Comparative statics in a simple class of
strategic market games
Rabah AMIRy, Francis BLOCHz
July 28, 2007
Abstract
This paper investigates the e¤ects of entry in two-sided markets where
buyers and sellers act strategically. Applying new tools from supermodular optimization/games, su¢ cient conditions for di¤erent comparative
statics results are obtained. While normality of one good is su¢ cient for
the equilibrium price to be increasing in the number of buyers, normality of both goods is required for equilibrium bids and sellers’equilibrium
utilities to be increasing in the number of buyers. When the economy is
replicated, normality of both goods and gross substitutes guarantee that
the equilibrium of the strategic market game converges monotonically (in
quantities) to the competitive equilibrium. Simple counter-examples are
provided to settle other potential conjectures of interest.
jel: D43, D51, L13
keywords: Strategic market games, two-sided markets, bilateral oligopoly,
supermodularity and comparative statics, market entry.
We are grateful to an anonymous referee of this Journal, Stefano Demichelis, Jean-Francois
Mertens and Ben Zissimos for helpful suggestions and/or feedback. Thanks also go to participants of the workshop in honor of Jean Gabszewicz (CORE, Louvain-la-Neuve, 2002) and of
the VIth SAET Meeting (Rhodos, 2003) for their comments.
y Department of Economics, University of Arizona, Tucson, AZ 85721 (E-mail:
[email protected])
z GREQAM, Aix-Marseille Universités. (E-mail:[email protected]) Francis Bloch
is also a¢ liated with the University of Warwick.
1
1
Introduction
The relationship between market structure and market performance is one of
the key cornerstones of microeconomic theory. Recently, this issue has received
renewed attention in a partial equilibrium framework, with the e¤ect of market
structure on oligopolistic markets being revisited using new tools from monotone
comparative statics (Amir and Lambson, 2000 and Amir, 1996a). These new
methods, based on the theory of supermodular optimization, clarify the relationship between the primitives of the model (demand and cost functions) and
the comparative statics results (changes in prices, quantities and pro…ts as the
number of …rms in the market changes exogenously), and allow for a uni…ed
view of various such results relying only on critical assumptions.
In the present paper, our objective is to apply the same tools to study
the e¤ect of entry in two-sided markets where traders on both sides of the
market act strategically. This is thus an attempt to investigate the e¤ects of
market structure in a simple general equilibrium setting where agents are not
price takers. The basic model of trade we consider is the strategic market
game introduced by Lloyd Shapley and Martin Shubik (Shubik 1973, Shapley,
1976, and Shapley and Shubik, 1977), wherein agents submit bids and o¤ers
on the market, and the price is given by the ratio of bids over o¤ers. This
model is a natural generalization of Cournot oligopolies to markets with strategic
buyers, and it embeds the partial analysis of oligopolistic markets in a general
equilibrium framework.
The analysis of comparative statics in simple strategic market games contributes to our understanding of the e¤ects of market structure on equilibrium
in three di¤erent ways. First, the analysis highlights conditions under which, in
a simple general equilibrium model, an increase in the number of traders a¤ects
equilibrium prices and quantities in a predictable manner. One important result
of our analysis provides conditions under which equilibria of the strategic market game converge monotonically to the competitive equilibria of the economy
as the number of traders is replicated. Second, the model enables us to test the
robustness of the partial equilibrium analysis of Cournot markets to situations
where both sides of the market act strategically. Our analysis provides famil2
iar conditions under which the classical results of oligopoly theory (e.g. more
competition results in lower prices and higher outputs) can be extended to an
exchange economy where buyers and sellers are strategic. Third, the use of
an abstract aggregated model with minimal restrictions on the utility functions
enables us to clarify results on the e¤ects of an increase in the thickness of the
market on the equilibria of strategic market games that are scattered in the
literature (Bloch and Ghosal, 1997, Bloch and Ferrer, 2001a-b, Koutsougeras,
2003, Peck and Shell, 1990, Goenka, Kelly and Spear, 1998).
We consider a simple version of a Shapley-Shubik game due to Gabszewicz
and Michel (1997). There are two goods in the economy (money and a private
good), and two types of traders, buyers who are endowed with money and sellers
who are endowed with the private good.1 We …rst analyze the e¤ects of entry
on one side of the market on equilibrium prices and quantities. Our analysis
focuses on three questions: When does an increase in the number of buyers
result in an increase in the total bids on the market? In an increase in the
equilibrium price of the private good? In an increase in the equilibrium utility
of the sellers? To these three comparative statics properties correspond di¤erent
su¢ cient conditions on the utilities of the traders. To show that equilibrium
prices increase as the number of buyers increases, we only need to assume that
the good that a trader is endowed with is a normal good. For equilibrium bids
and equilibrium utilities of sellers to increase with the number of buyers, we
need normality of both goods for the two types of traders.
We also address the e¤ects of a simultaneous increase in the number of buyers and sellers on the equilibrium quantities and utilities on the market. The
su¢ cient conditions under which equilibria of the strategic market game converge monotonically (in quantities) to competitive equilibria are quite stringent:
both goods have to be normal for both traders, and both utility functions have
to satisfy the gross substitutes property.
A key bene…t of the aggregated nature of the model at hand –one good on
each side of the market – is that the underlying analysis can conveniently be
1 As each trader is endowed only with one good, and there are only two goods in the
economy, all the di¤erent versions of the Shapley-Shubik games (buy-or-sell or simultaneous
bids and o¤ers, multiple trading posts or trading using a medium of exchange) are equivalent.
3
cast in the recently developed framework of monotone comparative statics based
on the theory of supermodular optimization and games2 . As quasi-concavity is
a natural assumption in our model, and a critical condition for many of our
results, it turns out that the full power of supermodularity analysis is invoked
mostly when dealing with the comparative statics of Nash equilibrium points
(Milgrom and Roberts, 1994). One noteworthy aspect of the analysis developed
in this paper is that the minimal assumptions that drive the comparative statics
conclusions turn out to be familiar notions from classical microeconomic theory,
such as normality and gross substitutes of goods. This allows a simple and
natural economic intuition behind the derived conclusions.
