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WALRASIAN BARGAINING
Muhamet Yildiz, MIT
Working Paper 02-02
October 2001
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
WALRASIAN BARGAINING
Muhamet
Yildiz,
MIT
Working Paper 02-02
October 2001
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssm.com/paper.taf7abstract id=
XXXXX
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAR
7 2002
LIBRARIES
Walrasian Bargaining
Muhamet
Yildiz*
MIT
October, 2001
Abstract
Given any two-person economy, we consider a simple alternating-offer bargaining
game with complete information where the agents
conditions under which the outcomes of
all
We
offer prices.
provide
stationary subgame-perfect equilibria
converge to the Walrasian equilibrium (the price and the allocation) as the agents
become
infinitely patient.
Therefore, price-taking behavior can be achieved with
only two agents.
Key Words: Bargaining, Competitive
JEL Numbers: C73, C78, D41.
Introduction
1
It is
in
equilibrium, Coase Conjecture
commonly
believed that, in a sufficiently large, frictionless economy, trade results
an approximately competitive (Walrasian)
economy
allocation.
In fact, the core of such an
consists of the approximately Walrasian allocations. Moreover, in a bargaining
model with a continuiun of patient agents, Gale (1986) shows that we
petitive allocation in
In contrast,
monopoly
when
case,
have a com-
any stationary subgame-perfect equilibrium (henceforth, SSPE).
there are only two agents,
and the outcome
in that case, the core
will
is
will
it is
believed that
we
will
have a bilateral
be indeterminate (Edgeworth, 1881). Accordingly,
very large, and the
SSPE outcome wiU
typically differ
from the
competitive allocation in the usual bargaining models, in which the agents are allowed
*1
cially
would
like to
thank Daron Acemoglu, Abhijit Banerjee, Glenn
Robert Wilson
for helpful
the earlier discussions with
comments.
Murat
Sertel
I
Ellison,
Paul Milgrom, and espe-
also gratefully acknowledge that
when we have worked on the
1
topic.
I
have benefitted from
to offer any allocation.^ In this paper,
where the agents
discount rates approach to
petitive allocation
We
offer prices.
and the
1,
we analyze a simple two-agent bargaining model
show
SSPE
the
under certain conditions, as the agents'
that,
allocations
and the
prices converge to the
price, respectively. Therefore, the
com-
Wahasian equilibrium may
have stronger foundations than commonly thought.
Considering a pure-exchange economy with only two agents, we analyze the following
alternating-offer bargaining
Agent 2 either demands a
game with complete
is
reahzed, and the
when Agent
is
offers
2 offers
a
game
a price.
he demands
ends. If he rejects the offer,
price,
and Agent
either
1
we
demands a
This goes on until they reach an agreement. Each
feasible trade or rejects the offer.
agent's utility fimction
1
feasible trade at that price, or rejects the offer. If
a trade, the demanded trade
proceed to the next date,
Agent
information:
normahzed by
at the initial
setting to
endowment. Agents
cannot consume their goods until they reach an agreement, and each discounts the future
with a constant discount rate.
Notice that
we
restrict the
proposer to
and allow the other agent to
offer prices,
choose the amoimt of trade at that price. For instance, in the case of wage bargaining,
how much
the luiion sets the wage, the firm has right to choose
if
labor to hire. Likewise,
the firm sets the wage, the workers or the union have the right to choose the
of labor they provide.
offer
In contrast, in the standard models, the proposer
any allocation, while the other agent can only accept or
if
is
amount
allowed to
reject the offer.
This
is
the only major difference.
When
the agents are restricted to making price offers, while their trading partners
are allowed to choose the
emerges.
To
price, the
game
amoimt
see this, consider an
ends, hence he
of trade at the offered prices, price-taking behavior
economy with two goods. When an agent
demands the optimal trade
this restricts the other agent to offer a payoff vector
payoff vectors that can be achieved
rates are close to
1,
in
any
SSPE
when
in
z
i
accepts a
at that price. Effectively,
the offer curve of agent
a price-taker. Moreover,
is
which
on
for
i
when the
i,
the
discount
the offers are accepted, both agents are
all
approximately indifferent between the allocations today and tomorrow. Note that the
payoffs associated with these allocations are
Therefore, the payoffs in such a
'For instance, as the
common
on the
SSPE must be
offer curves of
two
distinct agents.
very close to an intersection of the offer
discount rate approaches to
1,
the
SSPE outcome
in
Rubinstein (1982)
converges to the Nash (1950) bargaining solution (Binmore, Rubinstein, and Wolinsky, 1987), which
different
from the Wahasian outcome, as noted by Binmore (1987). (Rubinstein's model
and abstract, but the
set of feasible payoffs is typically
allowing the agents to offer any allocation.)
is
is
more general
taken to be the set of materially feasible payoffs,
curves.
many
payoff- vector
is
at such
an intersection. Moreover,
in
canonical economies with a unique Walrasian equilibrium, the Walrasian payoff-
vector
for
Of course, each Walrasian
is
the only intersection (of the offer curves) in the relevant region.
such economies, the payoffs in these
payoffs. Moreover, the
— as we have
SSPE must
Therefore,
be converging to the Walrasian
Walrasian payoffs are obtained only at the Walrasian allocation
By
strictly quasi-concave utihty fimctions.
continuity, this
shows that the
allocation at these
SSPE
utility fimctions are
continuously differentiable, there exists a continuous inverse-demand
also converge to the Walrasian allocation.
function, therefore the prices in these
We
further
show that
monopoly: he must
this to
when
the
also converge to the Walrasian price.
any other possible SSPE, an agent must act as a natural
in
offer his
be an equilibrium
SSPE
Finally,
monopoly
price,
and
must be accepted.
it
for large values of discount rates, the
monopoly outcome must
be Pareto-optimal under the constraint that one of the agents must be a
Ruling out the case that a monopoly
is
efficient in this sense,
In order for
price-taker.
we conclude that
all
SSPE
allocations (and the prices) converge to the Walrasian allocation (and the price) of the
static
pure-exchange economy at hand.
