 α (θ1 ) −
u (θ1 ) whenever u (θ1 ) + θ2 ≥ 0, and 0 otherwise. The measure of the participating types for
Z
a fixed θ1 is
f (θ1 , θ2 ) dθ2 , which we denote by g (u (θ1 ) , θ1 ).
u(θ1 )+θ2 ≥0
The seller’s problem is
Zθ
Problem III max g (u (θ1 ) , θ1 ) (θ1 α (θ1 ) − u (θ1 )) dδ
α,u(θ)
θ
subject to
£ ¤
• Feasibility: α (θ1 ) ∈ [0, 1] for every θ1 ∈ θ, θ ;
Zθ1 ³ ´
θ1 for
• Incentive Compatibility: (i) α is nondecreasing; (ii) u (θ1 ) = u (θ) + α e
θ1 de
θ
£ ¤
every θ1 ∈ θ, θ .
As in the model of substitutes we can set up the seller’s problem as an optimal control
problem and obtain the necessary conditions for the optimality.
Example 7 Let the distribution of the valuations be uniform on Θ = [c, c + 1]2 where c ≥ 0.
Solving the optimal control problem we find the optimal menu of the allocations and prices
(p1 , p2 , T ) depending on the parameter c.
Proposition 4 In Example 7 the optimal menu is
©
¡
¢
ª
(i) (0, 0, 0) , 1, 1, T , (1, α, T ) , (α, 1, T ) ,
where α is increasing with c, α = 0 when c = 0, and α = 1 when c = c
q
(ii) {(0, 0, 0) , (1, 1, T )}, where T = 23 c2 + 32 + 43 c
20
if c ∈ [0, c]
where c ≈ 0.0766;
if c ∈ (c, +∞).
Proof. See Appendix D.
The superiority of the mixed bundling strategy over the individual pricing of the goods in
this model was established long ago.26 For very small c the fully optimal mechanism turns
out to be stochastic. The revenue gain relative to the best mixed bundling mechanism is very
small: much less than 1%.27
7
Conclusion
We showed that under the standard regularity conditions a multiproduct monopolist selling
substitutable goods never finds it optimal to discriminate on the aggregate probability of
selling. This was shown to imply that the optimal mechanism is characterized be a significant
amount of pooling: all participating types with the same differences in the valuations receive
the same allocation. In the case of two goods we demonstrated how to apply the optimal
control technique to solve for the optimal mechanism. In symmetric problems it was shown
that the optimal allocation (for participating types) is in between the efficient allocation and
the allocation which optimizes the rent extraction from the differences in the valuations. The
solved examples suggest that: (i) generically the optimal menus have a simple structure; (ii)
the gain from using the fully optimal mechanisms as opposed to deterministic mechanisms can
be substantial. We also applied our approach to the problem of a multiproduct monopolist
when the buyer may desire more than a single unit of the good.
There seem to be a number of interesting economic application of the model we studied
other than those discussed in the paper. For example, we may interpret the model as a
problem of the product line design. The consumers do not value the goods per se, but derive
utility from a set of characteristics the goods possess.28 Assuming that the utility is separable
in the characteristics, the model of substitutes describes the case when the monopolist for
technological reasons is limited with respect to the intensities of different characteristics that
can be included in one good (e.g. speed vs safety in car design), while in the second model
26
See McAfee, McMillan and Whinston (1989).
See Appendix D for the formulas of the expected profits.
28
See for example Lancaster (1971).
27
21
there is no such constraint.
The model of substitutes may also be applied to study the question of the optimal auction
design in the presence of collusion between the bidders. The seller owns a single good and can
prohibit reallocations of the good between the bidders. The bidders form a cartel that allows
them to behave as a single buyer maximizing the sum of the bidders’ payoffs. The results of
this paper show how the seller can respond to collusion by allocating the good randomly in
certain situations.
8
Appendix A
Proof of Theorem 1. Consider some admissible mechanism (p1 , ..., pm , T ) which generates
the utility schedule U and does not satisfy the 0-1 property. We replace it with a 0-1 mechanism which keeps unchanged the utility schedules U (θ) for the types from the upper surface
¡
¢
∂Θ+ . Define the new allocations and payments p1 , ..., pm , T (θ) for the upper surface types
θ ∈ ∂Θ+ as follows.
³
T (θ) = T
´
P
Sni (θ)
1
−
j pj (θ) for every
n
(θ)
j j
³
´
m
P
P
θi ni (θ)
S
1
−
(θ) +
p
(θ)
.29
j j
j nj (θ)
i=1
P
pi (θ) = pi (θ) +
Notice that pi (θ) ≥ 0 (by Condition 1) and
i,
i pi (θ)
= 1. The utilities of the "upper
surface" types are the same as in the original mechanism:
m
m
P
P
U (θ) =
θi pi (θ) − T (θ) =
θi pi (θ) − T (θ) = U (θ) for every θ ∈ ∂Θ+ .
i=1
i=1
Next we derive the payoff of some type θ ∈ Θ from sending a message θ0 ∈ ∂Θ+ .
m
¡ ¢
¡ ¢
P
θ i pi θ 0 − T θ 0 =
i=1
³
³
m
m
m
¡ ¢ P
¡ 0 ¢´
¡ 0¢ P
¡ 0 ¢´
P
P
P
θ0i ni (θ0 )
θi ni (θ0 )
S
S
1
−
1
−
=
θi pi θ0 +
p
p
θ
−
T
θ
−
=
j j
j j θ
n (θ0 )
n (θ0 )
=
i=1
m
P
i=1
i=1
j
j
i=1
j
j
m
¡ ¢
¡ ¢ (1−S pj (θ0 )) £¡
¢ ¡ ¢¤ P
¡ ¢
¡ ¢
θi pi θ0 − T θ0 + S nj (θ0 )
θi pi θ0 − T θ0 .
θ − θ0 · n θ0 ≤
j
j
i=1
The above inequality follows from Condition 1 and the definition of the normal n (θ):
S
¢ ¡ ¢¤
pj (θ0 )) £¡
(1−
S j
θ − θ0 · n θ0 ≤ 0.
n (θ0 )
j
j
Hence the payoff from sending a message θ0 ∈ ∂Θ+ under the new mechanism is not higher
29
If there exist multiple normals at θ (i.e. the surface has a kink at θ) then we can pick any one of them.
