JOURNAL
OF ECONOMIC
THEORY
38,
371-379
(1986)
On a General Existence
Marginal
Cost Pricing
Theorem for
Equilibria*
J. BROWN
DONALD
Department
of Economics,
Yale Universify.
New Haven. Connecticut
06520
M.
GEOFFREY
Graduate
School of Business, Columbia
University,
New York, New York 10027
M.
Department
HEAL
qf Economics,
ALI
University
KHAN
of Illinois,
Champaign,
Illinois
61820
AND
RAJIV
VOHRA
Department
of Economics,
Brown University.
Providence.
Rhode Island 02912
Received
December
1. 1983; revised
September
17. 1985
We report a generalization
of recent results on the existence of marginal
cost
pricing
equilibria
(MCPE)
in economies
with an increasing
returns
to scale
industry.
Our result makes no ad hoc assumptions
which force the equilibrium
to
be on the efficiency
frontier
of the aggregate
production
possibility
set. We also
present an additional
condition
under which our MCPE
are productively
efficient
in the aggregate.
Journal of Economic
Literature
Classification
Numbers:
021. 022,
614.
0 L986 Academic Press, Inc
* This paper was given limited circulation
in January
1983. We thank Andreu
Mas-Cole11
for his comments and also Alok Aggarwal
and Salim Rashid for many valuable conversations.
We would also like to acknowledge
the comments
of three referees and an Associate Editor of
this Journal. Errors are solely ours. This research has been supported
in part by NSF Grants
SES-8319611
(Brown),
SES-8409857
and SES-8420244
(Heal), and SES-8410229
(Khon and
Vohra).
371
I-0nvri.M
0022-0531/86
$3.00
iT‘i 19M h” *ranemw Prrw 1°C
372
BROWNETAL.
1. INTRODUCTION
There have been several recent papers on the existence of marginal cost
pricing equilibria (MCPE) in economies with an increasing returns to scale
industry; see Beato [l], Brown and Heal [4], and Cornet [7]. All of these
papers make assumptions which guarantee that the equilibria
satisfy
aggregate productive efficiency. However, it is clear from Beato and MasCole11 [2] that such results have to be based on ad hoc assumptions since
there exist examples of “well-behaved” economies none of whose MCPE
exhibit aggregate productive efficiency. The existence of MCPE in
economies which are general enough to accommodate possibilities of
aggregate productive inefficiency has remained an important and elusive
open question. We answer this question in the context of an economy with
many convex firms but a single non-convex firm.
Beato and Mas-Cole11 [3] and Dierker, Guesnerie, and Neuefeind [S]
have independently obtained results similar to those reported here and we
begin by making brief comments on each.
The Beato and Mas-Cole11 result assumes: (i) upper hemicontinuity
and
convex-valuedness of the marginal cost correspondence and (ii) that each
production set can be written as the negative orthant plus a compact
set. The result that we report here dispenses with (i) through the use
of the Clarke-Rockafellar
tangent cones and with (ii) through the
Hurwicz-Reiter theorem on the compactness of attainable sets of nonconvex economies. The income hypothesis we use is very similar to that of
Beato and Mas-Colell. It is worth pointing out, however, that Beato and
Mas-Cole11 allow for many increasing returns to scale firms.
The Dierker, Guesnerie, and Neuefeind [8] result when specialized to
marginal cost pricing does apply to many firms but rules out the important
cases of firms with a fixed cost or nonconvex input requirement sets. Such
firms violate their assumption (P2). These cases are covered by our result.
In addition to our existence result, Theorem 1 below, we also present a
sufficient condition under which our MCPE are productively efficient in
the aggregate. This condition, though technically not surprising, is attractive from the intuitive point of view. It is essentially the requirement that
the aggregate production plan be “non-tight” in the sense that there exist
corresponding individual production plans such that any feasible e-perturbation of the aggregate production plan can be attained by suitable feasible
s-perturbations of the individual plans.
The model and results are presented in Section 2, and the proofs are
relegated to Section 3.
MARGINAL
COST PRICING
2. THE MODEL
EQUILIBRIA
313
AND RESULTS
We consider an economy consisting of m consumers and (I + 1)
producers. Consumers are indexed by i and each of them has a consumption set’ xi G R” and a utility function u’(. ) defined over Xi. Each
producer has a production set Yj where j runs over Z and 1 to 1 where Z
denotes the increasing returns industry. Aggregate endowment w is
assumed to be strictly positive. The income distribution is given in terms of
a fixed structure of revenues (a’ ), where a’ > 0 for all i and xi ai = 1, so that
consumer i’s income at prices p is M’( p, (Lj), y’) = a$. (Cj= 1JJ + y’ + w ).
We shall make the following assumptions on the underlying parameters of
our economy.
