Reaching Cournot-Walras Equilibrium
S. D. Flåm and K. Gramstad
Reaching Cournot-Walras Equilibrium
S. D. Flåm and K. Gramstad
March 2, 2015
Abstract.
Considered here are Cournot oligopolists who exchange
some production factors - say, pollution permits. On one hand, as a producer,
each player is construed as foresighted, swift, and strategic in exploiting his
market power. On the other hand, as factor owner, the same agent is viewed
as somewhat shortsighted, slow and non-strategic in adapting his holdings.
Thus, production and imperfect competition become parts of an exchange
economy. The disparate features of the composed economy motivate a solution
concept which blends Cournot equilibrium with that of Walras.
The novelty is to show that integrated equilibrium may obtain by repeated
play along two totally di¤erent time scales: a rapid one for quick attainment
of (contingent) Cournot outcomes, and a slow one for stepwise emergence of
factor prices. The only vehicle is bilateral exchange of factor endowments.
Key words: Cournot oligopoly, Walras equilibrium, coupling constraints, repeated play.
JEL Classi…cation: C72, D51, L13
1. Introduction
Various notions of equilibrium are chief concepts in economic theory. For good reasons, a quali…ed outcome is seen as a steady state - a con…guration from which no
concerned party wants to deviate. Ex post, if equilibrium already prevails, it appears natural to view its intrinsic stability in the said manner. Ex ante, one wonders
though, how might equilibrium be attained?1
Motivated by that question, this paper considers instances that reside at the
intersection of game and market theory. Serving since long there are the models
of Cournot and Walras. In the sequel, these are linked as follows. On top of a
Cournot oligopoly for commodity outputs is imposed a Walrasian exchange economy
for factor inputs. Accordingly, equilibrium acquires two disparate features. One is
that producers, conditioned by their factor holdings, choose a Cournot-Nash strategy
pro…le. The other is that, having chosen such a pro…le, all agents should value the
said holdings in the same manner. That is, they should see equal factor prices.
Economics Department, University of Bergen. Flåm thanks: Finansmarkedsfondet, The Arne
Ryde Foundation, and the NILS program for partial support - and Professor M. A. López at the
University of Alicante for great hospitality.
1
Game theorists have long wrestled with this pressing question; see Fudenberg (1998), Hart &
Mas-Collel (2013), Peyton Young (1998, 2004). Most studies deal with …nite-strategy noncooperative
games. It seems fair to say that there are no unifying results on learning to play Nah equilibrium.
1
Reaching Cournot-Walras Equilibrium
2
Equilibrium of this sort can hardly be brought down by a single shot. Agents
need time; they must learn to play - and to do so without assistance of any director.
Moreover, as long as factor holdings change, so does the stage game as well.
It’s presumed here that producers be so competent and experienced as to …nd
Cournot equilibrium quickly. Also by presumption, each player values diverse production factors on line, always using personal bid-ask prices. Repeatedly, two random
players meet - or are matched - to contemplate exchange of production factors. If their
valuations di¤er, they make a direct deal, most often facilitated by side payments.
Eventually, when valuations coincide across players, the factor market settles.
Will full equilibrium …nally obtain? That question is explored below. In doing
so, this paper di¤ers from many others studies on learning of strategic behavior
(Hart and Mas-Colell 2013), (Peyton Young 2004). Notably, it goes beyond …nitestrategy instances, and it allows coupling constraints. Neither …ctitious play nor
regret matching is an issue. No player needs know his rivals or their actions - present
or past. And nobody must form forecasts or statistics. Instead, it su¢ ces that
everyone be driven by own margins. Broadly, an oligopolist should adapt until he
sees zero net margins for products and common margins for inputs.
Other studies also emphasize the chief role of margins in pushing adjustments
of play. Rosen’s (1965) paper is closest to ours in that players are guided by …rstorder di¤erential data. Here, however, classical di¤erentiability is not needed. And
most important: exchange makes coordination super‡uous. At every stage, players
maintain feasibility by themselves.
