North-Holland. EXISTENCE OF EQUILIBRIA WHEN FIRMS FOLLOW BOUNDED. LOSSES PRICING RULES. Jean-Marc BONNISSEAU and Bernard CORNET.
Journal
of Mathematical
Economics
17 (1988) 119-147.
North-Holland
EXISTENCE OF EQUILIBRIA WHEN FIRMS FOLLOW BOUNDED LOSSES PRICING RULES Jean-Marc BONNISSEAU
and Bernard CORNET
CORE, Vniversite Catholique de Louvain, Louvain-la-Neuve, Belgium CERMSEM,
UniversitP Paris I PanthCon-Sorbonne, Paris, France
Submitted
February
1987, accepted
January
1988
We consider a general equilibrium model of an economy with increasing returns to scale or more general types of non-convexity in production. The firms are instructed to set their prices according to general pricing rules which are supposed to have bounded losses. This includes the case of loss-free pricing rules hence, in particular, profit maximizing and average cost pricing. As for the marginal (cost) pricing rule, the bounded losses assumption for a firm is shown to be equivalent to the ‘star-shapedness’ of its production set. This paper reports a general existence result in this model.
1. Introduction This paper studies the existence problem of equilibria in an economy with increasing returns to scale or more general types of non-convexities. There is now a growing literature on this problem which was first considered by Beato (1979, 1982), Brown and Heal (1982), Cornet (1982) and Mantel (1979) for marginal (cost) pricing equilibria, i.e., when each firm is instructed to follow the marginal (cost) pricing rule. They exhibited equilibria which had in common to be aggregate productive efficient, under assumptions on the aggregate production sector which were shown to be very limiting by Beato and Mas-Cole11 (1985). In this last paper, in Brown et al. (1986), Bonnisseau and Cornet (1985), Dierker et al. (1985) and Kamiya (1988), existence theorems were then presented to accommodate possibilities of aggregate productive inefficiency, with possibly more general pricing rules. The purpose of this paper is to provide a general existence result when each firm follows a pricing rule with bounded losses but also to clarify the relationship between the above existence results. In our model the non-convex firms follow general pricing rules and the convex firms may behave competitively or not. In fact we shall present a symmetric treatment of the firms without distinguishing a priori the convex firms from the non-convex ones or the price-taking firms from the price-setting ones. The technological possibilities of the jth firm (j= 1,. . .,n) are represented by a subset 5 of R’ which is assumed to be non-empty, closed, and to satisfy free-disposal, i.e., 03044068/88/$3.50
0
1988, Elsevier
Science
Publishers
B.V. (North-Holland)
120
J.M. Bonnisseau and B. Cornet, Equilibria and bounded losses pricing rules
Yj- R’+ c Yj. We point out that under our assumptions, aYj, the boundary of the production set Yj, is exactly the set of (weakly) efficient production plans. A pricing rule for this firm is then a correspondence 4j, from a? to R$, the closed positive orthant of R’, and we assume that ~j has a closed graph and that, for all y, ~j(Y) is a closed convex cone of vertex 0. Then the jth firm is in equilibrium at the pair (p, yj), of a price vector p and an efficient production plan yj, if p E bj(yj). An equilibrium of the whole economy is then a list of consumption plans (x:), a list of production plans (y;), and a price vector p* such that (a) every consumer maximizes his preferences subject to his budget constraint, (b) every firm follows his pricing rule, i.e., p* ~#~(yj*) for every j, and (c) the excess of demand over supply is zero. We are now ready to present our main existence result (Theorem 2.1). We posit standard assumptions on the consumption side of the economy, on the revenues of the consumers together with a boundedness assumption. Then Theorem 2.1 states that the economy has an equilibrium if we further assume the so-called survival assumption and that the pricing rule of each firm has bounded losses. The survival assumption guarantees that in a non-convex production model, where firms may exhibit deficits, at production equilibria, the total wealth is above the subsistence level, i.e., formally if w denotes the vector of total initial endowments and X denotes the total consumption set, then p*(~;=,yj+o)>infp.X for efficient production plans yj~ a? (j=l , . . . , n) which are ‘supported’ by the same non-zero price vector p, i.e., such that p E cbj(yj) for all j. The bounded losses assumption can be formally defined by saying that, for some real number c(~, p~4~(y~) n S implies