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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Knife-Edge or Plateau:
When Do Market Models
Tip?
Glenn Ellison
Drew Fudenberg
Working Paper 03-07
February 2003
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network
Paper Collection
http://papers.ssm.com/abstract_icN380061
at
i
KNIFE-EDGE OR PLATEAU:
WHEN DO MARKET MODELS TIP?
Glenn Ellison
MIT
Drew Fudenberg
Harvard University
First draft: July
10,2002
This draft: February 5,
2003
This paper studies whether agents must agglomerate
at a single
location in a class
models there is an increasing returns effect
crowding or market-impact effect that makes agents
prefer to be in a market with fewer agents of their own type. We show that such models
do not tip in the way the term is commonly used. Instead, they have a broad plateau of
equilibria with two active markets, and tipping occurs only when one market is below a
critical size threshold. Our assumptions are fairly weak, and are satisfied in Krugman's
[1991b] model of labor market pooling, a heterogeneous-agent version of Pagano's
[1989] asset market model, and Ellison, Fudenberg and Mobius's [2002] model of
competing auctions.
of models of two-sided
interaction. In these
that favors agglomeration, but also a
The authors thank the NSF for financial support. Bob Anderson, Abhijit Banerjee, Ed Glaeser, and
Vayanos for helpful comments, and Dan Hojman for careful proofreading.
Dimitri
1
Introduction
.
Many economic
agglomerated: people are crowded into a small
of the Earth's land mass; individual industries are geographically concentrated;
fraction
trading
activities are
is
concentrated in a few marketplaces. The standard
way
to
account for
concentration (and the arbitrariness of where activity concentrates) has been to propose
"tipping models" with three equilibria: one with
most
activity at B;
most
activity at location
A; one with
and an unstable "knife-edge" equilibrium with exactly half of the
activity in each market.
At
the core of
most models of agglomeration
is
some type of
increasing returns or "scale effect" that favors the emergence of a single dominant
In
some of these models,
all
agents are ex-ante identical, and they
the larger market. In such cases,
exactly the
with
51%
same
size
is
it is
clear that an equilibrium
of the agents than
and while
all
where
prefer to be part of
all
markets are
an unstable knife-edge; every agent would rather be in a market
a
market with 49%, so any departure from exactly equal
sizes leads to "tipping" to a single site.
agents,
all
site.
In other models, there are differences
between the
agents prefer larger markets, they also prefer markets where the
other agents are less like themselves. For example, firms like markets with an excess of
workers, upstream firms like markets with
assets like markets with
many
In this two-sided case, there
may
buyers, and
is
a potential
many downstream
men
firms, sellers of financial
prefer a dating site that has
many women.
"market impact" or "competitive" effect
discourage agents from switching markets.
We
that
find that this can turn the "knife-
edge" of exactly equal shares into a "plateau" of many stable equilibria with unequal
market
sizes, thus generalizing
competing auctions
in Ellison,
For example,
a
in
an observation that
in the
context of a model of
Fudenberg, and Mobius [2002].
Krugman's [1991b] labor pooling model,
market raises the average wage and so lowers the
market impact
we made
utility
effect, the equal-sizes configuration is not
a firm that switches into
of all firms. Because of this
only an equilibrium, but a
equilibrium: If a buyer or seller were to switch to the other market he or she
that there
other,
in
were now more participants on
which would make
many
it
his or her side of the
strictly less attractive.
It is
would
strict
find
market and no more on the
true that in
others, the market impact effect vanishes as the market
Krugman's model,
becomes
large.
as
However,
the scale effect that favors large markets also vanishes as the market
is
becomes
large, so
it
misleading to retain one of these effects and ignore the other unless one knows more
about the rates
To
of agents,
at
which the two
effects vanish.
we
investigate the importance of these effects,
who we
will call "buyers"
"sellers."
At the
is
study; in such cases there are always equilibria in
When
the
numbers of buyers and
point
is
this is a
which
Our
location.
all
property of the examples
agents concentrate in
sellers are even, there is also
equilibrium where the two markets are exactly the same
Our main
their payoffs
voluntary and a market with only
one agent provides no opportunity for trade; and indeed
either location.
of the period, buyers and
by the numbers of each type of agent who chose the same
assumptions are consistent with models where trade
we
start
choose between two possible locations or markets;
sellers simultaneously
are determined
and
study a model with two kinds
an
size.
that the equal-sizes configuration
need not be a knife-edge.
We
provide sufficient conditions for there to be a wide range (a "plateau") of size ratios for
the
two markets
at
which
all
of the incentive constraints for equilibrium are
Roughly speaking, these conditions
are that as the
number of agents
satisfied.
increases, the payoff
functions converge to well-defined limits that are continuous and differentiate, that the
derivatives of these limit payoff functions with respect to the ratio of various types of
agents be non-zero, and that the convergence to the limit occurs at rate
where
TV
is
the total
at least
1
/
TV
,
number of participants.
Throughout the paper,
we
simplify by ignoring the restriction that the numbers of
each type of agent in each market should be integers. Anderson, Ellison, and Fudenberg
[2003] studies the additional complications caused by the restriction to integer values.
They show
that for "typical"
economies the conditions of this paper suffice
of equilibria, but that there are examples where the integer
any split-market equilibrium
Section
II
restriction is inconsistent with
at all.
of the paper states and discusses our general conditions, and gives our
theorem on the lower bound on the width of the plateau.
We then
show
conditions are satisfied in a series of examples. Specifically, Section
Krugman
for a plateau
III
that the
analyzes the
labor-pooling model mentioned above, Section IV analyzes a two-population
version of Pagano [1989]'s model of competing financial markets. Section
examples of how the assumptions can
We are agnostic
agglomeration, but
as to
we do
facts.
in the
activity has tipped to
one rather than
the industry's
It
use as examples account for actual
Consider, for example, patterns of industry
most concentrated
famous
for
one rarely finds
industries
to several locations.
its
concentration
219 large plants are located
in California.
we
believe that models with an equilibrium plateau are needed to
agglomeration. Even
is
gives
fail.
whether the models
account for some of the stylized
industry, for example,
V
there.
in
that
The upholstered
furniture
North Carolina, but only 74 of
Another 52 are
in Mississippi
does not seem reasonable to claim the 52 furniture plants
are consistent with a tipping
model by arguing
most
that they are serving local
in
and 27 are
Mississippi
demand; nor
can one reasonably argue that tipping toward North Carolina has reached an upper bound
due
to congestion, etc.
-
the upholstered furniture industry
employs
Carolina's workforce and the density of large furniture plants
is
less than
1%
of North
about one per 658 square
miles of land." While "tipping" does not seem to be occurring between industry centers,
there
do appear
to
be threshold effects
at the
bottom end: Nineteen
zero large plants making upholstered furniture.
A
states
have exactly
model with an equilibrium plateau
could account for the coexistence of multiple centers and for
many
locations having
tipped to having almost no activity.
The reader should note
same
in the "split
that,
under our assumptions, per-capita
markets" equilibria as
when
the
economy has tipped
Nevertheless, the aggregate welfare loss need not be negligible, and
which influences the
our focus
is
rents that
about the
to a single market.
is this
aggregate
might be earned by consolidating the markets. Moreover,
not on welfare per se, but rather on understanding the positive question of
whether we should expect agglomeration economies
By
it
utility is
"large plant"
we mean
a plant with at least 100 employees.
Business Patterns for 2000. The upholstered furniture industry
to lead to tipping.
The plant counts
is
approximately
are taken from
at the
90
th
County
percentile in
Ellison and Glaeser's [1997] tabulation of industry agglomeration.
'
It
would
market
also be difficult to argue that the secondary industry centers are a disequilibrium feature of a
that
is in
the process of tipping.
Dumais,
Ellison, and Glaeser [2002] find that in the typical
mean reversion in the state-industry employment shares.
Similar threshold effects appear in many less concentrated industries. For example, the pharmaceutical
manufacturing industry has about the mean level of agglomeration in Ellison and Glaeser's [1997]
agglomerated industry, there
is
2.