The paper is planned as follows. Setting the frames, Section 2 recalls games with
coupling constraints. It characterizes what is called a normalized equilibrium and
inquires about its stability. Section 3 specializes the said constraints to be purely
additive. Motivated by environmental games, many of which feature exchange and
externalities, that section de…nes a solution concept which integrates Cournot-Nash
equilibrium with that of Walras. To prepare for repeated play, Section 4 deals with
just one exchange between just two agents. Section 5 models play as a discrete-time
process driven by di¤erences in agents valuations of factor inputs. Section 6 provides
numerical illustrations by considering workhorse oligopolies. Section 7 concludes by
relating to the literature.
2. Games with Coupling Constraints
This section …xes the frames and the notation. It also recalls some material concerned
with description and stability of solutions.
Henceforth accommodate a …xed …nite ensemble I of noncooperative economic
agents. Member i 2 I makes a choice xi ; codi…ed as a vector in some Euclidean space
Xi . When facing the pro…le x i := (xj )j6=i of his rivals, he gets pecuniary payo¤
i (xi ; x i ).
Choice is subject to coupling constraints in that each pro…le x = (xi ) must belong
3
Reaching Cournot-Walras Equilibrium
to a non-rectangular subset X of the product space X := i2I Xi .2 That is, X is a
strict subset of the box i2I fxi : (xi ; x i ) 2 X for some x i g :
Since payo¤s are transferable, the bivariate function
X
x^; x 2 X 7! (^
x; x) :=
xi ; x i )
i (^
i2I
records aggregate payo¤.
De…nition. Declare x = (xi ) 2 X a Nash normalized equilibrium i¤
(
)
X
(x; x) = max
xi ; x i ) : x^ 2 X :
i (^
i2I
Such an equilibrium tends to select among ordinary Nash outcomes.
Proposition 2.1 (Existence of normalized equilibrium [10], [33]). Suppose the constraint set X is non-empty compact convex. Also suppose aggregate payo¤ (^
x; x) is
concave in x^, jointly upper semicontinuous in (^
x; x), and separately lower semicontinuous in x: Then there exists a normalized equilibrium.
The hypotheses of Proposition 2.1 are henceforth in vigor. We shall need a differential characterization of normalized equilibrium. Recall that X is a Euclidean
space, with some inner product h ; i : A vector x 2 X is called a supergradient of a
function f : X ! R[ f 1g at x; if f (x) is …nite and
f (^
x)
f (x) + hx ; x^
To signal this, write x 2 @f (x) or x 2
X X has a normal cone
N (x; X) := fx 2 X : hx ;
xi for all x^ 2 X.
@
f (x).
@x
xi
At any point x 2 X, the subset
0 for each
2 Xg ;
composed of all vectors which point right out of X there. It’s convenient to refer to
members of the partial superdi¤erentials
Mi (x) :=
@
@ x^i
xi ; x i ) jx^i =xi
i (^
and M (x) :=
@
(^
x; x) jx^=x =
@ x^
i2I Mi (x)
as margins. Members of the set di¤erence M (x) N (x; X) are called essential margins. In the spirit of neoclassical economics, all essential margins should be nil in
normalized equilibrium. For succinct statement of this fact, let PC [ ] denote the orthogonal projection onto a non-empty closed convex subset C of a Euclidean space
2
Few non-cooperative games are modelled with coupling constraints. Debreu (1952) is an important exception.