This limiting behavior
where agents can
offer
is
strikingly different
any allocation. In that model,
agents share the gains from trade in the limit
make
agents
1).
offers, or
from that of the Rubinstein (1982) model
is
for a quasi-hnear
solely
economy, how the
determined by the frequencies the
the logarithmic ratio of the discount rates (as they approach to
All these are irrelevant for our limit, the Walrasian allocation of the static economy.
Therefore, our small change in the allocation of the rights (embedded in the bargaining
procedure) has a profound impact on the bargaining outcome, while the bargaining
outcome under the new allocation of
Oirr result
is
rights is very robust to other procediual details.
related to the hterature
on the Coase Conjecture. In the context
durable-good monopoly, Coase (1972) argued that,
price,
then he
will price his
good at the marginal
if
a monopoUst cannot commit to a
cost.
(For the consimaers expects the
monopoUst
to lower the prices in the future. In a sense, the monopolist
peting with
its future-selves.)
would be com-
In an asymmetric information model, Gul, Sonnensrhein,
and Wilson (1986) and Fudenberg, Levine, and Tirole (1985) have shown that
jecture
it is
is
true in any
SSPE,
as the discoimt rate approaches to
essential that the monopolist does not
valuations were
ation.
Our
common
know
Now, there
1.
this con-
In this formalization,
the buyers' valuation. If the buyers'
knowledge, the monopolist would charge each buyer his valu-
result extends Coase's notion to a two-person
information.
of a
are
two agents (and
economy without asymmetric
their future selves) setting prices.
The
outcome becomes Walrasian as the discount
There
rates approach to
1.
a large literature on the relation between the bargaining outcome an the
is
competitive equilibria (see Osborne and Rubinstein (1990) for a survey).
of
my
To
the best
knowledge, this literature assrunes a very large economy, except for Rubinstein
and Wohnsky
Rubinstein and Wolinsky (1990) obtain a similar result to that
(1990).
of Gale (1986 and 1987) with only finitely
economy
special
in the
theorem
is
we
unrelated to the competitive
is
In Section
true.
we explain why
lay out our model. In Section 3,
standard model
still
agents. However, they consider only a
which the core consists of the competitive allocations.
in
hi the next section,
outcome
many
relegated to the Appendix.
we
4,
SSPE
outcome, and why our
derive our result; the proofs of
Section 5 contains
the
lemmas
are
some coimter examples that show that
our assiunptions are not superfluous. Section 6 concludes.
Model
2
X
Let
=
p
= R"
(p\
.
>
p''
.
.
G
,p")
for all
where Good
IR"^.
k?
^ N. Assume that
«, (xi)
We
=
is
will
=
for
be the
aU price vectors
set of
taken to be the numeraire,
i.e.,
((ui,Xi)
,
where
(u2,^2)),
x^
of agent
i,
G
X
and
Ui
:
X
^f
R
each
i
N=
E
{1,2}. Write lu
=
Xi
+ X2.
wish to understand the relation between the Walrasian equilibrium of e and the
the set of
all
At date
dates.
demands a consiunption
that X2
is
feasible
is
X2
t
E {x
=
Agent
0,
E X\{x — X2)
game ends
At
G {x G X\{x
vector [6\ui (xi)
,
1
offers
-pi
=
Let
T=
{0, 1, 2,
0,x
< w]
—
^
=
Xi)
6\u2 [w
Agent
1,
p2
—
We write R for the set of real
=
0,x
—
X2,X2)-
2 offers a price p2,
<
numbers, M"
.}
.
be
either
or rejects the offer. (Note
—
pi.)
If
X2) ,6\u2 (3:2)),
she rejects, we proceed to
and Agent
1
either
demands
w], when the game ends yielding the payoff
Xj)), or rejects the offer,
space, K",,. for the interior of W\..
If
.
Agent 2
a price pi G P.
yielding the payoff vector [6\ui {w
associated with the allocation {w
the next date.
offering prices.
game G {61,62),
and can be reached by reallocating the endowments at price
she d .mands X2, the
^
and
Ui is strictly quasi-concave, continuous, monotonically-increasing,
where these two agents bargain altematmgly
Xi
1,
respectively, for each
stationary subgame-perfect equihbria of the following perfect-information
which
=
p^
simply say prices instead of price vectors.)
endowment and the utiUty function
are the initial
and
1
we
(Henceforth,
Consider a 2-person economy e
i
P
be a commodity space with n goods, and
for
when we proceed
to the next date,
the non-negative orthant of a n-dimensional Euclidean
when Agent
ment.
they never reach an agreement, each gets
If
€
{61,62)
Any
a price again. This goes on indefinitely until they reach an agree-
offers
1
approaches to
(0, 1)
(1, 1).
stationary subgame-perfect equilibrium (henceforth,
hst (Pi,ii,p2,i2) where, for each
whenever he
make an
to
is
^ N,
i
and
offer,
Xi
:
rejects the offer
if
Xi {pj)
=
P
—y
:
Di
at each
(p)
=
p E P. Since
Since w,
function.
allocation (xi, X2)
Assumption
X
[J
{Reject}
agent
i
and x*
is
Together with
arg
max {u^
u^ is
For each
{x)\{x
—
p <
x^)
0,
x
<
any pair
is
=
Xi
Di
(p,
i
is
made
in the interior of
(p) for
X for each
strict quasi-concavity,
only to
X}
w, x E
Di
each
i.
i
make
is
each
i,
and xi
-|-
also continuous.
