- Email: [email protected]
On the existence of equilibria in economies with increasing returns
Journal of Mathematical
Economics 17 (1988) 179-192. North-Holland
ON THE EXISTENCE OF EQUILIBRIA IN ECONOMIES INCREASING RETURNS*
WITH
Rajiv VOHRA BrownUniversity, Providence, RI
52912, USA lndian Statisticai Institute, New Delhi 110016, India
Submitted March 1986, accepted January 1988 We provide a result on the existence of equilibria in economies with non-convex production sets. The principal assumption is a boundary condition and an income hypothesis on a subset of the production sets. As corollaries of our main result, we obtain the results of Beato and Mas-Cole11 (1985), Brown et al. (1986) and the classical result on the existence of Walrasian equilibria in convex economies. The result of Bonnisseau and Cornet (1988) is similar, but not directly comparable, to ours.
1. Introduction There is now a substantial literature on the existence of equilibria in economies with increasing returns, or more generally, with non-convex production sets. The economic environment this literature analyzes is one in which firms with non-convex production sets, perhaps as a result of regulation, follow pricing rules which do not necessarily guarantee maximum profits. The profits are distributed among the consumers according to exogenously given rules, as in the Arrow-Debreu model. The notion of a marginal cost pricing equilibrium is a natural one to study in this context and the existence of such an equilibrium has been established by Beato (1976, 1982), Mantel (1976), Brown and Heal (1982), Cornet (1982), Beato and Mas-Cole11 (BM) (1985) and Brown, Heal, Khan and Vohra (BHKV) (1986). Of these, BM and BHKV are more general than the others. The existence of equilibrium with a wider class of pricing rules, which includes marginal cost pricing, has been shown by Dierker, Guesnerie and Neuefeind (DGN) (1985) and Kamiya (1984).1 However, these results are not directly comparable. The result in BM is derived from less primitive assumptions than in BHKV but does have the advantage of considering *The first version of this paper was circulated in 1985, as Brown University working paper no. 85-30. I am indebted to J.M. Bonnisseau, Pierre Dehez, Kayuza Kamiya, M. Ah Khan, Andreu Mas-Cole11 and an anonymous referee for helpful comments of an earlier version. Support from N.S.F. grants SES-8410229 and SES-8605630 is gratefully achnowledged. ‘The more recent results of Bonnisseau and Comet (1985, 1988) and Bonnisseau (1988) are discussed later in this section. 03044068/88/$3.50 0
1988, Elsevier Science Publishers B.V. (North-Holland)
R. Vohra, Existence of equilibria
180
many increasing returns firms. While the result in DGN is more general in terms of pricing rules and relates to many firms with increasing returns, it rules out firms with fixed costs or concave isoquants. In considering economies with non-convex technologies all these results make some crucial assumptions which are not required in proving the existence of Walrasian equilibrium in convex economies. BHKV rely on an income hypothesis along with a boundary condition on marginal costs. DGN use a different kind of income hypothesis and a boundary condition. While these assumptions are, in some sense, similar, the method of proof used by DGN is quite different from that of BM and BHKV. The former exploits the assumption that the input requirement correspondence is convex and upper-hemicontinuous. The latter rely on a homeomorphism between the boundary of a production set and the simplex - an approach that can be traced to Beato (1976) and Mantel (1976). The aim of this paper is to present a framework which facilitates a comparison of the above results. We provide an existence theorem for general pricing rules which yields as corollaries the results of BM, BHKV and the classical result on the existence of Walrasian equilibria in convex economies. While the assumptions used by Kamiya (1984) are not directly comparable to those of this paper, his result seems to be better suited for pricing rules which ensure each firm a non-negative profit [see Theorem 1 of Kamiya (1984)]. In that case, his result is also a corollary of the one presented below. It is also possible to derive from our result a variant of the DGN result. For a more recent and also more successful attempt at deriving the DGN result from a general existence result, see Bonnisseau (1988). Apart from the standard assumptions, the principal assumption we make is a modification of the boundary condition and income hypothesis of DGN. The method of proof, however, is simply a refinement of the approach used in BM and BHKV. Since this paper was first written, there have been additional papers on this subject which have further clarified the existence issue. Imposing the additional condition that all firms have production sets with smooth boundaries, Bonnisseau and Cornet (1985) have shown the existence of a marginal cost pricing equilibrium under the income hypothesis of BM. Bonnisseau and Cornet (1988) provide a result which also yields corollaries similar to those presented here. Their result is similar, but not directly comparable to ours, and it is not clear that there is a direct way to go from one to the other. We defer to section 3.3 some additional remarks on this comparison. 2. The model and result We consider
an economy
with m consumers,
indexed i= 1,. . . ,m, each
R. Vohra, Existence of equilibria
181
consumer i having a consumption set Xi and a utility function ui. There are n firms indexed j=l,..., n, with production sets J$ The aggregate endowment is denoted CU.A consumption plan is x = (xi,. . . ,x,) E X = niXi and a production plan is y =(yl, . . . , yJ E Y= njT* There are 1 commodities indexed h=l , . . .,I and all consumption and production sets are subsets of R’, the 1 dimensional Euclidean space. For any vector ZER’ we shall use z,, to denote the hth coordinate of z. 2>>z means that h,>z,, for all h, =l,...,I; i>z means that ihtz,, for all h, = l,... , 1 and at least one of these inequalities is strict; izz means that i,,zz,, for all h, = 1,. . . , 1. The positive orthant of R’ is denoted R’+ = {x E R’lx 2 O}. For x, y E R’, x. y denotes the scalar product. e is the vector in R’ all of whose coordinates are 1. For a set B E R’, aI(B) denotes its asymptotic cone and a(B) its boundary. Let the unit simplex of R’ bedenoted S={R:I~~=,s,,=l}. Given a vector of reference prices PCS, and an efficient production plan YE nj a( YJ, a pricing rule for firm j is defined as a mapping 4j: S x njc3( 5)~s. In many cases C#J~ will actually depend on Yj. In fact, we shall assume that this is the case for firm 1. A vector of reference prices will be a price vector which is consistent with the pricing rule of firm 1, i.e., p E c#J~(Y,).There is some abuse of notation in writing 4,(y,) rather than 4r(s,y) but this should not cause any confusion when the pricing rule of the firm actually depends only on its production plan. If the economy has a convex firm which maximizes profits we will usually consider it to be firm 1 and then, q5,(y,)={p~S(p-y,~p*y’ for all y’~Yi}. Given a production plan YE nj a( y) and p E 41(y,), the income of consumer i is defined by a continuous function ri(p, y). We shall assume that 17’ 1 ri(p, y) = p’ (xi”= 1yj + CO).The budget correspondence is denoted yi(p, y) = {xi E Xi Ip. Xi 5 ri(p, y)}. In a private ownership economy a la Arrow-Debreu, where each consumer i has a share Bij in firm j and endowment oi, ri(p, y) = cJ= 1 t3ijp* yj+p. Wi. We shall refer to this income distribution rule as one with a fixed structure of shares. The income distribution rule is said to be given by a fixed structure of revenues if ri(p, y) = ai(p, y) = ai(p. (xi”= i yj + w)), where ai > 0 for all i and cy= 1 a, = 1. Definition I.
- - _ An equilibrium is defined as (x, y, p) E X x Y x S such that
(i) for all i, xiEyi(p,y) and ui(Xi)zui(xi) for all xiEYi(ES,y), - (ii) for all j, PE bj(P, Y), (iii) C$i Xi~Ci”=1 jj+O. We shall make use of the following assumptions. A.l.
For all i, Xi c R$ is closed, convex and contains 0; ui( .) is continuous, quasi-concave and satisfies local non-satiation.
182
A.2. 5-R:
A.3.
R. Vohra, Existence
For
all j, 5 is closed, contains
of equilibria
0 and satisfies free disposal, i.e.,
s 5.