22
than under the original mechanism.
¡
¢
Augment this new mechanism with a null message ∅ which gives p1 , ..., pm , T (∅) =
(0, ..., 0, 0), and offer it to all types of the buyer. Notice that it is not a direct revelation
mechanism since the set of messages is just M = {θ for θ ∈ ∂Θ+ } ∪ {∅}. Different types
self-select into messages according to some strategy σ : Θ → ∆M .
Consider some type θ ∈ Θ and his strategy σ (θ). The utility in the new mechanism is:
µm
½
¶ ¾
m
¡ ¢
¡ ¢
P
P
U (θ) =
θi pi (σ (θ)) − T (σ (θ)) = max 0max
θi pi θ0 − T θ0 , 0 ≤
µm
½i=1
¶ ¾θ ∈∂Θm+ i=1
¡ 0¢
¡ 0¢
P
P
θ i pi θ − T θ
θi pi (θ) − T (θ) = U (θ).
≤ max 0max
,0 ≤
θ ∈∂Θ+
i=1
i=1
The first inequality follows from the result that the payoff from sending a message θ0 ∈
∂Θ+ is not higher than under the original mechanism, the second inequality comes from the
admissibility of the original mechanism.
Finally we show that U (θ) < U (θ) for a positive measure of types because the original
b be an open convex subset of Θ such that
mechanism does not satisfy the 0-1 property. Let Θ
Z
P
b
f (θ) dθ > 0 and i pi (θ) ∈
/ {0, 1} for almost every θ ∈ Θ.
e
Θ
¡
¢
b such that U (θ) > 0. There exists θ+1·t = θ1 + t, ..., θ m + t ∈
Consider some type θ ∈ Θ
Θ for some t > 0. By the incentive compatibility we can represent the utilities for the original
and the new mechanism as follows:
Zt
¡
¢
P
U (θ) = U θ + 1 · t −
i pi (θ + 1 · t) dt,
0
¡
¢
U (θ) = U θ + 1 · t −
Zt
P
i pi (σ (θ
+ 1 · t)) dt.
0
P
P
Notice that by the incentive compatibility both i pi (θ + 1 · t) and i pi (σ (θ + 1 · t))
P
are nondecreasing in t. Moreover i pi (σ (θ + 1 · t)) ∈ {0, 1}. If U (θ) = 0 then we are done.
P
If U (θ) > 0 then i pi (σ (θ)) = 1. Hence
Zt
¡
¢
¡
¢
P
U (θ) − U (θ) = U θ + 1 · t − U θ + 1 · t − (1 − i pi (θ + 1 · t)) dt <
0
¡
¢
¡
¢
< U θ + 1 · t − U θ + 1 · t ≤ 0.
b and the second inequality
The first inequality follows from the definition of the subset Θ,
was established earlier.
23
Hence U (θ) ≥ U (θ) for every θ with a strict inequality for a positive measure of types.
Using equation (2) we conclude that the new mechanism brings a strictly higher revenue than
the original mechanism.
9
Appendix B
9.1
Necessary Conditions
We deal with the monotonicity constraint in a standard way by introducing an auxiliary
£ ¤
·
control variable z : δ, δ → <+ such that α (δ) = z (δ), and in addition allow the state
variable α to have upward jumps.30 Write down the seller’s problem as an optimal control
problem.31
Zδ
Problem I max g (u (δ) , δ) (δα (δ) − u (δ)) dδ
z,α,u
δ
subject to
Feasibility
α (δ) ≥ 0
[η (δ)]
1 − α (δ) ≥ 0
[η (δ)]
Incentive Compatibility
u (δ) = α (δ)
·
[λ1 (δ)]
α (δ) = z (δ)
·
[λ2 (δ)]
z (δ) ≥ 0
[μ (δ)]
Transversality conditions
¡ ¢
¡ ¢
α (δ), α δ , u (δ) and u δ are free
Form the Lagrangian
¡
¢
L z, α, u, η, η, λ1 , λ2 , μ; δ = g (u, δ) (δα − u) + λ1 α + λ2 z + ηα + η (1 − α) + μz.
First we maximize L with respect to z.
30
See for example Guesnerie and Laffont (1984).
Notice that the function g (u, δ) may not be continuously differentiable at the point (u, δ) such that u = −e
θ2 ,
where e
θ2 =
min
θ2 . The left derivative with respect to u at this point is 0, while the right derivative is
31
θ∈Θ:θ 1 −θ 2 =δ
f (δ − u, −u). We can still apply the optimal control technique if this happens on at most a countable set of
points (see Note 6 in Chapter 2 in Seierstad and Sydsæter (1987)).
24
L∗ = g (u, δ) (δα − u) + λ1 α + ηα + η (1 − α),
·
with the conditions μz = 0, μ = −λ2 ≥ 0 and α = z ≥ 0.
Next we get the system of Hamiltonian equations together with the boundary require¡ ¢
¡ ¢
ments,
⎧ which follow from the transversality conditions: λ1 (δ) = λ1 δ = λ2 (δ) = λ2 δ = 0.
·
⎪
⎨ λ1 = − ∂L∗ = g (u, δ) − ∂ g (u, δ) (δα − u)
∂u
∂u
·
∗
⎪
⎩ λ2 = − ∂L = −g (u, δ) δ − λ1 − η + η
∂α
λ1 , λ2 are continuous throughout.32 The other conditions are ηα = 0, η ≥ 0 and α ≥ 0;
η (1 − α) = 0, η ≥ 0 and α ≤ 1.
Define a marginal revenue function V (δ) = g (u (δ) , δ) δ + λ1 (δ). Next we rework the
control theoretic necessary conditions for optimality into the conditions for the marginal
revenue function V . The condition for the separation of types on an interval is in part I, the
ironing conditions for pooling of the types on an interval are in parts II-IV.33
¡
¢
Lemma 3 Let z, α, u, η, η, λ1 , λ2 , μ satisfy the necessary conditions. Then
I. If α is strictly increasing on (δ 1 , δ 2 ), then V = 0
Zδ2
II. If α = 0 on (δ 1 , δ 2 ), then δ 1 = δ, V (δ 2 ) = 0, V
δ1
Zδ2 ³ ´
and V e
δ de
δ ≤ 0 for any δ in the interval.
on this interval.