3
The rest of the paper is organized as follows. Section 2 presents the model
and introduces simple conditions on utility functions. Section 3 forms the core
of our analysis and derives the comparative statics properties of entry. Section
4 concludes, and discusses the limitations of our analysis. All proofs, and some
intermediate lemmas of independent interest, are given in the appendix.
2
The Model
We consider an economy with two types of traders: buyers who own one unit
of money, and sellers who are endowed with one unit of a divisible commodity.
We index the buyers by i = 1; 2; :::; m and the sellers by j = 1; 2; :::; n. Letting
x denote the quantity of money and y the quantity of the commodity allocated
to a trader, we de…ne utility functions U (x; y) for the buyers and V (x; y) for the
sellers. The following assumptions are made throughout the paper.
Assumption 1 The utility functions U and V are twice continuously di¤ erentiable, strongly increasing (i.e. Vx > 0; Uy > 04 ), strongly quasi-concave5 and
2 This theory was developed by Topkis (1978), and extended by Vives (1990), Milgrom and
Roberts (1990) and Milgrom and Shannon (1994) among others.
3 By contrast, the conditions in Bloch and Ghosal (1997) are overly restrictive in terms of
mathematical generality as well as unnatural from the point of view of economic interpretation.
4 These inequalities are assumed to hold for all (x; y) in the domain of U and V . The same
convention holds whenever we consider partial derivatives of the utility functions.
5 U is strongly quasi-concave if the determinant of the bordered Hessian is strictly positive.
This condition is slightly stronger than the strict quasi-concavity of U , de…ned by
U [ x + (1
2
)x0 ] > minfU (x); U (x0 )g; 8x; x0 2 R+
; 8 2 (0; 1):
4
satisfy the boundary conditions: limx!0 Ux = limx!0 Vx = +1; limy!0 Uy =
limy!0 Vy = +1:
Every buyer chooses a bid bi 2 [0; 1] and every seller selects an o¤er gj 2
[0; 1]. Following standard practice in the literature on market games (e.g. Shapley and Shubik, 1977), the price of the commodity is de…ned as the ratio of total
P
P
bids B = i bi over total o¤ers G = j gj ,
P =
8
<
:
B
G
if G > 0 and B > 0
0 if B = 0 or G = 0
The …nal allocations of buyers and sellers are thus given by:
xi
= 1
bi ; yi =
xj
= gj P; yj = 1
bi
P
gj
De…nition 1 A market equilibrium is a vector of bids and o¤ ers (bi ; gj ) such
that, for any buyer i;
U (1
bi ;
bi G
bi + B
)
U (1
bi ;
i
bi G
bi + B
i
) for all bi 2 [0; 1];
and for any seller j,
V(
gj B
gj + G
;1
j
gj )
V(
gj B
gj + G
;1
j
gj ) for all gj 2 [0; 1]:
A market equilibrium is interior if and only if bi > 0 for all i, and gj > 0 for
all j.
Standard arguments can be used to prove the existence of a market equilibrium – which is just a Nash equilibrium of the strategic market game. The
existence of an interior market equilibrium requires a more complex argument
given by Bloch and Ferrer (2001, Proposition 2 p. 205). We recall this existence
result:
Remark 1 (Bloch and Ferrer, 2001b). Under Assumption 1, in a bilateral
oligopoly, an interior market equilibrium exists.
5
As a preliminary to the study of comparative statics in strategic market
games, we recall some well-known concepts and properties from standard consumer theory with two goods. Consider the basic problem
max U (x; y) subject to p1 x + p2 y = m
where the prices p1 and p2 and income m are …xed. Good x is said to be
a normal good if @x (p1 ; p2 ; m)=@m > 0, i.e. if its demand increases with
income for all p2 ; p1 > 0. Goods x and y are said to be gross substitutes (gross
complements) if @x (p1 ; p2 ; m)=@p2 > (<)0 for all p1 ; p2 ; m > 0; or equivalently
if @y (p1 ; p2 ; m)=@p1 > (<)0 for all p1 ; p2 ; m > 0.
The assumptions on utility functions described in the following de…nitions
are well-known to be su¢ cient conditions for normality and gross substitutes,
respectively, in the above partial equilibrium setting. It turns out that these
conditions play a crucial role in the comparative statics of our general equilibrium strategic setting. (To relate to our market game setting, these properties
are de…ned for the utility of buyers, U , and are similarly de…ned for the utility
function of sellers.)
De…nition 2 Good x is a normal good if
a normal good if
y
, Ux Uxy
x
, Uy Uxy
Ux Uyy > 0. Good y is
Uy Uxx > 0:
Observe that this notion of normality requires the property that the demand
for a good is increasing in income to hold at all prices, a strong but useful
notion in our world of endogenous prices. As a consequence, whenever the
two goods are normal, the utility function U is always strongly quasi-concave.
Indeed, recall that the utility function U is strongly quasi-concave if and only
if Ux
x
+ Uy
y
> 0:
De…nition 3 The utility function U satis…es the gross substitutes property if
Ux Uy y
x
if Ux Uy
y
> 0 and Ux Uy x
x
< 0 and Ux Uy
y
> 0: It satis…es the gross complements property
x
y
< 0:
The fact that the concepts of normality and gross substitutes are captured by
simple su¢ cient conditions is known to hold only in economies with two goods.
It is easy to see that if the utility function satis…es the gross complements
property, then the two goods are necessarily normal.
6
3
Comparative statics of entry
In this Section, we …rst study the e¤ect of entry on one side of the market
(when the number of buyers m increases). Our …rst results show that total bids
B increase with the number of buyers. The second set of results shows that the
equilibrium price P increases when the number of buyers increases.