Reaching Cournot-Walras Equilibrium
4
Proposition 2.2 (Characterizing normalized equilibrium [16]). x is a normalized
equilibrium i¤ the following three equivalent statements hold for one and the same
margin m 2 M (x) :
The margin is nil or normal, meaning m 2 N (x; X):
For each deviation, individual or joint, along any feasible direction d in the cone
D(x; X) := R+ (X
x)
clR+ (X
x) =: T (x; X);
(1)
the margin indicates an overall loss. That is, hm; di 0:
The orthogonal projection PT (x) [ ] onto the tangent cone T (x; X) (1) yields
(2)
0 = PT (x;X) [m] :
Equilibrium condition (2) immediately invites use of the di¤erential system
dx(t)
=: x_ 2 PT (x;X) [M (x)] ;
dt
x(0) 2 X;
(3)
as an idealized model of continuous-time out-of-equilibrium play. In view of such
dynamics, a normalized equilibrium x is declared asymptotically stable if
x 2 X & x 6= x =) hm; x
xi < 0 for each margin m 2 M (x):
(4)
Proposition 2.3 (Uniqueness and asymptotic stability of normalized equilibrium
[11], [16]). Suppose x is asymptotically stable (4). Normalized equilibrium is then
unique and globally attractive for system (3).3
Its stability notwithstanding, system (3) can hardly depict practical play. The main
reasons are two. First, real agents make discrete steps in discrete time. Accordingly,
as brought out in Section 5, their moves will be depicted in such manner. Second, for
implementation - and for the narrative - it’s essential that behavior be agent-based,
directed by no outsider or extra agent. On precisely this point, as model of noncoordinated interaction, (3) appears self-defeating. To wit, who makes the projection
PT (x;X) [ ] if any? To circumvent this hurdle, we ask: can players, by themselves,
dispense with joint projection? It turns out that indeed, they can. Speci…cally, the
subsequent sections address instances which render it super‡uous.
3. Games with Additive Coupling
This section specializes the joint constraints to have additive form, and it de…nes a
solution concept which blends Cournot-Nash equilibrium with that of Walras.
Henceforth the strategy xi = (yi ; zi ) of any player i 2 I has two quite di¤erent
components. The …rst, yi ; is seen as a production plan, chosen without any concern for
3
Proposition 2.3 adds to the pioneering paper of Rosen’s (1965). Related studies, all using various
forms of monotonicity, include [10], [11] and [13].
5
Reaching Cournot-Walras Equilibrium
overall e¢ ciency or cooperation. In contrast, the second component, zi ; is construed
as a perfectly transferable commodity bundle, contingent claim, or resource vector.
Alternatively, it de…nes user rights to natural resources.4
While purely additive, the coupling constraint a¤ects only the transferable items
zi . Speci…cally, players act in the domain
)
(
X
X
X := x = (xi ) : xi = (yi ; zi ) 2 Yi Zi &
zi
ei :
i2I
i2I
Throughout Yi ; Zi are non-empty compact convex subsets of Euclidean spaces Yi ;
Z respectively. Agent i holds an endowment ei 2 Zi
Z. The ambient endowment space Z is common and ordered in the customary componentwise manner. By
assumption, transferable items create no externalities in that
i (x)
=
i (y; z)
=
i (y; zi )
=
Further, by assumption,
P
Pat least one agent i has5
the equality i2I zi = i2I ei will always hold.
i (yi ; y i ; zi ):
i (yi ; y i ; zi )
increasing in zi : Hence
As announced, the solution concept is a mixed one. It embodies features found
in the equilibria of Cournot, Nash and Walras:
De…nition (Cournot-Nash-Walras equilibrium). A pro…le x = (xi ) 2 X; alongside a
price vector p 2 Z; constitutes an equilibrium if nobody regrets his choice. That is,
given factor price p;
i (yi ; y i ; zi )
hp; zi i = max f i (^
yi ; y i ; z^i )
Further, all factor markets clear in so far as
p
0;
X
zi
i2I
X
ei ;
i2I
and
hp; z^i i : y^i 2 Yi ; z^i 2 Zi g 8i:
*
p;
X
i2I
ei
X
i2I
zi
+
= 0:
(5)
(6)
In principle, equilibrium is easily recognized. It prevails when agents’essential margins are nil or equal to common prices. Indeed, it follows directly from (5):
Proposition 3.1 (Characterizing equilibrium). Let Xi := Yi
alongside a price p 0 is an equilibrium i¤ (6) holds and
(0; p) 2
@
@xi
i (x)
N (xi ; Xi ) for each i:
Zi . A pro…le x 2 X
(7)
4
Important instances comprise transfers of …sh quotas, production allowances, pollution permits,
or rights to water usage. In either case, reallocation of initial endowments may improve payo¤s and
entail partial e¢ ciency.