Zδ ³ ´
³ ´
e
e
δ dδ = k ≤ 0. Also V e
δ de
δ≥k
δ1
δ
Zδ2 ³ ´
δ de
δ = 0. Also
III. If α = α
b ∈ (0, 1) on (δ 1 , δ 2 ), then V (δ 1 ) = V (δ 2 ) = 0, V e
δ1
Zδ
δ1
Zδ2 ³ ´
³ ´
V e
δ de
δ≥0≥ V e
δ de
δ for any δ in the interval.
δ
Zδ2 ³ ´
Zδ ³ ´
IV. If α = 1 on (δ 1 , δ 2 ), then V (δ 1 ) = 0, δ 2 = δ, V e
δ de
δ = k ≥ 0. Also V e
δ de
δ≥
δ1
Zδ2 ³ ´
0 and V e
δ de
δ ≤ k for any δ in the interval.
δ1
δ
·
Proof. I. Notice that η = η = λ2 = 0 on this interval. Hence 0 = −λ2 = V .
32
33
Theorems 7 and 8 in Chapter 3 in Seierstad and Sydsæter (1987).
See for example Guesnerie and Laffont (1984), Myerson (1981).
25
II. By monotonicity δ 1 = δ. Notice that η ≥ 0, η = 0 and λ2 ≤ 0 on this interval. Also
λ2 (δ 2 ) = 0 since α changes the value at δ 2 , and λ2 (δ) = 0 from the transversality condition.
Zδ2 · ³ ´
Zδ2 ³ ´
Zδ2 ³ ´
e
e
e
e
δ de
δ = k ≤ 0. Also 0 ≥
Hence 0 = λ2 δ dδ, which implies V δ dδ = − η e
δ1
δ1
δ1
Zδ · ³ ´
Zδ ³ ´
Zδ ³ ´
e
e
e
e
δ de
δ ≥ k. Finally 0 ≥ λ2 (δ) =
λ2 (δ) = λ2 δ dδ, which implies V δ dδ = − η e
δ1
δ1
δ1
Zδ2 · ³ ´
Zδ2 ³ ´
Zδ2 ³ ´
δ de
δ=− η e
δ de
δ, which implies V e
δ de
δ ≤ 0.
− λ2 e
δ
δ
δ
III and IV. The proofs are similar to part II and therefore omitted.
¡
¢
Corollary 3 Let z, α, u, η, η, λ1 , λ2 , μ satisfy the necessary conditions. Then
Zδ
V (δ) (α (δ) − α
e (δ)) dδ ≥ 0 for every admissible α
e.
δ
9.2
Sufficient Conditions and Uniqueness
To show the sufficiency of the necessary conditions and to establish uniqueness we need one
extra result. Denote the expected revenue from the 0-1 mechanism (α, u) by Π (α, u).
Lemma 4 (i) Π is concave in (α, u).
¡
¢
¡
¢
(ii) If for at least one of the two different mechanisms α1 , u1 and α2 , u2 the exclusion
¡
¢
¡
¢
¡
¢
¡
¢
region is nondegenerate, then Π αt , ut > tΠ α1 , u1 + (1 − t) Π α2 , u2 , where αt , ut =
¢
¡
¢
¡
t α1 , u1 + (1 − t) α2 , u2 and t ∈ (0, 1).
¡
¢
¡
¢
Proof. Consider two different mechanisms α1 , u1 and α2 , u2 . Denote the associated
utility schedules by U 1 and U 2 . Define a new mechanism U t to be the weighted average
U t = tU 1 + (1 − t) U 2 for some t ∈ (0, 1). It is admissible by the convexity of the set of the
admissible utility schedules. Since the expected revenue is linear in the utility schedule U ,
we observe that the revenue from from U t is just a weighted average of the revenues from U 1
and U 2 with the weights t and 1 − t.34
34
See Section 3 for the representation of the expected profit in terms of the utility schedule.
26
There are two cases to consider. If the exclusion region is of zero measure for both
¡
¢
¡
¢
mechanisms α1 , u1 and α2 , u2 , then it is obvious that U t satisfies the 0-1 property since
pt1 (θ) + pt2 (θ) = 1 for almost every θ ∈ Θ. Moreover it is easy to see that the mechanism U t
¡
¢
¡
¢
¡
¢
¡
¢
is equivalent to αt , ut , and thus Π αt , ut = tΠ α1 , u1 + (1 − t) Π α2 , u2 .
¡
¢
If the exclusion region is nondegenerate for at least one of the two mechanisms α1 , u1 ,
¡ 2 2¢
α , u , then we show that U t does not satisfy the 0-1 property. Since U 1 and U 2 are different
Z
b ⊂ Θ such that
mechanisms, then there exists k ∈ {1, 2}, and a subset Θ
f (θ) dθ > 0,
pk1
(θ) +
pk2
(θ) = 1 and
p3−k
1
(θ) +
p3−k
2
e
Θ
b Since pt = tp1 + (1 − t) p2 we have
(θ) = 0 for θ ∈ Θ.
b By Theorem 1 we know that there exists a 0-1 mechanism
pt1 (θ) + pt2 (θ) = t ∈ (0, 1) on Θ.
which yields a strictly higher revenue. Following the proof of Theorem 1 it is easy to see that
¡
¢
¡
¢
this mechanism is αt , ut . Hence αt , ut brings a strictly higher revenue than U t , and thus
¡
¢
¡
¢
¡
¢
Π αt , ut > tΠ α1 , u1 + (1 − t) Π α2 , u2 .
Proposition 5 (i) The necessary conditions are sufficient.
(ii) If the exclusion region is nondegenerate then the optimum is unique.
Proof. (i) Let (α∗ , u∗ ) be the candidate for the optimum which satisfies the necessary
¡
¢
conditions and (α, u) be some other mechanism. Define αt , ut = t (α, u) + (1 − t) (α∗ , u∗ )
for t ∈ (0, 1). From Lemma 4 we know that the revenue is concave and thus
Π (α∗ , u∗ ) − Π (α, u) ≥
¡
¢¢
1¡
Π (α∗ , u∗ ) − Π αt , ut .
t
(7)
Define a Hamiltonian H (α, u, δ) = g (u, δ) (δα − u) + λ1 α.