The Model and Result
sellers
We
examine a simple two-stage model of location choice.
and
B
buyers simultaneously choose whether to attend market
second stage, they play some game with the other players
same market,
may
they
e.g.
trade at prices set
who have
and
B
buyers attend market
i
i,
1
or market 2. In the
chosen to attend the
by competing market makers or play some
wage-setting game. Rather than specifying the market game,
sellers
then the market
game
we
simply assume that
i
I
A pure
strategy
if 5,
gives the sellers in market
an expected payoff of w j (5,.,5 .) and the buyers an expected payoff of u b (S ,B
treat the utility functions u s
S
In the first stage
i
)
.
We
(S,B) and u b (S,B) as the primitives of our analysis.
Nash equilibrium of this model
such that each of the buyers and sellers receives
is
a profile of location choices
at least as
high a payoff in the market
they attend as they would if they had deviated and instead gone to the other market.
There will be two types of pure-strategy equilibria: equilibria where the market has
tipped and
all
buyers and sellers attend a single market, and equilibria in which both
markets are active. The goal of this section
that are possible in equilibria with
Proposition
1
Let
:
B +B2 = B
.
5,
,
S2
There
5,
,
,
two
to characterize the range
active markets.
Our
of market sizes
first result is
and B2 be positive integers with S
a pure strategy
is
is
l
immediate.
+S =S
2
Nash equilibrium with S
and
sellers
and
t
]
buyers choosing market
i
if
and only
if
B
t
the following four constraints hold.
(B\)ub (S ,B )>ub (S2 ,B2 +l)
l
1
(B2)u b (S 2 ,B2 )>u b {S ,B +\)
l
i
(SlKOSpB^O^+l,*,)
(S2)ut (S2 ,B2 )>u,{S +l,B
1
1
)
tabulation with 37 of the 231 large plants in
Carolina, etc.
Assuming
Still,
twenty-two
states
New Jersey, 31
that these are well defined implicitly
equilibrium or that
we have chosen
in
have one or fewer large
means
New
York, 21
in California, 17 in
plant.
either that the
a fixed equilibrium selection.
market game has a unique
North
i
B
,
)
)
As we remarked
5,
,
S,
B and B
,
]
the
2
be integers.
model with S
above are
sellers
and
first
u s (S 2
')
We will
B
say that
buyers
if 5,
+
S,
5,
,
S\ 5,
,
= S
.
B +
5,
= B and
,
the four constraints
}
,
+l)^« 6 (52 ,52 )-« t (5„
-u t (S 2 +l,B2 )> u5 (S 2
2 )
,
B
)
2
J
-«, (S, 5,
,
ud (5I>J B,)-«4 (5„3i+l)^tt t (51> 5i)-Mt (S2 ,B 2 )
(S2')
h, (5,
The
,
5,
-u s {S, +
1,
5,
)
>
left-hand sides of the
u, (5,
,
)
5,
,
B
2 )
that the agents
most applications one would expect
to a labor
-k, (52
two "stay-in-market- 1" conditions (Bf) and (ST)
measure the detrimental "market impact"
one extra firm
have when they move
that the expressions will
be positive,
right-hand sides measure the degree to which market 2
is
more
all sellers
larger markets are
more
As
efficient.
right-hand sides of (Bf) and (ST) will usually be positive
market
show
1,
that
but this
is
market
e.g.
adding
not necessary for our results.
whether a particular
split
whether the market impact effect
is
of buyers and
The
receive.
The
attractive given the
current division of buyers and sellers. Agglomeration models typically
make
to
market raises the equilibrium wage and thereby reduces profits
or adding another seller to a financial market reduces the price that
assumptions that
to note that they
is
Bi)
(B2')
2. In
and 5, are a quasi-equilibrium for
,
as:
u i (52 ,5 l )-a6 (S2 ,52
1
will usually ignore the requirement that
step in analyzing the implications of these constraints
can be rewritten
(BIO
we
satisfied.
Our
(S
earlier,
a result,
employ
one or both of the
when market
2
is
larger than
rewritten equilibrium conditions
sellers is
an equilibrium depends on
sufficiently large so as to
outweigh the current
differences between the utilities in the two markets.
The main assumption of our general theorem concerns
as the
number of agents
think of
it
the behavior of the market
increases holding the seller-buyer ratio y
as imposing three restrictions.
The
first is that
= SIB
fixed.
the sellers' and buyers' utility
functions in the large finite economies converge to well-defined limits as the
agents increases.
The requirement
that payoffs
One can
number of
converge rules out some potential
applications. In
Kingman's [1991a] model of agglomeration due
to increasing returns in
product variety, for example, each worker's payoff increase without bound as the number
of agents grows." The second restriction
buyer's
versa.
utility is strictly
While we view
that in the continuum-of-players limit, each
is
who
increasing in the proportion of agents
requirement as not being very demanding, there are several
this
V
ways
in
finite
markets, and an example where the monotonicity condition
finite
economy but
which
converge
can
it
Section
fail.
and vice
are sellers
gives examples where
The
fails in the limit.
third restriction
to the large population limit at a rate
allowed but slower rates such as
1 /
of
strict
monotonicity
is
fails in all
each
satisfied in
is
that the utility functions
1 /
B
at least
;
faster rates like 1 /
\[B are not. This third restriction holds for
must
B
all
are
of the
models we have analyzed.
Assumption Al There
:
is
non-empty
a
J d'y
>
=
F ,F ,G
continuously differentiable functions
dFb
T
interval
s
h
s
,
[7,7]
C
(0,oo) and twice
and
G
on Twith
()
dFs
/d'y
<
and
such that the approximations
F (y)-G
us (yB,B) =
s
s
(y)/B + o(\/B)
and
u b (yB,B)
=
F (y)-Gb (y)/ B + o(\/ B)
h
hold uniformly in y
One might expect
when B
G
s
and
is large.
G
b
to
6
be positive,
markets provide higher payoffs, but even
if larger
B, so that larger
at least for large
markers are more
efficient this
need
5
In the equilibrium of Krugman's model, the number of "varieties" of manufactured goods that are
produced increases linearly with the number of workers. The workers' utility is assumed to be a CES
aggregate of their consumption of each variety. It increases without bound as workers are able to divide
their consumption into smaller and smaller portions of a larger and larger number of goods. One could, of
course, write down many other reasonable specifications of a preference for product variety in which utility
would remain bounded even as the number of varieties produced grew without bound. For example, this
would happen in a model where each consumer has an ideal point in a fixed product space and varieties just
fill
6
up the product space as
More
their
number becomes
large.
formally, the assumption on the seller's utility function
lim m s (B) =
such that
|
B(u s (yB,B)-Fs (/))
~G
S
(B)
|<
that there exists a function
is
m s (B)
for all
y e
T and
all
m
s
(B) with
integers B.
not be true, as the size of the market can influence the division of the gains from trade.
For
we
this reason,
the signs of
G
faster than
/
1
should emphasize that our results do not require any restrictions on
and
s
D
Our next
,
G
l
h
.
In particular, in cases
the functions
G
and
s
G
effects,
is
the
same
in
1
/
B
The
for large B.
sizes but the
scale effect
7 S T and
that a,,
and S2 =
(a)
yB2
or,
.
Let
a
case the
same seller-buyer
1 /
the difference in
is
ratios;
B as
Part (b) of the
the population size
increases, and also goes to zero as the fractions of agents in market
Assume Al.
this
inequalities in part (a) of the
proposition shows that the size of this effect declines at rate
2:
at
both markets. In
which are the left-hand sides of the
two markets of different
PROPOSITION
Al
proposition explores the implications of Assumption
proposition, are proportional to
utility in
to the limit is at rate
will both be identically equal to 0.
h
configurations where the buyer-seller ratio
market-impact
where convergence
1
approaches
1/2.
be any positive constant. Suppose that
e[«,l-«] with
a,
+a2
=1 Let
.