P
P
5
Alternatively, absent such monotonicity, one may require that Pi2I zi = i2I ei in the de…nition of X: In either case, i might depend on the aggregate holding i2I zi : Since the latter sum is
constant, it has no e¤ect on behavior or welfare.
Reaching Cournot-Walras Equilibrium
6
The above de…nition of equilibrium would comply with the preceding one on normalized instances if some extra (exogenous and …ctitious) player chooses p
0 so
as to minimize the value of excess supply. In that optic, under reasonable assumptions, existence of a solution is guaranteed; see [16]. We shall side-step the attending
mathematical issues and rather ask right to the point: could the agents themselves by way or repeated exchanges - make a clearing price vector p emerge endogenously?
The subsequent two sections provide constructive and positive answers.
4. Bilateral Exchange
For preparation of the main model of play, this section isolates a recurrent episode
in which two agents - by themselves and between themselves - exchange production
factors.6 Their direct deal is driven only by di¤erences in their local and personal
valuations of marginal transfers. For that reason, exchange is moderate and somewhat
myopic. Recall that each factor bundle belongs to a common Euclidean space Z,
endowed with inner product h ; i and associated norm k k :
Fix two agents i; j who hold respective factor bundles zi 2 Zi and zj 2 Zj .
Suppose a vector 2 Z be transferred to i from j: In case is non-zero; it de…nes a
unit direction d := = k k, and step-size s := k k along that direction. Consequently,
with no loss of generality, a transfer = sd yields updated holdings
zi+1 := zi + sd 2 Zi
and zj+1 := zj
sd 2 Zj
(8)
where s kdk > 0 =) kdk = 1: The …rst inclusion in (8) implies that d belongs to the
cone
Di (zi ) := fr(^
zi zi ) : r 2 R+ & z^i 2 Zi g =: R+ (Zi zi )
composed of all feasible directions for agent i at zi : Quite similarly, the second inclusion tells that d 2 Dj (zj ): Hence, for the feasibility of exchange, it’s necessary
that
d 2 Dij (zi ; zj ) := Di (zi ) \ Dj (zj ):
Which directions in Dij (zi ; zj ) are desirable? To address that question, call
pi = gi
ni with gi 2
@
@zi
i (y; zi )
and ni 2 N (zi ; Zi )
(9)
an essential margin or personal factor price as it applies for agent i at (y; zi ). Since
y is …xed here, it simpli…es notation to write i (zi ) := i (y; zi ) and j (zj ) := j (y; zj ).
Proposition 4.1 (On bilateral exchange). When agents i; j own respectively zi 2 Zi
and zj 2 Zj , they cannot make a proper trade in case the cone of feasible directions
is degenerate, meaning Dij (zi ; zj ) = f0g. Moreover, they ought not make any trade
6
In a seminal paper Feldman (1973) modelled bilateral exchange as the solution of a cooperative two-person game. Our approach, being less stringent, allows that interlocutors contend with
betterments.
7
Reaching Cournot-Walras Equilibrium
if some personal prices pi and pj coincide. The reason is that (8) and (9) then yield
+1
+1
i (zi ) + j (zj )
i (zi ) + j (zj ):
Proof. For the second statement, suppose partial supergradients gi 2 @
gj 2 @ j (zj ) and normals ni 2 N (zi ; Zi ), nj 2 N (zj ; Zj ) satisfy
pi = gi
Since
i
ni = gj
i (zi );
nj = pj :
is concave, zi+1 = zi + d and d 2 Di (zi ) imply
+1
i (zi )
i (zi )
+ hgi ; di
Quite likewise, zj+1 = zj + ( d) and
+1
j (zj )
+ hgi ; di
hni ; di :
d 2 Dj (zj ) imply
hgj ; di
j (zj )
i (zi )
j (zj )
hgj ; di + hnj ; di :
Adding theses inequalities yields
+1
i (zi ) +
+1
j (zj )
i (zi ) +
j (zj ) +
hgi
ni ; di
hgj
nj ; di =
i (zi ) +
j (zj ):
This completes the proof.