Zδ
¡
¢¢
¡
¢¢
¡
¡
¢¡
1
1
g (u∗ , δ) (δα∗ − u∗ ) − g ut , δ δαt − ut dδ =
Then t Π (α∗ , u∗ ) − Π αt , ut = t
δ
=
1
t
Zδ
δ
¡
¡
¢¢
¡
¢
H (α∗ , u∗ , δ) − H αt , ut , δ − λ1 α∗ − αt dδ.
¡
¢
We add and subtract H αt , u∗ , δ under the integral and observe that αt = α∗ +t (α − α∗ )
and ut = u∗ + t (u − u∗ ). Then taking the limit as t goes to zero and using Dominated
27
Convergence Theorem35 we get
Zδ
Zδ
Zδ
∂
∂
∗ ∗
∗
∗ ∗
∗
λ1 (α∗ − α) dδ.
∂α H (α , u , δ) (α − α) dδ +
∂u H (α , u , δ) (u − u) dδ −
δ
δ
∂
∂u H
δ
(α∗ , u∗ , δ)
·
From the necessary conditions
= −λ1 . Also observe that
Zδ ·
Zδ ³
´
·∗
·
∗
λ1 (u − u) dδ + λ1 u − u dδ = λ1 (δ) (u∗ (δ) − u (δ))|δδ = 0,
δ
δ
¡ ¢
since by the boundary requirements λ1 (δ) = λ1 δ = 0.
∂
H (α∗ , u∗ , δ) = V (δ), then by Corollary 3
Finally since ∂α
Zδ
∂
∗ ∗
∗
∗ ∗
∂α H (α , u , δ) (α − α) dδ ≥ 0. Hence Π (α , u ) ≥ Π (α, u).
δ
(ii) If (α∗ , u∗ ) and (α, u) are two different mechanisms, and the exclusion region associated
with the mechanism (α∗ , u∗ ) is nondegenerate, then according to Lemma 4 the inequality (7)
is strict. Hence (α, u) cannot be optimal unless it is identically equal to (α∗ , u∗ ).
9.3
Bounds on the Allocation
Proof of Proposition 1.
By feasibility we have α0 (δ) =
δ <b
δ. What is left to show is that α (δ) = 1 when δ > b
δ.
1
2
≤ α (δ) ≤ 1 = α00 (δ) when
Zδ ³ ´
The marginal revenue in the first benchmark problem is V (δ) = g (δ) δ − g e
δ de
δ, where
g (δ) =
Z
δ
θ: θ1 −θ2 =δ
f (θ) dθ.36 First we show that V (δ) ≥ V (δ) for every δ.
·
·
¡ ¢
¡ ¢
Since V (0) = − 12 < 0 = V (0) and V δ = V δ , it is enough to prove that V (δ)− V (δ) ≥
0 for every δ. If V (δ) = V (δ), which happens when there is full participation of all types
·
·
above δ, then V (δ) − V (δ) = 0. Alternatively,
·
·
∂
∂
∂
V (δ) − V (δ) = 2g (δ) + ∂δ
g (δ) δ − 2g (u (δ) , δ) − ∂u
g (u (δ) , δ) u (δ) − ∂δ
g (u (δ) , δ) δ =
−u(δ)
−u(δ)
Z
Z
=2
f (θ2 + δ, θ2 ) dθ2 +
f1 (θ2 + δ, θ2 ) δdθ2 + f (−u (δ) + δ, −u (δ)) (−u (δ)) =
θ
35
36
θ
See Theorem 1, Section 30 in Kolmogorov and Fomin (1970).
See the proof of Lemma 2.
28
=
−u(δ)
Z
(3f (θ2 + δ, θ2 ) + f1 (θ2 + δ, θ2 ) (θ2 + δ) + f2 (θ2 + δ, θ2 ) θ2 ) dθ2 + f (θ + δ, θ) θ.
θ
·
Zδ
e
δ
·
By Condition 2 the expression under the integral is positive. Hence V (δ) − V (δ) ≥ 0.
´
³
Now assume there exists δ 1 > b
δ such that α (δ) < 1 on b
δ, δ 1 . By Lemma 3 we have
Zδ ³ ´
´
³ ´
³
V e
δ de
δ ≥ 0 and V e
δ de
δ ≤ 0 for every δ ∈ b
δ, δ 1 . Since V is pointwise weakly higher
e
δ
´
³
than V , this implies V (δ) = V (δ) for every δ ∈ b
δ, δ 1 . Hence we can set α (δ) = 1 on
³
´
b
δ, δ 1 .
10
Appendix C
Proof of Proposition 2.
Define δ ∗ ∈ [0, 1] such that c + u (δ ∗ ) = 0. It is straightforward
to show that such δ ∗ exists and
⎧ is unique.
⎪
⎨ c + 1 − δ + u (δ)
Notice that g (u (δ) , δ) =
⎪
⎩
1−δ
The equation (4) is as follows:
Zδ∗
(δα (δ) − u (δ)) dδ −
0
Zδ∗
if
δ ∈ [0, δ ∗ )
if
δ ∈ (δ ∗ , 1]
(c + 1 − δ + u (δ)) dδ −
0
Z1
.
(1 − δ) dδ = 0.
(8)
δ∗
Zδ∗
Zδ∗
Using u (δ) dδ = u (δ ∗ ) δ ∗ − δα (δ) dδ and c + u (δ ∗ ) = 0 we simplify (8):
0
0
Zδ∗
1
3 δα (δ) dδ + cδ ∗ − = 0.
2
0
The derivative
of the marginal revenue (see equation (6)):
⎧
⎪
⎨ 2c + 2 + 3u (δ) − 3δ if δ ∈ [0, δ ∗ )
·
V (δ) =
,
⎪
⎩
2 − 3δ
if δ ∈ (δ ∗ , 1]
29
(9)
⎧
⎪
⎨ 3 (α (δ) − 1)
··
and thus V (δ) =
⎪
⎩
−3
if
δ ∈ [0, δ ∗ )
if
δ ∈ (δ ∗ , 1]
.
Hence the marginal revenue V is described by two (weakly) concave functions: V1 on
·
[0, δ ∗ ) and V2 on (δ ∗ , 1]. Notice that V is discontinuous at δ ∗ unless c = 0:
·
·
V 1 (δ ∗ ) = 2 − c − 3δ ∗ ≤ 2 − 3δ ∗ = V 2 (δ ∗ ).
Case 1. The optimal α (δ) = 1 for every δ.