3.1
E¤ects of entry on equilibrium bids
In a strategic market game, the optimal choice of a buyer depends both on
the aggregate o¤ers of the sellers, G, and on the total bids of the (m
1)
other buyers, B i . By contrast, in the Cournot oligopoly analyzed by Amir
and Lambson (2000), the optimal quantity choice of an oligopolist depends only
on the total output of the (n
1) remaining …rms. As a starting point of our
analysis, we …x the total o¤ers of sellers G, and following the same steps as
Amir and Lambson (2000), we analyze the Nash equilibria of the game played
among buyers.
Formally, consider the auxiliary symmetric game, (G); played amongst the
buyers when the sellers’ aggregate bid is …xed at G. The payo¤ to a type 1
player, say i, is then U (1
G
): With G …xed, we may de…ne a type 1
bi ; bi bi +B
i
player’s reaction curve in the game (G) as
r(B i ) , arg maxfU (1
b; b
G
b+B
i
) : b 2 [0; 1]g:
Our …rst result shows that the pure-strategy Nash equilibria of the game (G)
are all symmetric.
Lemma 1 Assume that good x is normal for the buyers. Then, for any m, the
game (G) possesses Nash equilibria, all of which are symmetric.
Lemma 1 shows that for a …xed G, in equilibrium all buyers adopt the same
strategy b. The proof of the Proposition is based on the fact that when good
x is normal for the buyers, the reaction function r(B i ) has slopes everywhere
greater than
1. This property guarantees that all Nash equilibria are symmet-
ric, as in the standard Cournot oligopoly (Amir and Lambson, 2000).
7
We now present a stronger result, establishing that when both goods are
normal, the game (G) admits a unique symmetric Nash equilibrium. Letting
Bm denote the value of aggregate bids at equilibrium, we also show that Bm is
increasing in the number of buyers.
Lemma 2 Assume that both goods are normal for the buyers. Then, for any
m, the game (G) admits a unique Nash equilibrium. Furthermore, equilibrium
total bid Bm is increasing in the number of buyers m.
Lemma 2 concludes our analysis of the auxiliary game (G) –when the o¤ers
of sellers are …xed –and shows that given any G; the aggregate optimal reaction
of buyers to the sellers’o¤ers de…nes a continuous single-valued mapping Bm (G);
which is increasing in m for …xed G. This key fact will be invoked in an essential
way in the analysis below.
We now turn to the original strategic market game, where buyers and sellers
choose endogenousy their bids and o¤ers. We establish that the extremal equilibrium aggregate bids of the strategic market game are nondecreasing in the
number of buyers m by adapting results on the …xed points of shifting mappings
(Milgrom and Roberts, 1990 and 1994, and) to our setting. In view of the fact
that ”no trade” is always a Nash equilibrium of the strategic market game (as
is the case in most models of the Shapley-Shubik variety), we know that the
minimal equilibrium corresponds to B m = 0; which is clearly independent of
m.6 Accordingly, we obtain the following result for the maximal equilibrium.
Proposition 1 Assume that both goods are normal for the buyers and the sellers. Then the maximal equilibrium aggregate bid of the buyers, B m is nondecreasing in m. Furthermore, at this equilibrium, the utility of sellers is nondecreasing in m.
The last part of Proposition 1 establishes that an increase in the number
of buyers results in an increase in the equilibrium utility of the traders on
the other side of the market. The mechanism underlying this result is easy
to grasp: an increase in the number of buyers results in an increase in total
6 In order to avoid cumbersome notation, we may use the same symbol with di¤erent
meanings in the paper, but care will always be taken to clearly de…ne the attached meaning.
8
equilibrium bids, thereby improving the terms of trade for the sellers. One may
wonder whether similar comparative statics results hold for the equilibrium
utility of buyers: Does an increase in m have a determinate e¤ect (positive or
negative) on the equilibrium utility of the traders whose number increases on
the market? We argue that the answer to this question is ambiguous. On the
one hand, an increase in the number of buyers increases the total bids on the
market, which may result in higher o¤ers when bids and o¤ers are strategic
complements. Through this e¤ect, total trade increases on the market, and the
utility of buyers may go up. On the other hand, an increase in the number of
buyer reduces the market power of each buyer, which results in a lower share of
total o¤ers, and the utility of buyers may go down. Cordella and Gabszewicz
(1998) provide an example where the …rst e¤ect dominates: when all agents
have linear utility functions, an increase in the number of buyers may induce a
switch from an equilibrium with no trade to an equilibrium with trade.7 The
following example provides a setting where the second e¤ect dominates.
Example 1 Let U (x; y) = V (x; y) = log x + log y:
Simple computations show that, in equilibrium, B = m(m 1)=(2m 1); G =
n(n
1)=(2n
1): As B=m increases with m and G=m decreases with m, the
equilibrium utility of buyers, U = log(1
3.2
B=m) + log(G=m) decreases with m.
E¤ect of entry on the equilibrium price
In order to study the e¤ect of entry on equilibrium prices, we adopt a di¤erent
approach than in the study of equilibrium bids of the previous subsection. This
approach relies on …rst-order conditions. For any 1 > P > 0; de…ne the
auxiliary functions b(P; m) and g(P; n) as the solutions to the equations8 :
b
b m 11
) + Uy (1 b; )
= 0
P
P
m P
n 1
Vx (gP; 1 g)
P Vy (gP; 1 g) = 0:
n
Ux (1
b;
(1)
(2)
7 Formally, this example does not …t our framework because the utility functions do not
satisfy our boundary conditions.
8 Given Assumption 1, a solution to these equations is guaranteed to exist.
9
Consider next the mapping
m (P )
m
de…ned by
= mb(P; m)=ng(P; n):
(This notation is meant to emphasize the dependence of
on m, because n
will be kept …xed throughout the analysis.) It is easy to see that there exists
a one-to-one correspondence between the symmetric interior equilibria of the
strategic market game and the …xed points of the mapping
m:
Lemma 3 Let (b ; g ) be a symmetric interior equilibrium of the strategic market game, then mb =ng is a …xed point of the mapping
is a …xed point of the mapping
m,
m:
Conversely, if P
then (b(P ; m); g(P ; n)) is a symmetric
interior equilibrium of the strategic market game.
