ARTICLE IN PRESS
INDOR-01721; No of Pages 15
International Journal of Industrial Organization
xx (2007) xxx – xxx
www.elsevier.com/locate/econbase
Creating monopoly power ☆
Harborne W. Stuart, Jr.
Columbia Business School, New York, NY 10027, United States
Abstract
The standard analysis of monopoly assumes that the seller has price-setting power. Without this
assumption, it is not clear what monopoly power is or whether it exists. To address this question, this paper
uses the core of a TU game in place of price-setting. The core provides a model of competition in which
outcomes can be interpreted as the consequences of free-form bargaining or negotiations. This paper shows
that if a core analysis is embedded in a capacity choice context, monopoly power can be created with credible
undersupply. But in monopoly, a core analysis also shows that competition only partially determines the
outcomes. Using a biform game analysis, this paper shows that the monopolist's view of this indeterminacy
directly affects its capacity choice. A moderate view of the indeterminacy generates outcomes that are more
socially beneficial than setting a uniform price. The two extreme views of the indeterminacy are shown to
generate results consistent with uniform pricing and perfect price-discrimination.
© 2007 Elsevier B.V. All rights reserved.
JEL classification: C7; L12
Keywords: Biform game; Monopoly power; Cooperative game
1. Introduction
In the standard analysis of monopoly, it is usually assumed that the seller can set a take-it-orleave-it price for its goods. This price-setting, when credible, is often interpreted either as
evidence of monopoly power or as monopoly power itself. But in situations in which the seller is
not a price-setter—in business-to-business negotiations for instance — it is not clear if monopoly
power exists, much less what form it might take if it does. At first glance, this question has been
answered, both formally in the literature and informally in many classrooms. The formal answer
☆
This paper owes much to joint work with Adam Brandenburger. The author thanks Rena Henderson for her help and
two anonymous referees for their comments and advice. Financial support from Columbia Business School is gratefully
acknowledged.
E-mail address: [email protected].
0167-7187/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijindorg.2007.05.001
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
+ MODEL
ARTICLE IN PRESS
2
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
can be found in core analyses of monopoly. One of the interpretations of the core is as a
procedure-free model of competition.1 In this interpretation, the outcomes are consequences of
free-form bargaining or negotiation. There are no preliminary assumptions about the bargaining
power of any of the players, and, in particular, no player is assumed a priori to have any pricesetting power. Muto et al. (11, 1988) provide general core results for monopoly, and their results
suggest the following conclusions.2 First, if the seller has the capacity to supply all of the buyers,
the outcome is almost completely indeterminate. The seller effectively faces a bilateral bargaining
problem with each buyer. (Treating the bilateral problem as indeterminate has deep historical
roots. See, for example, Von Neumann and Morgenstern (14, 1944, Section 1.3.1)) Second, if
there is undersupply, competition between the buyers arises, and this competition guarantees that
the seller will capture some value. Thus, if undersupply can be credibly created, we can say that
monopoly power can be created.
These conclusions suggest that capacity is the relevant decision variable in an analysis of
monopoly without price-setting. This leads to the informal answer to our question. In elementary
treatments of Cournot competition, it is often noted that price-setting and quantity (or capacity)
competition yield different results with two or more firms. For contrast, it is then noted that these
two approaches yield the same result for monopoly. Thus, it would appear that the situation is
quite clear. The seller undersupplies to create monopoly power, and the chosen capacity
corresponds to the quantity it would sell if it could set a price. But when a core analysis is formally
embedded in a model with capacity choice, the situation takes on an added dimension. Although
the competition created by undersupply guarantees some profits to the seller, this competition
does not completely determine the outcome: there will usually be a residual bargaining problem.
How the seller views this residual bargaining problem becomes central to the analysis. If the seller
is extremely pessimistic about how the residual bargaining will be resolved, it will choose a
capacity that corresponds closely to the quantity sold under uniform price-setting. At the other
extreme, total optimism about the residual bargaining problem will lead the seller to act like a
perfect price-discriminator: it will install capacity to supply the whole market. Moderate views
about the residual bargaining lead to capacity choices between these two extremes.
To obtain these results formally, this paper models monopoly with a biform game.3 The biform
game formalism is a generalization of both a non-cooperative game and a cooperative game, and
we use it here for two reasons. First, the analysis requires elements of both kinds of game. As in a
non-cooperative game, there are specified moves, in this case the seller's capacity decision. Unlike
a non-cooperative game, the consequences of these moves are not payoffs. They are the results of
price competition between the seller and buyers, as modeled by the core of a cooperative game.