I;= 1 5 is closed and &‘(c;= 1 I;) n -d(&
1 rj) = (0).
In the classical setting of convex economies, these assumptions are standard, except for the fact that A.3 can be derived from more primitive assumptions. It should also be clear that if there exists a firm which maximizes profits and has a convex production set satisfying free disposal, then by the usual argument, as in Debreu (1959, pp. E&87), the weak inequality in condition (iii) of equilibrium can be replaced by an equality. We shall also assume that the pricing rule for each firm is well behaved. A.4 For all j, ~j: upper-hemicontinuous.
Sx
I-Ii
c?(QwS
is non-empty,
convex
valued
and
For k>O let K be the cube with edge 2k, i.e., K = {ZE R’IIz,,lI k for all h=l , . . . , l>, and let Z?= - {ke) + R’+ = K + R’+. Given A.l-A.3 it can be shown that there exists a k>O such that K contains in its interior all the attainable consumption and production sets (see, for example, Lemma 1 of BHKV). Moreover, it can also be shown [see Brown and Heal (1982)] that Condition 1. There exists k>O such that K contains all attainable sets in its interior and for all z E R’+, (I$+ {z]) n R$ is compact for all j.
Notice that (5 n I?) + (ke} = (q+ { ke)) n R$ . Our income hypothesis relates to production plans in some such R. We shall assume, roughly speaking, that given reference prices and production plans which are not too far from the attainable production sets, if the income of any consumer becomes non-positive, the pricing rule of some firm instructs it to raise the price of some output above its reference price or to lower the price of some input (being used at the highest level possible within R) below its reference price. A.5. There exists k>O which satisfies Condition 1 and the following: Suppose y1 E a( Y,), yj E a( 5) n R for j = 2,. . . , n and p E 4,(yi). If for some i, ri(p, y) 5 0, then there exists a firm j such that for all qj E ~j(p, y), there exists a commodity h for which yj,> -k and qj,,>ph or there exists a commodity h for which yjh= -k and qj,
with the others
183
R. Vohra, Existence of equilibria
We can now state the main result of this paper. Theorem 1. lf A.l-A.5.
are satisfied, then there exists an equilibrium.
Remark 1. Notice that A.5 is the only assumption which appears not to be standard. However, as we shall see in section 3, it is weaker than a correpsonding assumption which is implicitly made in many of the other existence results. It will also be clear from the proof of the theorem that the existence of a ‘quasi equilibrium’ can be established if ‘ri(p, v) S 0’ is weakened to ‘ri(p, y) < 0’ in A.5. Remark 2. Following Hurwicz and Reiter (1973), the irreversibility assumption, A.3, can be used to show that the attainable set is compact. And, as Bonnisseau and Cornet (1988) point out, assuming the attainable set to be compact is weaker than assuming A.3. Nevertheless, it is worth emphasizing that the irreversibility assumption is an assumption on the basic data of the economy, unlike the assumption that the attainable set is compact. In our result, A.3 is used only to justify Condition 1. Our result, therefore, is valid even if A.3 does not hold but Condition 1 and A.5 are satisfied. Proof of Theorem I. Suppose k satisfies Condition hypothesis as specified in AS. Let
1 and the income
f=(m+n)ke+w, Ri=Xin
K
for
i=l,...,m,
E(q)=(a(q+{ke}))nR$ For j=l,...,n,
for
j=2,...,n.
let
E(~):-)={ZEE(~)l~Z’EE(~) such that z’ 5 z and zi < z,, for all h for which z,, >O}. From Condition 1 it follows that E( 5) is compact for all j = 1,. . . , n. We can now appeal to Lemma 2 of BHKV to assert that E(y) is homeomorphic to S for all j. Moreover, the homeomorphism can be defined by a function vi: ShE( I;:, such that Vj(Sj)= tSj for some t >O. Thus Vj(Sj)h>O if and only if Sj,,>O. Let JJ~(s~)=v,(s,)--~ and yj(sj)=vj(sj)-k for j=2,...,n. Notice that
184
R. Vohra, Existence of equilibria
Let s=(s~,...,s,)ES” where S” for any j=2,..., n and sj~S, yj(sj)~a(~)nk denotes the n-fold Cartesian product of S. Correspondingly, y(s)= (Yl(sl), . . . , y,(s,)). If ~j(P,y) satisfies A.4, SO does 4j(p,y(s)) which maps from S“+I to s. Let yi(p, y)=y,(p, y) n Xi. We can now define for each consumer a modified demand correspondence ti: S”+ ‘-Xi as 5i(P, s)= {xiEYi(P3Y( =
YAP,
Sj)I U,4XiI=>U, 4 X,0 f or
all
X;ETi(P,y(S))}
Y(s))
=o
if
ri(p, y(S))> 0,
if
rdp,Y(S))= 0,
otherwise.
By A.l, this correspondence is non-empty, convex valued and upperhemicontinuous for all i. To ensure that all producer prices are identical to the reference prices we define for all firms, except firm 1, a mapping fi: S3t+S which adjusts its production according to the deviation between its price qi and the reference price p. For j=2,..., n, pi is defined as follows: sj, +
max(0,Ph-
qjJ
Certainly, bj is a continuous function for all j = 2,. . . , n. The production of firm 1 is adjusted in accordance with the aggregate excess demand of the rest of the economy. Let rc: X x S”HS be defined by
From the definition of k and f it is clear that (I?= I xi --cj”= 2 Yj-0 + f) >>O. Thus, 7cis a well-defined, continuous function. Finally, let q = (qz, . . . , q.) ES”- 1 and let @IS’”x X be defined by @(S, p, 4, X)
=x(x, S)X
fi j=2
Bj(sj,
P? 4j) x n
4j(P3
Yts))
’ 7
li(Py
s)*
j
Given A.4, @Jsatisfies the conditions of Kakutani’s fixed point theorem _-_and therefore has a fixed point (s,p,q, x), where SI E X(X,9, SjE fij(gj, p, gj), j=2 ,..., n, PE$J~(Y(~), 4jE~j(P,y(~), j=2 ,..., n and xiE[i(Pyg, i=l,...,m. Let Y= y($. We shall now show that (X,Y,p) is an equilibrium. From the mapping /I?>it follows that for all j = 2,. . . , n,
R. Vohra, Existence of equilibria
S;.hi
(max(O,&-qj,))=max(O,L%-Lfj,),
185
h=l,...,
1.
(1)
h=l
This yields the following condition: If p # qj,
then ph > gjh for h such that S;.,,> 0, ph 5 sj, for
h
such that Sjl,= 0.
and (2)
- We can now show that ri(p, y) >O for all i. Suppose not. Then, given that - qj~~j(@,j$ Yjoa(Yj)n R for all j=2,..., n and FE 4r(p, v), we can use A.5 to assert that there exists a tirm j for which ~?#@j and there exists a commodity h such that ~ji,> -k and 4j,>Ph or there exists a commodity h such that -j5.,,=-k and Qjhcph. In either case this would contradict (2), since from the mapping vj we know that yjh > -k, if and only if gj,,> 0. Since r&, j) >O for all i, from the definition of & we get p.(cKr Xi)5 0. (xi”= 1jj+ co), which can be rewritten as
(3) From the construction
(2
(Yl+f)=t
i=l
where t is c;=z jj-w-f).