Building on the equivalence of Lemma 3, we establish that an increase in m
results in an increase in the extremal equilibrium prices, P m and P m .
Proposition 2 Assume that good x is normal for the buyers and good y is
normal for the sellers. Then the extremal interior equilibrium prices Pm and
Pm are nondecreasing in m.
The proof of this result, whose details can be found in the Appendix, relies on
three steps. We …rst show that equations (1) and (2) admit a unique solution
for any P; m and n, so that the mapping
step, we prove that
m
m
is single-valued. In a second
is an increasing function of m: Finally, performing
equilibrium comparative statics, we prove that the extremal …xed points of
m
are necessarily nondecreasing in m, in the framework of Milgrom and Roberts
(1990) and Echenique(2002).
3.3
Entry on both sides of the market
We now suppose that entry occurs simultaneously on both sides of the market.
More precisely, we consider replicas of an initial economy where k denotes the
number of replicas. It is well-known that for all the varieties of Shapley-Shubik
games (see e.g. Dubey and Shubik, 1978 and Amir et. al. 1990), interior
Nash equilibria converge to Walrasian equilibria as the number of replicas goes
10
G
B(G)
A
C
G(B)
B
Figure 1: Nonmonotonic reaction functions
to in…nity. What is not known though is whether this convergence to competitive equilibrium can be monotonic, in quantities, prices or utilities. While
convergence to competitive equilibrium clearly holds in the present setting,9 we
focus attention on whether equilibrium quantities, prices or utilities follow a
monotonic pattern as the number of replicas increases, under some conditions.
We …rst analyze under which conditions replication leads to an increase in
equilibrium bids and o¤ ers of the two types of traders. Building on the analysis of
subsection 3.1 (in particular Lemma 2), we observe that an increase in k results
in a simultaneous increase in the reaction curves of aggregate bids to aggregate
o¤ers, Bk (G), and aggregate o¤ers to aggregate bids, Gk (B). Without any
further restriction on the shape of these reaction curves, nothing can be said
about the comparative statics of equilibrium bids and o¤ers. This is illustrated
in the following Figure:
Figure 1 shows that, as both reaction functions increase, the equilibrium level
of o¤ers decreases. (The solid lines represent the initial reaction functions, with
equilibrium at A and the dotted lines the reaction functions after the increase,
9 A formal proof for the present model would only replicate the steps of the more complex
models in this theory, and is thus left out.
11
with equilibrium at C.) The di¢ culty illustrated in Figure 1 is rather robust,
if the only property satis…ed by the two curves is continuity. By contrast, a
su¢ cient condition to guarantee that equilibrium bids and o¤ers increase with
an increase in k is that both reaction functions are monotonically nondecreasing.
It turns out that monotonicity of the aggregate reaction functions Bk (G) and
Gk (B) is directly related to the gross substitutes property of the utility functions
of the traders. Hence, in addition to normality, we need to assume the gross
substitutes property to obtain monotonic convergence in quantities.
Proposition 3 Suppose that both goods are normal for the buyers and the sellers and that both utility functions satisfy the gross substitutes property. Then
the aggregate reaction functions are monotonically nondecreasing, the interior
market equilibrium is unique, and total equilibrium bids and o¤ ers Bk and Gk
are nondecreasing in k.
Proposition 3 shows that total bids and o¤ers increase when the number of
replicas increase. This does not guarantee that equilibrium prices or utilities
increase as the economy is replicated. Whether prices or utilities increase depends crucially on the number of buyers and sellers in the initial economy, m
and n (and the shape of the utility functions U and V ). This is illustrated by
means of the following example.
Example 1 (continued)
We have already computed the equilibrium total bids and o¤ers in this example as
kn(kn 1)
km(km 1)
;G =
:
(2km 1)
(2kn 1)
Equilibrium bids and o¤ers are clearly increasing in k. Individual bids and o¤ers
B=
are given by
(km 1)
(kn 1)
and g =
(2km 1)
(2kn 1)
and converge to the competitive equilibrium level, b = g = 1=2 as k goes to
b=
in…nity. The equilibrium price is given by:
P =
m(km 1)(2kn
n(kn 1)(2km
12
1)
:
1)
Simple computations show that P is increasing in k if n > m, decreasing in k
if n < m and constant (equal to 1) if n = m. Furthermore, P converges to the
equilibrium price 1 when k goes to in…nity. The equilibrium utility of a buyer
is computed as :
U=
kn(kn 1)
(2kn 1)(2km 1):
It is easy to verify that U is increasing in k if m
n and decreasing in k if
m < n.
4
Extension: Heterogeneous traders
We have conducted our analysis in a model with homogeneous traders, where
entry can be parametrized by the number of active traders on the market. We
have chosen this approach mainly for its clarity, as it allows for transparent
comparative statics exercises, and enables us to use techniques that were developed earlier by Amir and Lambson (2000) for Cournot oligopolies to achieve a
signi…cant generalization of earlier results on the model at hand. However, the
intuitions gained in the simple model with symmetric agents are quite robust,
and some of our results carry over to general models with heterogeneous traders.
In this Section, we outline how our analysis can be extended to a model with
heterogeneous traders. Rather than replicating the entire analysis of the model
with symmetric traders, we establish three Claims. These claims constitute
some of the key steps needed to extend our results. Combined with the arguments of the proofs given in the Appendix, they allow us to generalize some,
but not all, of the propositions of the two previous Sections.
Claim 1 Suppose that both goods are normal for all the buyers. Then the game
(G) admits a unique Nash equilibrium.
Proof. Suppose by contradiction that the game admits two equilibria with
di¤erent aggregate bids B < B 0 . Because both goods are normal,
@
Uxi
Uyi
@bi
=
i
x
+
(Uyi )2
13
i
y
> 0:
Hence, for any player i for whom bi < b0i ;
Uxi
Uyi jbi
<
Uxi
0
Uyi jbi .
Using the …rst order
conditions,
Uxi
B bi
=
:
Uyi
B2
B 0 b0
B bi
< B 02 i .