Second, when core outcomes describe the consequences of a decision, a subtlety arises: core
outcomes generally are not unique. This is particularly true in the case of monopoly. The seller,
when deciding between different capacity choices, will not be comparing specific payoffs. Instead,
for each capacity choice, it will be faced with a range of payoffs. The minimum of this range is
usually interpreted as the amount guaranteed to the seller due to competition, but the range itself is
considered indeterminate with respect to the effects of competition. Consequently, this range is
often interpreted as a residual bargaining problem between seller and buyers, as is done above.
For a notable example, consider Aumann's (1, 1985, pg. 53) statement that “the core expresses the idea of unbridled
competition.” Note that with this interpretation, a cooperative game is often treated as a primitive, not as an object derived
from a non-cooperative game.
2
For the specific case of unitary demand, see Kaneko (8, 1976), with earlier work on identical buyers in Shapley and
Shubik (12, 1967). For non-unitary demand, see Stuart (13, 2007).
3
Brandenburger and Stuart (2, 2007).
1
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
3
This implies that a monopolist's optimal choice will depend upon how it evaluates different
residual bargaining problems. At a formal level, this can be modeled by specifying a player's
preference over intervals of payoffs. The biform formalism encompasses such a specification.
In Section 2, we use a simple monopoly example with unitary demand and constant marginal
capacity cost to demonstrate the main ideas of this paper. With this example, we show how the
core models competition and how the non-uniqueness of the core is central to the monopoly
problem. In particular, we show how the seller's view of the indeterminacy in the core can lead to
dramatically different capacity choices, and we preview the formal results. Section 3 then presents
the model and results. We first show that inefficiency (i.e., undersupply), if it occurs, is due to
seller pessimism.
To make comparisons to price-setting monopoly, we then consider a unitary demand model. To
guarantee that the seller chooses a capacity corresponding to the traditional price-setting capacity,
we show that the seller must be extremely pessimistic. Assuming that this is generally not the
case, and if uniform pricing is taken as a benchmark, the model suggests the following conclusion: monopoly with bargained (or negotiated) outcomes will generally lead to greater social
efficiency. In Section 4, we discuss some of the model assumptions. First, we consider what
happens when there is complementarity in the buyers' preferences. Second, we comment on a
modeling implication of the seller's preference over intervals. Finally, we emphasize the role of
the core as a ‘competitive’ solution concept by contrasting it with the Shapley Value. We show
that with the Shapley Value, the seller never undersupplies the market when capacity is costless.
2. Example
Consider a game with one seller and 12 buyers. Each buyer j (j = 1,…, 12) has a willingness-topay of wj = 14 − j dollars for only one unit from the seller. Assume that the seller has a constant
cost of capacity of one per unit and a marginal production cost of zero. Fig. 1 below depicts a
situation in which the seller has installed six units of capacity. In the core, the seller is guaranteed
to capture at least area B, and it is indeterminate how much of area A it receives. We sketch the
argument why this is the case. In the core of any game, a player can never receive more than its
marginal contribution, for if it did, the other players would do better by excluding it. (This follows
from the definition of a marginal contribution.) For a monopoly game, it is convenient to consider
the marginal contributions of the buyers first.
Fig. 1. Installed capacity of six units.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
4
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
Start with the marginal contribution of buyer 1 (the buyer with the highest willingness-to-pay).
If it were not in the game, the seller would sell to buyer 7 instead. Because the seller would
continue to sell to buyers 2 through 6, buyer 1's marginal contribution is just the difference
between its willingness-to-pay and the willingness-to-pay of buyer 7, namely a difference of 6.
We can repeat this reasoning for buyer 2. Without buyer 2 in the game, the seller would again turn
to buyer 7 to take its place. But since buyer 2's willingness-to-pay is one less than buyer 1's,
buyer 2's marginal contribution is 5. Repeating this reasoning for buyers 3 through 6 gives
marginal contributions of 4, 3, 2, and 1, respectively. Using the fact that a player cannot receive
more than its marginal contribution, it follows that area A represents the most value that the
buyers can collectively receive. Thus, the seller is guaranteed the value in area B.
In Fig. 1, note that area B is bounded above by the willingness-to-pay of the just-excluded buyer,
namely buyer 7. The seller's decision to supply only six units has created competition, where the
source of the competition is the just-excluded buyer.4 In general, a seller might expect to receive
more than area B, but the core says nothing about how area A will be divided between the seller and
the buyers (other than each buyer receiving, at most, its marginal contribution). In any game, a
player's minimum core allocation is interpreted as the amount it is guaranteed due to competition
(area B in this example). The difference between the minimum and maximum (area A in this
example) is interpreted as beyond the influence of competition. (This interpretation for the lack of
uniqueness in core outcomes has roots in Edgeworth (3, 1881, e.g., pg. 20).) In Fig. 1, the core
analysis gives the seller any amount between 36 (area B) and 57 (areas A and B). Table 1 below
shows how this interval changes with different capacity choices. Notice that since there is no obvious
choice for the seller, we will need to consider the seller's preference over intervals like these.