Al. For all i, Xi is a closed, convex subset of R”, and contains 0. ui(. )
are quasi-concave, continuous, and exhibit local non-satiation.
A2. For all j, Yj is closed, contains 0, and Y’- R"+ E Y’. For all j # Z, Y’
is convex.
Y’, 8= Y’ + Y. Then
A3. Let Y=c,,,
(i)
of B(ii)
y is closed.
d(- P)nd(F)=
{0},
where G!(P)
denotes asymptotic
cone
Before we introduce our equilibrium concepts, we need to develop formally the notion of marginal rates of transformation corresponding to a
point on the efficiency frontier of an arbitrary closed production set. For
this we shall need the cone of normals as developed by Clarke [6] and
Rockafellar [ 111. This concept was first applied to our problem by Cornet
[7] but our presentation, in relying on tangent cones as the primitive concept, will be marginally different from his.* The use of tangent conesallows
US to formalize
the notion of marginal rates of transformation
even at
inward kinks of a production set. The cone of interior displacement as used
by Beato [l] does not share this property.
For any x E CC R", let xk + c x denote a sequence chosen from C which
tends to x and tk JO denote a sequence of positive numbers converging to
zero. We now have
Dl. For any non-empty closed subset C of R”, and for x E C, the tangent
cone T,(x) consists of all y E R” such that for all sequences xk + c x, tk LO,
there exists a sequence y, + y such that for all k large enough,
’ R" is an n-dimensional Euclidean space whose non-negative orthant is denoted by R; and
the strictly positive orthant by R",+ We shall use the following convention for ordering the
elements in R": 9, >, 2.
’ Compare, for example, his proof and his Lemma 4(i) with ours.
374
xk + tk yk E C. The normal
vz E T,(x)}.
BROWN
ET AL.
cone is given by N,(x) = { y E R” 1(z, y) 6 0
PROPOSITION.
(a) Let C be a C’ manifold. Then N,(x) coincides with the
usual space of normals at x.
(b) Let C be convex. ThenN,(x) coincides with the cone of normals to
C at x in the sense of convex analysis.
(c) Let C have convex cones of interior displacements k(x, C), at each
point x. Then N,(x) is identical to the polar of k(x, C).
(d) T,(x) is non-empty and convex.
Proof: T,(x) is identical to the tangent cone originally and differently
defined by Clarke [6]; see the first paragraph of Rockafellar [ 111. Given
this, (a), (b), and (c) follow from Proposition 3.3 in Clarke [6]. Part (d)
follows from Theorem 1 in Rockafellar [ 111.
D2. A quasi-marginal cost pricing (QMCP) equilibrium is a 4-tuple ((Xi),
(fl), y’, p) such that p E A and3
(i) For all i, x’~A’(p, (y), j’)= {xi~Xij
px’<ap(&j’+j’+
w)}
and u(?)> u(xi) for all x’~A’(p, ($), y’) unless p(&j?++‘+
w)=O.
(ii) For all j= l,..., 1, I, y E Bdry ( Y’) and p E N&p’).
y+y’+
w.
(iii) Cy!, Xi<&
Our equilibrium concept is clear; Xi are quasi-demands as conventionally
defined in the literature; by (b) of the proposition above, j? are profitmaximizing bundles for all non-increasing returns to scale industries; p are
the marginal cost prices corresponding to the production plan jj’ for the
increasing returns industry; and aggregate excess demands are non-positive.
For our main result we shall need one final assumption. This can be
traced to Meade [9, paragraph 123 and simply says that for any efficient
production plan of the increasing returns industry, marginal cost pricing is
viable in the sense that the aggregate revenue from production is not
negative. More formally,
A4. For all y’~ Bdry (Y’), p E N,,(y’),
implies that p(cj, 1 fl+
y’+ w) 2 0, where 9 E Arg Max,. P,py and Ei is the jth attainable production set j = l,..., 1.
Note that this income hypothesis relates to profits being maximized over
the attainable production sets rather than the production sets.
‘Bdry
(A) denotes the boundary
of the set A, Arg Max,,,f(x)
f(Z) Lf(x)
for all x in A; B,(x) is the open ball of radius E centered
dimensional
simplex.
denotes X such that
at X; A denotes the n - 1
MARGINAL
THEOREM
1.
COST PRICING EQUILIBRIA
Under Assumptions Al
375
to A4, there exists a QMCP
equilibrium.
COROLLARY 1. If in addition to Al-A3, y’ is assumedto be convex and
consumers have endowments wi, C, wi= w, and shares 0” in all firms
j = l,..., 1,I, then there exists a quasi-Walrasian equilibrium. In this case, A4
is automatically satisfied.