p. yjzUi (where S denotes the simplex of R’). In this paper we shall also particularize our existence result to the two following cases of particular economic importance: (i) loss-free pricing rules, and (ii) marginal cost pricing, which have motivated the present study. A pricing rule 4j is said to be loss-free if, for every yj~a~;-, PE ~j(yj) implies p. ~~20. This includes the case of average cost pricing and also, when 0 E 5, profit maximizing, for which the pricing rule can be defined by PMj(Yj) = fp E R’(p. yj 2 p .y>, for every y$E Y$. Thus, our existence result for loss-free pricing rules (Theorem 3.1) will allow us to generalize the existence result of Walras equilibria of Debreu (1959). As for the marginal (cost) pricing rule MR,, following Cornet (1982), it is formalized by letting MRj(yj)=N,j(yj), the normal cone to 5 at Yj~ aq in the sense of Clarke (1975) and we refer to Bonnisseau and Cornet (1985) for the relationship between this rule and the standard marginal cost pricing rule. In Lemma 4.2, we shall give a geometric characterization of the bounded losses assumption for the marginal rule MRj. Namely, under the above assumptions on Yj, MR, satisfies the bounded losses assumption if and only
J.M. Bonnisseau and B. Cornet, Equilibria and bounded losses pricing rules
121
if the production set q is star-shaped, i.e., for some y, E q, for every YE q one has [yo, y] c 5. We further point out that 5 is star-shaped in the two following cases where, for some non-empty compact subset Kj of R’, (C.l) T=Kj-R’+ and (C.2) y\Kj is convex. The first condition (C.l) is assumed by Beato and Mas-Colell(l985) in their existence result. Condition (C.2) allows to cover, in a two-dimensional space, the case of a firm with a fixed cost and then a convex one. We end this introduction by discussing more precisely the link between this paper and other ones on the same subject. Firstly, we do not assume here that the boundary of the aggregate production set cj”=i 5 is smooth as in Beato (1979, 1982), Brown and Heal (1982), Mantel (1979) or that CT=, 5 has no ‘inward kinks’ as in Cornet (1982). From the example of Beato and Mas-Cole11 (1985) we know that this type of assumption is far from being innocuous in a non-convex world and actually amounts to assume implicitly that there is a single firm in the economy (see Corollary 3.7). As for marginal (cost) pricing equilibria, our existence theorem (Theorem 4.1) extends the result of Beato and Mas-Cole11 (1985) by only assuming that the production sets are star-shaped [instead of (C.l)] and also the result of Brown et al. (1986) who were allowing only one non-convex firm in their model which in fact will be shown to be star-shaped (see Corollary 4.2). However, in Bonnisseau and Cornet (1985), we prove an existence result of marginal cost pricing equilibria in which we remove the bounded losses assumption but under the additional assumption that the boundaries of the production sets are smooth. Neither this result implies the present one, nor the converse is true, and we point out that the proof in Bonnisseau and Cornet (1985) uses more elaborate arguments than in the present one, which essentially relies on Kakutani’s theorem. The model of Dierker et al. (1985) is more specific than the one considered here. Indeed, they assume that for each nonconvex firm, one can distinguish a priori between the inputs and the outputs of the firm which is instructed to minimize its cost and to set the outputs according to a special pricing rule. We refer to Bonnisseau (1988) for a proof of the result of Dierker et al. as a consequence of the present existence result. The model considered by Kamiya (1988) and Vohra (1988) is very close to the one presented here but their two existence results are not directly comparable to ours; see, however, section 5.2 for Vohra’s result and Bonnisseau (1988) for a proof of Kamiya’s result as a consequence of Theorem 2.1. The paper is organized as follows. In the next section, we present more precisely the model we consider and we state our main existence result: Theorem 2.1, and a slight extension of it, Theorem 2.1’. In section 3, we give some consequences of these results to the case of loss-free pricing rules (section 3.1) and we also state and prove a further result for general pricing rules (Theorem 3.4) which will allow us to deduce some of the results quoted
122
J.M. Bonnisseau and B. Cornet, Equilibria and bounded losses pricing rules