In a biform game, a player's preference over closed intervals is represented by a single
parameter α ∈ [0, 1], called the player's confidence index.5 This index is an adaptation of
Hurwicz's (6, 1951) optimism–pessimism index. In the biform formalism, the use of the confidence interval is based on four axioms, three of which are standard: order, dominance, and
continuity. The fourth axiom is a positive affinity axiom. In the biform context, it is equivalent
to ensuring that a player's preference over cooperative games is invariant under strategic
equivalence. Based on these four axioms, a seller evaluates an interval of payoffs as an α: 1 − α
weighted average of the maximum and minimum. For example, in Fig. 1, if the seller had a
confidence index of .5, its evaluation for the capacity choice of 6 would be 57α + 36(1 − α) = 46.5.
(Table 1 provides the α = .5 calculations for all the capacity choices.) It is important to emphasize
that this weighting is a representation. The seller evaluates this choice as equivalent to a choice in
which it would receive 46.5, but it does not necessarily expect to receive 46.5.
By considering the two extreme cases of a confidence index, we can make connections to pricesetting models. First, in the case of extreme pessimism, namely α = 0, note that the seller will choose
to maximize only area B. From Fig. 1, this looks like the profit-maximizing decision of a pricesetting monopolist. Allowing for the discreteness of the demand curve, this is, in fact, the case.
Corollary 1 of the next section provides the details. When α = 1, namely when the seller is extremely
optimistic, it will choose to maximize areas A and B. In other words, it will choose to supply the
whole market. In a price-setting model, this would correspond to the case of perfect pricediscrimination. For intermediate values of a confidence index, we will show that a seller's capacity
choice is increasing in its confidence index. Thus, for α N 0, the seller will typically choose a capacity
4
More accurately, the source of competition is a just-excluded unit. (See Stuart (13, 2007).) In the case of unitary
demand, a just-excluded unit corresponds to a just-excluded buyer, so either wording is acceptable here.
5
See Brandenburger and Stuart (2, 2007).
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
5
Table 1
Consequences of capacity choices
Capacity choice
Core projection for seller
Midpoint (α = 0.5)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
[11, 12]
[20, 23]
[27, 33]
[32, 42]
[35, 50]
[36, 57]
[35, 63]
[32, 68]
[27, 72]
[20, 75]
[11, 77]
[− 12, 78]
0
11.5
21.5
30
37
42.5
46.5
49
50
49.5
47.5
44
33
greater than the quantity sold under uniform price-setting. Fig. 2 below shows the core allocations for
each of the seller's potential capacity choices. A curve traced from the bottoms of the intervals
describes the case in which α = 0, and it peaks at 6; this is the choice that maximizes area B and
corresponds to a uniform price-setting result. A curve traced from the tops of the intervals describes
the case in which α = 1, and it peaks at full supply, corresponding to a perfect price-discrimination
result. A curve traced from the midpoints of the intervals describes the case where α = .5, and it peaks
at an intermediate value of 8. As noted above, this suggests the conclusion that the absence of
uniform price-setting will generally lead to more units sold — e.g. 8 vs. 6. (For a textbook treatment,
see Gans (4, 2005, Chap. 6), where α is implicitly taken to be one-half.)
As previously noted, within the biform formalism, the seller's confidence index is based on a
representation theorem. One possible interpretation for the seller's index is that it describes the
seller's self-assessment of its inherent bargaining power. We can use this interpretation to provide a
perspective on the differences between a biform analysis and price-setting analyses. In a biform
model, an extremely pessimistic seller relies solely on the power it derives from creating
competition. Thus, undersupply results from a need to create bargaining power. In contrast, in the
usual price-setting story, the decision and ability to undersupply the market is often considered an
exercise of the seller's monopoly power. In this latter case, undersupply is the consequence, rather
than the source, of monopoly power. At the other extreme of the biform model, if the seller is
Fig. 2. Seller payoff ranges vs. capacity choice.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
6
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
extremely confident about its bargaining power, it has no need to create power from competition. It
supplies the whole market. Comparing this with the relevant price-setting model, namely perfect
price-discrimination, there is less of a contrast. In both models, the seller anticipates capturing all of
the value. The reason why the seller believes this differs: in the biform analysis, it is due to bargaining
ability, and in the price-discrimination model, it is due to discriminatory pricing power. But one
aspect of the result is the same: in both models, the seller supplies the whole market. There is one
significant difference, though. In the price-discrimination model, the seller can be sure of capturing
all of the value. In the biform model, the seller anticipates capturing all of the value, but could easily
end up with much less. In fact, there is a similar difference in the case of the extremely pessimistic
buyer, as well. Although it anticipates capturing the same value as the uniform price-setting seller, it
could easily end up with more. Finally, note that the cases between these two extremes provide the
conclusion mentioned in the Introduction. Since a seller with a moderate view of its bargaining
power will choose a capacity greater than the quantity sold under uniform price-setting, the absence
of uniform price-setting can be said to generate greater social efficiency.