It is clear that if in A4 p(cj,, JJ + y’+ w) > 0, the equilibrium referred to
in Theorem 1 will be such that for all i, Xi belong to the demand correspondence rather than the quasi-demand correspondence.
The reader should take note of there being no presumption that the
equilibria whose existence is established in Theorem 1 are productively
efficient in the aggregate. A natural question arises as to the conditions
under which there exist productively efficient marginal cost pricing
equilibria.
As mentioned in the Introduction,
the existence theorems
presented by Beato, Brown and Heal, and Cornet can be viewed as answers
to this question. We present another sufficient condition here which
appears not to have been noticed. We shall need
D3. An aggregate production plan .ij E y is said to be non-tight if there
exist Jo E Yj, cj= ,, I JJ = J, such that for all E> 0, for all z E B,(j) n P there
exist z-lE B,( JJ’) n Yj such that 2 = cj= ,, i i-j . If a production plan is not nontight, it is said to be tight.
The above concepts can be given a very simple economic interpretation.
A non-tight aggregate production plan p is such that any other production
plan “arbitrarily
close” to J can be obtained by individual
plans
“arbitrarily close” to JJ. Put differently, the economy can accommodate
marginal changes in aggregate production levels as a result of marginal
changes in individual production levels.
We can now weaken A4 to A4’ where
A4’. For all J E Bdry( P), p E NP( p) implies p( jj + w) > 0.
Cornet [7] has shown that under Al-A3 and A4’ there exist ((xi), $, p)
such that $e Bdry( y), FE Np(%) and conditions (i) and (iii) of D2 are
satisfied. It follows from Lemmas 2(a) and 2(b) of Brown et al. [S] that if
5 is not tight there exist y E P, j= l,..., 1, Z such that cjg =; and
jje NV(y) for all j, i.e., ((?), (-$), p) is a MCPE. Non-tightness can,
therefore, be seen as a sufficient condition under which an aggregate
production plan can be decentralized through marginal cost prices.
376
BROWNETAL.
3. PROOFS
Our proof of Theorem 1 relies on the fact that the attainable consumption and production sets are compact, and that a suitably truncated
boundary of the production set of the increasing returns industry is
homeomorphic
to the simplex. We present these essentially known results
first.
Let the set of attainable
states be denoted
by A where
and x’EX’, yip Yj for all i,j}. Let
A = {WL W)lC~~n_l ~~+~/=~,,yi=w,
the projection of A on x’ and Yi be denoted respectively as J? and i”. We
can now state
LEMMA 1 (Hurwicz and Reiter).
p are compact for all i, j.
Under Assumptions Al to A3, J?’ and
See Lemma 3 in Brown and Heal [4].
Since all attainable consumption and production sets are compact, we
can assume that there exists a cube C, centered at zero and with length k,
such that all J? and P are in its interior. Thus, there exist, for all i and j,
compact sets P c (Xi n C), P c ( Yjn C) which contain J? and ?j in their
interiors and, for i = l,..., m, j= l,..., 1, F and Yj are convex. Now let4
Proof:
K=((l+fk)e+w)ER”,,
and Ek( Y’) = Bdry ( Y’ + {K} ) n R”, . We shall focus on the following subset of Ek( Y’):
I?,( Y’) = {Z E EK( Y’) 13Z E EK( Y’), Z < z with
strict inequality for all i with zi > 0 >.
The reason for this apparently arbitrary truncation will become clear in the
proof of Theorem 1. For the moment, we can state
LEMMA 2 (Moore).
Zf Y’- R”, E Y’, then gK( Y’) is homeomorphic to
the simplex A, i.e., there exists a continuous mapping v: A -+ EK( Y’) which is
bijective with a continuous inverse and such that 6 $0 implies v(6)$0.
See the Appendix in Brown et al. [S].
Proof of Theorem 1. Let N(z) denote the normal cone to the set gK( Y’)
at a point z in the set and define the correspondence 4: I?,( Y’) + A such
that
b(z) = N(z) n A
for all z E R”++
Proof:
otherwise.
=A
4 e is an element
of
R",+ , all
of whose
components
are unity
MARGINAL
COST
PRICING
377
EQUILIBRIA
Next, we develop our concepts of pseudo-demands
p E A let
and supplies. For any
For any p E A, for any 6 E A and v(6) as in Lemma 2, and for all i = l,..., m,
let
M’(p,6)=a’
A’(p,6)=
pws t: d(p)+p(v(S)-K)
(
,= I
{xEF~pxdM’(p,~)}
d’b 6) = $&iy
= A’(p, 6)
= {Oj
ed
>
if M’(p, 6) > 0
if M’(p,&)=O
if M’(p, 6)<0.