Zδ ³ ´
δα e
δ de
δ = 12 δ 2 .
In this case u (δ) = u (0) + δ, and e
Using equation (9): δ ∗ =
1
3
0
³√
´
³
´
√
2
c + 3 − c . Using c+u (δ ∗ ) = 0: u (0) = − 23 c + 13 c2 + 3 .
Using V ⎧
(0) = V (1) = 0 we get
³
´
√
⎪
⎨ 2 − c2 + 3 δ
if
V (δ) =
⎪
⎩ 1 (1 − δ) (3δ − 1) if
2
δ ∈ [0, δ ∗ )
.
δ ∈ (δ ∗ , 1]
The function V is nonnegative on [0, 1] when the parameter c is no higher than one, and
thus by Lemma 3 the candidate α is indeed optimal.
The optimal menu of (p1 , p2 , T ) is deterministic: {(0, 0, 0) , (1, 0, T ) , (0, 1, T )}.
Every participating type δ ∈ [0, 1] chooses (1, 0) contract. The payment is t (δ) = δα (δ) −
√
u (δ) = −u (0) = 23 c + 13 c2 + 3 = T .
The expected revenue is Π (c) = Pr {max (θ1 , θ2 ) ≥ T } · T ,
³√
´2
where Pr {max (θ1 , θ2 ) ≥ T } = 1 − 19
c2 + 3 − c .
Case 2. The optimal α (δ) is not identically equal to one on [0, 1].
In this case both V1 and V2 are strictly concave. First we argue that δ ∗ <
δ∗ ≥
1
3,
V2 (δ) =
·
then concavity together with the facts V (0) = V (1) = 0, V 1 (δ ∗ ) ≤
1
2
1
3 . Assume
·
V 2 (δ ∗ ) and
(1 − δ) (3δ − 1) imply that V is strictly positive almost everywhere on (0, 1), and
thus α (δ) 6= 1 cannot be optimal.
Since δ =
1
3
is the only place where V (δ) crosses zero from below, by Lemma 3 we have
£
¢
¡
¢
α (δ) equal to some constant α ∈ 12 , 1 on the interval 0, 13 .
Zδ ³ ´
¢
£
Notice that u (δ) = u (0) + αδ, e
δα e
δ de
δ = 12 αδ for δ ∈ 0, 13 .
0
30
Using equation (9):
3 2
1
αδ ∗ + cδ ∗ − = 0.
2
2
(10)
1
By Lemma 3 we must have V
1
Z3
V (δ) dδ ≤ 0 if α = 12 .
0
Case 2.1. α ∈
¡1
¡1¢
3
= 0 together with
Z
0
V (δ) dδ = 0 if α ∈
0
¢
.
2, 1
¡1¢
Using integration by parts and V (0) = V
1
3
Z3
3
1
3
¡1
¢
,
1
, and
2
= 0:
Z ·
¡¡
¢
¢ ¡2
¢
V (δ) dδ = − V (δ) δdδ = − c + 1 + 32 u (0) δ 2∗ + (α − 1) δ 3∗ − 27
− δ 2∗ + δ 3∗ =
0
= 12 αδ 3∗ + 12 cδ 2∗ −
2
27 .
For the last equality we used c+u (δ ∗ ) = 0, which implies u (0) = −c−αδ ∗ . Using equation
(10) we substitute out αδ 2∗ =
1
3
1
Z3
0
− 23 cδ ∗ to get:
µ
¶
1
4
2
cδ ∗ + δ ∗ −
= 0.
V (δ) dδ =
6
9
(11)
¢
¡√
1
The solution to the system of equations (10) and (11) is δ ∗ = 6c
16c + 9 − 3 and
¡9
¢√
1
α = 27
16c + 9. Notice that α is strictly decreasing in c. Also α = 1 when
32 + 32 − 4 c
1
2
when c = c ≈ 1.3721.
©
¡
¢ ¡
¢ª
The optimal menu of (p1 , p2 , T ) is (0, 0, 0) , (α, 1 − α, T ) , (1 − α, α, T ) , 1, 0, T , 0, 1, T .
¡
¢
Every participating type δ ∈ 0, 13 chooses (α, 1 − α) contract. The payment is t (δ) =
¡
¤
δα (δ) − u (δ) = −u (0) = c + αδ ∗ = T . Every participating type δ ∈ 13 , 1 chooses (1, 0)
¡
¢
contract. The payment is t (δ) = δα (δ)−u (δ) = δ−u (0)−α 13 − δ − 13 = c+αδ ∗ + 13 (1 − α) =
c = 1, and α =
T.
Solving for T and T we have T = 13 c +
¡9
¢√
¢
¡5
T = T + 13 32
− 32
− 14 c 16c + 9 .
3
8
+
1
8
√
16c + 9 and
The expected revenue is
©
ª ¡
¢
Π (c) = Pr {α max {θ1 , θ2 } + (1 − α) min {θ1 , θ2 } ≥ T } · T + Pr |δ| ≥ 13 · T − T ,
31
where Pr {α max {θ1 , θ2 } + (1 − α) min {θ1 , θ2 } ≥ T } =
¡
¡9
¢√
¢ ¡ 1 ¡√
¢¢2
1
1 − 27
16c + 9 6c
16c + 9 − 3 ,
32 + 32 − 4 c
ª
©
and Pr |δ| ≥ 13 = 49 .
Case 2.2. α = 12 .
´
³q
c2 + 32 − c .
¢
¡√
1
16c + 9 − 3 , which solves
It is straightforward to verify that δ ∗ is no higher than 6c
This is the case when c ≥ c. Using equation (10): δ ∗ =
2
3
the equation (11). Then
1
Z3
¡
¢
V (δ) dδ = 16 cδ 2∗ + δ ∗ − 49 ≤ 0 as required by Lemma 3.
0
¢ ¡
©
¡
¢ ¡
¢ª
(0, 0, 0) , 12 , 12 , T , 1, 0, T , 0, 1, T .
q
¡
¢
1
2
1
Solving for T and T gives T = c + 6 δ ∗ = 3 c + 3 c2 + 32 and T = T + 13 1 − 12 = T + 16 .