5,
=a,5
5,
,
-a^B
,
S,
= yB
]
,
Then,
The market impact
approximated uniformly
effects can be
in
y,a
,
and
x
us {Si ,Bi )-us (Si
+%B = -F
i
)
\l)/a
s
i
a
2
by
B + o(l/B)
and
MSM - u^.B, +1) = jF \j)/a,B +
b
(b)
The
scale effects can be approximated uniformly in
o(l/ B).
y,a
,
and
or,
by
t
us (jB2 ,B2 )- us ( 1 Bl ,B1 )
= G {j)^^'-^-^ + oO./B)
s
a a2
x
B
and
ub ( 1B2 ,B2 )-ub ( 1 B1 ,B1 )
= Gb
(
7)
{
Qi> l
+ o(l/i?)
a a2
B
°C2
x
For example, Schwartz and
to
show
that
when
Ungo (2002)
generalize an example in Ellison, Fudenberg and
the sellers in a market conduct an
S+ P'
Mobius (2002)
price auction and buyers values are
independently drawn from a distribution F with the monotone hazard rate property, then buyers
one market are made worse off by the combination of two formerly separate markets.
in at least
Proof: The
first
u^B.B,) -
u,( 7 5,
approximation
+
1,5,)
=
in part (a)
follows immediately from
us (lBv B,) - «,(( 7
+ 1/B,)B„B,)
-F (j + 1/B + F (j) - (G
s
= -F
t
'(
J
)
s
s (
7) -
G
5(
7
+ 1/B,))(l/B +
)
1
0(1/5,
7 )(l/a,B) + o(l/5).
The second follows from
+l) = u h (7^,^) -
u^B^B,
Uti-yB^BJ -
= 7^ 7 )(5 + I)" - Gb (-y)m +
= 1 F \ 1 )la B + o{l/B).
1
'(
I)"
1
7
1
^+
- B~') + Gb '(7)7(5, +
for the first approximation in part (b)
u s { 1 B2 ,B2 )-u s { 1 Bl ,Bl )
= (B2 - 5
1
The argument
l)"
l)"
1),B
2
+
;
+
1)
o{\ / B,)
1
b
The argument
^(7^ +
52 )- G
= -G
1
)(fi1
second
for the
is
s (
7)
+
s {
1 ){llB2
o(l/5)
=
is
-l/B + o{\/B)
x )
G,( 7 )(Q'2
-
a,) /
a,a 2 B
+
QED
the identical but with different subscripts.
One way of analyzing agglomeration
in a
model
o(l/B).
like this is to ignore the
market
impact effect and define a split-market equilibrium as an allocation of buyers and sellers
that
makes them exactly
symmetric model
will
indifferent
between the two markets. With
this definition,
have an equilibrium where exactly half of the agents are
in
any
each
market, but this will typically be the only split-market equilibrium, and one can argue that
the
model would
tip
away from
this
unstable knife-edge. This sort of argument implicitly
supposes that the market impact effect
number of agents
is
large. Part (a)
it is
small enough to be ignored, at least
of Proposition 2 shows
small in the sense of being order 1/
be ignored since
is
B
,
is
Al has
it
cannot
is
a "plateau" of quasi-equilibria
not a "knife-
where
all
of the
incentive constraints are satisfied, and the size of this plateau (as a fraction of the space
8
Note
that
when both G and
s
G
b
is
effect.
a demonstration that the 50-50 equilibrium
edge." Instead, a model satisfying
the
market impact effect
but part (b) of the proposition shows that
no smaller than the scale
Our main theorem
that the
when
are negative, larger markets are less efficient,
which tends
to favor
of possible market divisions) does not go
For example,
it
may be
that
no matter how large B
with one-third of the buyers in market
THEOREM
B
>
1
Assume Al Then
.
:
B and any
BJBe[a
any £ >
s
(r)
+
We want to
to
:
show
show
=ajB
5,
this for all a,
Our
first
in
step
is
always
there exists a
B such
model with
the
market
1
for
to infinity.
a quasi-equilibrium
in
market
that for
B buyers
2.
any integer
and S
sellers
has
every 5, with
(y)
1
1
2
2
2r(y)
= max{0,
for
2Gh (r)
1
i
rK(r)
that there exists a
any 7 G T, the four constraints (ST),
t
there
,
2G
number of agents goes
and two-thirds of the buyers
with 5, buyers
-K(r)
B =a B and
is
(S/B) + £,l-a (S I B) - s ] where a
r{y) = max
Proof:
for
1
S with S / B G f
integer
a quasi-equilibrium
to zero as the
for every
(S2'),
B such
that for
(BT) and
any integer
and
(B2') are satisfied at the allocation
s[a'\y) + s ,\ - a''(/) - s]
or,
B>B
.
By symmetry
it
suffices
e[a'(y) + sA/2].
is
to note that
it
suffices to
show
that four simpler constraints
obtained by applying first-order approximations to both sides of (SI'), (S2'), (Bf), and
(B2') are satisfied for
all
«,,«,, and y with 7 G
T and a, e[«* (y) + e, 1/2]. The
four
simpler constraints are:
-F
(AS1)
s
'(7)>G,( 7 )(a2
(AS2)
-F \ 1)>G
(AB1)
jF 7)
(AB2)
7iV(7)>G
s(
s
'(
b
The
1 )(a1
-a
]
)/«,
-a2 )/a2
>G (7)(o2 -«i)/<*i
6
h
(7)(a
1
-a
2
)/a2
sufficiency of these four constraints
is
a straightforward
consequence of
Proposition 2 and our various continuity and differentiability assumptions; Since the
arguments for each of the constraints are similar
equilibria with
two active markets.
we show
this
only for (AS1
)
.
Suppose
—
(AS1) holds
continuous
for
all
7 G T and
a and y
in
,
1
e[a'(y) + s,l/2]. Because both sides of (AS1) are
alia,
and these parameters
compact
lie in a
set,
there
is
5>
a
for
which
-Fs \ 1 )>Gs ('y)( a2 -a )/a +8
(AS10
for all
1
7 G T and
^
(AS1")
for all
all
7 G T and
m,(5) and
m
2
cr,
e[a*(y) + s,\/2]. Dividing by
{l )/
all a,
1
of (AST)
RHS
is at
is at
B
a q2
a^B
e [a* (/) + £",1/2]. Proposition 2 implies
2
- ua (£> 2 +
us (S2 ,B2 )
least
- «S (51 ,S ) +
most us (S2 ,B2 )
£ > 5(w, (5) + m
get that
2
aB >G,{ 1 )^^)- + -^-
lim^^
(B) that are independent of 7 with
LHS
a B we
1
(B)) for
B>B
all
.
1,52 )
«i2 (5)
.
fi
m
|
- m^B)
(B)
such that the
|=
:
and the
5 so
Choose
that there are functions
first
term on the
that
Then, since
us (S2 ,B2 )^ s (S2 +\,B2 )>us (S2 ,B2 )^s (S ,B
1
)
1
+ -^- + m
]
(B) + m 2 (B)
a-,B
for all
7 G T and
whenever
2?
all a,
5 7G T
>
,
6 [a* (» + £•, 1/2], (ST)
and
,
We now show that the
7 G T and
satisfied for all
showing they are
Case
positive,
(
Suppose
1:
AS2)
<
Fs '(7)
to r
so
l
2
!
a -a
2
is
-
ol
<2
satisfied
or,
(7)
>
e [a* (/) + e,\/2].
.
satisfied for all a,
—
.
Defining r5
=
e [a* (7) + e, 1 / 2]
(
=
=
-F,
- 2a,
'(7)
.
2
t
and (AS l)
a, is bigger than
2
are
cases.
Al
or a. >
l
(AB1) and (AB2)
We begin with the seller constraints,
Since Assumption
Qj
i-l
=a,7#
S,.
t
i
e [a* (7) + £,1/2].
by considering three
s
B =a B and
satisfied at
four constraints (AS1), (AS2),
all «,
G
is
.
2
AS
implies that —^.(7)
1
) is
equivalent to
—
~F
S
is
+
1
,
this is
equivalent
'(7)
We have chosen
r'(y) so that r
(r)> r
2r5
is
satisfied
2r
10
whenever
a, e [a'(y)
+ s,\
2].