The above concepts are clear once we recognize the fact that in a privateownership economy with an increasing returns industry, incomes may well
be negative.5 In this case we let each consumer obtain a zero commodity
bundle. Note also that it is through the incomes M’(p, 6) that the production plan of the increasing returns industry, as parameterized by 6, affects
demands.
Our proof of existence of QMCP equilibria will revolve around the
following mappings in which (Xx Y) denotes the compact convex set
cc:, xi) x (C&, F):
p:A+A
where ~(6) =&v(6)),
n:XxY+A
where ni( x, y ) =
v(.) as in Lemma 2;
(Xi-yi-
x7= 1 (Xj-yip
Wj+ Ki)
wj+ Ki)’
The economics behind our mappings are clear once the reader takes note
of the fact that the (n - 1)-dimensional simplex A is representing two different concepts, viz., the prices p and the production plans 6 of the increasing returns industry. The fact that the latter are equivalently represented by
a point on the simplex is precisely the content of Lemma 2. Given any 6, p
‘This
operate
is assuming,
of course, that
under marginal
cost pricing.
the increasing
returns
industry
is being constrained
to
378
BROWN
ET AL.
determines the corresponding normalized marginal rates of transformation
for the increasing returns industry. Given any (p, a), 4 determines the consumer demands and competitive supplies. Finally, 7-cprojects an arbitrary
non-negative vector in R”, onto the (n - l)-dimensional
simplex. Note that,
given our truncation,
x-y-w+K>e$O
(*)
and hence the mapping rc(. ) is well defined.
Let us now consider the composite map c1(.) which takes A x A x Xx Y
into itself such that
a4 4 x3Y) = /4@ x 5(P, 6) x 44 Y).
We now show that tl satisfies all the conditions of Kakutani’s fixed point
theorem. It is clear that given continuity of v( .), all arguments are standard
except for the fact that 4 is a non-empty, upper-hemi-continuous,
convexvalued map. The fact that N(z) is non-empty and convex follows from the
proposition in Section 2. To show that N(z) n A # 4, we first observe that,
as a consequence of free disposak6 (-R”,) G T(z). For any y E (-R”,),
simply consider the sequence yk = y for all k, corresponding to arbitrary
sequences zk -+ z, zk E b,J Y’), and t, LO. This implies that N(z) c R”, . It is
also clear that T(z) #R” so that N(z) # (0). Finally, upper-hemi-continuity
follows from Rockafellar [ 11, Corollary 2, Sect. 21 given that Int T(z) # 4
for any z. Hence a(. ) has a fixed point (p, 6, X, F).
It follows from (*) that S=rr(X,y)gO
so that, by Lemma 2, v(S)%O.
Hence, from the mapping pL(.), p are indeed marginal cost prices
corresponding to the production plan Jr = (v(S) - K) of the increasing
returns industry. Since, from the mapping <, we know that there exist
y E Yj such that cj= 1 y= j and j+j, s’(p), j= l,..., 1, we can appeal to
Assumption A4 to assert that M’(p, 6) > 0 for all i.’ From 4 we know that
for all i, there exist X’ E d’(p, 8) such that Cy!, X’ = X. Clearly,
((Z’), (J?), J?, ~7) would be a QMCPE if we can verify that X <.V +.V’+ u’.
Given that M’(& 8) > 0, we can write
pT<f M’(&s)=pw+py+g.
;=1
w
From the construction of rc(. ) we know that (X -- jj - w + K) = L(j’ + K),
where I is a positive real number. By appealing to (W) we can write this as
6 T(z) is the tangent space corresponding
to N(z), i.e., N(Z) is the polar cone of T(Z).
’ We are using the fact that, since P 2 ?J, aggregate income obtained by maximizing
profits
over P, is at least as high as that obtained
by maximizing
profits over P.
MARGINAL
COST PRICING
EQUILIBRIA
p(j’+K)>@(y’+K).
S’mce J’ + K = v(d)>>0 by Lemma
p( j’ + K) > 0. Thus, ,I < 1 and the proof is complete.
Proof
Theorem
379
2, we have
1. Define all the mappings as in the proof of
1 but with consumer incomes M’( .) modified as follows:
of Corollary
/=I
As in the proof of Theorem
1 we can obtain a fixed point (p, 6, X, j). Let
j’ = v(S) - K. Th en, it is clear that j’ E s’( p) and there exist J’ E s’(p),
j = l,..., i, such that xj=, g = j. Also, there exist X’ E A’( p, 6) such that, if
M’(p, 8) > 0, then xi6 Arg Max,,E,-,(P,gj u’(x’), for all i. The mapping rr
ensures, as before, that x <<y + j’ + w. Thus, ((Xi), ( y), j’, p) is a quasiWalrasian equilibrium. It is clear that A4 is automatically satisfied since Y’
is convex and 0 E fi, j = l,..., 1.
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