a
of rc we get Xi-
i j=t
_Vj-U+f
1 3
(4)
positive number. This yields p*(yr +f) = t~~(~~= 1 XiSubstituting (3) in this equation we have
from the definition of f and k that (IF= I xiit fo 11ows from the construction of rc and the definition of y1 that SE= t’(jjl + f) ~0, where t’>O. Now (5) implies that t 2 1. Substituting this in (4), we get Since
we
know
C;=zYj-O+f)>>O,
(6) which is simply the feasibility condition (iii) of equilibrium. Since FE or and ~~~ (P, p) for all j = 2,. . . , n, to verify condition (ii) of equilibrium we need to show that for all j=2,. ..,n, ~j=~. Suppose not, i.e.,
186
R. Vohra, Existence of equilibria
for some j, 4j #p. Since we have already shown that all production plans are feasible, jj>> - ke for all j, which means that gjij>>O. We can again use (2) to assert that this implies j&,> qj,, for all h. But since, p, 4j~ S, this is impossible. Thus condition (ii) of equilibrium is satisfied. Since r,(& jj))>O for all i, given A.1, we can apply the usual argument, as in Debreu (1959, p. 87), to the mappings ti to show that condition (i) of equilibrium is also satisfied by (X, y, j). 0 3. Relationship with the literature In this section we relate our result to the other existence results in the literature. In drawing out this relationship we shall need to suitably interpret the pricing rule and we begin, therefore, by considering the formal specilication of pricing rules that are of particular interest. Firm j is said to follow marginal cost pricing if ~j(P, y) = S n N( q, yj), where N( q, Yi) is Clarke’s normal cone of rj at yj [see, for example, Clarke (1983)]. Since this pricing rule does not depend on the reference prices or on the production plans of the other firms, there should be no confusion if we write $j(y) or ~j(yj) instead of cbj(p,y). Clarke’s normal cone was first applied to our problem by Cornet (1982) and was also used in BHKV for defining marginal cost prices. A justification for this definition of marginal cost pricing may be found in section 2 of Khan and Vohra (1987). We can, therefore, consider a marginal cost pricing equilibrium to be an equilibrium with ~j(y) = S n N( I$ yj) for all j. If q is convex and yj E a( k;), then N( 5, yj) coincides with the cone of normals to Yj at yj in the sense of convex analysis (see, for example, the Proposition in BHKV). In other words, in the classical context of convex production sets S n N(I$yJ is precisely the set of normalized prices for which yj is a profit maximizing production plan for firm j. Therefore, if all production sets are convex, we can define a Walrasian equilibrium to be a marginal cost pricing equilibrium. It can also be shown that if Yj satisfies A.2, then S n N(Yj, yj) satisfies assumption A.4 [see for example BHKV nor Cornet (1982)] and we can, therefore, deduce from our theorem that a marginal cost pricing equilibrium exists if A.l-A.3 and A.5 are satisfied. As we shall verify in Corollaries 3 and 4, this is a generalization of the BM and BHKV results. Firm j is said to follow a loss-free pricing rule if qj. yj 20 for all qj E 4j(p, y). One example of the loss-free pricing rule is average cost pricing. Another example is marginal cost pricing for a firm with a convex production set containing 0. 3.1. Existence of equilibrium with loss-free pricing rules
As a straightforward
consequence
of our theorem,
we can show the
R. Vohra, Existence of equilibria
187
existence of equilibrium when all firms follow pricing rules which ensure nonnegative profits. The classical result on the existence of a Walrasian equilibrium in an economy with convex production sets is a special case of this result. Corollary 1. Suppose A.l-A.4 are satisfied and the income distribution is given by a fixed structure of shares with wi>>O,for all i. If all firms follow loss-free pricing rules, then there exists an equilibrium. Proof Since there exists k>O satisfying Condition 1, it suffices to show that, under the conditions of Corollary 1, A.5 is satisfied for any k which satisfies Condition 1. Consider any such k and suppose y, E a( Y,), YjE a( 5) n R for j=2,... ,n, p~q5t(y,) and for some i,ri(p,y)=xj”=I Bijp.yj+p.wi~O. Since w,>>O, this implies that there exists a firm j for which p. YjO. Since qj,pES, (qj-p)*ke=O and we get (qj-p).(yj-ke)>O. Certainly, for all h=l,...,l, yjh~ -k since Yj~R. But this must imply that there exists a commodity h such that yjh> -k and qjh>ph. Since this holds for all qjEr#Jj(p,y) it completes the proof that A.5 is satisfied. 0
A slightly weaker version of Corollary 1 can also be deduced from Kamiya (1984). It should also be clear from the proof that Corollary 1 can be strengthened to allow firms to make losses, provided these are restricted to be less than mini,Jwih). Since the marginal cost pricing rule satisfies A.4 and, in an economy with convex production sets, it also ensures each firm a non-negative profit we can also deduce the classical result on the existence of a Walrasian equilibrium from Corollary 1. Corollary 2. Suppose A.&A.3 are satisfied and the income distribution is given by a fixed structure of shares with Wi>>O,for all i. If all firms have convex production sets, then there exists a Walrasian equilibrium. 3.2. Existence of a marginal cost pricing equilibrium
We begin by showing that the BM result can be derived as a special case of ours, All firms are assumed to follow the marginal cost pricing rule. Instead of assumptions A.3 and A.5 above, BM make assumptions which are essentially equivalent to the following: BM.1 For all j, I$=Kj-R’,., where Kj is non-empty, exists k > 0, such that z 2 - ke for all z E Kj.
compact and there
188
R. Vohra, Existence of equilibria
BM.~.
For all j, if yjhs -k and 4jE 4j(Y), then qj,,=O*
BM.3.
If y E nj a( 5) and p E 4j(Y) for all j, then p *(Cy= 1 yj+ O) > 0.
BM.4.
If p.(syj+o)
>O, then ri(p,y)>O for all i.
Corollary 3. Suppose A.1 and BM.l-BM.4 marginal cost pricing equilibrium.
are satisfied.
Then there exists a
Proof. Clearly, BM.l implies A.2 and A.3. Since all firms follow marginal cost pricing, A.4. is satisfied and it only remains to be shown that BM.2BM.4 imply AS. Choose k’> k such that Condition 1 is satisfied for k’. We shall show that A.5 holds for Z?‘. Suppose y1 E a(YJ, yj~a(~) A R’ for j=2,. . . ,n, p E +I(yI) and for some i, ri(p, y) 5 0. By BM.4, this implies that ~~=I ri(p, y) 5 0. By BM.3, this means that there exists a firm j such that P$~~(Y). Clearly then, given that PES and $j(Y) s S, for any qjE~j(y) there must be a commodity h such that qj,,>ph. By BM.2, it cannot be the case yjh= -k, i.e., yj, > -k > - k’ and this completes the proof that A.5 is satisfied. 0 Remark 3. In the context of marginal cost pricing, it is possible to deduce BM.l and BM.2 from more primitive assumptions. Suppose the production sets satisfy A.2 and A.3. Let K be large enough so that the cube R, which has edge 2(k- l), contains in its interior all the attainable production sets and define Kj= Yjn Z?. Certainly, ~j=Kj- R’+ satisfy BM.l. It is also not difficult to check that if yjh 5 -k and qj~ N( q, yj), then q,,=O, i.e., for the marginal cost pricing rule, Yj satisfy BM.2. Thus, by considering q rather than 5 we can drop BM.l and BM.2. However, in that case, BM.3 would also have to relate to ~j rather than rj.
We now turn to the BHKV result. They assume that there is only one firm which has a non-convex production set. Let this be firm 1. All firms follow the marginal cost pricing rule, income distribution is given by a fixed structure of revenues and instead of A.5 consider the following assumption. BHKK 1. Suppose yr~a(Y,), p~$,(y,) and p*~.~Zp.yj for all y.\~q, j= 2, . . . , n, where c is the jth attainable production set. Then p *(cJ!=1 yj+ co)> 0.
The only difference between this and the corresponding assumption in BHKV is that they assume that income is non-negative while we assume it to be strictly positive. This is explained by the fact that they consider a quasi equilibrium. Given this, the following can be seen as the BHKV result.