X2
=
p E
P
A
and an
w.
{p* {xl, X2)) ; (xj,X2) 7^
,
E N.
the assumption that {xl,X2)
In that case, the
a weU-defined
is
(xi,X2)) of a price
initial allocation is
hence there are gains from trade. The assimiption that x*
each
€
E N, define the (constrained) de-
i
Maximum Theorem,
continuous, by the
for
offers
Xi {pj) if Xi {pj)
continuous and strictly quasi-concave, Di
E X^ such that
>
tees that Ui (x*)
i
the function that
is
demands
There exists a unique Walrasian equilibrium
1
represented by a
the price that the agent
is
pj, the
is
X by setting
Walrasian equilibrium
(constrained)
(xi,X2),
is
-^
SSPE)
Reject.
Basic Definitions and Assumptions
mand function Di P
P
E
pi
determines his responses: when offered a price
X, and
We are interested in the case when
0.
sure that (p*,(x^,X2))
is
is
7^
(xi,X2) guaran-
not Pareto optimal,
in the interior of
X
for
an "miconstrained" Wahasian
equilibrium.
Define the offer curves of Agents
OCi =
1
and 2 as
{(ui {D, {p)),U2 {w
- D,
{p)))
\pEP]
and
OC2 =
respectively.
OCt
the set of
all
who wiU then maximize
agent
i,
Vi for
an agent
agent j
is
{(«i [w
is
i,
- D2
{p)),U2 {D2 (p)))
utihty pairs that can be reached by offering a price to
his payoff given the price. In general, given
there might be multiple pairs (^1,^2) in
to choose
\pEP},
OQ.
between these pairs by offering different
In that case,
prices,
he
will
any payoff
if
the other
choose the
.
maximum
pair with
To formulate
Vj.
functions Ui and
this, define
f/2
(on appropriate
domains) by
Ui {V2)
=
max{?'i|
U2
=
max{i.'2| (^'i,f'2)
e OC2}
(t'i,t'2)
and
satisfies
is,
=
=
{ui (xj) ,U2 (3^2))
=
U2
{ui (xj))
Ui {u2 (X2)) and U2 {xD
in
OCi
(1
(1)
payoff'- vector.
For our main
result,
we
will
assume that
this
the only intersection in the relevant region (see Assumption 4 below).
Throughout the paper, we wiU make the following assumption, which
many
Since U^
some
is
satisfied
by
economies, such as the Cobb-Douglas economies in the Edgeworth box.
Assumption 2 For
at
is
the graphs of Ui and U2 (which are typically the offer cvirves themselves) intersect
each other at the Walrasian
is
v*
the equations
ui (xj)
That
e OCi},
Note that the Wabrasian payoff- vector
respectively.
OC2, and
(vi)
Vj
any
is
each
>
Uj [Dj {pi))
Vj.)
=
E
N
,
and
Ui is continuoxis
single-peaked.
single-peaked and cannot be monotonically increasing,
G R. (Note that
Vj
i
Vj
f/j
is
strictly increasing at
By a monopoly price of an agent
and u^ {w - Dj {p,)) = U^ (vj).
i,
it is
maximized at
< Vj and strictly decreasing
we will mean any price p, E P with
any
Vj
An Example
3
In this section, using a canonical example,
we
will explain
our formulation, and show
how
the Walrasian outcome typically differs from the subgame-perfect equilibriiun (henceforth
tions.
SPE) outcome
We
will
in Rubinstein's
then explain
model where the agents bargain by
why our theorem
Consider a quasi-linear economy e
=
"1 {'m, y)
m + y" —
1,
U2 {m, y)
quasi-hnear utihty functions,
the
first
m
will
-f-
true.
{{ui, (0, 1))
y°
,
(^2, (Af, 0)))
- M, a €
take A"
=
IR
(0, 1),
and
with two goods where
M>
0.
Since
we have
x IR^, allowing negative amounts of
good — money.
For a
single
we
=
=
is
offering alloca-
=
0.5,
the offer curves are plotted in Figiire
peaked and continuous. The
offer
curves
1.
Notice that both Ui and U2 are
OCi and OC2
are simply the graphs of
\J.t
^
-7x
f
1
/^^S^
0.35
N
0.3
\
1
.
V
[Walrasian payoffs]
\he-
-
^sX
M
/"'
Vv\
\\^^\
(
^^iW
0.25
1
'
1
I
1
I
1
\ ^
\
>:^
\
\
j
\
^2
f
^C
2
^\
^\
^v
V
\.
/
y\
-
U
5
2
/
Pareto Frontier
/
0.2
^^^\^
/
/
/
y
1
0.15
y\
0.1
,
5
\
U
\
\ \^
0.05
/
V
1
0.05
1
1
0.15
0.1
Figure
1:
1
1
0.25
0.2
Offer curves for
q
1
=
1/2, 6i
0.3
=
^2
=
0.9.