,
-
Case
Suppose
2:
-F
Moreover, since
a
F
s
> -Gs (i)
'(7)
s
< Gs (-y) <
'(7)
g [a (y ) + €,]/ 2]. Hence, (AS2)
l
Case
3: Finally,
suppose
obviously satisfied. (AS2)
is
G
s
s
<
'
,
Fs '(7)
2-^ +
-F,
the
first
case, this
is
(~f)
< Fs
equivalent to
'(7)
t
a,
a
.,
1
]
2
2k
,
.
(y)
f
tor
>
1
all
Qj
As
—
in the
second case, (AS1)
—
In this case,
-
a2 -
'(7)
.
>
Ql
/;<2—^
we have chosen
and
is
1=
——
!
.
As
in
l-2ar,
r (y) so that this
is
2rs
n
= 2-^Tl +
shows
l
that
(AB1) and (AB2)
are satisfied if
Again, r*(j) was chosen so that
lFb '(7)
]
=
-
a,- a,
or,
nearly identical argument
>
1
2
obviously satisfied.
e[a*(y) + £, 1/2].
or,
A
.
so (AS2) becomes
2
true for
<
—
F
equivalent to
while -
,
is
is satisfied.
= 2 —allL-i
Fs \j)
1
'(7)
1
Oi
s
/;
(AS1)
In this case
.
1
>
;
2r(y)
2
11
2
2rh
QED
Remarks
1
.
The
ratios
:
size
of the quasi-equilibrium plateau identified
-2G (7)/F
5
5
(^)+1 and
whichever of the two
is
2Gi (/)/xF6 (7)+l
.
in
theorem
(More
1
precisely,
is
decreasing in the
it is
decreasing in
the binding constraint provided that the equilibrium plateau
the entire space.) Intuitively, the scale effect,
while the market impact effect
is
which
is
ofF and makes
plateau larger. Inspection of the formula in the theorem reveals that the quasifull interval
II
as
not
proportional to G, favors tipping,
proportional to the derivative
equilibrium plateau converges to the
is
G
s
(^)/F5 (7) —»
and
the
.
k'/X/V-^C/) ~^
!
here the scale effect
absent.
is
The equilibrium plateau shrinks
to
zero in the limit as either ratio goes to infinity.
2.
The theorem provides
sufficient but not necessary conditions for the existence
quasi-equilibria, and the proof considers only allocations with the
in
each market.
large the
We
can establish a partial converse: If
model with B buyers and S
sellers
is
equilibrium
allowed
binding
at
somewhat
is
a{y) and
larger
when
In general, though, only
and Mobius [2002]
we
is
a quasi-
two markets are
give an exact
characterization for the quasi-equilibrium set for the case of competing auctions
buyers' valuations have the uniform distribution.
the size of the smaller market
two lower bounds converge
but that the
3.
is strictly
The theorem applies
constraint that the
that this integer constraint
in
turn out to be fairly complex. Ellison,
]
B
sellers has
as the seller-to-buyer ratio increases towards
opposed
to equilibria
when
the
number of buyers
is
a
7
'
is
large
is
S
n
sellers
the results
reflect
9
To
,
its
sufficiently large
ignores the
expect
implications
that in the
yB
,
then for any
model with B
many
or even most
Anderson, Ellison, and Fudenberg [2003] show that there
B
1
close to the given point. This, however,
whenever
but that there are nongeneric sequences (S
n
,B
n
)
BIS approaches
for
is
indeed a
almost any
which the models with
and B" buyers have no equilibria with two active markets.
we
it
One might
and S =
close to y for which the
an equilibrium that
broad plateau of equilibria for
value of 7
because
each market be integer- valued.
leaves open the possibility that there might not be equilibria for
values of B and S.
lower bound on
that the actual
Fudenberg and Mobius (2002) show
point in the quasi-equilibrium set there
buyers and 7
show
not very demanding in large economies, but
is
case of competing auctions,
We
when
smaller than the lower bound established here,
to quasi-equilibria as
numbers of sellers
B
one of the four
which there
the seller/buyer ratios in the
In Ellison, Fudenberg,
to differ.
£[a"\y),\ - a (y)] then for
the range of splits for
,
seller/buyer ratio
does not have an equilibrium with exactly
equal seller/buyer ratios in the two markets.
constraints
a
same
of
We conclude that
derive in this paper about the sizes of the quasi-equilibrium plateaus do
what would we would find
see this, observe that
when a
(y) >
in a full
,
at
equilibrium analysis in generic economies.
any allocation with the same seller-buyer
markets, at least one of the buyer or seller constraints
is
12
violated for sufficiently large
ratio in
B when a
both
<
a
(y)
1
Krugman's Labor Market Pooling Model
3.
Our
example
first
Kxugman's [1991b] labor market pooling model of industry
is
agglomeration. In this model, the advantage of industry agglomeration
can better adjust
employment
its
We treat the model
game. In the
first
stage,
F firms and L
choose between two possible locations. In the second stage each firm
2
a
e.
firm
L workers
hires
i
wage of
at a
I
n =a + (j3 + £
i
I
2
)L - {5 1 2)L -
Suppose
we suppose
= (/? +
that
F firms
from the
£,.
At the
assumed
to
w
."
is
their
i's
labor
w') =
then
Workers
start
£
,
receives an
have mean zero and variance
are risk-neutral
at their location. If
and supply one unit of
money wage.
in a
given market.
Following Krugman,
market. Each firm's labor
first-order condition for profit
-w)/S.
workers
profits are
its
and L workers are
The market-clearing wage
demand
demand
at the
+S{LI F)-^T
is
w =/?-c>(L/F) + V
_
£ .IF.
we
is
= E(w) = J3-S(L/F)
market-clearing
e.
is
maximization, yielding
of the second stage, before the productivity shocks are revealed,
u w (L,F)
Firm
,
i
10
i
see that a worker's expected utility
L](
is
that firms are price takers in the labor
easily derived
(w)
wL
I
labor inelastically; their utility
Z,*
which
,
Firms observe these shocks and then hire workers from the pool
.
each firm
that
level in response to idiosyncratic productivity shocks.
as a two-stage
independent productivity shock,
is
F\l 5
.
wage
is
Substituting this into firm
z's profit
function and
\
taking expectations
we
find after
some algebra
that
Kmgman cites Marshall's [1920] discussion (which notes both that larger markets may provide workers
with insurance and that firms may benefit from access to skilled labor) as the inspiration for his model.
Our notation
departs slightly from
use y for the ratio
Krugman 's.
We write 8 for the parameter he called
y so
that
we
can
.
some firms may employ a negative number of workers; we follow Krugman in not worrying
and thereby keeping the model tractable. Note also that the labor demands need not be integers;
can be interpreted as workers splitting their time between several jobs.
Note
about
this
LI F
that
this
1
I
8 L
a
-—
+—
2
2
u f {L,F)
Writing ^for LI
=
E(?r,(L'
i
F
(which
(wlw)) = a +
is
f.
i
13
,
F
analogous to the seller-buyer ratio
in the labor
market), these utility functions have the form
u w (L,F)
= Fw {y)-Gw (r)/F
u (L,F)
= Ff (y)-Gf (y)IF
f
2
2
jj
2
torFw (r) = 0-Sr, Gw (y) = 0,Ff (y) = a+—+-y
Fw \y) = —8 <
Now we
we suppose
Ff \y) = <5y > 0.
and
this contrasts
Hence, Assumption
analyze the choice of location in the
that agents take their
,
and G, (r)-~- Note
1
is
satisfied for
first stage.
As
that
any y
14
.
model,
in our general
market impact into account when choosing a location;
with Krugman's analysis, which directly defines equilibrium to
expected profits and wages are equal
in the
two markets.
mean
that
Although our solution concept
does not require that wages be equalized, expected wages are the same in both markets
we
the equilibria
Theorem
in
analyze.
1
now
implies that this model has a plateau of equilibria with two active
markets, and gives a characterization of how unequal in size the two markets can be.
Corollary
1
.
Fix y and
such that for
all
a =
let
F>F
,
—
2S
2
r-=
2
2
y
.
+2a
2
Then, for any e >
the quasi-equilibrium set of the
13
This formula corrects an error in Krugman's equation (CIO).