R. Vohra, Existence of equilibria
189
Corollary 4. Suppose A.l-A.3 and BHKVIl are satisfied. If the income is given by a fixed structure of revenues, then there exists a marginal cost pricing equilibrium. Proof: Given A.l-A.3, the attainable production sets q are compact. Consider the economy with F., j=2,. . . , n as defined in Remark 3. Since $cI;., and all firms j=2 ,..., n have convex production sets, given that they follow marginal cost pricing, BHKV.l is now the same as BM.3. As mentioned in Remark 3, the marginal cost pricing rule also satisfies the boundary condition BM.2 for the economy so constructed. Given a fixed structure of revenues and BHKV.l we can follow the same argument as in the proof of Corollary 3 to prove that this economy has a marginal cost - pricing equilibrium (x, y,p). Since j belongs to the attainable sets and jj E nj a( q), it follows that jj E flj a( q). Thus, (X,J, p) is an equilibrium of the original economy. 0
3.3. Existence of equilibrium with bounded losses pricing rules While a direct comparison of our result with that of Bonnisseau and Cornet (1988) is not possible, here we provide an example of an economy in which the bounded losses assumption of Bonnisseau and Cornet (1986)2 is not satisfied but, given Remark 2, it is possible to use our result to deduce that an equilibrium exists. Consider an economy in which there are two firms with the following production sets:
Firm 1 cannot produce any output while firms 2’s technology is the same as that of firm 2 in the BM example. There are two consumers such that their preferences satisfy A.1 above but, unlike the consumers of the BM example, both consumers have strictly positive endowments. Let o1 = (7,3) and o2 =(l, 1). Suppose consumer 1 owns both the firms and that both firms follow the marginal cost pricing rule. It is easy to see that in this example the losses of the increasing returns firm are not bounded, as postulated in A.3(ii) of Bonnisseau and Cornet (1988). With respect to our result, it is easy to see that A.l, A.2 and A.4 are satisfied. Let k= 10. Notice that the cube K with edge 2k contains the production possibility curve in its interior. It is then easy to see that Condition 1 is satisfied for k= 10. We can now verify that A.5 holds for k= 10. Since both firms follow the marginal cost pricing rule, *The assumption, all (P,Y) ES X nja(I;),
A.3(ii) of their paper, infdJj(P,Y)
‘YjLaj
is that for every firm j there exists ~(~60 such that for
R. Vohra, Existence oJ equilibria
190
4d~J=K4
1) if
YC~
=s
if
= (LO)
otherwise.
y,=O,
h(P1y)=t(-~y2,,1) wheret=
_kyi +1
if
Y,,
I
=s
if
=(LO)
otherwise.
y,=O,
Suppose y, E Y,, y, E a( Y,) n I? and PE i(yi). It is clear that r,(p, y) = p*(l,l)>O. Moreover, for any qE+*(p,y) and y2Ea(Y,)nR, q-y,2 c#I&,(- 10,25/4)).(10,25/4)= -y> -3. Since p*(7,3)23, this implies that q.y,+p.y,+p.o,>O. Thus, if r,(p,y)=p.(y,+y,+o,)~O, then for any q E 4&, y), q. y, >p’yz. We can now derive the conclusion of A.5 by following the same argument as in the proof of Corollary 1. A.3 is, therefore, the only assumption of our theorem which is not satisfied. Since Condition 1 and A.5 are also satisfied, given Remark 2, we can use our result to assert that equilibrium exists. Existence of equilibrium in this particular example can also be deduced from Bonnisseau and Cornet (1985). The latter result, however, relies crucially on the smoothness of the production functions and would not apply if the example was modified to make the production functions non-differentiable. 3.4. Existence of equilibrium under special pricing rules ci la DGN It is possible to derive a variant of the DGN result from our main result. Since we have not been able to obtain the exact result of DGN we shall only indicate briefly the progress in this direction. A more detailed version of our reformulation of the DGN result is available from the author upon request. For a more recent attempt in this direction the reader is referred to Bonnisseau (1988) which uses the methods of Bonnisseau and Cornet (1988) to derive a result which is much closer to the DGN result. DGN assume that there are two kinds of firms in the economy; C-firms with convex production sets which maximize profits and ZZ-firms which follow special pricing rules. The C-firms are described by their aggregate production set and we may, therefore, assume that firm 1 is the only C-firm. In our notation firms (2,. . . , n} are the U-firms. A n-firm j produces a set of commodities rtj which are not produced by any other firm in the economy. Let the set of commodities produced by the H-sector be denoted K and rL={l,..., I,}. The U-firms are assumed to have convex isoquants and
R. Vohra, Existence of equilibria
191
minimize costs given input prices and a vector of outputs. Each ZI-firm then sets output prices in accordance with a pricing rule I$~:R’ x fljd(lJ~R7, which is assumed to be non-empty, upper-hemicontinuous and compact and convex valued. Notice that the prices set according to this rule are not normalized. They also make the following assumption: DGN.2.
For
any j, j=2 ,..., n, qjE~j(P,y),
yj,=O
implies that
(qj,,...,
t4j,9.. .3qj,n)E 4j(P,Y) for all t E I3 11. In their concluding section DGN replace this assumption by another assumption on non-inferiority. Neither of these assumptions fits in naturally with the price normalization which is used in our basic model. Consider the following asumption. A.6.
For any j, j=2 ,..., n and any qjo4j(PtYL Chprrqjh=&EnPh.
This assumption is an important distinguishing feature of our reformulation of the DGN model. It allows us to construct a pricing rule for each firm which takes values in S. The income hypothesis of DGN is very similar to A.5 above and by assuming A.6 instead of DGN.l it is possible to use our result to establish the existence of an equilibrium with special pricing rules a la DGN. References cost pricing equilibria with increasing returns, Ph.D. dissertation Beato, P., 1976, Marginal .._. (University of Minnesota, Mmneapohs, MN). Beato, P., 1982, The existence of marginal cost pricing equilibria with increasing returns, Quarterly Journal of Economics 79, 669-688. Beato, P. and A. Mas-Colell, 1985, On marginal cost pricing with given tax-subsidy rules, Journal of Economic Theory 37, 356365. Bonnisseau, J.M., 1988, On the existence of equilibria results in economies with increasing returns, Journal of Mathematical Economics 17, this issue. Bonnisseau, J.M. and B. Cornet, 1985, Existence of marginal cost pricing equilibria in an economy with several non convex firms, CORE discussion paper no. 8723 (Universiti CathoIique de Louvain, Louvain-!a-Neuve). Bonnisseau, J.M. and B. Cornet, 1988, Existence of equilibria when firms follow bounded losses pricing rules, Journal of Mathematical Economics 17, this issue. Brown, D.G. and G.M. Heal, 1982, Existence, local uniqueness and optimality of a marginal cost pricing equilibrium with increasing returns, Social science working paper no. 415 (California Institute of Technology, Pasadena, CA). Brown, D.J., GM. Heal, M. Ah Khan and R. Vohra, 1986, On a general existence theorem for marginal cost pricing equilibria, Journal of Economic Theory 38, 371-379. Clarke, F.H., 1983, Optimization and nonsmooth analysis (Wiley, New York). Cornet, B., 1982, Existence of equilibria with increasing returns, in: B. Cornet and H. Tulkens, eds., Contributions to economics and operations research: The XXth anniversary of CORE (The MIT Press, Cambridge, MA) (forthcoming). Dierker, E., R. Guesnerie and W. Neuefeind, 1985, General equilibrium when some firms follow special pricing rules, Econometrica 53, 1369-1393.
192
R. Vohra, Existence of equilibria
Hurwicz, L, and S. Reiter, 1973, On the boundedness of the feasibility set without convexity assumptions, International Economic Review 14, 58Ck586. Kamiya, K., 1984, Existence and uniqueness of equilibria with increasing returns, Journal of Mathematical Economics 17, this issue. Khan, M. Ah and R. Vohra, 1987, An extension of the second welfare theorem to economies with nonconvexities and public goods, Quarterly Journal of Economics 102, 223-241. Mantel, R., 1976, Existence of equilibria with Pareto optimality in a general equilibrium model with non-convex production possibilities, Mimeo. (Yale University, New Haven, CT).