B2
B 0
B 0 bi . Obviously, if
Hence,
bi >
As B < B 0 , this implies that
bi
all players i, we obtain B >
b0i , then
B
0
B0 B =
we also have bi >
B bi
B
B 0
b
B0 i .
<
B 0 b0i
B0
so that
Summing up over
B, a contradiction, which shows that the
equilibrium must be unique.
Claim 2 Suppose that both goods are normal for the buyers. Then, in the game
(G), if a new buyer enters the game, total bids will (weakly) increase.
Proof. Let B denote the aggregate bids after the entry of player j, and B 0
the aggregate bids before entry, and suppose (towards contradiction) that B <
B 0 . By the same computations as above, we know that for all i 6= j; bi >
Summing up over all traders, B
B 0
B 0 bi .
bj > B, a contradiction, which shows that
the entry of a new buyer cannot reduce the aggregate bids on the market.
Claim 3 Suppose that both goods are normal for the buyers and that all buyers’
utility functions satisfy the gross substitutes property. Then aggregate bids are
monotonically nondecreasing in G.
Proof. Consider any buyer i. Direct computations show that
Uxi Uyi y i
@bi
=
Ui
Ui
@G
Uxi B(2 Gy + (U iy)2
x
i
y
i
y
+
1
Uxi
0
i)
x
in view of the normality and gross substitute properties. Hence, individual bids
are nondecreasing in G, so that aggregate bids are nondecreasing in G:
Claims 1 and 2 parallel Lemma 2, and can be used to show that, in a model
with heterogeneous traders where both goods are normal, the entry of a new
buyer necessarily increases total bids on the market (the …rst part of Proposition
1). Claim 3 provides the key step towards showing that when both goods are
normal and all traders’utility functions satisfy the gross substitutes property,
simultaneous entry on both sides of the market results in an increase in total
bids and o¤ers (as in Proposition 3).
14
Two of the results we obtained in the model with symmetric players (the fact
that sellers’utility increases with the entry of a new buyer, and the uniqueness of
equilibrium when traders’utilities satisfy the gross substitutes property) rely on
the fact that the elasticities of the relations between aggregate bids and o¤ers,
=
G @B
B @G
and
G
=
B @G
G @B
are bounded above by one. We suspect that the
P @bi
G
corresponding elasticities in models with heterogeneous traders, B = B
i @G
P @gj
B
and G = G j @B satisfy the same property, but have not yet been able to
B
prove it, and must leave it as an open problem. Finally, our construction of the
mapping b(P; m) and g(P; n) relies on the fact that traders on the same side of
the market are symmetric, so that bid and o¤ers can be de…ned only as a function
of the prices and the total number of active traders. As this construction cannot
be replicated in a model with heterogeneous traders, the comparative statics
e¤ect of entry on equilibrium prices in general models remains an interesting
open question for future work.
5
Summary and Conclusions
The following table summarizes our results in the model with symmetric traders
for the di¤erent su¢ cient conditions on the utility functions. We list these
conditions by order of increasing strength.
Su¢ cient Condition
Results
Good x normal for buyers,
good y normal for sellers
Both goods normal for
the two types of traders
only symmetric equilibria
P is increasing in m, decreasing in n.
aggregate response functions B(G),G(B) single-valued
B is increasing in m, G is increasing in n
V is increasing in m, U is increasing in n
unique interior equilibrium
B(G) is increasing, G(B) is increasing
B and G are increasing in k .
Both goods normal,
gross substitutes property
Table 1.
It is instructive to compare the results of Table 1 with the results on comparative statics in strategic market games proposed in the earlier literature.
15
Bloch and Ghosal (1997) analyze the comparative statics e¤ect of an increase
in the number of buyers on the utility of sellers. They focus on situations
where the interior equilibrium is unique, and impose a set of unduly restrictive
assumptions (symmetry between buyers and sellers, complementarity in the utility function, gross substitutes property). As our results show, these conditions
can be considerably weakened, and the only relevant su¢ cient condition is the
normality of the two goods in the utility function. Bloch and Ferrer (2001a)
study monotonicity of the aggregate reaction functions in an example with CES
utility functions. They show that the reaction functions are increasing if the
goods are substitutes and decreasing if the goods are complements. From our
analysis, it appears that this result is a special case of a more general result
(aggregate bids and o¤ers are monotonically increasing if the gross substitutes
property holds, and montonically decreasing if the gross complements property
holds.)
It is well-known that the gross-substitutes poperty is a su¢ cient condition
for a competitive equilibrium to be unique. (See, for example, Mas-Colell, Whinston and Green (1995, Proposition 17F3, p. 613). Our results show that this
equilibrium is also unique in a strategic market game, and furthermore, that
the equilibrium quantities increase monotonically to the competitive equilibrium quantities as the economy is replicated. This result of monotone convergence is signi…cantly stronger than the classical results on the convergence of
equilibria of strategic market games to competitive equilibria (see, for example,
Postlewaite and Schmeidler, 1978 or Amir et. al., 1990).
We conclude by pointing out that some of our results (namely those on the
comparative statics of entry on equilibrium bids) can be extended to general
models with heterogeneous traders. Other results, in particular those dealing
with comparative statics of entry on equilibrium prices and utilities, rely on
constructions that cannot be extended to models with hetereogeneous traders
in a straightforward manner. However, this does not mean that these results
necessarily fail when traders are not identical, and more work is needed to fully
characterize the comparative statics of entry in general strategic market games.
16
6
References
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Games and Economic Behavior 15, 132-148.
Amir, R. (1996b), ”Sensitivity Analysis of Multisector Optimal Economic Dynamics”, Journal of Mathematical Economics, 25, 123-141.
Amir, R. and V. Lambson (2000), ”On the E¤ects of Entry in Cournot Markets,”
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Amir, R., S. Sahi, M. Shubik and S. Yao (1990), ”A Strategic Market Game
with Complete Markets”, Journal of Economic Theory 51,123-141.