14
The example
'
This
is
satisfies
Assumption Al even though
there exists an
model with
the firm profit function
F firms
and
need not be monotone
equivalent to assuming that both firms and workers ignore their market impact.
An
F
in F.
alternate
way
Krugman's solution would be to consider a game where firms choose locations first, workers then
choose locations, and there is a continuum of workers so each worker has no market impact. A problematic
aspect of this justification is the contrast between assuming that workers migrate instantly across the two
markets to arbitrage away any expected wage differences due to unexpected plant location choices, but are
immobile when productivity shocks are realized and wages become different. Our analysis corresponds to
the case where workers are stuck in the locations they choose and can neither relocate in response to
expected wage differences if they are surprised by firms' location decisions, nor in response to realized
wage differences; our qualitative findings should carry over to models with anything less than perfect
to justify
mobility.
16
Wages
are equal in equilibrium, but not following a deviation
14
from the equilibrium.
.
8
.
I
yF
workers includes
\
F
with
splits
,
firms in market
1
every
for
}
F
with
]
F / F e[a* + £,l-a* - e]
t
G K (y) - 0,
Proof: Since
1
firms
and G, (y) >
we
incentive constraint, and
,
the width of the plateau
2G (y)/F
r'(y)=
find that
2
2
a = a /2S y
2
2
/(l
V
2
+ cr /S y
2
)
;
=
—
^^
2S
+2a
2
2
determined by the
{y)y +
/
t
and
is
\
2
=
2
I5 y +1
cr~
-.
2
y
Remarks
:
The equilibrium plateau can cover almost
1
the
whole space,
fraction of the firms can coexist with a large one. This occurs
are very small
(
5 —>
oo
),
or
2
(a ~
when
),
when
a
is,
when
market with
a tiny
productivity shocks
there are strong decreasing returns at the firm-level
the worker-firm ratio
large
is
—>
(y
oo
).
When
down
the opposite extremes, the equilibrium plateau shrinks
result
that
the parameters are at
to a point. Intuition for
can be obtained by comparing the market impact of a firm
that
moves
each
to the larger
market, thus bidding up wages, to the advantage of larger markets, namely that a firm
has a smaller wage impact
when
raising or lowering labor
productivity shock. For example,
there
is little
or decrease
advantage to being
its
labor
would bid up wages
labor
2.
demands
demand very much
substantially if
to try to
it
in
response to
there are strong decreasing returns
in a large
(S
market because a firm won't want
its
large),
to increase
response to a productivity shock, and a firm
in
moved
to the larger
market because the other firms'
are inelastic.
Although we would be hesitant
way
when
demand
do so would be
to put
much
stock in any calibration of this model, one
of parameters that determines the size
to note that the ratio
of the equilibrium plateau also determines the variability of firm-level employment.
Specifically, Var(L]
was
1
when
F = 10
I
,
E(L', ))
= ((F -
then a' ~ 0.28
1) /
;
F)
2
a
2
2
2
1
y
.
If
one assumed
(
this says that the
Corollary
large.
1
I
E(L] ))
smaller market needs to be about
three-eighths of the size of the larger market to be viable (0.28/0.72
3.
that Var L]
= 0.38).
does not include an assumption that the number of firms be sufficiently
Theorem
1
required an assumption about the
15
number of buyers being
large only
.
we assumed just
because
that the utility functions
F{y)-G{y)l B when B was
were approximately equal
to
large. In this application, the expressions for the utility
functions are exact, and hence the formula for the bounds on the equilibrium plateau
holds for
F.
all
17
In the quasi-equilibria described in Corollary
4.
utility in
both markets. Thus, they would
1
the workers receive exactly equal
be quasi-equilibria
still
if workers
consider their market impact. This would be the appropriate assumption
the workers as a
In the
5.
continuum of agents of mass
Krugman model with
the
model with
market
1
F firms
and
any integer
for
alternate
way
all
of the quasi-equilibria described
to state the corollary
would be
mass L of workers has an equilibrium with
F
F
with
F F e [a
l
x
The generalization of the
to say that
workers in
result to a
+e,l — a'—e].
model with
N locations (and a continuum of
workers) would be that there exists an equilibrium with
F
firms in market
1
,
]
market
2,
.
.
.,
or at least a*
4.
and
FN
firms in market
l(\-a) times
as
many
F
2
firms in
N provided that every market has either zero firms
firms as the largest market.
Pagano's Security Market Model with Heterogeneous Traders
Our next example
is a
two-population version of Pagano's [1989] model of
competing securities markets. In Pagano's model markets serve
risk. Specifically,
he considered a two-stage game. In the
to diversify
first stage,
endowment
N agents
simultaneously choose which of two markets to attend. In the second stage, each agent
receives a
i.i.d.
trade
random endowment
K =K +e
Qt
of a single risky
by simultaneously submitting demand curves
The bounds
are exact
bounds
asset,
The
i
where the e are
i
?
draws from a symmetric distribution with mean
trades at the market-clearing price.
17
in
x
x
6.
number of
a continuum of workers of mass L, only the
An
a
one modeled
L.
firms in each market must be an integer, and hence
the corollary are equilibria.
if
did not
.
Agents then
market maker
who
executes
random dividend
d, so
an agent
to a
asset pays a
and variance ae
who
for the existence of an equilibrium with equal worker-firm ratios in the
two
markets. There are equilibria with unequal worker-firm ratios (with a higher ratio in the smaller market) for
smaller values of
a
16
keeps A" shares has random
final
market price of the asset and
R
wealth
all
KQ = —
some
1
this sort
;
is
assumption that S
may
them.)
sell
where the dividend has
means
a
To
more
and
B
K
Q
is
same
the
for
"buyers" have
"seller" are
only
endowment shock may end up purchasing
simplify the algebra,
we
because there
shares,
will specialize to the case
symmetric distribution with mean
efficient
1
names "buyer" and
and variance
a
18
2
This
.
key aspect of the model, which
that the asset is a "bad," but preserves the
larger markets are
Q
that
expected purchases seems reasonable for
in
will be clear shortly, the
suggestive, as a "seller" with a negative
K =
have
"sellers"
of ex-ante asymmetry
(As
applications.
and a "buyer"
not free disposal.
Pagano's model by replacing the assumption
JV agents with the
demands
the risk-free rate of return. Both/7 and asset
is
are allowed to be negative, and there
We modify
w = dK + Rp(K — K), where/? is the
is
•
that
less undiversifiable social risk;
is
allowing for a positive dividend would only complicate the algebra without altering the
nature of our conclusions.
As
in
we
Pagano,
suppose
that agents" preferences
over distributions of wealth are
described by a mean-variance utility function defined directly on the space of wealth
distributions
a
V(w
l
)
= E{w — (b / 2)Var(w
l
symmetric equilibrium
functions
D (p)
.'
9
in
)
we
= K0i /(JV -1) - R(N - 2)p/(/V -
K* =
K
We
19
/(JV-
1)
+
(JV
- 2)Z/(N -
have already simplified by assuming
D (p) is the
D (p)
will
depend on
expectation over
all
A'
.
Q
first
.
To
l)ba
1)JV
that the
agent's desired asset holding, e.g.
variance utility in the
Further following the original,
find firm
D,(p)
0i
.
which the agents simultaneously submit
In the equilibrium
z
)
l
,
i
Z =
S-B + J2^=1 e
endowment of cash
if
D (p) = K
replicate Pagano's results
0i
is
the realized
i
is 0.
then agent
we assume
stage, but not in the second. In the
demand
Equilibrium asset holdings are
.
where
look for
submits the demand curve
20
2
linear
we
/
makes no
that agents
trades.
maximize
Note
their
that
mean-
second stage, they instead maximize the
possible realizations of the vector of endowment shocks of their expected utility
conditional on the shocks. This
would be equivalent
to
expected
utility
maximization
if
mean-variance
preferences were an expectation of a utility function defined on realized wealth, but they are not.