Economics 17 (1988) 179-192. North-Holland
ON THE EXISTENCE OF EQUILIBRIA IN ECONOMIES INCREASING RETURNS*
WITH
Rajiv VOHRA BrownUniversity, Providence, RI
52912, USA lndian Statisticai Institute, New Delhi 110016, India
Submitted March 1986, accepted January 1988 We provide a result on the existence of equilibria in economies with non-convex production sets. The principal assumption is a boundary condition and an income hypothesis on a subset of the production sets. As corollaries of our main result, we obtain the results of Beato and Mas-Cole11 (1985), Brown et al. (1986) and the classical result on the existence of Walrasian equilibria in convex economies. The result of Bonnisseau and Cornet (1988) is similar, but not directly comparable, to ours.
1. Introduction There is now a substantial literature on the existence of equilibria in economies with increasing returns, or more generally, with non-convex production sets. The economic environment this literature analyzes is one in which firms with non-convex production sets, perhaps as a result of regulation, follow pricing rules which do not necessarily guarantee maximum profits. The profits are distributed among the consumers according to exogenously given rules, as in the Arrow-Debreu model. The notion of a marginal cost pricing equilibrium is a natural one to study in this context and the existence of such an equilibrium has been established by Beato (1976, 1982), Mantel (1976), Brown and Heal (1982), Cornet (1982), Beato and Mas-Cole11 (BM) (1985) and Brown, Heal, Khan and Vohra (BHKV) (1986). Of these, BM and BHKV are more general than the others. The existence of equilibrium with a wider class of pricing rules, which includes marginal cost pricing, has been shown by Dierker, Guesnerie and Neuefeind (DGN) (1985) and Kamiya (1984).1 However, these results are not directly comparable. The result in BM is derived from less primitive assumptions than in BHKV but does have the advantage of considering *The first version of this paper was circulated in 1985, as Brown University working paper no. 85-30. I am indebted to J.M. Bonnisseau, Pierre Dehez, Kayuza Kamiya, M. Ah Khan, Andreu Mas-Cole11 and an anonymous referee for helpful comments of an earlier version. Support from N.S.F. grants SES-8410229 and SES-8605630 is gratefully achnowledged. ‘The more recent results of Bonnisseau and Comet (1985, 1988) and Bonnisseau (1988) are discussed later in this section. 03044068/88/$3.50 0
1988, Elsevier Science Publishers B.V. (North-Holland)
R. Vohra, Existence of equilibria
180
many increasing returns firms. While the result in DGN is more general in terms of pricing rules and relates to many firms with increasing returns, it rules out firms with fixed costs or concave isoquants. In considering economies with non-convex technologies all these results make some crucial assumptions which are not required in proving the existence of Walrasian equilibrium in convex economies. BHKV rely on an income hypothesis along with a boundary condition on marginal costs. DGN use a different kind of income hypothesis and a boundary condition. While these assumptions are, in some sense, similar, the method of proof used by DGN is quite different from that of BM and BHKV. The former exploits the assumption that the input requirement correspondence is convex and upper-hemicontinuous. The latter rely on a homeomorphism between the boundary of a production set and the simplex - an approach that can be traced to Beato (1976) and Mantel (1976). The aim of this paper is to present a framework which facilitates a comparison of the above results. We provide an existence theorem for general pricing rules which yields as corollaries the results of BM, BHKV and the classical result on the existence of Walrasian equilibria in convex economies. While the assumptions used by Kamiya (1984) are not directly comparable to those of this paper, his result seems to be better suited for pricing rules which ensure each firm a non-negative profit [see Theorem 1 of Kamiya (1984)]. In that case, his result is also a corollary of the one presented below. It is also possible to derive from our result a variant of the DGN result. For a more recent and also more successful attempt at deriving the DGN result from a general existence result, see Bonnisseau (1988). Apart from the standard assumptions, the principal assumption we make is a modification of the boundary condition and income hypothesis of DGN. The method of proof, however, is simply a refinement of the approach used in BM and BHKV. Since this paper was first written, there have been additional papers on this subject which have further clarified the existence issue. Imposing the additional condition that all firms have production sets with smooth boundaries, Bonnisseau and Cornet (1985) have shown the existence of a marginal cost pricing equilibrium under the income hypothesis of BM. Bonnisseau and Cornet (1988) provide a result which also yields corollaries similar to those presented here. Their result is similar, but not directly comparable to ours, and it is not clear that there is a direct way to go from one to the other. We defer to section 3.3 some additional remarks on this comparison. 2. The model and result We consider
an economy
with m consumers,
indexed i= 1,. . . ,m, each
R. Vohra, Existence of equilibria
181
consumer i having a consumption set Xi and a utility function ui. There are n firms indexed j=l,..., n, with production sets J$ The aggregate endowment is denoted CU.A consumption plan is x = (xi,. . . ,x,) E X = niXi and a production plan is y =(yl, . . . , yJ E Y= njT* There are 1 commodities indexed h=l , . . .,I and all consumption and production sets are subsets of R’, the 1 dimensional Euclidean space. For any vector ZER’ we shall use z,, to denote the hth coordinate of z. 2>>z means that h,>z,, for all h, =l,...,I; i>z means that ihtz,, for all h, = l,... , 1 and at least one of these inequalities is strict; izz means that i,,zz,, for all h, = 1,. . . , 1. The positive orthant of R’ is denoted R’+ = {x E R’lx 2 O}. For x, y E R’, x. y denotes the scalar product. e is the vector in R’ all of whose coordinates are 1. For a set B E R’, aI(B) denotes its asymptotic cone and a(B) its boundary. Let the unit simplex of R’ bedenoted S={R:I~~=,s,,=l}. Given a vector of reference prices PCS, and an efficient production plan YE nj a( YJ, a pricing rule for firm j is defined as a mapping 4j: S x njc3( 5)~s. In many cases C#J~ will actually depend on Yj. In fact, we shall assume that this is the case for firm 1. A vector of reference prices will be a price vector which is consistent with the pricing rule of firm 1, i.e., p E c#J~(Y,).There is some abuse of notation in writing 4,(y,) rather than 4r(s,y) but this should not cause any confusion when the pricing rule of the firm actually depends only on its production plan. If the economy has a convex firm which maximizes profits we will usually consider it to be firm 1 and then, q5,(y,)={p~S(p-y,~p*y’ for all y’~Yi}. Given a production plan YE nj a( y) and p E 41(y,), the income of consumer i is defined by a continuous function ri(p, y). We shall assume that 17’ 1 ri(p, y) = p’ (xi”= 1yj + CO).The budget correspondence is denoted yi(p, y) = {xi E Xi Ip. Xi 5 ri(p, y)}. In a private ownership economy a la Arrow-Debreu, where each consumer i has a share Bij in firm j and endowment oi, ri(p, y) = cJ= 1 t3ijp* yj+p. Wi. We shall refer to this income distribution rule as one with a fixed structure of shares. The income distribution rule is said to be given by a fixed structure of revenues if ri(p, y) = ai(p, y) = ai(p. (xi”= i yj + w)), where ai > 0 for all i and cy= 1 a, = 1. Definition I.
- - _ An equilibrium is defined as (x, y, p) E X x Y x S such that
(i) for all i, xiEyi(p,y) and ui(Xi)zui(xi) for all xiEYi(ES,y), - (ii) for all j, PE bj(P, Y), (iii) C$i Xi~Ci”=1 jj+O. We shall make use of the following assumptions. A.l.
For all i, Xi c R$ is closed, convex and contains 0; ui( .) is continuous, quasi-concave and satisfies local non-satiation.
182
A.2. 5-R:
A.3.