3. Results
We start with an abstract definition of a monopoly game. As a preliminary, recall that a TU
(transferable utility) cooperative game consists of a player set N and a mapping v : 2N Yℝ (the
characteristic function). For any S ⊆ N, the term v(S) denotes the maximum economic value that
the players in S can create among themselves. An outcome of a TU cooperative game is described
by an allocation xaℝjN j , where component xi denotes the value captured by player i. The core of
a TU cooperative game (N; ν) is the set of allocations satisfying ∑i ∈ N xi = v(N) and for all S ⊆ N,
∑i ∈ S xi ≥ v(S).
Now, consider a super-additive, TU cooperative game in which there is one player (the
monopolist) without whom value cannot be created. The core is trivially non-empty (consider the
allocation in which the monopolist receives the total value). Muto et al. (11, 1988) show that with
a third condition, condition (1) below, the core is easily characterized. Their result follows, with a
slight modification: super-additivity is used instead of a non-negativity condition.
Proposition 1. (Muto–Nakayama–Potters–Tijs (11, 1988)) Fix a super-additive TU cooperative
game where N = {m, 1,…, b} and v(N \ {m}) = 0.
Suppose that the game satisfies, for all A ⊆ N \{m},
X
½vðN Þ vðN nfjgÞVvðN Þ vðN nAÞ:
ð1Þ
jaA
Then the core is the non-empty set of allocations xaℝbþ1 given by
b
X
xm þ
x j ¼ vðN Þ;
ð2Þ
j¼1
0 V x j V vðN Þ vðN nf jgÞ for j ¼ 1; N ; b;
vðN Þ b
X
½vðN Þ vðN nfjgÞ V xm V vðN Þ:
ð3Þ
ð4Þ
j¼1
All proofs are in the Appendix. Condition (1) provides structure to the core in this game.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
7
By bounding the sum of the buyers' marginal contributions, it ensures that the minimum core
allocation for the monopolist is easily characterized. The condition holds in typical monopoly
settings, including, for instance, situations with downward-sloping demand and non-decreasing
unit costs. The condition can fail when buyers are complements, either indirectly (e.g., decreasing
unit costs) or directly (e.g., a buyer ‘network effect’). In a core analysis, complementarity in the
buyers usually implies that the core has very little structure from the seller's perspective. (We
provide an example of this in Section 4.) But in other contexts, buyer complementarity does
provide meaningful results. In particular, in the literature on buyer power, buyers are often assumed
to be complements as well as substitutes. (For representative examples, see Horn and Wolinsky
(5, 1988) and Inderst and Wey (7, 2003).)
We now define a biform monopoly game.
Definition 1. A biform monopoly game is a collection (N, S, V, α), where N = {m, 1,…, b}, S =
{0, 1,…, D} for some D ≥ b, 0 ≤ α ≤ 1, and V is a map from S to the set of maps from 2N to ℝ,
with V(s)(t) = 0 for every s ∈ S, and
(a) for every s ∈ S and A ⊆ N \ {m}, V(s)(A) = 0 and
X
½V ðsÞðN Þ V ðsÞðN nf jgÞ VV ðsÞðN Þ V ðsÞðN nAÞ;
jaA
(b) for every 0 ≤ s ≤ D − 1, V(s + 1)(N) N V(s)(N).
This game can be interpreted as a two-stage game. The first stage is the monopolist's (player
m's) decision: a capacity choice s ∈ S. The second stage describes the consequences of the
decision: for any s ∈ S, (V(s); N) is a TU cooperative game. The monopolist and buyers (players
1,…, b) compete, and the result of this competition is described by the core of (V(s); N). Condition
(a) states that for any capacity choice s, condition (1) from above will hold for the resulting
cooperative game, (V(s); N). Condition (b) puts some structure on the monopolist's strategy
choice: increasing capacity increases the total value created — i.e. the social welfare. The term α
is the monopolist's confidence index, as discussed in Section 2.6
To answer the question of what capacity the monopolist will choose, our first result
characterizes the relationship between the monopolist's confidence index and social efficiency.
To simplify the statement of the result, let gðsÞ ¼ V ðsÞðN Þ Rbj¼1 ½V ðsÞðN Þ V ðsÞðN nfjgÞ,
namely the value guaranteed to the monopolist. The monopolist then chooses s ∈ S to maximize
π(s) = αV(s)(N) + (1 − α)g(s). Let B(α) denote the set of optimal strategic choices for the
monopolist, as a function of α. The following proposition characterizes this set.