N
X
1
0.35
0.4
U2 and
L'^i,
The Walrasian
respectively.
where the
payoffs are at the unique point
offer
curves intersect each other. Notice also that the Walrasian payoffs are very asymmetric.
One can compute
that the Walrasian price
is
p*
=
(1,
a2^~°), yielding the Walrasian
payoff vector
+Q
1
—a
1
2°
'
2^
Hence, at the Walrasian equihbrium, the ratio of Agent
—
(1
q) /
On
(1
+Q—
model
As
share to Agent
I's
share
is
determined by a.
2"),
the other hand,
cation, the payoffs
2's
we have a
add up to 2^~°
transferable utihty case: in any Pareto-optimal allo-
—
Then, the imique
1.
SPE outcome
in Rubinstein's
is
=
(^1,^2) -^ (1,1) at the rate r
log
((5i)
/log
(62),
Therefore, in the limit, the ratio of Agent 2's share to Agent I's share
by the discount
solely
rates, or the frequencies at
well-known fact imphes that, in the hmit, the
SPE
is r,
which the agents make
payoffs in Rubinstein's
determined
offers.
Tliis
model carmot
possibly be related to the Walrasian payoffs, which are determined by a. In particular,
when
there
is
a
common
discount rate, in the hmit, the
SPE
distributes the gains from
trade equally, while the Walrasian payoffs are very asymmetric.
Now
that,
Di
consider the bargaining procediu-e in
whenever an agent
[pj )
—
for the
G {61,62)- Lemma
accepts a price pj, he
i
game ends
there,
as the Rubinstein's bargaining
1
below estabhshes
demands the optimal consimiption
and thus our bargaining procedure can be considered
model where each agent
is
restricted to offer payoffs
the other agent's offer curve. Therefore, as in Rubinstein (1982), there
is
determined by the intersection
1).
In this SPE, an agent
i
{61,62) of the graphs of 61U1
accepts an offer
agent j offers a price that gives
is
v^
v]'^ {61,
iff
62) to
he gets at
i
is
and 62U2
least v]^ {61, 62),
and Uj [v^
a
SPE
on
that
(see Figure
and the other
{61,62)) to j. But, since v*
the unique intersection of the graphs of Ui and U2, as {61,62) -^ (1,1),
v"^ {6^,62) -^v*.
In the limit, the
rate at
SPE
payoff- vector itself
which the discount rates go to
1.
is v*.
This convergence
is
independent of the
Tliis also suggests that this
8
convergence to
the Walrasian equilibrium
may
hold even
the agents would
if
make
offers at different
frequencies.
As one can
easily see
from Figure
1,
the changes in the agents' discoiint rates do
change the bargaining outcome — in a predictable way.
When we
increase the discount
rate of an agent keeping everything else constant, his equiUbrium payoff
(if
anything)
increases, while the other agent's equilibrium payoff (if anything) decreases, provided
that there
a unique SSPE.
is
Notice in Figure
U2, respectively.
1
and
that [vi, U2 (^1))
That
is, if
an agent
(t/i (€'2)
,
C'2)
monopoly
offers his
of discount rates) the other agent can reject that offer
counter-offer.
Under
price,
then
property of Ui and U2 hold,
all
when
SSPE
the
and
(for large values
and make a Pareto-improving
this condition, for large values of discoimt rates,
there cannot be other SSPE. Therefore,
this
are below the graphs of Ui
this condition
we show
and the rmique
that
intersection
converge to the Walrasian outcome
— as
in
example.
Theorem
4
In this section,
we
will describe the
sufficient conditions
SSPE
of
G {61,62). We
game
under which the allocations and the prices at
the Walrasian allocation and the price, respectively, as {61,62)
Our
first
Lemma
describes the basic properties of
Given any 61,62 G (0,1), any
1
G{6i, 62),
lemma
under Assumption
e {Dj
Xj {pi)
2.
if
Xj {pi)
3.
if
Xj {pi) y^ Reject, then
Part
1
^
,
Reject, then Ui {w
states that,
at that price, for the
if
u-j
- Dj
{Dj
game ends
—hence the second
part.
SSPE
converge to
(1,1)-
SSPE.
E N, any
SSPE
(pi,
£1,^2,^2) of
{pi))
G P;
{pr))
>
there.
=
Ui {uj {Dj (p,)));
Vj.
price,
he demands his optimal consumption
Hence, in terms of equiUbria, our game
game where each agent
in the other agent's offer curve.
of Ui
all pi
an agent accepts a
equivalent to a bargaining
j
then provide
the following are true:
Reject} for
1.
{pi)
2,
^
i
—>
all
will
is
restricted to offer a payoff vector
In that case, each agent
The proof
is
i
offers
a point on the graph
of this part uses the continuity of
the availability of the prices that allow the other agent to
demand consumptions
t/,
and
better
than his equihbrium demand. Part 3 simply states that each agent's
offer is at least as
generous as his monopoly price.
Our next lemma
some necessary conditions
lists
towards proving our Theorem. The basic argument
2, if
i
an agent j
offering a price pj that will
is
SSPE. This
for a
is
is
the main step
the following. Under Assimiption
be accepted and that allows the other agent
to obtain a higher payoff than his continuation value, then pj
must be a monopoly
price
of j. For, otherwise, j would offer a less generous price that would be accepted and would
yield a higher payoff for j. This
imphes that either
Lemma
1
either
(i)
2 Under Assumptions
(i)
for
or
G
i,j
all, distinct
for any
=
some
distinct
(2)
= 6,u,{w-D,{p;)) =
i
(4)
(5)
Each equilibrium-offer
(3), similar to
in Rubinstein (1982).