R. Vohra, Existence
For
all j, 5 is closed, contains
of equilibria
0 and satisfies free disposal, i.e.,
s 5.
I;= 1 5 is closed and &‘(c;= 1 I;) n -d(&
1 rj) = (0).
In the classical setting of convex economies, these assumptions are standard, except for the fact that A.3 can be derived from more primitive assumptions. It should also be clear that if there exists a firm which maximizes profits and has a convex production set satisfying free disposal, then by the usual argument, as in Debreu (1959, pp. E&87), the weak inequality in condition (iii) of equilibrium can be replaced by an equality. We shall also assume that the pricing rule for each firm is well behaved. A.4 For all j, ~j: upper-hemicontinuous.
Sx
I-Ii
c?(QwS
is non-empty,
convex
valued
and
For k>O let K be the cube with edge 2k, i.e., K = {ZE R’IIz,,lI k for all h=l , . . . , l>, and let Z?= - {ke) + R’+ = K + R’+. Given A.l-A.3 it can be shown that there exists a k>O such that K contains in its interior all the attainable consumption and production sets (see, for example, Lemma 1 of BHKV). Moreover, it can also be shown [see Brown and Heal (1982)] that Condition 1. There exists k>O such that K contains all attainable sets in its interior and for all z E R’+, (I$+ {z]) n R$ is compact for all j.
Notice that (5 n I?) + (ke} = (q+ { ke)) n R$ . Our income hypothesis relates to production plans in some such R. We shall assume, roughly speaking, that given reference prices and production plans which are not too far from the attainable production sets, if the income of any consumer becomes non-positive, the pricing rule of some firm instructs it to raise the price of some output above its reference price or to lower the price of some input (being used at the highest level possible within R) below its reference price. A.5. There exists k>O which satisfies Condition 1 and the following: Suppose y1 E a( Y,), yj E a( 5) n R for j = 2,. . . , n and p E 4,(yi). If for some i, ri(p, y) 5 0, then there exists a firm j such that for all qj E ~j(p, y), there exists a commodity h for which yj,> -k and qj,,>ph or there exists a commodity h for which yjh= -k and qj,
with the others
183
R. Vohra, Existence of equilibria
We can now state the main result of this paper. Theorem 1. lf A.l-A.5.
are satisfied, then there exists an equilibrium.
Remark 1. Notice that A.5 is the only assumption which appears not to be standard. However, as we shall see in section 3, it is weaker than a correpsonding assumption which is implicitly made in many of the other existence results. It will also be clear from the proof of the theorem that the existence of a ‘quasi equilibrium’ can be established if ‘ri(p, v) S 0’ is weakened to ‘ri(p, y) < 0’ in A.5. Remark 2. Following Hurwicz and Reiter (1973), the irreversibility assumption, A.3, can be used to show that the attainable set is compact. And, as Bonnisseau and Cornet (1988) point out, assuming the attainable set to be compact is weaker than assuming A.3. Nevertheless, it is worth emphasizing that the irreversibility assumption is an assumption on the basic data of the economy, unlike the assumption that the attainable set is compact. In our result, A.3 is used only to justify Condition 1. Our result, therefore, is valid even if A.3 does not hold but Condition 1 and A.5 are satisfied. Proof of Theorem I. Suppose k satisfies Condition hypothesis as specified in AS. Let
1 and the income
f=(m+n)ke+w, Ri=Xin
K
for
i=l,...,m,
E(q)=(a(q+{ke}))nR$ For j=l,...,n,
for
j=2,...,n.
let
E(~):-)={ZEE(~)l~Z’EE(~) such that z’ 5 z and zi < z,, for all h for which z,, >O}. From Condition 1 it follows that E( 5) is compact for all j = 1,. . . , n. We can now appeal to Lemma 2 of BHKV to assert that E(y) is homeomorphic to S for all j. Moreover, the homeomorphism can be defined by a function vi: ShE( I;:, such that Vj(Sj)= tSj for some t >O. Thus Vj(Sj)h>O if and only if Sj,,>O. Let JJ~(s~)=v,(s,)--~ and yj(sj)=vj(sj)-k for j=2,...,n. Notice that
184
R. Vohra, Existence of equilibria
Let s=(s~,...,s,)ES” where S” for any j=2,..., n and sj~S, yj(sj)~a(~)nk denotes the n-fold Cartesian product of S. Correspondingly, y(s)= (Yl(sl), . . . , y,(s,)). If ~j(P,y) satisfies A.4, SO does 4j(p,y(s)) which maps from S“+I to s. Let yi(p, y)=y,(p, y) n Xi. We can now define for each consumer a modified demand correspondence ti: S”+ ‘-Xi as 5i(P, s)= {xiEYi(P3Y( =
YAP,
Sj)I U,4XiI=>U, 4 X,0 f or
all
X;ETi(P,y(S))}
Y(s))
=o
if
ri(p, y(S))> 0,
if
rdp,Y(S))= 0,
otherwise.
By A.l, this correspondence is non-empty, convex valued and upperhemicontinuous for all i. To ensure that all producer prices are identical to the reference prices we define for all firms, except firm 1, a mapping fi: S3t+S which adjusts its production according to the deviation between its price qi and the reference price p. For j=2,..., n, pi is defined as follows: sj, +
max(0,Ph-
qjJ
Certainly, bj is a continuous function for all j = 2,. . . , n. The production of firm 1 is adjusted in accordance with the aggregate excess demand of the rest of the economy. Let rc: X x S”HS be defined by
From the definition of k and f it is clear that (I?= I xi --cj”= 2 Yj-0 + f) >>O. Thus, 7cis a well-defined, continuous function. Finally, let q = (qz, . . . , q.) ES”- 1 and let @IS’”x X be defined by @(S, p, 4, X)
=x(x, S)X
fi j=2
Bj(sj,
P? 4j) x n
4j(P3
Yts))
’ 7
li(Py
s)*
j
Given A.4, @Jsatisfies the conditions of Kakutani’s fixed point theorem _-_and therefore has a fixed point (s,p,q, x), where SI E X(X,9, SjE fij(gj, p, gj), j=2 ,..., n, PE$J~(Y(~), 4jE~j(P,y(~), j=2 ,..., n and xiE[i(Pyg, i=l,...,m. Let Y= y($. We shall now show that (X,Y,p) is an equilibrium. From the mapping /I?>it follows that for all j = 2,. . . , n,
R. Vohra, Existence of equilibria
S;.hi
(max(O,&-qj,))=max(O,L%-Lfj,),
185
h=l,...,
1.
(1)
h=l
This yields the following condition: If p # qj,
then ph > gjh for h such that S;.,,> 0, ph 5 sj, for
h
such that Sjl,= 0.
and (2)
- We can now show that ri(p, y) >O for all i. Suppose not. Then, given that - qj~~j(@,j$ Yjoa(Yj)n R for all j=2,..., n and FE 4r(p, v), we can use A.5 to assert that there exists a tirm j for which ~?#@j and there exists a commodity h such that ~ji,> -k and 4j,>Ph or there exists a commodity h such that -j5.,,=-k and Qjhcph. In either case this would contradict (2), since from the mapping vj we know that yjh > -k, if and only if gj,,> 0. Since r&, j) >O for all i, from the definition of & we get p.(cKr Xi)5 0. (xi”= 1jj+ co), which can be rewritten as
(3) From the construction
(2
(Yl+f)=t
i=l
where t is c;=z jj-w-f).