Bloch, F. and H. Ferrer (2001a), ”Strategic Complements and Substitutes in
Bilateral Oligopolies,” Economics Letters 70, 83-87.
Bloch, F. and H. Ferrer (2001b), ”Trade Fragmentation and Coordination in
Strategic Market Games,” Journal of Economic Theory 101, 301-316.
Bloch, F. and S. Ghosal (1997), ”Stable Trading Structures in Bilateral Oligopolies,”
Journal of Economic Theory 74, 368-384.
Dubey, P. and M. Shubik (1978), ”The Non-cooperative Equilibria of a Closed
Trading Economy with Market Supply and Bidding Strategies,”Journal of Economic Theory 17, 1-20.
Echenique, F. (2002), ”Comparative Statics by Adaptive Dynamics and the
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G. Lipsey” (B.C. Eaton and R.G.Harris eds.) , Cheltenham, UK: Elgar.
Goenka, A., D. Kelly and S. Spear (1998), ”Endogenous Strategic Business
Cycles,” Journal of Economic Theory 81, 97-125.
Koutsougeras, L. (2003), ”NonWalrasian Equilibria and the Law of One Price,”
Journal of Economic Theory 108, 169-175.
Mas-Colell, A., M. Whinston and J. Green (1995), Microeconomic Theory, Oxford: Oxford University Press.
17
Milgrom, P. and J. Roberts (1990), ”Rationalizability, Learning and Equilibrium
in Games with Strategic Complementarities,” Econometrica 58, 1255-1278.
Milgrom, P. and J. Roberts (1994), ”Comparing Equilibria,” American Economic Review 84, 441-459.
Peck, J. and K. Shell (1990), ”Liquid Markets and Competition,” Games and
Economic Behavior 2, 362-377.
Postlewaite, A. and D. Schmeidler (1978), ”Approximate E¢ ciency of NonWalrasian Nash Equilibria,” Econometrica 46, 127-135.
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7
Proofs
This section provides all the proofs for the results of this paper. Some intermediate results, of some independent and general interest, are also stated and
proved here. We begin with a result about a key property of the reaction curves
in the game (G) that ensures that only symmetric equilibria exist.
Lemma 4 Assume that good x is normal for buyers, then in the game
for any B
i
0
6= B i ; we have
r(B 0 i ) r(B
B0 i B i
i)
>
(G),
1:
Proof. Through the change of variable B = b + B i , one may view the
decision variable of player i as being B 2 [B i ; B
i
+ 1] instead of b 2 [0; 1],
with the corresponding payo¤ given by
e (B; B i ) = U [1
maxfU
B + B i ; (1
18
B i
)G] : B 2 [B i ; B
B
i
+ 1]g:
Denote the optimal response by B (B i ): As B (B i ) = r(B i ) + B i , we
clearly have
r(B 0 i ) r(B
B0 i B i
i)
>
1 if and only if B (B i ) is strictly increas-
ing. To establish the latter conclusion, we start with the …rst-order condition
e (B; B i )=@B = 0, which reduces to
@U
e (B; B i )=@B =
@U
Ux + Uy
GB
B2
i
=0
It can be easily veri…ed that
e (B; B i )
@2U
=
@B@B i
G2 B i
G
Uyy + 2 Uy
B3
B
h 2e
i
U (B;B i )
Evaluating along the …rst-order condition, we see that @ @B@B
i
Uxx +
1
B i
(1 +
)Uxy
B
B
e =@B=0
@U
the same sign as
B if
Uy
(Ux Uxy
Ux
Uy Uxx ) + (Uy Uxy
Ux Uyy )g + b(Uy Uxy
has
Ux Uyy ) + Uy2 ; (3)
which is > 0 since the …rst (bracketed) term in (3), having the sign of the determinant of the bordered Hessian of U , is > 0 by Assumption 1, and the second
expression is
0 due to the normality of good 1. By the Implicit Function
Theorem, we can conclude that B (B i ) is strictly increasing whenever it is interior. As B ( ) is also globally continuous and the constraint set [B i ; B
is ascending in B
i + 1]
and has continuous boundaries in B i , it follows that B ( )
i
is globally strictly increasing (even if it lies on the boundary for some values of
B i ). For the only way B ( ) might fail to be increasing is to switch from the
upper boundary to the lower boundary of the feasible set [B i ; B
i
+ 1]; which
would contradict the fact that B ( ) is continuous (if the switch is a jump) or
the fact that it is increasing when interior (if it switches continuously).10
Proof of Lemma 1 Existence of a pure-strategy Nash equilibrium of the
game
(G) follows from standard arguments. As U is jointly strictly quasi-
G
concave and b b+B
i
is a strictly concave function of b, each payo¤ function in
the game (G) is strictly quasi-concave in own bid b. Each payo¤ is also jointly
continuous. Hence, the reaction curves are continuous single-valued functions,
and a pure-strategy Nash equilibrium exists by Brouwer’s …xed point theorem.
1 0 It is instructive to compare this proof to the usual argument showing strict monotonicty
of an argmax using the supermodular approach, as in Amir (1996b) or Edlin and Shannon
(1998). See also the treatment in Topkis (1998).
19
To …nish the proof, suppose, towards eventual contradiction, that the game
(G) admits an asymmetric equilibrium, with total bid B and two buyers having
di¤erent bids, say b1 6= b2 : Then clearly, B
B (B
b1 6= B
b2 . Yet, B (B
b1 ) =
b2 ) = B, a contradiction to the fact that B ( ) is strictly increasing (as
shown in Lemma 4). QED
The proof of Lemma 2 requires the following result about equilibrium comparative statics. In words, it says that an upward shift in a continuous function
coupled with a rightward expansion of its domain will always increase the function’s extremal …xed-points.11
Lemma 5 Let b
a and f : [0; a]
! [0; a] and g : [0; b]
! [0; b] be two
continuous functions. Let xf and xf be the largest and smallest …xed points of
f; and xg and xg be those of g. Then if f (x)
xg
xf and xg
xf .