The upshot is that i and j ought trade only when they see some leeway but no
common prices - that is, when
Dij (zi ; zj ) 6= f0g &
@
@zi
i (y; zi )
N (zi ; Zi ) \
@
@zj
j (y; zj )
N (zj ; Zj ) = ;:
(10)
On such an occasion, it appears reasonable that a transfer, to i from j; be aligned
with a direction
pi pj
d=
,
(11)
kpi pj k
which features the di¤erence in personal prices pi and pj (9). To argue in supplementary manner for format (11), note that the joint payo¤ i + j has steepest slope
Sij (zi ; zj ) := sup
0
i (zi ; d)
0
j (zj ;
+
d) : d 2 Dij (zi ; zj ) & kdk
1 ;
(12)
0
i (zi ; d)
denoting the standard directional derivative. Let Pij be the orthogonal
projection (in the factor endowment space Z) onto the closed convex tangent cone
Tij (zi ; zj ) := clDij (zi ; zj ): That operation links (11) to (12) - as is clari…ed next:
Proposition 4.2 (On steepest slopes and minimal deviations of margins [17]). The
steepest slope (12) equals
Sij (zi ; zj ) = min fkPij [gi
= min fkpi
gj ]k : gi 2 @
i (zi );
gj 2 @
j (zj )g
pj k : pi ; pj being personal prices (9)g :
Reaching Cournot-Walras Equilibrium
8
Thus, a best direction (11) obtains when personal prices pi ; pj minimize the last expression. The steepest slope partly justi…es players’use of format (11). It also serves
to identify how and when trade might bene…t both parties:
Proposition 4.3 (On bene…cial barters [17]). For any factor 'ij 2 (0; 1), whenever (zi ; zj ) 2 Zi Zj and Sij (zi ; zj ) > 0; there are updates (8) which solve the strict
inequalities
+1
i (zi )
+
+1
j (zj )
i (zi )
j (zj )
'ij sSij (zi ; zj ) > 0
and
+1
i (zi )
+
i
with monetary side-payments
>
i;
i (zi )
j
and
+1
j (zj )
+
j
>
j (zj )
that sum to zero.
Together, Propositions 4.2&3 open up for manifold modes of direct exchange at any
stage. Trade could unfold in parallel or asynchronously, be guided by steepest slopes,
and be governed by diverse protocols on who will meets next whom. For more on this,
see [17]. However, from here onwards, to accommodate humanlike players, nobody
are compelled to compute steepest slopes.
5. Repeated Play
Using bilateral exchange as vehicle and workhorse, this section casts repeated play as
a discrete-time process. Time ‡ows on two di¤erent clocks. One ticks after attainment
of Cournot type equilibrium; the other ticks after completion of a bilateral exchange.
The two clocks are put in separate motion because condition (5), as it applies to
agent i; involves two choices, namely: yi and zi :
For motivation, suppose (the oligopolistic variable) yi already be chosen, and that
zi be (re)considered thereafter. More precisely, prior to any (re)consideration of zi ;
let
yi ; y i ; zi ) : y^i 2 Yi g for all i:
(13)
i (yi ; y i ; zi ) = max f i (^
(13) …xes a crucial feature of repeated play as modelled here. Speci…cally, by hypothesis, given any endowment pro…le (zi ), the agents need negligible time to …nd a
Cournot-Nash equilibrium (13). The latter is contingent and partial, however, being
restricted to (yi ) and depending on (zi ):
In contrast, while still out of full-‡edged equilibrium, players adapt their factor
holdings stepwise - with notable caution and moderation. To capture this behavioral
feature, suppose agents i; j; meet - or are matched - on some platform, at some stage.