The
in the Appendix. The calculations show that this is the
which agents' demand curves are linear and downward sloping. We do not know if
there are other equilibria. Klemperer and Meyer [1989] discuss conditions sufficient for equilibrium
uniqueness in a loosely related game where agents submit supply functions.
details
of this and subsequent calculations are
unique equilibrium
in
17
F
,
net supply of the asset. Thus, equilibrium allocations are better diversified
endowment shocks more completely because
larger; agents offset their
other TV -1 agents'
share
To
demand curves
also smaller. This
is
simplify notation,
is
is
steeper and the aggregate
when
N is
sum of the
the
endowment
risk they
the source of the larger market's efficiency advantage.
=
let g( 7 )
(7
— l)/(7 +
1).
A series of calculations shows that the
equilibrium payoffs are approximated by
ba
q{if
2
\a
q(i,
+
2(7
2
)
ba
lilf
2
-^—
+
+ 9(7)
2(7
Thus, both the
F(j)
—
G(-y) /
B + o(l / B)
2
F {l) =
bo
=
bo
=
6a \o\
F
'(7)
s
G.(i)
\q{
2
seller's
1 f/2-
\o]
+
F
q ( 1 )\,
2
2
\q(ci) b
2
b (
2
2
2
2
2
o oe +
b
[a
2
)
e
+ 4g( 7 )Wa 2 + b 2 a 2 (a 2 + a 4 + 2q{ 1)\±- +
)
e
1
in
1)
utility functions
'B
have the form
assumption Al. Specifically,
=
ba
2
\q{
1 f/2
+
q{ 1 )\,
+ b 2 a 2 (a 2 + a 4 - 2g( 7 )]/2( 7 +
o oe
V
a\a 2 + a 4 - 2g( 7 )l- + o
B
[B,
1)
and buyer's
assumed
\q{ 1 ) b
1)
2
=
uh ( 7 B B)
2
+
e )
1)
and
G
h (
7)
Thus
Al
s
2
+ 4g( 7 2 6Va 2 + b 2 a 2 (a 2 + a 4 +
= -46a 2 (7 + If 3 <
is satisfied for
sell shares,
goes
and
any 7
that
converges
.
The
0, and
seller's
F
+ 1).
= 4 76cr 2 ( 7 + l)" 3 >
'(7)
h
2g( 7 )]/2( 7
market impact
0, so assumption
effect reflects that sellers expect to
adding a seller to a market lowers the expected price. Note that as
to infinity, the total
sellers)
e )
)
to
B
expected welfare in the population (summing over buyers and
— bo
g( 7 )
/2
,
which
is
the utility of holding the average
endowment.
We can now immediately apply Theorem
Pagano's model has a plateau of equilibria
algebra
have
is
G
s
in
1
to
which
show
that
our two-sided version of
different-sized markets coexist.
simplest if there are equal numbers of buyers and sellers
(r)J-
s
\y) =
G (y)lyF
h
h
2
\y) = (a] + b cj\a]
18
+ae4 ))/2
.
(i.e.
7
The
= 1). We then
This gives
1_
B,
.
.
COROLLARY
2.
Letr* =
1
Consider our Pagano model with equal numbers of buyers and
+ cr + b
2
2
a (a
2
+<?*), and
N>N
exists an Af such that for all
and
N
with
]
Remarks
1.
The
N buyers
I
,
-a -
1
N
buyers and sellers
in
model with
market
>0
there
N sellers
for every
1
]
e\
]
:
size of the equilibrium plateau
shocks.
Then, for any e
.
the equilibrium set of the
includes the splits with
N N e [a' + s
a' = 1/2 — l/2r*
let
sellers.
When
the
is
endowment shocks
inversely related to the size of the
are trivial
(
a
2
«
),
endowment
efficiency differences are
When
unimportant and a market need only have a tiny fraction of the traders to be viable.
endowment shocks become extremely
extremely large
(a —» co )
2
or agents
large
become extremely
coexist only if they are almost equal in size.
that
we have assumed
of buyer
is
-1. If
a
2
that the
=1 and b
(a 2 —> oo ),
To
2
a = 0.1
risk-averse
interpret the scale
mean endowment of a
2
or dividend shocks
seller
is
1
(
b
—> oo )
to
markets can
of the variances, note
and the mean endowment
then the interval in the corollary
approximately (0.27, 0.73), so a market would need
become
is
have about one-quarter of the
traders to be viable."'
2.
The second remark
after
Theorem
implies that these bounds are tight. If
1
outside the specified interval, then for sufficiently large
with
5.
aB
buyers
Some Models
in
market
that
we
Fit the
discuss a few models of agglomeration that don't
modified so that an analysis
21
there will not be an equilibrium
Assumptions
framework, explain why they don't
A.
is strictly
1
Don't
In this section
B
a
fit,
like ours
and suggest ways
in
fit
into our
which they might be
would apply.
A Matching Model
Calibrating a mean-variance utility function can be problematic. For example, even with b
decision-maker would prefer getting
for sure to getting a lottery ticket that
and 100 with probability p for any p < 0.8
.
19
pays
=
0.
1
a
with probability
1
- p
,
Consider
Each
(type w).
a
matching model with two types of agents,
exogenous probability q > Oof becoming unable
participant has
suppose that each type of agent gets
other type, and utility
be matched
women
M agents of type m
1
/ 7)
and
Then
uw (l)
=
>
0)
two binomials with
the ratio
7
lim^^
Pr(x
continuum
+
and
and
let
< 0)E
=
<
0)
=
W
x
—
g)/(l
\M
M'—W
<
0)
M)
= I/7 + o(l /M)
22
,x<0
1
Thus
.
=
q)
1
and
,
sure to
in general,
,
.
different
and sending
and uw (M,W)
=1-
satisfy
0)E
\x
The excess supply
for the case
on the "short side" of the market does not
M/W
=
.
be the realized excess supply of
+ Pr(x >
same success probability but
o(l /
—
is
be the realized numbers of agents of each
}
.=
Pr(.T
be fixed and greater than
to
we
W agents of type w, the utilities of the men and
M'
um {^)
Pr(x
the
(1
who
an agent
limit,
,
the payoffs are
Pr(x
um (M,W)
to
matched with an agent of the
and min(l, 7) respectively, where 7
are able to participate,
men.
in the
q) if
he participates has expected payoff of
if
are min(l,
who
—
utility 1 /(l
Thus
otherwise.
In the finite markets, let
type
m) and women
(type
and these chances are independent across agents. To simplify the algebra,
participate,
if there are
men
and
the difference of
is
sample
>
sizes, so
holding
M and W to infinity,
7 >
1
o(l/M), and
Assumption
A 1,
the utility of the agents
as
it
is
insensitive to
7-
Two
limit
with
model
are noteworthy. First, convergence to the continuum
a faster rate than the 1/ TV rate required by
is at
A 1,
aspects of the
as
Assumption Al. This
although both players care about the "buyer-seller" ratio
of agents,
is
22
this is not true in the
one for any y>\. This
and
F
b
is
continuum
limit.
The
s
and
7 when
limit
<
1
,
b
are both 0. Second,
there are a finite
of the women's
have non-zero derivatives. One way to modify the model
p
G
consistent
utility,
number
FK {y),
not consistent with Al's requirement that the functions
In fact, Chernoffs theorem (Billingsley 1995, p. 151)
constant
G
corresponds to the case where the functions
it
is
so that the convergence
is (at
least)
shows
that the
exponentially
20
F
s
make
it
compatible
lim,,^ Pr(x <
0)
<
fast.
to
p
h'
for
some
B
S
with Al might be to assume that a
in
,
woman's expected payoff from
a
match
is
increasing
M IW because the women are able to select from the available men.
B. Preference for Variety
Consider the following greatly simplified version of Gehrig [1998]. In the
stage,
S firms and B consumers choose one of two
locations.
The firms
first
each location
at
are then placed uniformly around a Hotelling circle of possible products and the
consumers draw
ideal points
from a uniform
distribution.
Firms play
a standard
competition-on-a-circle pricing game.
Assume
a market with
5"
firms
is
p =
c
+
1
/
S Each
would be
u h (S,B) =
his or value,
minus
u s (S,B) =
is
is
such that the equilibrium price
firm in expectation sells to
.
consumers, so the firm's profit function
utility
parameter
that the transportation cost
2
BI
.