Proof. We …rst show that xg
xf . Since f
xf . Consider the function G(x) , g(x)
Since g(xf )
g(b)
b
g(x) for all x 2 [0; a], we have
xf and g(b)
g, we have g(xf )
f (xf ) =
x on the restricted domain [xf ; b].
b, we have G(xf ) = g(xf )
xf
0 and G(b) =
0: By the intermediate value theorem applied to the continuous
function G on [xf ; b], there is some x
e 2 [xf ; b] such that G(e
x) = 0. This is
equivalent to g(e
x) = x
e. Since x
e
xf and xg is postulated to be the largest
…xed-point of g, we have, a fortiori, xg
The proof that xg
x
e
xf .
xf is similar and thus left out.
Proof of Lemma 2 Since the pure-strategy Nash equilibria of the game (G)
are symmetric, for any m; they are the intersections of the reaction curve r
with the line B i =(m
1). Indeed, at such an intersection, and only there, the
responding buyer’s bid is equal to every other buyer’s bid:
In view of the symmetry of every Nash equilibrium (Lemma 1) and the
di¤erentiability of the reaction curve r (by the implicit function theorem), to
show uniqueness of Nash equilibrium for every m, it su¢ ces to show that at
every Nash equilibrium, we have jr0 (B i )eq j < 1=(m
1). Indeed, since r is
1 1 The closely related results on the comparative statics of …xed-points derived in Milgrom
and Roberts (1994) deal with functions that are de…ned on a common domain, but are not
necessarily continuous.
20
continuous, for the reaction curve r to cross the line B i =(m
1) twice, as is
entailed by nonuniqueness of equilibrium, it is necessary that r have some slopes
> 1=(m
1) between the two crossings.
By the implicit function theorem, we have
0
r (B i ) =
bB i G2
bG
Uyy + b BB3 i GUy
B 2 Uxy
B2
G2 B 2 i
B iG
Uxx + 2 BBi2G Uxy
B 4 Uyy + 2 B 3 Uy
Substituting the …rst-order condition for a Nash equilibrium in, we see that
r0 (B i )eq < 1=(m
1) if and only if
Uxx +
m 1
Ux
Uxy +
Uy > 0;
Uy
m2
which is clearly implied by the normality of good y to buyers (i.e.
y
> 0).
Hence, there is a unique (and symmetric) Nash equilibrium for every m.
To prove that total bids B are nondecreasing in the number of buyers, consider the mapping Rm : [0; m
1] ! [0; m
Rm (B i ) =
m 1
[r(B i ) + B i ]:
m
It is easy to verify that Rm maps [0; m
creasing in m for each B
i
1] de…ned by
1] into itself, and that Rm is nonde-
and nondecreasing in B
i
for each m (by Lemma
2 and its proof). Furthermore, a short calculation reveals that the …xed-points
of Rm are Nash equilibria of the game
(G), and vice-versa. Hence, by the
…rst part of this proof, Rm has a unique …xed point, which, by Lemma 3, is
nondecreasing in m (note here that the domain [0; m
1] also expands in m as
allowed in Lemma 3). In other words, the equilibrium value of B i ; call it B mi ;
is nondecreasing in m. Since Rm is nondecreasing in B
that Bm =
Rm (B mi )
i
for each m; it follows
also is nondecreasing in m:
Proof of Proposition 1. Given Lemma 2, we can de…ne, for any G, the
em (G) that assigns to each G the equilibrium aggregate
single-valued mapping B
bids of buyers in game (G), called Bm in Proposition 3. From Assumption 1
em (G) is a continuous function.
and Proposition 3, we know that the mapping B
e n (B) as the equilibrium aggregate o¤ers of the sellers
Similarly, we may de…ne G
when total bids are equal to B. Now, consider the mapping
21
: [0; m] ! [0; m]
where
em G
e n . As G
e n is independent of m and B
em is nondecreasing in m,
=B
is also nondecreasing in m. Given that
is continuous and that its domain is
of the form [0; m], we can invoke Lemma 3 to conclude that the extremal …xed
points of
are nondecreasing in m.
em G
e n (B0 ), it is clear that the
As a …xed point B0 of satis…es B0 = B
e n (B0 )) form aggregate equilibrium bids for the market
elements of the pair (B0 ; G
game. Conversely, every Nash equilibrium of the market game is a …xed point
of . Thus the minimal …xed point of
must be zero, and by Remark 1, there
exists at least one strictly positive …xed point of
of
. The maximal …xed point
, call it B, is then interior, and increasing in m by the argument of the
previous paragraph.
To show that the equilibrium utility of sellers corresponding to the aggregate
e n (B)) is increasing in m, note that
equilibrium (B; G
V =V(
e n (B)
G
):
n
B
;1
n
Hence, dropping all subscripts for clarity
1 @B
dV
=
(Vx
dm
n @m
@G
Vy ):
@B
By the …rst-order conditions,
Vx =
n
n
1B
Vy :
G
Hence,
dV
1 @B
Vy (1
=
dm
n @m
n
n
1 B @G
):
G @B
By implicit di¤erentiation, we compute:
@G
=
@B
Vy
B
Vy
G
+
y
nVx
x
nVx
:
Given that both goods are normal for the sellers,
that (@G=@B) (B=G)
y
0 and
x
0, so
1: This implies that the sign of dV =dm is identical
to the sign of @B=@m. Hence, as Bm increases with m, the utility of sellers is
nondecreasing in m.