On that occasion, they hold respective factor endowments zi , zj and personal prices
(9):
@
@
N (zi ; Zi ) and pj 2
N (zj ; Zj ):
(14)
pi 2
i (y; zi )
j (y; zj )
@zi
@zj
If pi = pj ; inclusions (14) and Proposition 4.1 rationalize their actual holdings. That
is, no direct exchange is worth their while. Otherwise, if (10) holds, a transfer to
9
Reaching Cournot-Walras Equilibrium
i from j appears justi…able. Moreover, let the transfer be aligned with the price
di¤erence pi pj ; see (11). To appreciate this suggestion, consider an instance where
zi 2 intZi and zj 2 intZi . Then, both normal cones N (zi ; Zi ); N (zj ; Zj) in (14)
reduce to f0g : If moreover, di¤erentiability be granted, the price di¤erence equals
@
@zi
i (y; zi )
@
@zj
j (y; zj )
When zi resides at the boundary of Zi , agent i wants to suppress normal components
of @z@ i i (y; zi ), if any, these pointing right out of Zi at zi .7 In short, d is aligned with
the di¤erence in (partial but) essential margins. Proposition 4.2 provides more detail
about a best direction d of transfer. And Proposition 4.3 clari…es that side payments
may lubricate the exchange mechanism. The essence is that the net recipient of a
factor is the one who appreciates it most.
Synthesizing these aspects, we cast repeated play as a discrete-time process:
P
PStart at any allocation i 7! zi 2 Zi of the transferable goods such that i2I zi =
i2I ei For example, let zi = ei :
Find a contingent Cournot equilibrium in y = (yi ) (13), depending on the actual
allocation z = (zi ):
Randomly select two agents i; j with uniform probabilities. Actually, these agents
hold transferable endowments zi 2 Zi and zj 2 Zj respectively.
Given y = (yi ); update their holdings by (8), using a step-size s 0 and a direction
d = pi pj with pi ; pj satisfying (14).
Return to …nd a contingent equilibrium (13).
Continue until convergence.
Out of equilibrium, during repeated play, instead of (7), it always holds
(0; pi ) 2
@
@xi
i (x)
N (xi ; Xi )
(15)
with personal prices pi which do not all coincide. At each stage, since yi is conditionally optimal, each agent i sees a partial margin @y@ i i which is normal to Yi at yi - or
it’s nil; that is
@
02
N (yi ; Yi ):
i (x)
@yi
Simultaneously, in terms of factors, he has an essential margin
pi 2
7
@
@zi
i (y; zi )
N (zi ; Zi ):
In geometric terms, regarding zi as his actual origin or point of reference, agent i would project
@
any candidate gradient gi 2 @z
i (y; zi ) onto Zi :
i
Reaching Cournot-Walras Equilibrium
10
It’s tacitly understood that together these partial margins margins satisfy (15). This
requirement is super‡uous when i is di¤erentiable xi .
Theorem 5.1 (Convergence of non-coordinated play [16]). Suppose repeated play, as
modelled above, proceeds at discrete stages k = 0; 1; ::: with step-sizes sk 0; chosen
by the agents themselves, which satisfy
1
X
sk = +1 and
k=0
1
X
s2k < +1.
(16)
k=0
Then, under asymptotic stability (4), the generated sequence (xk ) converges to the
unique normalized equilibrium.
Corollary 5.2 (On common factor pricing). Besides the hypotheses in Theorem
5.1 suppose equilibrium x = (y; z) is such that some agent i has zi 2 intZi and
i (y; ) Gâteaux di¤erentiable at zi : Then there is a unique equilibrium price in the
factor market.