At
61
=
62
is
accepted,
between accepting (and
offers leave the other agents indifferent
demanding the optimal consumption) and
the equation system
(3)
= A(P,),
= V,.
describes a type of equilibrium:
and the equihbrium
6,U,{u,{D,{p,)))-
and j such that
u,(A(ft))
(i)
G {81,82),
{p\,Xi,p2,X2) of any
D,{p,),
x,{p,)
Condition
SSPE
below must hold.
(ii)
A'',
u,{D,{p,))
there exist
2,
or
true:
(ii) is
x,{p,)
(ii)
and
(i)
This indifference yields
rejecting the offer.
the equation system that characterizes the equihbrium
=
1,
this equation
system
is
identical to (1), the
system
of equations satisfied by the Walrasian prices.
Condition
who always
(ii)
describes another type of equihbrium:
offers his
agent's offer
is
monopoly
price,
typically rejected. This
which
is
is
Now, there
when
j
is
an agent
i
accepted by the other agent. The other
an equihbrium
iff
there
curve of j that gives each agent at least the continuation value
of payoffs
exists
a monopoly. In that case, the other agent
that must be accepted by the sequentially- rational agent
is
no point on the
offer
— the discounted value
{i)
cannot
offer
any price
j?
'in a bargaining model where the agents bargain over trades (rather than prices), an agent can
always offer the trade that will take place in the next date, guaranteeing the continuation value to each
agent, but this
is
not true in our model.
10
Lemma
3 Under Assumptions
if (ii)
Lemma
of
2
i.e.,
[w
- Dj
(p,))
the inequality 6^U^ {6jUj [vi))
We
will
ruling out
now assume
all
Assumption
is,
SSPE
for any
{pi,Xi,p2, X2) of any
P
>
,
>
Uj [Dj (p,)))
3 For
{v^,6jUj [vi))
(6)
,
does not hold.
v^
that (6) holds for
all i,j
an agent
i
G N,
G [61,62),
such that
some
when
price in the limit
the equilibria of the second tjrpe for large values of 61 and
Pi
E
Uj {Dj {p^)))
>
(l*„
- Dj
(p^))
can
offer
a price that
,
P
some
there exists
(W
[U,
That
2,
then there does not exist any Pt E
is true,
{6^u,
and
1
=
62
=
1,
62-
such that
Uj
{Vi))
better than the
is
61
.
monopoly
price of the
other agent j for both of them, provided that j maximizes his payoff given the price
offered
by
z.
monopoly
In other words, any
that the agents trade through prices.
By
is
Pareto-inefficient even under the constraint
continuity, this assumption implies that (6)
holds for large values of {61,62)-
Lemma
(5,1),
4 Under Assumption
and for
all i,j
G
3,
some
there exists
we have 6iUi{6jUj
A'',
E
6
(vi))
(0, 1)
>
Vi,
such
that,
i.e.,
there exists Pi
for
all
61,62
G
P
E
satisfying (6).
The
following
Lemma
is
an immediate corollary to
2, 3,
5 Under Assumptions 1-3, there exists some 6 G
(5,l), for all
SSPE
{pi,Xi,p2,X2) of
x,{pj)
Ui
The
Lemmas
(A
{Pj))
G {61,62),
=
D,{pj),
=
6^Ui
and for
and
(0, 1)
all
4:
such
{W - Dj
{pi))
There exists a unique (^1,^2)
=
1^1
all
61,62
G
(7)
=
6^U^ {Uj {Dj (p,)))
and
intersection in the relevant region.
Ui {V2)
for
G N, we have
distincti,j
following assumption guaranties that the graphs of Ui
Assumption 4
that,
>
and U2
11
(^1,^2) such that
{vi)
=
V2.
.
f/2
(8)
has a unique
At the Walrasian eqmlibrium, each agent
6 Under Assumption
4,
U, {u, [D, (p,)))
gets at least
we have the
U2 intersect each other. Therefore,
Lemma
i
and the graphs of Ui and
Vt,
lemma.
following
given any pi,p2 £ P, we have
=
u,
(A
{Pj))
>
^jeN)
(Vi
vr
This gives us our main theorem.
Theorem
{61,62)
Under Assumptions
1
1-4, if (pf ,xf,p2,X2) is a
SSPE
of
G {6)
for each 5
=
e{0,lf, then
1™,^*(A0=^;
(9)
*-»(!, 1)
for each distinct i,j G N. Moreover,
lim
That
if
ui
p*
=
and U2 are continuously
lim
pi
as the discoimt rates approach to
is,
= p\
1,
(10)
imder Assumptions
allocations converge to the unique Walrasian allocation.
continuously differentiable, this implies that the
rasian price.
Notice that the
Assumptions
1-4,
A = (A
f
:
=
[0, l]'^
:
D2
(F))
A -^M?
Xl, X2)
and the correspondence ^
i{6^M) =
prices also converge to the
existence of
Wal-
SSPE. (Under
n {{xi, X2) G
X^IVz G
A'',
Ui (x,)
>
v^}.
Define
6^Ui {Uj {Xj))
-
U, (x,)
[l
^
j
^ N)
-^ 2^ by
{{Xx,X2) G^|/(<5i,(52,Xi,X2)
=0}.