MS
of the
The consumer's expected
minus an expected mismatch
the price,
cost,
v-{c + ]/S)-\/{2S).
Neither of these functions satisfies Al.
On
S —>
This prevents the application of theorem
zero as
in
BIS
with
co
fixed so
F (y) =
.
s
the firm side, profit
is
converging
to
1
but a direct analysis of the seller's incentive constraints (ST) and (S2') shows that they
are satisfied with equal buyer-seller ratios in the
just as in the proof of Theorem
On
the
consumer
1
.
is
1
= S2
.
5,
/
modify
(B
Fb {y) = v-c
to
,
which
case buyers don't care about
1
')
is
independent
how many
and (B2') constraints are satisfied only
if
Substituting this into the sellers' constraints yields
+ 2 / 5, +
a=
converging
in the finite
other buyers are in the market, and the
5,
interval of sizes,
.
side, utility
of y More fundamentally, even
two markets over an
B
2
1
2
this
/
S,
>
B
I
]
2
> 1 - 2 / 5, - 1 / S 2 so
,
that are consistent
model so
that
it
if
there
S goes
to infinity, the set
with equilibrium collapses to the knife-edge
had an equilibrium plateau,
would be necessary. For example, adding
the other buyers
that as
a
buyer
were congestion costs or
21
to a
if
a
a=
of ratios
1
.
To
buyer market-impact effect
market would reduce the
utility
of
firms had increasing marginal costs.
.
C. Preference for Variety II
Fujita [1988] develops a general equilibrium
a continuum of locations) in
products locally
is
offset
which
a preference for
by higher land prices
model of spatial agglomeration (with
being able to buy a variety of
in the city center.
A
greatly simplified
two-location version of this model with a sufficient amount of land per agent would tip to
having
all
It
consumers and firms
would
fail to satisfy
in
our assumptions for a couple reasons.
that the preference for the variety
x of each good
i
at a
price of
one location.
p
t
is
receives utility u h
function increases like log(S) as the agent
among
a larger
bound, the
set
number of goods
is
=
-x
purchases a small quantity
\og{x
i
i
)-x p
i
.
B - (1 / 2 + e)B
has the form u s (S,B) =
and the market impact
us (S2 ,B2 )-u s (S2 +l,B2 )
When utility
increases without
*2K(B2 )F
s
K(B) (Fs (y)-Gs (y)f B), then when
effect is
\y)/ B(l + 2e)
* 2(K(B/2) + sK \B I2))FS \y)/B(l + 2e)
and the scale effect
is
us (S2 ,B2 )-u s (S ,B
)«(K(B2 )-K(B ))Fs (y)^2eBK\B/2)Fs (y). Holding
1
}
diverges. This ratio
is
B/\og(B)
if
K(B) =
log(fl),
and
is
aB
if
B
2
if
he increases the land
rent. If
land
is
its
alternate use, agriculture,
assumed constant. Hence, there would be no market impact
impact
effect,
one could modify the model so
that
when moving
to a
sufficiently plentiful, then the
equilibrium rent would always be the value of land in
is
B K\B)/ K(B)
K(B) = B"
Second, the primary market impact that a firm or consumer has
is that
s fixed, the
]
market impact effect will be smaller than the scale effect for large
market
This utility
i
able to divide his or her consumption
constant price).
(at a
/,._
assumes
of interior equilibrium will typically collapse to a knife-edge. For example,
if the seller's utility
2
who
such that a consumer
First, Fujita
effect.
To
which
create a market
consumers and firms had
to outbid a
heterogeneous population of farmers for land.
6.
Conclusion
This paper shows that Assumption
plateau of quasi-equilibria.
Al
The assumption
is
sufficient for
market models
also yields an easily
provides a lower bound on the "width" of this plateau.
22
to
have a
computed formula
In our opinion.
that
Assumption Al
is
not very restrictive;
it
Krugman [1991b] and
fails are
when
in the
many
applies to
a
if
not most market models, including those of
two-type version of Pagano [1989]. The leading cases where
continuum
some agents
limit,
are indifferent about the ratio of
"buyers" to "sellers," and when per agent payoffs converge
number of agents goes
to zero or infinity as the
to infinity.
That said, the generality of Assumption Al and the simplicity of Theorem
been obtained by leaving our
sufficient conditions in
results
Theorem
1
incomplete
in a
couple ways.
quasi-equilibrium plateau
when one
is
First
have
1
of all, the
apply to quasi-equilibria with exactly the same
buyer/seller ratio in each market; this leaves open the question of
which the buyer-seller
it
how much broader
also considers the possibility of quasi-equilibria in
ratios differ. Ellison,
Fudenberg, and Mobius [2002] provides a
detailed analysis of the case of competing uniform-price auctions, in
purchases a single unit, and the price
the
in a
which each buyer
market with k goods for sale
is
the (k+l)st
highest buyer value.
Secondly, Theorem
does not reveal
when
1
concludes that the incentive constraints are satisfied, but
these constraints can be satisfied along with the constraints that the
numbers of each type of agent
addressed
in
We
each market should be integers; these constraints are
in
Anderson, Ellison and Fudenberg [2003].
have written
this
efficient there will typically
paper to emphasize that even
still
in
which
larger markets are
in
which there are no scale
effects,
larger markets give lower payoffs to both buyers
because of crowding
effects.
more
be a plateau of equilibria with two active markets. Our
assumptions also encompass models
models
when
One
implication of our theorem
crowding effects are no stronger than
is
is
and
and even some
sellers, e.g.
that as long as the
allowed under assumption Al there will be an
equilibrium plateau that includes splits in which
some markets
than others. Hence, the observation that an activity
is
are substantially larger
concentrated in a small
number of
locations need not imply that there are increasing rather than decreasing returns to
agglomeration.
23
.
Appendix
We
follow Pagano in assuming agents have mean-variance preferences over
wealth distributions and maximize their
first stage.
demand
we suppose
Also following Pagano,
D(p) that maximizes
function
preferences would be if they
when choosing between markets
utility
knew
that in the
in the
second stage, agents choose the
the expected value of what their conditional
the full vector of
endowments but did not know
the
random dividend.
Write Z_ = T]
;
K
0/
for the aggregate
Consider the possibility of an equilibrium
demand curve Dj(p) =
i's
opponents
is
AK 0/ -mp
in
;'
the
power
to
.
the ability to submit a
choose the quantity of the risky asset
25
conditional on every realization of Z_
If agent
.
receives allocation
i
_.
t
/
;
endowment shocks of agent
z's
he will receive
K
in state
Z_ then
;
t
Bi
implies that the price
+K -Kj/iN-^m. Hence, the expectation over the
0i
mean-variance
shock when he receives allocation
Ez
gives
i
AZ - (N - \)mp + K = Z_ + K
mustbeP(A^ ,Z_ .) = -((l-^)Z_,
the linear
demand curve
K^Z^) that
t
the market-clearing condition
i.
The sum of the demand curves submitted by agent
.
{
agent
other than
which each agent/ submits
AZ_ -(JV — Y)mp Thus,
then
endowment of all agents
K
(Z_
i
i
utility conditional
on the endowment
is
)
b
£(^,.(Z_J + /?(#0f -K,.(Z_,.^
2
where we have written ^(Z.^for
23
.P(/^
(Z^ ), Z_. )
Since the equilibrium prices are fully revealing, this would be
with standard expected-utility preferences. Mean-variance
V(w) =
Ez (V(w
|
Z))
,
so the assumed behavior
utility
utility,
maximizing behavior
not utility-maximizing;
is
for agents
however, does not have the property that
we
regard
it
as a "behavioral"
assumption.
4
!5
We will see that there is always an
This
is
true provided that
allocation
K
(Z
)
agent
i
equilibrium of this form.
K.(Z __) and (l-A)Z
.
~K(Z
)
are
monotone
can submit the downward-sloping demand curve
24
increasing.
To
obtain
)
To maximize
this
expectation over
expression within the brackets,
K
,Z_
)
i
i
-K )P(K ,Z_
= R(K 0i
I
I
—K a
2
)
I
2
.
