existence of equilibria when firms follow bounded...

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 non- convex 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


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 <i on his consumption set Xi. The relation of strict preference x<~x’ is then defined by [xdix’ and not x’ lix]. Finally, the wealth of the ith consumer is defined by a function Ti: R: x ny= 1 al;. + R. This abstract wealth structure clearly encompasses the case of a private ownership economy for which ri(p, yr, . . . , y,) = p. wi + C;=l eijp.yj for some 8ij~O (i= l,..., m; j=l,..., n) such that, for all j, CT= r fIij= 1 and, f or some WOE R’ (the initial endowment of the ith consumer) such that I:! 1 Oi=O. We now posit the following standard assumptions which describe the general framework of the paper.

Assumption C. For all i: (i) Xi is a non-empty, closed, convex, bounded below subset of R’; (ii) & is continuous, convex and locally non-satiated? and (iii) li: R: x ny= 1 a? + R is continuous, IF= 1 ri(p, (Yj)) = p. (cJ= r yj + o), and ri(rPy (Yj)) = rri(P, (yj)), for all (P, (Yj), r) E R’+ x ny= r a I;- x R + .

Assumption P. For all j, 5 is non-empty, closed, and 5-R: c rj (free disposal).

Assumption B. For every w’ 2 w, the set A(@‘) = {((Xi), (Yj)) E ny= 1 Xi X n;=r ~;.)~~~Ixi~~~~lyj+w’} is bounded.

LNotation. If x=(xh), y=(y,) are vectors in R’, we let x.y=~~=r xhyh be the scalar product of R’ and llxll =(x.x)“’ be the Euclidean norm. The notation x>,y (resp. x>>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<~x~} and {x~X~IX~<~X} are closed; (ii) the set {xeXilxi six} is convex; (iii) for all EPO, there exists xeXi n B(x,,E) such that xi xix.

J.M. Bannisseau and B. Cornet, Equilibria and bounded lasses pricing rules 123

The first two assumptions need no special comments and the third one is implied by the more standard one: A(x;= i 5) n -A(cj”,i I;) = {0}, where A(x,“,i q) denotes the asymptotic cone of ~~=r Yj. [See Hurwicz and Reiter (1973).]

We now describe the behaviour of the agents of the economy. The ith consumer maximizes his preferences under his budget constraint. Thejth firm follows the pricing rule $j, which is a correspondence from 85 to R: satisfying the following homogeneity assumption: #am is a cone of vertex 0. (At the end of this section we shall in fact consider a more general framework.) Then the jth firm is in equilibrium at the pair (p, yj) of a price vector p E R: and an efficient production plan yj E aYj if p E ~j(yj). The above formalization takes into account the case where the jth firm maximizes its profit, given the price vector p, for which the pricing rule PM, is defined by

PMj(yj)={pE R’Ip.yj~p’yji for all Y~E yj),

noticing that PMj(yj) c R:, under Assumption P. We now formally define our notion of equilibrium.

Definition 2.1. An equilibrium of the economy d = ((Xi, ii, ri), (5, +j), w) is an element ((xr),(yT),p*) in R’” x R’” x R’ satisfying

(a) for all i, x: is a greater element for ii in his budget set {xi~Xilp* .xiS

‘i(P*, (Y~))l; (b) for all j, y; E al; and p* E ~J~(yj*);

(c) ~~~“=x~=~~=lyj*+o.

Remark 2.1. We require in Condition (c) the equality between demand and supply whereas, it is usually considered the weaker notion of a free-disposal

equilibrium, i.e., ((xF),(yj*),p*) satisfies the above conditions (a) and (b), with (c) replaced by

(c’) ~Li~~~~;=~yj*+o, p*zO, and p*.(c~=“=xX:-c~=,y,*--)=O.

It is worth pointing out that the standard argument [Debreu (1959)] which allows to deduce the existence of a Walras equilibrium from the existence of a free-disposal Walras equilibrium is not working, in general, when the firms are no longer convex and profit maximizing. Hence, we shall need later to prove directly the existence of equilibria.

2.2. The existence result

From the above homogeneity assumptions of ri and +j and the local


non-satiation assumption in Assumption C, we can assume that the equilibrium price vector p* belongs to S= {pi R’+ Ix;= 1 p,+= l}, the simplex of R’. We now let the normalized pricing rule ij be the correspondence from a? to S defined by

6j(Yj)=cbj(Yj) n s. We also let

be the set of elements (p,(yj)) satisfying condition (b) of Definition 2.1 and such that PCS. An element of PE is called a production equilibrium of the economy 67.

Theorem 2.1. The economy S=((Xi, ii, ri),( rj, cbi),O) has an equitibrium if Assumptions C, P, B hold and ij!