6,
Moreover, since /
is
SSPE
hy
C(l,l)
that ^
the
the utility functions are
If
Theorem does not claim the
(F) x
{0,lf x
f^ ((5i, 62,
By Lemma
SSPE
1-4,
one can easUy show that there does exist a SSPE.)
Proof. Define
also the function
differentiable, then
is
=
{(x*,xa)}.
continuous, ^ has a closed graph. Since
upper semi-continuous. Hence, given any
that, for each 61, 62
>
6, for
each
(xj, X2)
G ^
(^i, <52),
||X,-X*||<6.
12
e
>
and
0,
A
is
compact, this imphes
there exists b
for
each j G
A'',
G
(0,1) such
we have
(11)
On
the other hand, by
e
(61,62)
Lemma
there exists a
5,
<5
G
such that, for each S
(0, 1)
(6,1)2,
(f*(p^),x*(p*))G {(61,62).
Therefore, by (11) and (12), given any
each 6
=
(61,62)
G
(6*, 1)^,
we have
Towards proving the second
e
>
part, define
> max{6, 6}
there exists 6*
0,
-
||x^ (pf)
(12)
<
x*||
X=
proving
e,
{x G X|0 <
P
if Uj is
exists
x''
<
w'' \/k
any
X* [pf) -. X*, by continuity, pf
An
intuition for the
= D;^
Theorem
maximizing his utihty at the price
utihty at the price set by Agent
rates converge to
similar,
is,
D~^
continuously differentiable, then inverse-demand function
and continuous, where Di {D~^
1,
[x] [pf)) ->
set
by Agent
2,
The markets
will
become
approximately, each agent
is
i
indifferent
D;^
(x*)
same
by
distinct
XnD^
i,jE
{P)
set
-^•
(P). But, since
1
must be
and Agent 2 must be maximizing
definition.
by
different agents will
j
his
As the discount
by
become
between today and tomorrow.
Therefore,
i.
n}, the set
between he himself maximizing
indifferent
price set
XnDi
:
<
In equilibrium. Agent
clear
for
= p\ m
by the other agent j and the other agent
at approximately the
G
for each Xi
'^i
one would expect that the prices
and each agent
at the price set
=
the following.
is
1.
i^i))
such that,
(9).
of consumption bundles corresponding to the interior allocations. For
N,
=
That
his utility
maximizing her payoff
we must be approximately
at a
Walrasian equihbrium. (The logic of our proof is clearly difTerent from this intuition, for
proving convergence to a Walrasian equilibrium turns out to be more straight-forward
than proving that the prices converge to the same
What
is
price.)
the essential aspect of the Walrasian equihbrium
—
is it
competition or price
taking? In the tradition of Coase (1972), but relying on different forces, our
suggests that,
later
may be
when the goods
are diirable, the competition between the selves
sufficient for the price-taking
Theorem
now and
behavior to emerge, provided that the agents
are restricted to trade using prices as in this paper.
In sequential bargaining models, the outcome
and
offer
when.
traditional
rights
find this a
determined by which agent makes
weakness of the model and adhere to the
view that the bargaining outcome should be determined by the property
— not by the
Harsanyi,
it
Some authors
is
1977)."*
details of the procedure (see
Aumann
Our theorem has two imphcations on
(1987),
this issue.
Nash
On
(1950),
and
the one hand,
establishes that a change in the procedure (that the proposers are required to offer
who develop a bargaining model with endogenous timing of offers,
who analyze the equilibria of the continuous time game that are not
"See also Perry and Reny (1993),
and Smith and Stacchetti (2001),
determined by the temporal monopoly of the proposer.
13
prices rather
than allocations
wliile the
responders are allowed to choose the size of the
trade) has a profound effect on the outcome.
On
rights matters.
Therefore, the allocation of procedural
the other hand, the limiting behavior under the
procedural rights does not depend on
can easily show that
it
how
new
allocation of the
the discoimt rates approach to
would not depend which agent makes an
that each agent makes offers sufficiently frequently.
offer
1.
Hence, one
when, provided
Therefore, such procedural details
do not matter for the limiting behavior under the new allocation of procedural rights.
Early mechanism design literature had an interest in implementing the Walrasian
Nash environment
allocation in a
Theorem
(see
Our
Hurwicz (1979) and Schmeidler, 1980).
establishes that our simple bargaining procedure approximately implements
SSPE under
the Walrasian allocation in
endowments
certain conditions, provided that the initial
known by the mechanism
are
designer.
Along with the simphfying assumptions
monopolies are
inefficient
and
1
our Theorem assumes that the
2,
even under the constraint that one of the agents must be a
price-taker (Assumption 3)
and the graphs of
the relevant region (Assumption
4).
f/j
and U2 have a unique intersection in
Under these assumptions
estabUshes that
it
SSPE
outcome converges to the Walrasian outcome. Are these assumptions superfluous? Are
there non-stationary subgame-perfect equilibria? These questions are answered in the
next section.
Counter-examples
5
We
will
there
now
may
present two examples.
SSPE
exist
example shows
first
as described in condition
to the Walrasian outcome.
It
also
to the Walrasian equilibrium.
equilibria,
The
based on the
SSPE
(ii)
of
that,
Lemma
2,
if
Assumption 3
fails,
which do not converge
shows that there may be multiple SSPE that converge
Hence, there
may be
non-stationary subgame-perfect
that converge to the Walrasian equihbrimri.
example shows that, when Assiunption 4
fails,
SSPE may
The second
converge to a non- Walrasian
outcome.