(Z^
i
—
0i
)
,
one can maximize the
i
for each
{
The
RP(K r Z_ )-R(K - K,)—
l
K
utility inside the
dP
this expression is
K
maximize over
i.e.
expression for the conditional mean-variance
V(w\
functions
all
Z
.
For a fixed Z_ the
i
i
brackets simplifies to
first-order condition for
barK,
.
maximizing
Substituting
f'k
A',
+K -K
(\-A)Z
n
*-
for
nfV „
)
1
J
(N-\)m
l'Ul's
dK
i
i
we need
the derivation
to find the
D'(p) =
allocation and use the equilibrium condition
Writing^ for
the denominator in the equation for
corresponds with
K
demand curve
AK -mpio
0i
(Z_
I
I
)
produces
that
solve for/4 and m.
the price in state Z_
,
this
that
i
this allocation is
Q-A)Z_,+K -(2K +{l-A)Z_ )/X
_
-
+ {N-\)mba- !R
To complete
p{Z
dP
f
tor
and
;
2Kn ,+(\-A)Z_
2
N
P(A,,Z_
(N-\)m
K,(Z_,)
(Z.i
'
0i
0i
}
i
X{N-\)m
(N-\)m
The inverse of this function
^^
2-X
=
(\-A)Q-X)
z
X{N-\)m
is
X(N ~ l)m
iX ~ 2)
/*qo- (\-A)(\-X) *.,+ (\-A)(\-X) P
.
°'
From
Z>* (/>)
Note
X-2
= (!- A)P-\ P ) + K 0I +(N -\)mp =
that this
-1/(1- X)
is
indeed linear and
1
D(p) = P~
K(Z
it
\-X
.
X=N
(p){\ - A) +
.),
K
v
.
The
latter
+ (N - \)mp
,
implies
where
+K -K
P(Z_,) = -((l-/l)Z
i
ai
that
implements K'(Z_
K ,+
X
\-x
•+]
t
P
m
is
)
is
and only
solution to these equations
= ((N - 2) l(N - \))R I bo 2
if
is
.
the inverse of the market-clearing price consistent
(Z J)I{N -\)m.
25
I
(N-l)mp
satisfies the equilibrium condition if
= A and \/(\-X) = -\/(N -I) The
A = /(N - 1) and
with
demand curve
the expression in footnote 22, the
Agent
equilibrium asset holding as a function of the
;'s
2K + (l-A)Z_,
.
K ,{ K o) =
X
X
The equilibrium
K
+ A)K 0i + {\-A)Z
(\
0i
price
p (Z) = -
is
Agent f s wealth as
RpK
=
0i
(1)
N
N-l N
N-l
ba 2 Z
N
R
a function of the dividend
+KM- Rp) = -b°
-2)
d
N-l)
(N -
+
i
(N -
=
)
that
S-B
^—2-
N
E(Z 2 )
N
(
a'
7
\
(N
where
E
Uz
+
)
I
-2)ba
2)ba
2
)
,
—
so
J
N
.
,
N
.We
fo
±
ge
1)
TV
2=1
7-1
V
j
,17
2
(N -2)ba
)
\l~^
the "+"
now
i7
N
Oi'
kN
—
Y^
^—
- + ^' =1
-7
1
7+1
also have
+ Y.ti/N
1
2)6a
N'
2^
Oi
\N
TV
1
+ bo'
N-l
N
7
{d
2)
TV
N-l
N-l
where
(N -
Qi
±
"
7 7 +
1
1
the "+" corresponds to sellers. Hence,
\N
t
)
o„
'
E{w
N
Z = $ ~ B + ^'
} =1 e,
=
—1
7-1
7
N-l)
K d
(N-l) N
{
N-l
the variance of e
is
+
K,Oi
is
and dividends and endowments are independent.
is
(N -
7+1
+
2?
N'
N'
= 2—1
+ lj
l7
E(Z 2 )
1
j
-7
7-1
2
where
2
{N-l)
1
Note
2)ba
and endowment shocks
first
and variance of this expression.
Since the expected dividend
E(w
2
2)ba
agent maximizes in the
(N -2 Z
jfK0i + [N-1N
2
N-l
We now compute the mean
are
N-2Z
0i
We now compute the mean-variance utility that each
stage.
endowment vector
+
^
2
+
2
2N-2 °l ± 7~1
N-l (TV 7 + IJ
l
—
tV
1,
+ l;
X
2
"
+_7"11)
74 -1
corresponds to buyers.
26
.
N
,
e.
and
Now we
of four terms.
term
Var
is
need
compute
to
the variance of
w The
.
.
expression
(1
)
for w.
a
is
sum
(
Z and d are
—d
[7,
independent, and d has
72
=
hi
N
JV
K
C
-
T«
2
hi
— d\
7,
=
mean
0, so the
Z
X
2
E(d )E
\N
variance of the
2
1
2
first
7-1
N
h'
+
ij
Next,
t2\
=
Var
- E!li e
7-1
+ 2N
7 + 1
7 + 1J
y
Var
V
h-V
(7
The
third
+
trV
T~
N
1,
'
=
l/vj
1
a smaller variance,
>
V
v
;=i
TV
;
2
w="
_< Z ^(^;)
^-(e
;
;
The
first
N
random
covariance of the
J
2
a;
)
(
N
first
N
variable
and
is
N
2
^N ^ (—
uncorrected with the second and fourth, the
2
third is cr cr
2
2
/TV
,
the covariance of the second and fourth
because
N)
Z2
Z
N N
2
\
Var
2
TV
Tar
f
A^
kNj
\
Cov
I
'
•Far
A'
/
&=>
A
random variable has
fourth
V
o(l
'
1
.V
TV
Var
2
random variable has
Var
The
2
1
\
K
= ±Cov
(7
2
Z^
l^
')
^
= Cov
(z
^
N
2
z
(
+ Cov
NJ
Z
)
-^-e
N
)
2
—z —z a
N
r.
=
{N'-N',
+ Cov
(z
1
1
=
— Cov
v^ N) N
27
(7
2
Z + pL
,—
e
{N<
2
e \
+ Cov
(
{
—
Z2
A^'
,)
e,
)
\
N
)
=
rn
—
<N)
is
and the covariances of the second and fourth terms with the third term are o{X/
N)
because the products of the standard deviations are o(X/ N). Hence, the variance of w.
approximated to order XI
N-2
Var{w
)
i
f
d
N-X
{N-2)ba
2
^
(
first
N
a2a<
)
N
n.
72
(N-2)ba 2\
\
Var
N-
variances to order XI
f
by the sum of the variances of each term:
Var
Approximating the
Var(w
N
f
Var
t
UN, the other constants to order
l
,
>
\N)
and the
gives
y-V
+ *>
2\
+b
r +x
2
a4
J
a
2
+a
2
2
7-1
a
y+1
N
\\
f
2
a +4
7-1
~
+ b1.2 a
2
4
cr e
e
— + a- + o
4
/
i
]
N
N
Z-2_4_2
baa
+b a
,
7.2
4/
(a
2
— +o\n—
1
+a 4.
,
)
N
v7 + ly
i's
1
+o
°
N
N-l
k^j
constant to order
a' +
Agent
is
mean-variance
utility
before the
endowment shocks
\N
are realized
is
thus
7-1
{N - 2)6a 21 7-1
=
=
--Var(
E(w
±
V(«0
Wl
N-l
7 +1
.7 + 1,
z
6
)
)
'7-1]
2 .7
+
2
a
2
where
the
2 [7-1]
2
1,
(7-1
2 U + l
1
b
,
,7
+
±
7
6a
2
-
-L
+1
+ term corresponds
2
7-1'
+ a 1 a] + 4
17
1,
,
7-1
a
^- +
7
+
to buyers.
28
1
2
<x
+4
+ lJ
7-1
+ 1,
2
2
l'2
6
,7
+
a ae
cr
cr
e
a
1,2
+6
.
2
—
1
+a
[ae
e )
2
/
a (a
e
+
,
4
ae
N
-i
\
)
2N
+
References
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L.
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"
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2003
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