above. Section 4 is devoted to the marginal (cost) pricing rule and Lemma 4.2 shows the characterization of the bounded losses assumption in terms of the star-shapedness of the production set. The existence result of marginal (cost) pricing equilibria (Theorem 4.1) is proved in section 4.2 and we also give some consequences of it. Finally, the proofs of the existence results (Theorems 2.1’ and 3.1) are given in section 5. 2. The model and the existence results’ 2.1. The model and the definition of equilibria We consider an economy with 1 goods, m consumers and n firms. We let o in R’ be the vector of total initial endowments. The technological possibilities of the jth firm (j=l,..., n) are represented by a subset Yj of R’. We denote by Xic R’ the set of possible consumption plans of the ith consumer (i=l,..., m) and the tastes of this consumer are described by a complete, reflexive, transitive, binary preference relation >y) meansx,Ly, (resp. x,>y,) for all h, and we let R’+={x~R’lx~0} and R’++={x~R’(x>>0}. For AcR’, we denote by cl A, int A, 8A and co A, respectively, the closure, the interior, the boundary and the convex hull of A and for B c R’, for real numbers I, p, we let LA+pB={la+yb]a~A,b~B}. If A is non-empty, we let, d,(x)==inf{llx-all]aoA}, infx.A=inf{x.alaoA},supx.A=sup{x.alaoA} and for rLO,B(A,r)={xER’Id,(x)~r}. G’tven two topological spaces X and x a correspondence C$ from X to Y, associates with each element x in X, a subset 4(x) of Y, it is said to be upper hemi-continuous (or simply u.h.c.) if, for every open subset U of x the set {xoXI&x) c U} is open. *I.e., for every xioXi: (i) the sets {x~X~Jx inf p *Xi, for all i.
and
p~(~~, 1Yj+O) >infp*Cy!
r Xi
imply
2.3. Some general remarks Remark 2.2. Assumption PR is satisfied under Assumption P in the three important following cases: (i) a profit maximizing firm whose production set is convex; (ii) a firm following the marginal (cost) pricing rule (see section 4), or (iii) the average cost pricing rule under the additional assumption that q n R’+ = (0). Weakening of this assumption is discussed in Remark 2.8 below. Remark 2.3. Assumption BL (bounded losses assumption) has a clear economic interpretation and it is satisfied in two cases of particular economic importance. Firstly, in the case of loss-free pricing rules, i.e., a1 = . -. =c1, =O
J.M. Bonnisseau and B. Cornet, Equilibria and bounded losses pricing rules
125
and, in particular, average cost pricing (cf. section 3.1). Secondly, in the case of marginal (cost) pricing for star-shaped production sets (cf. section 4). In Kamiya (1988), a weaker assumption than Assumption BL is made, namely, it is assumed that, for all j, for all sequences ((p’, y;)} CS x aq such that IIyjl + + 00 and, for all v, p’~ ~$~(yj’),one has lim,, mp’.(yj/[[yjI)zO. However, Kamiya’s existence result is not directly comparable to ours since he only proves the existence of a free-disposal equilibrium and since his boundedness assumption is stronger than the above one, namely, he assumes that co A(-& i I;) n -co A(Ci”, 1 5) = (0). w e refer to Bonnisseau (1988) for a proof of Kamiya’s result as a consequence of Theorem 2.1. The example of Remark 2.7 will show that it is not possible, in general, to weaken in Theorem 2.1, Assumption BL by Kamiya’s. 2.4. Assumption SA (survival assumption) is a key assumption in Theorem 2.1. It means that the economy can ‘survive’, in the sense that at production equilibria, the total wealth may be distributed among the consumers in such a way that each one stays above his subsistence level. Assumption R is satisfied in two cases of particular economic importance. The first one is the case of a fixed structure of revenue pi, i.e., for some fixed 6,>0 (i=l,..., m) such that CT= I di= 1, for all i. Remark
Pi(P3(Yj))=
6i[P’(jl
Yj+m)-infP’$l
Xi]+infp’Xi
We notice that under the additional assumption that, for all i, Xi c R’+ and OEX~, it reduces to saying that the wealth of each consumer is proportional to the total wealth of the economy. The second case, under which Assumption R is satisfied concerns a private ownership economy whose firms follow loss-free pricing rules, i.e., ai = ... = c(,=0 and such that, for all i, coie Xi+ R: +. This case will be considered in section 3.1. We finally point out that Assumption R is a weakening of the analogue Assumption D, made by Dierker et al. (1985) who assume that the distribution of income ri(p, w) depends on the price vector p and the total wealth w=p.(c;=i yj+w), and that w>infp.~~=“=, Xi implies ri(p, w)> infp.Xi, for all i. In view of applications in the next sections, our weaker assumption will be needed. Remark 2.5. If we assume that, in Debreu (1959)], then the set empty. However, the fact that consequence of the assumptions Theorem 2.1).