Example 1 Consider the economy e =
2-ayi-Q _ go
^^^ ^^ (x, y) = x^~^y^ — 9^.
a=
0.6,
P=
0.1,
and
61
=
62
=
((uj, (9, 1))
0.9. Firstly, notice
Hence, for large values of (^1,^2), we have a
iff
ui {Di [p))
>
In Figure
61U1 {V2) and X2
(p)
SSPE
= D2 (p)
14
iff
,
(u2, (1,9)))
2,
that Ui
we
plot the offer curves for
(C'2)
> max {vi\U2
(vi)
>
0}.
=
Di
{p)
{pi,Xi,p2,X2) where Xi (p)
U2 (^2 (p))
=
where Ui{x,y)
>
0.
(Recall that p^
is
the monopoly price of
monopoly
and
price pi,
and that gives
i.)
it
In
is
tliis
SSPE, Agent
There
accepted.
positive payoff to
Agent
wiU be rejected. Second, observe that,
emerges as a monopoly: he
1
is
1
liis
would accept
so he offers the non-serious price p2, which
2,
=
for 61
62
two distinct points v and
intersect each other at
no price that Agent
offers
=
v',
0.9,
the graphs of SiUi and 62U2
two more SSPE. Both of
yielding
these equilibria converge to the Walrasian outcome.
Figure
The
following example
is
2:
The
offer curves for
Example
1.
taken from Sertel and Yildiz (1994). Using this economy,
they show that there cannot be an axiomatic bargaining solution that always pick the
Walrasian payoff- vectors.
graphs of
and U2
C/i
Here,
we
will
use the same economy to show that,
intersect each other at non- Walrasian payoff vectors, the
if
the
SSPE can
converge to non- Walrasian outcomes.
Example
2 (Sertel
where u{x,y)
=
and Yildiz, 1994)
Consider the economy e
(1/45) min{24x4-37/-f- 15,9x
-I-
18y,4x
-I-
23y
-I-
5}
=
((«, (0, 10)), (u, (10, 0)))
-1. The
indifference
curves and the offer curves (in utility space and in the Edgeworth box) are plotted in
Figure
3.
The
the diagonal,
indifference curves are tangent to each other only at the strip
when
the
common
slope of the indifference cin-ves
unique Walrasian price in this economy
vector V*
=
(4, 1).
In the
is
p*
=
Edgeworth box, the
15
(1,2), yielding
offer
is
aroimd
—1/2. Therefore, the
a unique Walrasian payoff-
curves of Agent
1
and Agent
2 are
-'
2'-''
~^^^^^~~^
^^^/
\\/Walras/
80U:
/v\
\ \
\cy
'
-^^ip*
/^
1
~
~
~
""-----V2 \pi
•
\"
9'/
h
X"
------.':
'
Figure
3:
4
The
offer
—
curves for economy e of Example 2
5
Ui
the Edgeworth box and in
in
the utihty space. (The hyperplanes are indicated by the associated prices.)
the hne segments connecting the points
But
[x, a, b, 6', c, d]
in the utihty space, the graphs of Ui
and Ui
{V2)
=
5
—
V2-
Then, the
the transferable utility case.
SSPE
For each
and U2
in this
^1, 62
and
coincide:
economy
€
(0, 1),
is
=
and
5
—
vi
as in Rubinstein (1982) for
62^/2
SSPE
—
— with payoff vector
—
SSPE payoffs converge to (5/2, 5/2), distinct
1). The SSPE prices p* and pj converge to
the
from the Walrasian payoff- vector v*
=
(4,
let 5
=
<52(l-<5i)
1,
5
(^i)
1-^2
1 — S162
+
^2
we have U2
respectively.
there exists a imique
determined by the intersections of the graphs of 61U1 and
When we fix 61 =
[x, a', b, b', g, h],
1
—
6162
=
(1,29/37) and P2 = (1)7/11)) respectively.
Pi
Even though the prices pi and p2 are distinct, the equilibrium allocation
is
Pareto-
optimal in the hmit, for the utility functions are concave but not strictly concave.
can shghtly modify the
utility fimctions in this
utihty functions, yielding the
same Walrasian
example to construct
payoffs
strictly
One
concave
and the same SSPE. In that
case,
the allocation in the hmit would be Pareto-inefficient, but there would not be a Pareto-
improving trade that can be achieved through price taking behavior.
16
.
Conclusion
6
Consistent with
common sense,
allocation in a large
economy
the
SSPE in usual bargaining models yield the Walrasian
(Gale, 1986), while they typically yield non- Walrasian out-
comes when there are only limited niunber
are allowed to offer any trade
among
of agents. In these bargaining models, agents
we
the bargaining parties. Here,
present a simple
bargaining procediu:e where the agents are restricted to offer prices, while their trading
partners optimize at these prices.
SSPE
the
of this
mechanism
As the discoimt
yield the
rates go to
1
,
under certain conditions,
Wahasian equilibrium. Therefore, the Walrasian
equilibrium does not necessarily require a large economy.
simply corresponds to price-
It
taking behavior, which can be achieved even with only two agents.
A
Lemmas
Proofs of
This appendix contains the proofs of the Lemmas.
[Lemma
Proof.
1]
hence we must have Xj
(Part 1) If Xj
(p,)
In order to prove Part
some p E
P
= Dj
2,
we
(pi)
Xj {pi) j^ Reject,
Assume that
i
will first
show
offers pi
,
it
Ui
{Dj
Xj (p,)
i.e.,
>
{pi))
pi
node,
Reject, then there exists
<
=
u, [D, iP,))
Dj
{pi). If
{pi)
^
>
0.