Proposition 2. The set B(α) satisfies:
(i) B(0) = argmaxs∈S g(s),
(ii) B(1) = {D},
(iii) for 0 ≤ α b 1, max B(α) ≤ min B(α + ε) whenever 0 b ε ≤ 1 − α.
6
In the formal definition of a biform game, (singleton) strategy sets and confidence indices would be specified for
players 1, …, b. We suppress these here.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
8
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
Part (iii) shows that capacity choice is increasing in a monopolist's confidence. Combined with
part (ii), this proves the claim in the Introduction that inefficiency, if it occurs, must be due to
some pessimism, namely α b 1.
To compare this result with standard price-setting results, we apply Propositions 1 and 2 to a
specific model with unitary demand. Let wj, for j = 1,…, b, be numbers satisfying w1 ≥ … ≥ wb ≥
0, where each wj is buyer j's willingness-to-pay for a unit of product. Let K(s) denote the cost of
installing s units of capacity, and let C(q) denote the production cost of q units. We assume that
both K and C are non-negative with non-decreasing marginal costs. For simplicity, we also
assume that wb N k + c, where k = K(b) − K(b − 1) and c = C(b) − C(b − 1). (In the example of
Section 2, K(s) = s and C (q) = 0 for all q.) Let D = b, so that S = {0, 1,…, b}. For each s ∈ S, the
characteristic function is given by:
V ðsÞðAÞ ¼
8
0
>
>
>
< KðsÞ
>
>
>
: KðsÞ CðqÞ þ
R
X
if mg A;
if A ¼ fmg;
vA ð jÞwj
ð5Þ
otherwise;
j¼1
where
q ¼ minfs; jAj 1g;
(
)
r
X
R ¼ max r :
vA ð jÞ V q ;
j¼1
and χA(j) is the characteristic function of A (i.e. χA (j) = 0 or 1 according as j ∉ A or j ∉ A).
Eq. (5) describes the maximum value that can be created in a coalition. The value of q is the
R
number of units that will be exchanged in the coalition, and the expression Σj=1
χA( j) collects the
points on the demand curve associated with the willingness-to-pay of the buyers in the coalition.
Thus, the value of any coalition containing the monopolist can be described as follows. If demand
exceeds the monopolist's supply, then the units are sold to the buyers who value them the most. If
supply exceeds demand, then units are sold to all of the buyers in the coalition. In this latter case,
note that production costs are incurred only for units that are sold. By contrast, capacity cost
remains fixed: it is a sunk cost with respect to the potential competition between seller and buyers.
Some comments on the parameter assumptions may be helpful. There is no restriction on the
shape of the demand curve, other than it must be (weakly) downward sloping. The assumption
that demand be unitary is, however, important to obtaining a relationship to a uniform pricesetting model. Without unitary demand, the characterization of the seller's minimum in the core
does not suggest a natural relationship to a price-setting model.7 The convexity of the production
cost is sufficient (but not necessary) for condition (1) of Proposition 1 to hold. The convexity of
the capacity cost, along with the convexity of the production cost, ensures that condition (b) of
Definition 1 is met.
Let (xm, x1,…, xb) ∈ ℝb+1 be an allocation. Then Proposition 1 yields:
7
Proposition 2 of Stuart (13, 2007) provides the details.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
9
Proposition 3. Consider the biform monopoly game described by Eq. (5).
(i) Suppose that s = 0. Then in the core, xm = 0 and x j = 0 for every j = 1,…, b.
(ii) Suppose that 0 b s b b. Then in the core,
xm þ
b
X
xj ¼
j¼1
s
X
wj ðKðsÞ þ CðsÞÞ;
j¼1
swsþ1 ðKðsÞmCðsÞÞ V x m V
s
X
wj ðKðsÞmCðsÞÞ;
j¼1
0 V x j Vðwj wsþ1 Þ for j ¼ 1; N ; s;
x j ¼ 0 for j ¼ s þ 1; N ; b:
(iii) Suppose that s = b. Then in the core,
xm þ
b
X
j¼1
xj ¼
b
X
wj ðKðbÞ þ CðbÞÞ;
j¼1
bc ðKðbÞ þ CðbÞÞ V xm V
b
X
wj ðKðbÞ þ CðbÞÞ;
j¼1
0 V x j V ðwj cÞ for j ¼ 1; N ; b:
This result (without a cost of capacity) was first shown in Kaneko (8, 1976). The proof in the
Appendix uses Muto et al. (11, 1988, Example 5.1).
Part (i) covers the trivial case in which the seller installs no capacity. No value can be created, so
no player can capture any value. Part (ii) covers the case discussed in the example of Section 2. The
seller installs a positive amount of capacity, but sufficiently little for there to be at least one
excluded buyer (buyers j = s + 1,…, b). Each included buyer (buyers j = 1,…,s) cannot capture more
than its willingness-to-pay (wj) minus the willingness-to-pay of the just-excluded buyer (ws+1).
s
(Thus, area A in Section 2 is equal to Σj=1
(wj − ws + 1).) Each excluded buyer captures no value.