©
©
ª
ª ¡
¢
The expected revenue is Π (c) = Pr 12 θ1 + 12 θ2 ≥ T · T + Pr |δ| ≥ 13 · T − T ,
³q
´2
ª
ª
©
©1
1
2
c2 + 32 − c and Pr |δ| ≥ 13 = 49 .
where Pr 2 θ1 + 2 θ2 ≥ T = 1 − 9
The optimal menu is of (p1 , p2 , T ) is:
Proof of Proposition 3. The revenue of the seller is
Z
Θ
"
#
³ ´
m ³ ´c−1
Q
m
e
θj
T e
θ c
de
θ.
j=1
θ = sθ for some
For every e
θ we can find a type θ on the upper surface ∂Θ+ such that e
s ∈ [0, 1]. We change variables e
θj = sθj , i.e. the integration is now performed over s from
³
n o
´
zero to one, and over the upper surface. Then if i = arg max e
θi , e
θ−i =
θj then θi = 1, T e
j
m ³ ´c−1
Q
Q
c−1
e
θj
= cm sm(c−1)
(θj ) , and the Jacobian transform is sm−1 .
T (s, sθ−i ), cm
j=1
j6=i
⎧
⎫
"
#
Z ⎨Z1
⎬
m
P
Q
1 m−1
The expected revenue is
T (s, sθ−i ) dsmc m
c
(θj )c−1 dθ−i .
⎩
⎭
i=1
j6=i
0
Θ−i
Notice that
Z1
0
U (sθ) = U (0) +
T
Zs
(sθ) dsmc
=
Z1
(sθ · p (sθ) − U (sθ)) dsmc . The envelope condition implies
0
d
dx
[U (e
sθ)] de
s, where
0
Z1
0
U
(sθ) dsmc
= U (0)
Z1
0
dsmc
+
d
dh
s [U
Z1 ∙ s
R
0
0
(e
sθ)] = θ · ∇U (e
sθ) = θ · p (e
sθ). Hence
¸
θ · p (e
sθ) de
s dsmc =
32
= U (0) +
∙s
R
0
= U (0) +
Z1
¸
¸
¸∙ s
∙s
Z1
R mc 1
R mc
ds =
θ · p (e
sθ) de
s
de
s
− θ · p (sθ)
de
s
0
0
0
0
sθ · p (sθ) 1s (1 − smc ) ds.
0
Thus the expected revenue is
Z1
0
Pointwise maximization yields:
¡
sθ · p (sθ) 1 −
1−smc
mcsmc
¢
dsmc − U (0).
1
pj (sθ) = 0 for all j if s < (mc + 1)− mc ;
1
pi (sθ) = 1 and pj (sθ) = 0 if s ≥ (mc + 1)− mc and i = arg max {θj },
j
which gives the result.
11
Appendix D
Proof of Proposition 4. Define θ∗ ∈ (0, 1) such that u (θ∗ ) + θ∗ = 0. It is straightforward
to show that such θ∗ exists and⎧is unique.
⎪
⎨ c + 1 + u (θ1 )
Notice that g (u (θ1 ) , θ1 ) =
⎪
⎩ c + 1 − θ1
if
θ1 ∈ [c, θ∗ )
if
θ1 ∈ (θ∗ , c + 1]
.
Zθ∗
Zθ∗
Optimizing with respect to u (c) and using u (θ1 ) dθ1 = u (θ∗ ) θ∗ −u (c) c− θ1 α (θ1 ) dθ1
c
c
we derive the following condition:
Zθ∗
2u (θ∗ ) θ∗ − 2u (c) c − 3 θ1 α (θ1 ) dθ1 + 12 (θ∗ )2 − 12 c2 +
c
Using u (θ∗ ) + θ∗ = 0 (which implies u (c) = −
R θ∗
c
1
2
= 0.
α (θ1 ) dθ1 − θ∗ ) we get:
Zθ∗
Zθ∗
3 ∗
1
− (θ − c)2 − c (θ∗ − c) + 2c α (θ1 ) dθ1 − 3 θ1 α (θ1 ) dθ1 + = 0.
2
2
c
Notice that
⎧
⎪
⎨ 2c + 2 + 3u (θ1 )
·
V (θ1 ) =
⎪
⎩ 2c + 2 − 3θ1
c
if
θ1 ∈ [c, θ∗ )
if
θ1 ∈ (θ∗ , c + 1]
33
,
(12)
⎧
⎪
⎨ 3α (θ1 )
··
and thus V (θ1 ) =
⎪
⎩ −3
if
θ1 ∈ [c, θ∗ )
if
θ1 ∈ (θ∗ , c + 1]
.
Hence V is described by two functions: a (weakly) convex V1 on [c, θ∗ ) and a strictly
·
concave V2 on (θ∗ , c + 1]. Notice that V is continuous at θ∗ . Also V (c) = (c + 1 + u (c)) c ≥ 0
since u (c) ∈ [− (c + 1) , −c], and V (c + 1) = 0. Notice that V2 (θ1 ) =
which is nonnegative for θ1 ≥
c+1
3 .
1
2
(c + 1 − θ1 ) (3θ1 − c − 1)
Combining these facts it is clear that unless c = 0 there is
exists at most a single point on (c, c + 1) where V (θ1 ) crosses zero from below.
Case 1. The optimal α (θ1 ) = 1 for every θ1 .
Zθ1
³ ´
θ1 =
θ1 α e
θ1 de
In this case u (θ1 ) = u (c) + (θ1 − c) and e
∗
Using equation (12): θ −c =
1
3 c.
Notice that θ∗ >
1
3
³q
c2 +
1
2
c
3
2
(θ1 − c)2 + c (θ1 − c).
q
´
− c . Using u (θ∗ )+θ∗ = 0: u (c) = − 23 c2 + 32 −
and thus V (θ1 ) = V2 (θ1 ) > 0 for θ1 ≥ θ∗ .
Zθ1 ³ ´
θ1 de
θ1 ≥ 0 for every
Then by an analog of Lemma 3 it is enough to establish that V1 e
c+1
3
c
θ1 ∈ (c, θ∗ ].
Rearranging the expression for V1 we get
V (θ1 ) =
3
2
(θ1 − c)2 + (2c + 2 + 3u (c)) (θ1 − c) + (c + 1 + u (c)) c.