(13)
Uj {Dj {pi))
and
Xj,
=
w—
= p*
0, pi
Dj
^
{pi)
satisfies
^
x
{w
- Dj
Now,
Xi.
must be that
{w
- D,
(pO)
(14)
the Separating-Hyperplane Theorem, there exists a price pi such that Di
Xi.For any
>
^
that, if x^ (pi)
Hence, Dj
0.
u,
By
final decision
such that
Assume that
since
then we are in a
(p,).
u, {D, (pO)
(13).
^ Reject,
Xi
with pi
{pi)
-
{x
=
Xi)
—
pi
Xi)
{xj
< 0,
- Dj
<
Ui {x)
0.
{pi)), i.e., p,
Together
{Dj
{pi)
(14), this
-
Xj)
<
{pi)
=
imphes that
0.
Since u^
is
monotonically increasing, this implies (13).
(Part 2)
Assume that
Xj (pi)
^
Reject,
i.e.,
Xj {pi)
=
Dj
{pi).
But, by stationarity,
the continuation values do not depend which offers are rejected, hence, for any pi with
Uj {Dj {pi))
>
Uj {Dj {pi)),
we must have
Xj {pi)
— Dj {p^)) < Ui {uj {Dj {pi)))Ui
then exists some e > and some price p' such that
Suppose that
{w
17
^
Reject,
Since Ui
and thus Xj
is
{pi)
=
Dj
{pi).
continuous, by (13), there
Ui {uj {Dj {pi))
+
e)
> Ui{w — Dj
(p^))
.
and
Uj {Dj [p'))
=
+ e. Now,
Uj {Dj (p,))
+
payoff Ui {uj {Dj {pi))
e),
^
stationarity, Xj (p,)
deviation for
Proof.
Case
i,
Reject,
Xj (p,)
i.e.,
two
Assmne
that xi (p2)
^
N
,
offer is
we have
Ui
{w
f/j
{Uj
{Dj
for
t,
any
p_,,
and demand Di
price Pj
be accepted, yielding higher
=
Dj
^
X2{p\).
<
(p,))
=
Vj
Thus, offering
{p,).
Then,
Uj {Dj {pi)).
a profitable
p^ is
cases.
Reject
— Dj
=
(p,))
i
{Dj
t/, {xlj
Lemma
Then, by
Since Xj
{pi))).
at the beginning of any date
i
L2, for each
(p,)
^
Reject,
+
at
which he
is
accepted,
1
This has two implications. First, since pj
{pi))).
«,
Second, at
will
a contradiction.
follows that the continuation value of
makes an
it
Reject and Uj {Dj
2] There are
6
distinct i,j
^
(p,)
[Lemma
1:
offers p',
i
contradiction.
a.
(Part 3) Suppose that Xj
by
if
(A
>
iPj))
(A
S,U, {u,
(15)
{Pj)))
(A (p^)) > 6iUi {Uj {Dj (p,))), player i must accept the
(see Lemma LI). Since j offers pj it must be true that
u^
if
{pj )
,
Uj{u,{Di{pj)))>Uj{v,)
each
for
Ui
(A
with
>
Vi
>
(P,))
Vi
>
6,U, {u,
(£>_,
(A
Now, assume that
(P;))).
Hence, by (16),
{?,)))
>
SiUi {ui {Di {pj))). Since Ui {Di {pj))
has a local
Cose
S^Ut {uj
maximum
at u^
Assume
that Xj
,2:
Reject, agents
(A
(Pj))-
{pi)
=
(i)
{u,
f/,
(16)
(A
>
{?:)))
Uj
Then, by
{v,) for
some
G N.
i,j
would never reach an agreement, thus
2, u^
(A
(Pj))
we
also
If
=
their continuation values
Therefore, Xi {pj)
6^Ui {Di (pj))
=
=
i.e.,
Vi
S'^Ui
{Di (Pj)).
Proof.
.
^
Reject
Since j offers pj,
Ui
{Di
As
in
On
{pj)).
Case
[Lemma
1,
if
=
Ui {Di {pj))
Proof.
Vi.
Since Ui
=
the other hand,
0,
3]Assume there
him only
Vi.
{vi)
=
Vi.
Di
v,
<
>
would be
would
must accept
who makes
>
>
0;
Ui {Di {pj))
>
offers Pi, it will
be
«,
(A
Uj (0) for each
0,
=
{Pj))
(jpj) if
then
Vi
m
G P. Then,
if i
Then, at the previous date,
i
should not
m
[Lemma 4] By Assumption 3,ior 6i =
is
=
Ui {Di {pj))
if
exists such pi
T'l/Si.
Xi {pj)
then Uj
this yields Ui {Di {pj))
accepted, yielding a payoff higher than
accept pj, which gives
Then,
Xj {pi).
x, (pj)
1), this
the Wahasian price p*, providing a profitable deviation (p*) for the agent
offer.
each
'^'t-
had
yield a contradiction: In that case, in a stationary equilibrium each agent
an
(15),
6iUi {ui {Di {pj))), this implies that Uj
Thus, by Assmnption
Reject for
not true.
is
Since the Walrasian payoffs are strictly positive (by Assumption
zero.
it
continuous, this imphes the
62
Lemma.
18
=
l,6iUi {6jUj {vi))
>
Ui
{w - Dj
{pi))
>
References
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/
20
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