The seller is guaranteed to receive revenue equal to the willingness-to-pay of the just-excluded
buyer (ws+1) times the number of units of capacity installed (s); the seller's cost is just its capacity
cost plus its production cost (K(s) + C(s)). (Area B in Section 2 is then sws+1 − (K(s) + C (s)).) Part
(iii) describes what happens if the seller installs capacity sufficient to serve all buyers. As noted in
the Introduction, the seller is simply involved in a series of bilateral bargaining problems, and the
outcome is almost completely indeterminate. Specifically, for each buyer j (for j = 1,…,b), the seller
and buyer divide the value (wj − c) between them in any way. The seller is guaranteed to receive
revenue of at least c from each buyer, but this does not guarantee a positive profit: it is entirely
possible that bc b K(b) + C(b), as in the example of Section 2.8
The following corollary provides the results that compare the biform analysis to the pricesetting analyses. As before, let B(α) denote the set of optimal strategic choices for the monopolist,
as a function of α.
8
For an example in which the revenue bc can be viewed as part of a competitive advantage, see Section 5.1 of
MacDonald and Ryall (9, 2004).
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
10
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
Corollary 1. Consider the biform monopoly game described by Eq. (5).
(i) B(0) = argmax0 ≤ s ≤ b−1 sws+1 − (K(s) + C(s)), and
(ii) B(1) = { b}.
Part (i) provides the connection between the biform model and a uniform price-setting model.
Consider the case of costless capacity. A price-setting monopolist chooses p to maximize
qp − C (q), where q is taken to be a function of p. From part (i) of the corollary, if α = 0, the seller
will maximize sws+1 − C(s).9 We can argue that this is a discrete version of qp − C (q). In a
discrete model, the intersection of the supply and demand curve is typically not unique. It is a
vertical line between ws and ws+1 (unless ws = ws+1). (See Fig. 1 in Section 2, for example.) If we
choose p to be the minimum of this intersection, then the expressions are the same. Thus, with
α = 0, we obtain our connection to a uniform price-setting model. In the case that capacity is
not costless, we can get a similar result as long as we assume that the price-setting monopolist
considers its future pricing decision at the time it installs capacity. Then the price-setting
monopolist chooses p to maximize qp − (K(q) + C(q)), and we have the same equivalence to part
(i) as before.10
Part (ii) provides the connection to perfect price-discrimination, and the result is immediate.
The seller supplies the whole market.
4. Discussion
4.1. Complementary buyers
In Section 3, we noted that with complementary buyers, the core could have little structure, from the perspective of the seller. We demonstrate this with an example. Consider a
game with a seller and three buyers, i.e. N = {m, 1, 2, 3}. Consider the following characteristic
function:
vðN Þ ¼ vðN nf3gÞ ¼ 3
vðN nf1gÞ ¼ vðN nf2gÞ ¼ 1
vðSÞ ¼ 0; otherwise:
In this game, we can think of the seller as having two units to sell, but for any value to be
created, the seller must sell both units. We can think of this as a ‘network’ effect between buyers.
The network effect is much stronger when buyers 1 and 2 are the buyers. From the seller's
perspective, the core of this game is completely indeterminate: the seller will receive between 0
and 3. From the buyers' perspectives, there is some structure. In the figure below, the closed area
9
For the case K(s) = 0, Moulin (10, 1995, p.58) describes the seller's minimum Core outcome as the “competitive
equilibrium profit.”
10
The author thanks an anonymous referee for clarifying this issue.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
11
bounded by the dark line represents feasible outcomes for buyers 1 and 2. For any point (x, y) in
that area, the seller receives 3 − (x + y).
4.2. Seller's confidence index
In the biform monopoly game with unitary demand, the confidence index can be interpreted as
an assumption that the buyers are homogeneous in their bargaining power. For instance, recall
Proposition 3, and note that for 1 ≤ s b b, the seller's anticipated share of the residual bargaining
problem is
a
s
X
ðwj wsþ1 Þ ¼
j¼1
s
X
aðwj wsþ1 Þ:
j¼1
The right-hand side of the equation suggests that the seller does equally well (or poorly) in
bargaining with each buyer. This is the argument for why the model is consistent with an
assumption of homogeneity in the bargaining ability of the buyers. Formally, this interpretation
should be used with caution. Since the confidence index is a representation of a preference over
s
intervals, the upper bound of the interval, namely ∑j=1
(wj − ws+1), is treated as a whole. Thus,
there is no formal basis for treating it as a collection of intervals of the form [0, wj − ws+1].
Informally, of course, this interpretation provides a useful perspective on the model.