Using integration by parts we have
Zθ1 ³ ´
Zθ1 · ³ ´ ³
´
e
e
e
e
V θ1 dθ1 = − V θ1 θ1 − c de
θ1 + V (θ1 ) (θ1 − c) =
c
c
Zθ1 ³
³
´´ ³
´
e
=−
2c + 2 + 3u (c) + 3 e
θ1 − c
θ1 − c de
θ1 +
c
+ 32 (θ1 − c)3 + (2c + 2 + 3u (c)) (θ1 − c)2 + (c + 1 + u (c)) c (θ1 − c) =
³
´
= 12 (θ1 − c) (θ1 − c)2 + (2c + 2 + 3u (c)) (θ1 − c) + 2 (c + 1 + u (c)) c =
q
q
´
´ ´
³
³
³
= 12 (θ1 − c) (θ1 − c)2 + c + 2 − 2 c2 + 32 (θ1 − c) + 2 23 c + 1 − 23 c2 + 32 c ,
where for the last step we used the expression for u (c).
q
q
´
³
´
³
Define Φc (x) = x2 + c + 2 − 2 c2 + 32 x + 2 23 c + 1 − 23 c2 + 32 c. It is straightfor-
ward to show that (i) Φc (0) ≥ 0 for every c ≥ 0; (ii) Φ0c (0) < 0 for every c ≥ 0; (iii)
34
³ ³q
³ ³q
´´
´´
1
0 1
2+ 3 −c
2+ 3 −c
c
T
0
if
and
only
c
T
0.025;
(iv)
Φ
c
T 0 if and only
c 3
3
2
2
³q
h
´i
√
¢
¡
c S 25 1 + 6 ≈ 1.3798. These facts imply that Φc (x) > 0 on 0, 13
c2 + 32 − c for
√ ¢
√ ¢¤
£
¡
¡
c > 25 1 + 6 . Next consider c ∈ 0, 25 1 + 6 . The equation Φc (x) = 0 has no real roots
³q
h
´i
1
on 0, 3
c2 + 32 − c when c > c ≈ 0.0766, where c is such that
q
q
³
´2 ³
´
1
3
2
2
3
2
2
= 3 c + 1 − 3 c + 2 c.
8 c+2−2 c + 2
Φc
Zθ1 ³ ´
Hence V1 e
θ1 de
θ1 ≥ 0 for every θ1 ∈ (c, θ∗ ] when c ∈ [c, +∞).
c
The optimal menu of (p1 , p2 , T ) is {(0, 0, 0) , (1, 1, T )}.
Every participating type θ1 chooses (1, 1) contract. The payment is t (θ1 ) = θ1 α (θ1 ) −
q
u (θ1 ) = θ1 − (θ1 − c) − u (c) = 23 c2 + 32 + 43 c = T .
The expected revenue is Π (c) = Pr {θ1 + θ2 ≥ T } · T where Pr {θ1 + θ2 ≥ T } = 1 −
³q
´2
2
2+ 3 −c .
c
9
2
Case 2. The optimal α (θ1 ) is not identically equal to one on [c, c + 1].
By an analog of Lemma 3, and due to the properties of V1 and V2 discussed above the
optimal α is ⎧
a step function
h ´
⎪
⎨ α if
θ
θ1 ∈ c, b
α (θ1 ) =
³
i ,
⎪
⎩ 1 if θ1 ∈ b
θ, c + 1
Zeθ ³ ´
³ ´
θ1 = 0.
θ1 de
where α ∈ [0, 1), and b
θ is such that V b
θ = 0 and V e
∗
∗
c+1
b
Let us guess that
⎧ θ < θ , which implies θ > 3 .
⎪
⎨
u (c) + α (θ1 − c)
if
Hence u (θ1 ) =
´
³
´ ³
⎪
⎩ u (c) + α b
θ
θ − c + θ1 − b
if
c
h ´
θ
θ1 ∈ c, b
³
i .
θ1 ∈ b
θ, c + 1
Zθ1
³ ´
h ´
θ1 = 12 α (θ1 − c)2 + αc (θ1 − c) if θ1 ∈ c, b
θ1 de
θ , and
θ1 α e
Also e
c
Zθ1
³ ´
e
θ1 =
θ1 α e
θ1 de
³ c
i
b
θ, c + 1 .
1
2
(θ1 − c)2 + c (θ1 − c) −
Using equation (12):
35
1
2
³
´2
³
´
(1 − α) b
θ − c − c (1 − α) b
θ − c if θ1 ∈
µ ³
´2
³
´¶
b
b
3 (θ − c) + 2c (θ − c) − (1 − α) 3 θ − c + 2c θ − c
−
∗
2
∗
1
2
1
2
= 0.
Denoting x = θ∗ − c and y = b
θ − c we get:
3x2 + 2cx −
¡
¢ 1
1
(1 − α) 3y 2 + 2cy − = 0.
2
2
(13)
Using u (θ∗ ) + θ∗ = 0 we have u (c) = −2x + (1 − α) y − c.
Rearranging the expression for V1 we get
3
2 α (θ 1
− c)2 + (2c + 2 + 3u (c)) (θ1 − c) + (c + 1 + u (c)) c.
Evaluating V1 at b
θ, substituting the expression for u (c) and changing variables to x and
y we get
³ ´ ¡
¢
V b
θ = 3 − 32 α y 2 + (2 − αc) y − 6xy − 2cx + c.
Next we find
Zeθ
c
Zeθ
c
³ ´
³ ´
θ1 using integration by parts and the fact that V b
V e
θ1 de
θ = 0.
Zeθ · ³ ´ ³
Zeθ µ
³
³ ´
´
´
³
´2 ¶
e
e
e
e
e
e
e
(2c + 2 + 3u (c)) θ1 − c + 3α θ1 − c
de
θ1 =
V θ1 dθ1 = − V θ1 θ1 − c dθ1 = −
c
³
´2 ³¡ c
³
´´
¢
¡
¢ ¢
¡
3
b
b
c + 1 + 2 u (c) + α θ1 − c = −y 2 1 − 12 c − 3x + 32 − 12 α y ,
= − θ1 − c
where for the last step we substitute the expression for u (c) and change variables to x
and y.
Zeθ ³ ´
³ ´
θ = 0. Together with
θ1 de
By an analog of Lemma 3 we must have V b
θ = 0 and V e
c
equation
(13) we get the following system:
⎧
¡
¢
⎪
2 + 2cx − 1 (1 − α) 3y 2 + 2cy − 1 = 0
⎪
3x
⎪
2
2
⎪
⎨ ¡
¢ 2
3
3 − 2 α y + (2 − αc) y − 6xy − 2cx + c = 0 .