for all j, 0 E k; and that o E cr= I Xi + R’+ [as of attainable allocations A(o) is clearly nonA(w) is non-empty is no longer a direct of Theorem 2.1 (but clearly a consequence of
126
J.M. Bonnisseau and B. Cornet, Equilibria and bounded losses pricing rules
Remark
2.6. The following example shows that it is not possible, in general, in Theorem 2.1 to replace Assumption BL by Kamiya’s (cf. Remark 2.3). We consider an economy with two goods, an arbitrary total initial endowment vector w E R: +, a consumption sector with m consumers satisfying Assumptions C and R and having each consumption set Xi equal to R:, a production sector with two firms with production sets Y, = {(y’, y2) E R2 1y' 5 0,y25(y’)‘}, Y, = -R: and normalized pricing rules $i and d;2 defined for (YI,~z)EW
xau,,
by
We first notice that the set PE of production equilibria is empty since (LO) $ $2(y2), hence the economy has no equilibrium. However, this economy is easily shown to satisfy all the assumptions of Theorem 2.1 with the exception of Assumption BL (noticing that the losses of the first firm are not bounded), and Kamiya’s assumption is satisfied by the two pricing rules. Remark 2.7. The following example shows that it is not possible, in general, in Theorem 2.1, to weaken the survival assumption by only assuming it on the set of attainable production equilibria APE = ((p, (yj)) E PE (I;= 1 Yj+ w ELF! 1Xi + RI+} as in the case of a single firm (cf. Corollary 3.7). However, this will be possible for Assumption R (cf. Corollary 2.2 below). We consider an economy with two goods, a total initial endowment vector w = (3, _5),a consumption sector with m consumers satisfying Assumptions C and R, with each consumption set Xi equal to R:, a production sector with two firms with production sets Yr = {(y’, y’) E R2 1y’ + y2 5 0}, Y, = -R: and normalized pricing rules qj defined, for (yl, y2) in 8Y, x aY2 by:
if
Y~E(-TOI,
=s
if
yk= -2,
={(O,l)}
if
y$E(-co,
~2(~2)={(1~0)~
-2).
We first notice that this economy has no attainable production equilibrium. Indeed, one easily sees that PE = {(p, y,, y2) E R6 1y, =(x, -x), for some and that (x,-x)+(-2,0)+($,$20 is never x~& ~2=(-2,0), ~=(t,;)>, possible. Then all the assumptions of Theorem 2.1 are easily shown to be satisfied but Assumption SA (we notice that one can take a1 = 0 and a2 = - 2).
J.M. Bonnisseau and 8. Cornet, Equilibria and bounded losses pricing rules
127
However, this economy has no equilibria (since it has no attainable production equilibria) but the survival assumption would be clearly satisfied if it was made on the (empty) set of attainable production equilibria.
2.4. A more general result In this section, we shall state a more general result than Theorem 2.1. We first notice that condition (b) of Definition 2.1 can be equivalently rewritten as follows:
(b’)
(yl ,... ,y,,)~a& x ... x aK and (p,...,p)E~(p,yl,...,y,),
whereNP,Y,, . . . ,Y,) dzf41(~1) x ... x &J,(Y,). We now consider a pricing rule 4 for the whole economy, which we define to be an arbitrary correspondence 4 from R: x n’j= I a? to (R:)” (and 4 is no longer assumed to be the Cartesian product of the pricing rules of firms). An equilibrium of the economy &=((Xi,