4.3. Shapley Value
In establishing the connections between a biform analysis and price-setting analyses, the
notion of creating competition was a central theme. This was particularly true for a pessimistic
seller, where the connection relied on the fact that the seller was relying solely on the power
gained from creating competition. We close by emphasizing that the use of the core is an
important modeling choice in achieving these connections. To see this, consider the unitary
demand model from above, and assume that the seller chooses a capacity to maximize what it
receives under another well-known solution concept for TU cooperative games, the Shapley
Value. If there is a constant marginal cost of capacity k and a constant marginal cost of production
c, we get the following result.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
12
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
Proposition 4. If wb ≥ kb + (k + c), the seller chooses s = b.
(The proof is in the Appendix.) To interpret this result, note that if k = 0, the seller will
always choose to serve the whole market. There is no notion of creating competition, no notion
of a seller's assessment of its bargaining power, and no connection to a uniform pricing model.
There is some similarity to a perfect price-discrimination model, but only in the capacity
choice and not in the value captured. With the Shapley Value, the seller never captures all of
the value. With perfect price-discrimination, the seller does capture all of it, and the optimistic
biform seller at least anticipates capturing all of it.
Appendix
Proof of Proposition 1. For any A ⊆ N, define x (A) = ∑i∈A xi. As noted in the text, the core is
clearly non-empty. Consider any allocation satisfying (2), (3) and (4). The conditions are
necessary for the core, so we show that they are sufficient. It is enough to consider an A ∋ m.
Inequality (3) implies
X
xðN nAÞ V
½vðN Þ vðN nf jgÞ
jaNnA
V vðN Þ vðAÞ;
where the second line uses condition (1). Eq. (2) says x(N ) = v(N ). Writing x(A) = x(N ) − x(N \ A) =
v(N ) − x(N \ A) then gives
xðAÞ z vðN Þ ½vðN Þ vðAÞ
¼ vðAÞ;
as required.
□
Proof of Proposition 2. Part (i) is immediate from Proposition 1, and part (ii) is immediate from
Proposition 1 and condition (b) of Definition 1. For part (iii), fix some ε such that 0 b ε ≤ 1 − α,
and note that
ða þ eÞV ðsÞðN Þ þ ð1 a eÞgðsÞ ¼ dV ðsÞðN Þ þ ð1 dÞpðsÞ
where δ = ε / (1 − α). Now, consider any s′ ∈ B(α) and s″ ∈ B(α + ε), and suppose, contra
hypothesis, that s′ N s″. Then, by condition (b) of Definition 1 again, V(s′)(N ) N V(s″)(N ), so that
dV ðsVÞðN Þ þ ð1 dÞpðsVÞNdV ðsWÞðN Þ þ ð1 dÞpðs VÞ:
But π(s′) ≥ π(s″) since s′ ∈ B(α), so that
dV ðsWÞðN Þ þ ð1 dÞpðs VÞ z dV ðsWÞðN Þ þ ð1 dÞpðsWÞ;
from which
dV ðsVÞðN Þ þ ð1 dÞpðsVÞNdV ðsWÞðN Þ þ ð1 dÞpðsWÞ;
contradicting s″ ∈ B(α + ε).
□
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
13
Proof of Proposition 3. First, note that for 1 ≤ s ≤ b,
V ðsÞðN Þ ¼ KðsÞ CðsÞ þ
s
X
wj :
j¼1
For 1 ≤ s ≤ b − 1 and j = 1,…, b,
V ðsÞðN Þ V ðsÞðN nfjgÞ ¼
wj wsþ1
0
if 1 V j V s;
otherwise:
For s = b and j = 1,…,b,
V ðsÞðN Þ V ðsÞðN nf jgÞ ¼ wj c:
The proof then follows from Proposition 1, provided condition (1) holds. To show this, consider
any 1 ≤ s b b. (The case s = 0 is trivial.) Consider a set N \ A ∋ m. Since
R
X
vN nA ð jÞ V
j ¼ sþ1
s
X
vA ðjÞ;
j¼1
define
qV¼
s
X
vA ð jÞ j¼1
R
X
vN nA ð jÞ:
j¼sþ1
R
Let q = ∑j=1
χN\A( j), and note that
q þ qV¼ s:
Then
V ðsÞðN Þ V ðsÞðN nAÞ ¼ KðsÞ CðsÞ þ
s
X
wj KðsÞ CðqÞ þ
j¼1
¼
s
X
vA ð jÞwj j¼1
z
s
X
s
X
!
vNnA ð jÞwj
j¼1
vN nA ð jÞwj CðsÞ þ CðqÞ
j¼sþ1
vA ð jÞwj j¼1
¼
R
X
R
X
R
X
vN nA ð jÞwsþ1 CðsÞ þ CðqÞ
j¼sþ1
vA ðjÞðwj wsþ1 Þ þ ½q Vwsþ1 ðCðsÞ CðqÞÞ
j¼1
z
s
X
vA ð jÞðwj wsþ1 Þ ¼
½V ðsÞðN Þ V ðsÞðN nf jgÞ:
jaA
j¼1
The argument for s = b is similar.