⎪
⎪
⎪
¡
¢ ¢
¡
⎪
⎩
−y 2 1 − 12 c − 3x + 32 − 12 α y = 0
We numerically check that the solution has the following property: α is increasing with c,
α = 0 when c = 0, and α = 1 when c = c.
©
¡
¢
ª
The optimal menu of (p1 , p2 , T ) is (0, 0, 0) , 1, 1, T , (1, α, T ) , (α, 1, T ) .
h ´
Every participating type θ1 ∈ c, b
θ chooses (1, α) contract. The payment is t (θ1 ) =
θ1 α (θ1 ) − u (θ1 ) = αθ1 − u (c) − α (θ1 − c) = −u (c) + αc = T . Every participating type
36
³
i
θ1 ∈ b
θ, c + 1 chooses (1, 1) contract. The payment is t (θ1 ) = θ1 α (θ1 ) − u (θ1 ) = −u (c) +
αc + (y + c) (1 − α) = T . Hence T = 2x − (1 − α) y + (1 + α) c and T = T + (y + c) (1 − α).
The expected revenue is
©
©
ªª
Π (c) = Pr max {θ1 , θ2 } + α min {θ1 , θ2 } − T ≥ max 0, θ1 + θ2 − T
· T+
©
ª
Pr θ1 + θ2 − T ≥ max {0, max {θ1 , θ2 } + α min {θ1 , θ2 } − T } · T ,
©
©
ªª
where Pr max {θ1 , θ2 } + α min {θ1 , θ2 } − T ≥ max 0, θ1 + θ2 − T
=
¡
¡
¢ ¢
= 2y 1 − 2x + 1 − 12 α y ,
©
ª
and Pr θ1 + θ2 − T ≥ max {0, max {θ1 , θ2 } + α min {θ1 , θ2 } − T } = (1 − y)2 −2 (x − y)2 .
Optimal Deterministic Bundling Menu
Let T be the price of (1, 0) and (0, 1) contracts and T be the price of a bundle (1, 1).
The seller has two options.
©
¡
¢ª
(i) Only a bundle is offered, i.e. the menu is (0, 0, 0) , 1, 1, T . Then the analysis is
exactly the same as in Case 1:
q
©
ª
2
T = 3 c2 + 32 + 43 c, the expected revenue is Πd1 (c) = Pr θ1 + θ2 ≥ T · T
³q
´2
ª
©
c2 + 32 − c .
where Pr θ1 + θ2 ≥ T = 1 − 29
(ii) Both individual goods and bundle are offered, i.e. the menu is
©
¡
¢
ª
(0, 0, 0) , 1, 1, T , (1, 0, T ) , (0, 1, T ) .
The optimal prices can be found from maximizing the expected revenue:
³¡
¢¢2 1 ¡
¢¢2 ´
¡
¢
¡
¡
Πd2 (c) = 2T (c + 1 − T ) T − T − c + T c + 1 − T − T
.
− 2 T − T −T
We did not derive the closed form solutions, but instead checked numerically that it is
optimal to offer just a bundle when c > c0 ≈ 0.05, and it is optimal to offer both a bundle
and individual goods when c < c0 ≈ 0.05.
References
Adams, W. and J. Yellen (1976): ”Commodity Bundling and the Burden of Monopoly,”
Quarterly Journal of Economics, 90(3), 475-498.
37
Armstrong, M. (1996): ”Multiproduct Nonlinear Pricing,” Econometrica, 64(1), 51-75.
Baron, D. and R. Myerson (1982): ”Regulating a Monopolist with Unknown Costs,”
Econometrica, 50(4), 911-930.
Guesnerie, R. and J.-J. Laffont (1984): ”A Complete Solution to a Class of a Principal
Agent Problems with an Application to the Control of a Self-Managed Firm,” Journal of
Public Economics, 25(3), 329-369.
Kolmogorov, A. and S. Fomin (1970): Introductory Real Analysis. Dover Publications,
Inc.
Lancaster, K. (1971): Consumer Demand: A New Approach. Columbia University Press.
Manelli, A. and D. Vincent (2004a): ”Bundling as an Optimal Mechanism by a MultipleGood Monopolist,” forthcoming in Journal of Economic Theory.
–––– (2004b): ”Multi-Dimensional Mechanism Design: Revenue Maximization and
the Multiple Good Monopoly,” mimeo, University of Maryland, College Park.
Maskin, E. and J. Riley (1984): ”Optimal Auctions with Risk Averse Buyers,” Econometrica, 52(6), 1473-1518.
McAfee, P. and J. McMillan (1988): ”Multidimensional Incentive Compatibility and Mechanism Design,” Journal of Economic Theory, 46, 335-354.
McAfee, P., J. McMillan and M. Whinston (1989): ”Multiproduct Monopoly, Commodity
Bundling, and Correlation of Values,” Quarterly Journal of Economics, 104(2), 371-383.
Myerson, R. (1981): ”Optimal Auction Design,” Mathematics of Operations Research, 6,
58-73.
Riley, J. and R. Zeckhauser (1983): ”Optimal Selling Strategies: When to Haggle, When
to Hold Firm,” Quarterly Journal of Economics, 98(2), 267-289.
Rochet, J.-C. (1985): ”The Taxation Principle and Multitime Hamilton-Jacobi Equations,” Journal of Mathematical Economics, 14, 113-128.
Rochet, J.-C. and P. Chone (1998): ”Ironing, Sweeping, and Multidimensional Screening,”
Econometrica, 66(4), 783-826.
Rochet, J.-C. and L. Stole (2002): “Nonlinear Pricing with Random Participation,” Review
38
of Economic Studies, 69(1), 277-311.
–––– (2003): ”The Economics of Multidimensional Screening,” in Advances in Economics and Econometrics: Theory and Applications, Eighth Wold Congress.
Schey, H. M. (1996): Div, Grad, Curl, and All That: An Informal Text on Vector Calculus.
W. W. Norton & Company.
Seierstad, A. and K. Sydsæter (1987): Optimal Control with Economic Applications. Elsevier Science & Technology Books.
Thanassoulis, J. (2004): ”Haggling over Substitutes,” Journal of Economic Theory, 117(2),
217-245.
39