Proof. Since the particular agent i has N (zi ; Zi ) = f0g ; posit
p :=
@
@zi
i (y; zi )
N (zi ; Zi ):
From Proposition 4.2 follows that Sij (zi ; zj ) = 0 i¤ p equals some personal price pj of
agent
i that all Sij (zi ; zj ) = 0. Consequently, the intersection
h j: Proposition 4.3 implies
\j @z@ j j (y; zj ) N (zj ; Zj ) reduces to a singleton, namely p:
6. A Cournot Oligopoly
For illustration, this section considers multi-output, multi-input Cournot oligopolies
which feature transferable production factors. Firm i 2 I produces marketable output
yi , using factor bundle zi ; at cost ci (yi ; zi ); to get payo¤
i (yi ; y i ; zi )
:= hP (yI ); yi i
ci (yi ; zi ):
P
All spaces Yi coincide, yI := i2I yi denotes total supply, and h ; i is the standard
inner product. The ”inverse demand curve” yI 7! P (yI ) reports the price vector at
which output markets clear. If factor bundles were traded at price p; a price-taking
…rm i would take home overall pro…t i (yi ; y i ; zi ) + hp; ei zi i :
For existence of equilibrium, note that rectangular domains of the type Yi =
[0; yi ]
Y and Zi = [0; eI ]
Z are compact convex. PThe vector yi
0 denotes
here the production capacity of …rm i, whereas eI := i2I ei
0 is the aggregate
endowment. For the other hypotheses in Proposition 2.1, let cost ci (yi ; zi ) be jointly
convex, and output pricing P ( ) be concave and di¤erentiable with P 0 + P 0T negative
Reaching Cournot-Walras Equilibrium
11
semide…nite. Under these conditions, i (yi ; y i ; zi ) becomes concave in xi = (yi ; zi );
see [33] or Lemma 7.5 in [18]. For a case with joint transportation, see [12].
As one might expect, cost sharing becomes (conditionally) e¢ cient - as in Montgomery (1972):
Proposition 6.1 (E¢ cient sharing). In equilibrium x = (xi ) factors are e¢ ciently
allocated. That is, the factor price regime p satis…es
@
ci (yi ; zi ) 2
@zi
p 8i:
Ni (zi ; Zi )
Cournot and CO2 emissions. Simplifying further, let there be just one marketable
product and one transferable factor. For additional simplicity, presume di¤erentiable
data and interior solutions. In equilibrium, there is
P
P
9
market balance:
=
i2I zi =
i2I ei ;
optimal output from each producer: P (yI ) + P 0 (yI )yi = @y@ i ci (yi ; zi );
(17)
;
@
and common factor pricing:
c (y ; z ) = p.
@zi i i i
While still out-of-equilibrium, the …rst two conditions in (17) hold, but factor pricing
isn’t uniform. Therefore, at any stage, two agents i; j, for which
:=
@
cj (yj ; zj )
@zj
@
ci (yi ; zi ) 6= 0;
@zi
undertake an exchange (8) with step-size s 0 and transfer direction
8
if zi ; zj > 0
>
>
<
minf ; 0g if zi > 0; zj = 0
d :=
maxf
; 0g if zi = 0; zj > 0
>
>
:
0 otherwise:
A suitable step-size takes the form s := min fsk ; s^g with
s^ :=
zi =d if d < 0
zj =d if d > 0
In line with the arguments above, for any feasible factor allocation z = (zi ); players
easily …nd Cournot equilibrium (13).
Some numerical simulations. Let Cournot oligopolist i 2 I = f1; 2; 3g put out
yi 2 R+ , using input zi 2 R+ , at cost
ci (yi ; zi ) =
i yi
1 + zi
:
12
Reaching Cournot-Walras Equilibrium
P
The inverse demand curve for products is linear: P (yI ) = 1 yI where yI = i2I yi .
To ensure interior solutions, let each i
1=3. Firms come immediately up with
Cournot output levels
!
X
1
i
j
1 3
+
:
yi (zi ; z i ) =
4
1 + zi
1
+
z
j
j6=i
Example 1 (Identical costs). Choose cost parameters
(3; 0; 0); and step-sizes sk = 200=k; to get
i
1
2
3
Table
i
= 1=4, endowments (ei ) =
zi
yi
p
1.000 0.2187 0.0205
1.000 0.2187 0.0205
1.000 0.2187 0.0205
1 Results for Example 1.
0.5
y1
0.4
y2
y3
0.3
0.2
0.1
0
5
10
15
20
Iterations
25
30
35
40
3
z1
z2
2
z3
1
0
0
5
10
15
20
Iterations
25
30
35
40
Figure 1. Evolution of holdings for Example 1.
Example 2 (Asymmetric costs). Choose cost parameters ( i ) = (1=3; 1=4; 1=5),
endowments ei = 1, and step-sizes sk = 200=k to get
i
1
2
3
Table
zi
1.1446
0.9990
0.8565
2 Results
yi
p
0.1916 0.0208
0.2220 0.0208
0.2393 0.0208
for Example 2.