22
Proof of Lemma 3. Due to the strong quasi-concavity of each payo¤ in own
action, any symmetric interior market equilibrium is characterized by the two
…rst order conditions:
Ux (1
b;
bG
) + Uy (1
B
b;
bG m 1 G
)
=0
B
m B
and
Vx (
gB
;1
G
g)
n
1B
n G
Vy (
gB
;1
G
g) = 0:
It is easy to verify that these conditions are equivalent to the equations de…ning
b(m; P ) and g(n; P ) for any …xed point P of the mapping
:
The proof of Proposition 2 requires the following lemma:
Lemma 6 Suppose that good x is normal for the buyers and good y is normal
for the sellers. Then the mappings b(P; m) and g(P; n) are single-valued for all
P; m and n:
Proof. We prove the lemma for b(P; m): De…ne
F (b) ,
Ux (1
b;
b
) + Uy (1
P
b;
b m 11
)
P
m P
Then,
F 0 (b)
As
x
Uxy
P
= Uxx
=
1
(
Uy
x
y
y
+
m 11
m 1 1
Uxy +
Uyy
m P
m P2
1
P
x ):
> 0;
y
+
1
P
+
1m 1
P m
x
=
1
(Uy
Uy
y
+ Ux
x)
> 0:
Hence, F 0 (b) < 0 whenever F (b) = 0, which implies that F crosses the horizontal
axis only once, as F is a continuous function. The proof for g(P; n) is similar.
This lemma establishes that the mapping
m
is single-valued, and we are
now ready for the proof of Proposition 2.
Proof of Proposition 2 As a …rst step, we need to de…ne the domain of
the function
m.
We …rst prove the existence of two bounds Am and Bm such
23
that, 8P < Am ;
m (P )
> P and 8P > Bm ;
m (P )
< P: To this end, we …rst
establish that
b(P; m)
= +1 for all m and lim P g(P; n) = +1 for all n.
P !1
P
lim
P !0
To prove the …rst equality, consider the reaction function of the buyers:
Ux
(1
Uy
b;
b
m 11
)=
:
P
m P
(4)
Suppose by contradiction that there exists a …nite bound M such that
lim
P !0
b(P; m)
P
M:
Given our assumptions on utilities, Ux =Uy is continuous and increasing in b:
Hence,
lim
P !0
Ux
(1
Uy
b;
b
)
P
=
Ux
( lim (1
Uy P !0
Ux
(1; M ),
Uy
b(P; m)); lim
P !0
b(P; m)
)
P
since, as a consequence of the contradiction hypothesis, we must have limP !0 b(P; m) =
x
0: Now, as M is …nite and 1 > 0; U
Uy (1; M ) is bounded. But then, as
m 1 1
m P
goes
to in…nity when P goes to zero, equality (4) cannot be satis…ed, a contradiction.
A similar argument shows that limP !1 P g(P; n) = +1 for all n.
As g
1 and b
lim
P !0
m (P )
P
1, this implies that
m
b(P; m)
P !0 ng(P; n)
P
= lim
b(P; m)
m
lim
= +1
n P !0 P
and
lim
P !1
m (P )
P
mb(P; m)
1
P !0
n
P g(P; n)
= lim
m
1
lim
= 0:
n P !1 P g(P; n)
These two equalities imply that there exist bounds Am and Bm such that
m (P )
> P; 8P < Am and
m (P )
< P; 8P > Bm :
We now show that P m and P m are nondecreasing in m: To this end, the
…rst step is to verify that implicit di¤erentiation of equation (1) with respect to
m shows that b(P; m) is strictly increasing in m. The details of this standard
step are left out. As ng(P; n) is independent of m,
increasing in m:
24
m (P )
is clearly strictly
The second step is to consider the mapping
of which are clearly the …xed-points of
the above argument that
m (P )
m (P )
m (P ).
=
m (P )
P , the zeros
For each m, we know from
is a continuous function that starts strictly
above the horizontal axis and ends strictly below it. By the Intermediate Value
Theorem, it has at least one zero. Let P m and P m denote its largest and
smallest zeros.
Clearly P m > 0. We now show that P m is increasing in m. Let m0 > m.
Since
m
m (P m0 )
<
m0 (P m0 )
= 0, applying the Intermediate Value Theorem to
on the restricted domain [0; P m0 ] yields the existence of a zero of
m,
call
it Pem ; that is thus clearly < P m0 . Since P m is by de…nition the smallest zero
of m , we must have P m0 > Pem P m : A similar argument applies to P m .
Proof of Proposition 3 If the utility functions satisfy the gross substitutes
property and both goods are normal for the two types of traders, the unique
symmetric equilibrium of (G) can be computed as the solution to:
H(B; G; m) ,
Ux (1
B G
; ) + Uy (1
m m
B G m 1G
; )
= 0:
m m m B
By implicit di¤erentiation with respect to G, one obtains that the sign of
@B=@G is the same as the sign of @H=@G: Now,
1
@H
=
fUx Uy
@G
GUy
G
m
x g:
Hence, if the utility function satis…es the gross substitutes property, @B=@G > 0.
Uniqueness of the interior market equilibrium results from the fact that
1
(G=B) (@B=@G)
0 (see the proof of Proposition 1) and @B=@G > 0.
Now consider the aggregate best-response mapping Ak : [0; km] [0; kn] !
e
e
e
[0; km] [0; kn] de…ned by Ak (B; G) = (B(G);
G(B)),
where the mappings B(G)
e
and G(B)
are as de…ned in Lemma 2. By the latter result, both aggregate
e ) and G(
e ) are increasing in k, so that Ak (B; G) is also
reaction functions B(
increasing in k. As @B=@G > 0 and @G=@B > 0, Ak (B; G) is a nondecreasing
mapping in (B; G). We know that the minimal …xed point of Ak (B; G) is 0 and
that its maximal …xed point is the unique interior market equilibrium (Bk ; Gk ):
To show that the latter is nondecreasing in k; let k 0 > k: Consider the restriction
of the mapping Ak0 (B; G) to the subset [Bk ; k 0 m] [Gk ; k 0 n] of its domain. Since
25
Ak0 (B; G)
Ak (B; G), we know that Ak0 (B; G) maps [Bk ; k 0 m]
[Gk ; k 0 n]
into itself. By Tarski’s …xed point theorem applied to this restricted map, we
conclude that Ak0 (B; G) has a …xed point that is
(Bk ; Gk ), by construction12 .
Since this …xed point is the only nonzero …xed point, the conclusion follows.
1 2 This result is closely related to Theorem 6 of Milgrom and Roberts (1990), the only novelty
here being the fact that the domains of the two monotone maps being compared are di¤erent.
26