X
□
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
14
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
Proof of Proposition 4. The Shapley Value gives player i ∈ N the amount
ui ¼
X ðt 1Þ!ðn tÞ!
½vðT Þ vðT nfigÞ
n!
TpN ;T ji
where t = |T|. Consider an s such that 1 ≤ s b b. Using Eq. (5), the seller's allocation can be written
as
um ðsÞ ¼
¼
X
ðt 1Þ!ðn tÞ!
V ðsÞðT Þ
n!
T pN;Tjm
bþ1
X
ði 1Þ!ðn iÞ!
n!
i¼1
¼ sk þ
V ðsÞðT Þ
Tjm;jT j¼i
X
bþ1
X
ði 1Þ!ðn iÞ!
i¼1
¼ sk þ
X
n!
vT ð jÞðwj cÞ
T jm;jTj¼i j¼1
sþ1
X
ði 1Þð w̄ cÞ
i¼2
RðsÞ
X
n
þ
RðsÞ
bþ1
X
ði 1Þ!ðn iÞ! X X
vT ð jÞðwj cÞ;
n!
i¼sþ2
T jm;jTj¼i j¼1
1
where w̄ ¼ Rbj¼1 wj . For any s b b, for any T ∋ m with |T | N s + 1, define Δ(s, T ) by
b
Rðsþ1Þ
RðsÞ
X
X
Dðs; T Þ ¼
vT ðjÞðwj cÞ vT ð jÞðwj cÞ:
j¼1
j¼1
Note that Δ(s, T) ≥ wb − c. We have
um ðs þ 1Þ um ðsÞ ¼ k þ
zkþ
bþ1
X
ði 1Þ!ðn iÞ! X
Dðs; T Þ
n!
i¼sþ2
Tjm;jT j¼i
bþ1
X
ði 1Þ!ðn iÞ! X
ðwb cÞ
n!
i¼sþ2
T jm;jTj¼i
¼ k þ
bþ1
X
ðwb cÞ
n
i¼sþ2
¼ k þ
ðb sÞ
ðwb cÞ:
bþ1
Thus, if (b − s) (wb − c) ≥ k (b + 1), we get φm(s + 1) ≥ φm(s). Setting s = b − 1 gives (wb − c) ≥ k
(b + 1), from which wb ≥ kb + (k + c). □
References
Aumann, R., 1985. What is game theory trying to accomplish? In: Arrow, K., Honkapohja, S. (Eds.), Frontiers of
Economics. Basil Blackwell, pp. 28–76.
Brandenburger, A., Stuart Jr., H.W., 2007. Biform games. Management Science 53, 537–549.
Edgeworth, F., 1881. Mathematical Psychics. Kegan Paul, London.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001
ARTICLE IN PRESS
H.W. Stuart Jr. / Int. J. Ind. Organ. xx (2007) xxx–xxx
15
Gans, J., 2005. Core Economics for Managers. Thomson.
Horn, H., Wolinsky, A., 1988. Bilateral monopolies and incentives for merger. RAND Journal of Economics 19, 408–419.
Hurwicz, L., 1951. “Optimality Criteria for Decision Making Under Ignorance,” Cowles Commission Discussion Paper,
Statistics, Mo. 370.
Inderst, R., Wey, C., 2003. Market structure, bargaining, and technology choice in bilaterally oligopolistic industries.
RAND Journal of Economics 34, 1–19.
Kaneko, M., 1976. On the core and competitive equilibria of a market with indivisible goods. Naval Research Logistics
Quarterly 23, 321–337.
MacDonald, G., Ryall, M.D., 2004. How do value creation and competition determine whether a firm appropriates value?
Management Science 50, 1319–1333.
Moulin, H., 1995. Cooperative Microeconomics: A Game-Theoretic Introduction. Princeton University Press.
Muto, S., Nakayama, M., Potters, J., Tijs, S., 1988. On big boss games. Economic Studies Quarterly 39, 303–321.
Shapley, L., Shubik, M., 1967. Ownership and the production function. Quarterly Journal of Economics 81, 88–111.
Stuart Jr., H.W., 2007. Buyer symmetry in monopoly. International Journal of Industrial Organizations 25, 615–630.
Von Neumann, J., Morgenstern, O., 1944. Theory of Games and Economic Behavior. Princeton University Press, Princeton.
Please cite this article as: Stuart, H.W. Creating monopoly power. International Journal of Industrial Organization
(2007), doi:10.1016/j.ijindorg.2007.05.001