13
Reaching Cournot-Walras Equilibrium
0.4
y1
0.3
y2
y3
0.2
0.1
0
0
5
10
15
20
Iterations
25
30
35
40
2
z1
1.5
z2
z3
1
0.5
0
0
5
10
15
20
Iterations
25
30
35
40
Figure 2 Evolution of holdings for Example 2.
Examples 1 and 2 con…rm the emergence of equilibria with symmetric and asymmetric
agents.
7. Links to Literature
This paper ties oligopolies for …nal products to exchange of production factors. A particular feature is that experienced producers adapt their outputs swiftly, at the spur
of the moment, but adjust their factor use slowly and stepwise. Nonetheless, granted
monotone margins (4), overall equilibrium may eventually obtain. Such monotonicity holds in some workhorse models of industrial organization, notably in Cournot
oligopolies.
The paper relates to a large literature. Regarding production strategies, there is
room for more generality than indicated above. Partial equilibrium (13) could come
à la Cournot, Bertrand, Stackelberg, Arrow-Debreu (1954), Gabszewicz-Vial (1972),
or Shapley-Shubik (1977). It imports in each case that the production component of
equilibrium be stable.
While adapting their factor use, agents move slowly, driven by di¤erences in personal valuations. Bilateral exchange is the only vehicle. It’s agent-based, decentralized, and non-coordinated. This optic has several consequences:
The Shapley-Shubik (1977) market game, in which players put payments and quantities on diverse tables, is a one-shot, strategic-form episode. Hence it doesn’t …t our
emphasis on repeated, small-size trades.
Hahn (1984) and Westskog (1996) consider an oligopoly, surrounded by a pricetaking fringe which generates a price curve. Identifying that curve might require
more knowledge than the oligopolists have [24]. Further, it’s intricate to discern or
Reaching Cournot-Walras Equilibrium
14
predict who comes forward as oligopolist and who will not [35].
The Cournot-Walras setting - as modelled by Codognato and Gabszewicz (1991) presumes that some scarce goods be owned by just a few agents. Whence only the
latter act as oligopolists. Such is not our setting. To good approximation, endowments are widely held. For that reason, factor owners are already ”in the game”.
Entry or exit becomes less of an issue, hence not modelled here [29].
Admittedly, owners of scarce rights may game the situation - say, by destructing,
withholding, or misrepresenting endowments [2], [34], [32]. Whatever be the moves
which precede the game proper, including maybe reallocation of endowments [25],
property rights are here construed as reliable and well de…ned.
The factor market, modelled above, is essentially Walrasian, like in Montgomery
(1972).8 However, during trade, no player is a price-taker. There are no prices to
take; they are all personal - and not necessarily posted. Moreover, quantities are
bounded. Consequently, trade may evolve as in double auctions or order markets [4].
Despite decentralization, equilibrium is apt to emerge all the same.
Attainment of equilibrium in exchange economies has long been studied; see [7],
[8], [9], [14], [15] and references therein. A novelty here is that exchange is superimposed on non-cooperative production. There are some similarities to [12], but two
major di¤erences: …rst, the aggregate endowment is …xed; second, out-of-equilibrium
behavior is explicitly modelled.
Since the product market is imperfect, Pareto e¢ ciency is rare, and overall payo¤ is apt to evolve in non-monotone (non-potential) manner. These features are at
variance with transferable-utility pure exchange models [7], [8], [9], [14], [15]. Also
noteworthy is that individual constraints can be general and criteria non-smooth.
For sake of organizing arguments, trades were modelled as execution of an algorithm. Being of gradient type the algorithm appears classical - except that it accommodates non-smooth functions and uses projected gradients. There are major
di¤erences though. Computationally and informationally the procedure is starkly
decentralized, invoking only small blocks of variables at each round. To analyze its
complexity and convergence rate is important, but outside the scope of this paper;
see [3], [37] and [38].
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15
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17
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