a
of rc we get Xi-
i j=t
_Vj-U+f
1 3
(4)
positive number. This yields p*(yr +f) = t~~(~~= 1 XiSubstituting (3) in this equation we have
from the definition of f and k that (IF= I xiit fo 11ows from the construction of rc and the definition of y1 that SE= t’(jjl + f) ~0, where t’>O. Now (5) implies that t 2 1. Substituting this in (4), we get Since
we
know
C;=zYj-O+f)>>O,
(6) which is simply the feasibility condition (iii) of equilibrium. Since FE or and ~~~ (P, p) for all j = 2,. . . , n, to verify condition (ii) of equilibrium we need to show that for all j=2,. ..,n, ~j=~. Suppose not, i.e.,
186
R. Vohra, Existence of equilibria
for some j, 4j #p. Since we have already shown that all production plans are feasible, jj>> - ke for all j, which means that gjij>>O. We can again use (2) to assert that this implies j&,> qj,, for all h. But since, p, 4j~ S, this is impossible. Thus condition (ii) of equilibrium is satisfied. Since r,(& jj))>O for all i, given A.1, we can apply the usual argument, as in Debreu (1959, p. 87), to the mappings ti to show that condition (i) of equilibrium is also satisfied by (X, y, j). 0 3. Relationship with the literature In this section we relate our result to the other existence results in the literature. In drawing out this relationship we shall need to suitably interpret the pricing rule and we begin, therefore, by considering the formal specilication of pricing rules that are of particular interest. Firm j is said to follow marginal cost pricing if ~j(P, y) = S n N( q, yj), where N( q, Yi) is Clarke’s normal cone of rj at yj [see, for example, Clarke (1983)]. Since this pricing rule does not depend on the reference prices or on the production plans of the other firms, there should be no confusion if we write $j(y) or ~j(yj) instead of cbj(p,y). Clarke’s normal cone was first applied to our problem by Cornet (1982) and was also used in BHKV for defining marginal cost prices. A justification for this definition of marginal cost pricing may be found in section 2 of Khan and Vohra (1987). We can, therefore, consider a marginal cost pricing equilibrium to be an equilibrium with ~j(y) = S n N( I$ yj) for all j. If q is convex and yj E a( k;), then N( 5, yj) coincides with the cone of normals to Yj at yj in the sense of convex analysis (see, for example, the Proposition in BHKV). In other words, in the classical context of convex production sets S n N(I$yJ is precisely the set of normalized prices for which yj is a profit maximizing production plan for firm j. Therefore, if all production sets are convex, we can define a Walrasian equilibrium to be a marginal cost pricing equilibrium. It can also be shown that if Yj satisfies A.2, then S n N(Yj, yj) satisfies assumption A.4 [see for example BHKV nor Cornet (1982)] and we can, therefore, deduce from our theorem that a marginal cost pricing equilibrium exists if A.l-A.3 and A.5 are satisfied. As we shall verify in Corollaries 3 and 4, this is a generalization of the BM and BHKV results. Firm j is said to follow a loss-free pricing rule if qj. yj 20 for all qj E 4j(p, y). One example of the loss-free pricing rule is average cost pricing. Another example is marginal cost pricing for a firm with a convex production set containing 0. 3.1. Existence of equilibrium with loss-free pricing rules
As a straightforward
consequence
of our theorem,
we can show the
R. Vohra, Existence of equilibria
187
existence of equilibrium when all firms follow pricing rules which ensure nonnegative profits. The classical result on the existence of a Walrasian equilibrium in an economy with convex production sets is a special case of this result. Corollary 1. Suppose A.l-A.4 are satisfied and the income distribution is given by a fixed structure of shares with wi>>O,for all i. If all firms follow loss-free pricing rules, then there exists an equilibrium. Proof Since there exists k>O satisfying Condition 1, it suffices to show that, under the conditions of Corollary 1, A.5 is satisfied for any k which satisfies Condition 1. Consider any such k and suppose y, E a( Y,), YjE a( 5) n R for j=2,... ,n, p~q5t(y,) and for some i,ri(p,y)=xj”=I Bijp.yj+p.wi~O. Since w,>>O, this implies that there exists a firm j for which p. Yj
A slightly weaker version of Corollary 1 can also be deduced from Kamiya (1984). It should also be clear from the proof that Corollary 1 can be strengthened to allow firms to make losses, provided these are restricted to be less than mini,Jwih). Since the marginal cost pricing rule satisfies A.4 and, in an economy with convex production sets, it also ensures each firm a non-negative profit we can also deduce the classical result on the existence of a Walrasian equilibrium from Corollary 1. Corollary 2. Suppose A.&A.3 are satisfied and the income distribution is given by a fixed structure of shares with Wi>>O,for all i. If all firms have convex production sets, then there exists a Walrasian equilibrium. 3.2. Existence of a marginal cost pricing equilibrium
We begin by showing that the BM result can be derived as a special case of ours, All firms are assumed to follow the marginal cost pricing rule. Instead of assumptions A.3 and A.5 above, BM make assumptions which are essentially equivalent to the following: BM.1 For all j, I$=Kj-R’,., where Kj is non-empty, exists k > 0, such that z 2 - ke for all z E Kj.
compact and there
188
R. Vohra, Existence of equilibria
BM.~.
For all j, if yjhs -k and 4jE 4j(Y), then qj,,=O*
BM.3.
If y E nj a( 5) and p E 4j(Y) for all j, then p *(Cy= 1 yj+ O) > 0.
BM.4.
If p.(syj+o)
>O, then ri(p,y)>O for all i.
Corollary 3. Suppose A.1 and BM.l-BM.4 marginal cost pricing equilibrium.
are satisfied.
Then there exists a
Proof. Clearly, BM.l implies A.2 and A.3. Since all firms follow marginal cost pricing, A.4. is satisfied and it only remains to be shown that BM.2BM.4 imply AS. Choose k’> k such that Condition 1 is satisfied for k’. We shall show that A.5 holds for Z?‘. Suppose y1 E a(YJ, yj~a(~) A R’ for j=2,. . . ,n, p E +I(yI) and for some i, ri(p, y) 5 0. By BM.4, this implies that ~~=I ri(p, y) 5 0. By BM.3, this means that there exists a firm j such that P$~~(Y). Clearly then, given that PES and $j(Y) s S, for any qjE~j(y) there must be a commodity h such that qj,,>ph. By BM.2, it cannot be the case yjh= -k, i.e., yj, > -k > - k’ and this completes the proof that A.5 is satisfied. 0 Remark 3. In the context of marginal cost pricing, it is possible to deduce BM.l and BM.2 from more primitive assumptions. Suppose the production sets satisfy A.2 and A.3. Let K be large enough so that the cube R, which has edge 2(k- l), contains in its interior all the attainable production sets and define Kj= Yjn Z?. Certainly, ~j=Kj- R’+ satisfy BM.l. It is also not difficult to check that if yjh 5 -k and qj~ N( q, yj), then q,,=O, i.e., for the marginal cost pricing rule, Yj satisfy BM.2. Thus, by considering q rather than 5 we can drop BM.l and BM.2. However, in that case, BM.3 would also have to relate to ~j rather than rj.
We now turn to the BHKV result. They assume that there is only one firm which has a non-convex production set. Let this be firm 1. All firms follow the marginal cost pricing rule, income distribution is given by a fixed structure of revenues and instead of A.5 consider the following assumption. BHKK 1. Suppose yr~a(Y,), p~$,(y,) and p*~.~Zp.yj for all y.\~q, j= 2, . . . , n, where c is the jth attainable production set. Then p *(cJ!=1 yj+ co)> 0.
The only difference between this and the corresponding assumption in BHKV is that they assume that income is non-negative while we assume it to be strictly positive. This is explained by the fact that they consider a quasi equilibrium. Given this, the following can be seen as the BHKV result.
R. Vohra, Existence of equilibria
189
Corollary 4. Suppose A.l-A.3 and BHKVIl are satisfied. If the income is given by a fixed structure of revenues, then there exists a marginal cost pricing equilibrium. Proof: Given A.l-A.3, the attainable production sets q are compact. Consider the economy with F., j=2,. . . , n as defined in Remark 3. Since $cI;., and all firms j=2 ,..., n have convex production sets, given that they follow marginal cost pricing, BHKV.l is now the same as BM.3. As mentioned in Remark 3, the marginal cost pricing rule also satisfies the boundary condition BM.2 for the economy so constructed. Given a fixed structure of revenues and BHKV.l we can follow the same argument as in the proof of Corollary 3 to prove that this economy has a marginal cost - pricing equilibrium (x, y,p). Since j belongs to the attainable sets and jj E nj a( q), it follows that jj E flj a( q). Thus, (X,J, p) is an equilibrium of the original economy. 0
3.3. Existence of equilibrium with bounded losses pricing rules While a direct comparison of our result with that of Bonnisseau and Cornet (1988) is not possible, here we provide an example of an economy in which the bounded losses assumption of Bonnisseau and Cornet (1986)2 is not satisfied but, given Remark 2, it is possible to use our result to deduce that an equilibrium exists. Consider an economy in which there are two firms with the following production sets:
Firm 1 cannot produce any output while firms 2’s technology is the same as that of firm 2 in the BM example. There are two consumers such that their preferences satisfy A.1 above but, unlike the consumers of the BM example, both consumers have strictly positive endowments. Let o1 = (7,3) and o2 =(l, 1). Suppose consumer 1 owns both the firms and that both firms follow the marginal cost pricing rule. It is easy to see that in this example the losses of the increasing returns firm are not bounded, as postulated in A.3(ii) of Bonnisseau and Cornet (1988). With respect to our result, it is easy to see that A.l, A.2 and A.4 are satisfied. Let k= 10. Notice that the cube K with edge 2k contains the production possibility curve in its interior. It is then easy to see that Condition 1 is satisfied for k= 10. We can now verify that A.5 holds for k= 10. Since both firms follow the marginal cost pricing rule, *The assumption, all (P,Y) ES X nja(I;),
A.3(ii) of their paper, infdJj(P,Y)
‘YjLaj
is that for every firm j there exists ~(~60 such that for
R. Vohra, Existence oJ equilibria
190
4d~J=K4
1) if
YC~
=s
if
= (LO)
otherwise.
y,=O,
h(P1y)=t(-~y2,,1) wheret=
_kyi +1
if
Y,,
I
=s
if
=(LO)
otherwise.
y,=O,
Suppose y, E Y,, y, E a( Y,) n I? and PE i(yi). It is clear that r,(p, y) = p*(l,l)>O. Moreover, for any qE+*(p,y) and y2Ea(Y,)nR, q-y,2 c#I&,(- 10,25/4)).(10,25/4)= -y> -3. Since p*(7,3)23, this implies that q.y,+p.y,+p.o,>O. Thus, if r,(p,y)=p.(y,+y,+o,)~O, then for any q E 4&, y), q. y, >p’yz. We can now derive the conclusion of A.5 by following the same argument as in the proof of Corollary 1. A.3 is, therefore, the only assumption of our theorem which is not satisfied. Since Condition 1 and A.5 are also satisfied, given Remark 2, we can use our result to assert that equilibrium exists. Existence of equilibrium in this particular example can also be deduced from Bonnisseau and Cornet (1985). The latter result, however, relies crucially on the smoothness of the production functions and would not apply if the example was modified to make the production functions non-differentiable. 3.4. Existence of equilibrium under special pricing rules ci la DGN It is possible to derive a variant of the DGN result from our main result. Since we have not been able to obtain the exact result of DGN we shall only indicate briefly the progress in this direction. A more detailed version of our reformulation of the DGN result is available from the author upon request. For a more recent attempt in this direction the reader is referred to Bonnisseau (1988) which uses the methods of Bonnisseau and Cornet (1988) to derive a result which is much closer to the DGN result. DGN assume that there are two kinds of firms in the economy; C-firms with convex production sets which maximize profits and ZZ-firms which follow special pricing rules. The C-firms are described by their aggregate production set and we may, therefore, assume that firm 1 is the only C-firm. In our notation firms (2,. . . , n} are the U-firms. A n-firm j produces a set of commodities rtj which are not produced by any other firm in the economy. Let the set of commodities produced by the H-sector be denoted K and rL={l,..., I,}. The U-firms are assumed to have convex isoquants and
R. Vohra, Existence of equilibria
191
minimize costs given input prices and a vector of outputs. Each ZI-firm then sets output prices in accordance with a pricing rule I$~:R’ x fljd(lJ~R7, which is assumed to be non-empty, upper-hemicontinuous and compact and convex valued. Notice that the prices set according to this rule are not normalized. They also make the following assumption: DGN.2.
For
any j, j=2 ,..., n, qjE~j(P,y),
yj,=O
implies that
(qj,,...,
t4j,9.. .3qj,n)E 4j(P,Y) for all t E I3 11. In their concluding section DGN replace this assumption by another assumption on non-inferiority. Neither of these assumptions fits in naturally with the price normalization which is used in our basic model. Consider the following asumption. A.6.
For any j, j=2 ,..., n and any qjo4j(PtYL Chprrqjh=&EnPh.
This assumption is an important distinguishing feature of our reformulation of the DGN model. It allows us to construct a pricing rule for each firm which takes values in S. The income hypothesis of DGN is very similar to A.5 above and by assuming A.6 instead of DGN.l it is possible to use our result to establish the existence of an equilibrium with special pricing rules a la DGN. References cost pricing equilibria with increasing returns, Ph.D. dissertation Beato, P., 1976, Marginal .._. (University of Minnesota, Mmneapohs, MN). Beato, P., 1982, The existence of marginal cost pricing equilibria with increasing returns, Quarterly Journal of Economics 79, 669-688. Beato, P. and A. Mas-Colell, 1985, On marginal cost pricing with given tax-subsidy rules, Journal of Economic Theory 37, 356365. Bonnisseau, J.M., 1988, On the existence of equilibria results in economies with increasing returns, Journal of Mathematical Economics 17, this issue. Bonnisseau, J.M. and B. Cornet, 1985, Existence of marginal cost pricing equilibria in an economy with several non convex firms, CORE discussion paper no. 8723 (Universiti CathoIique de Louvain, Louvain-!a-Neuve). Bonnisseau, J.M. and B. Cornet, 1988, Existence of equilibria when firms follow bounded losses pricing rules, Journal of Mathematical Economics 17, this issue. Brown, D.G. and G.M. Heal, 1982, Existence, local uniqueness and optimality of a marginal cost pricing equilibrium with increasing returns, Social science working paper no. 415 (California Institute of Technology, Pasadena, CA). Brown, D.J., GM. Heal, M. Ah Khan and R. Vohra, 1986, On a general existence theorem for marginal cost pricing equilibria, Journal of Economic Theory 38, 371-379. Clarke, F.H., 1983, Optimization and nonsmooth analysis (Wiley, New York). Cornet, B., 1982, Existence of equilibria with increasing returns, in: B. Cornet and H. Tulkens, eds., Contributions to economics and operations research: The XXth anniversary of CORE (The MIT Press, Cambridge, MA) (forthcoming). Dierker, E., R. Guesnerie and W. Neuefeind, 1985, General equilibrium when some firms follow special pricing rules, Econometrica 53, 1369-1393.
192
R. Vohra, Existence of equilibria
Hurwicz, L, and S. Reiter, 1973, On the boundedness of the feasibility set without convexity assumptions, International Economic Review 14, 58Ck586. Kamiya, K., 1984, Existence and uniqueness of equilibria with increasing returns, Journal of Mathematical Economics 17, this issue. Khan, M. Ah and R. Vohra, 1987, An extension of the second welfare theorem to economies with nonconvexities and public goods, Quarterly Journal of Economics 102, 223-241. Mantel, R., 1976, Existence of equilibria with Pareto optimality in a general equilibrium model with non-convex production possibilities, Mimeo. (Yale University, New Haven, CT).