Assumption PR. For all j, the correspondence ~j, from ayj to S, is upper hemi-continuous with non-empty, convex, compact values;

Assumption BL (bounded losses assumption). For all j, there exists a real number CLj such that, for all (p, yj) E S x 3 5, p E dj(yj) implies p. yj 2 ctj;

Assumption SA (survival assumption). inf p. CT= 1 Xi;

(P,(yj)) E PE implies p .(x= 1 Yj+ W) >

Assumption R. (p,(yj)) EPE and p~(~~, 1 Yj+O) >infp*Cy! r Xi imply ri(p, (yj)) > inf p * Xi, for all i.

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


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,, m p’.(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.

Remark 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.

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, for all j, 0 E k; and that o E cr= I Xi + R’+ [as in Debreu (1959)], then the set of attainable allocations A(o) is clearly non- empty. However, the fact that A(w) is non-empty is no longer a direct consequence of the assumptions of Theorem 2.1 (but clearly a consequence of Theorem 2.1).


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 Assump- tions 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 1 y' 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! 1 Xi + 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 1 y’ + y2 5 0}, Y, = -R: and normalized pricing rules qj defined, for (yl, y2) in 8Y, x aY2 by:

~2(~2)={(1~0)~ if Y~E(-TOI,

=s if yk= -2,

={(O,l)} if y$E(-co, -2).

We first notice that this economy has no attainable production equilibrium. Indeed, one easily sees that PE = {(p, y,, y2) E R6 1 y, =(x, -x), for some x~& ~2=(-2,0), ~=(t,;)>, and that (x,-x)+(-2,0)+($,$20 is never 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,), where NP,Y,, . . . ,Y,) dzf 41(~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, <i,ri),(~),~,O) is then an element ((x’),(yj*),p*) in R’“’ x R’” x R’ which satisfies conditions (a) and (c) of Definition 2.1, with (b) replaced by the above condition (b’), and a production equilibrium of E is an element of the set

We also let the normalized pricing rule 6 be the correspondence from let of S) defined by S x ny= r 85 to s” (the n-fold Cartesian prod1

$(P, (Yj)) =4% (Yj)) n S”.

Theorem 2.1’. The economy E = ((Xi, <i, ri), Assumptions C, P, B, SA, and R hold and if:

(I$~,o) has an equilibrium if

Assumption PR’. The correspondence 4 from S x n;= 1 a$ to S” is u.h.c. with non-empty convex compact values.

Assumption BL’ (bounded losses). There exists a real number CI such that for all (P, (Yj)) E S X ny= I aq, (4j) E f$<P, (Yj)) implies Cy= 14j. Yj 2 Cf.

In the following corollary, we weaken Assumption R by only assuming that it holds on the set of attainable production equilibria APE= {(p, (yj)) E PE(Cj"= , yj+ o E EYE 1 Xi + R:}, and we also weaken Assumption BL’ by only assuming that the losses are bounded for some (and not all) price vector satisfying the pricing rule. The first weakening will be needed in Theorem 3.4 below.


Corollary 2.2. The economy d =((Xi, 3 i, ri), (q), 4, o) has an equilibrium if all the assumptions of Theorem 2.1’ hold, but Assumptions R and BL’ which are replaced by the weaker ones:

Assumption WR. (p,(yJ) E APE and p.(& I yj+o) >infp*x”‘“=, Xi imply ri(p, (yj)) > inf p. Xi, for all i.

Assumption BL”. There exists a real number c1 such that for all (p, (yj)) ES x JJ=l aYj9 for some (qj)E$(P,(yj)), one has C;=r qj.Yj~a.

Proof: It is a consequence of Theorem 2.1’ by associating to the economy d the modified one d =((XJ, -&, ?i),( q), 6, w), for which the structure of revenue (pi) is defined below and the pricing rule 6 is defined by &p, (yj)) = $(p, (yj)) n {(qj) E S” ICY= 1 qiyj 2 a}. Indeed d satisfies the assumption of Theorem 2.1’ and each equilibrium of d is an equilibrium of d since

$(P, (YJ) c $(P, (yj)) and (“J r, coincide with (ri) on APE. We now define the functions (I”i). We first choose an open neighborhood Sz in S x ny=i a?, of the set APE, such that for (p, (yj)) E s2, ri(p, (yj)) > infp. Xi, for all i. This is possible from Assumptions SA and R’, from the continuity of the functions ri and of the functions p + infp. Xi [see, for example, Rockafellar (1970, Theorems 10-2 and 20-31. Then, we let Fi =( 1 -L)r,+ &+, where (pi) is the fixed structure of revenue as defined in Remark 2.4 and LS x n;= I a? -+ [0, l] is a continuous function such that J.=O on APE and L = 1 outside Sz. Such a function I clearly exists from Tietze-Urysohn’s theorem, noticing that the set APE is closed by Assumption PR. 0

Remark 2.8. In Assumption PR’ one can weaken the assumption that d, is convex-valued in two ways for which the proof given below can be simply adapted. Firstly, by assuming that (is is upper hemi-continuous with non- empty acyclic compact values and in the proof below, one only needs to replace Kakutani’s theorem by Eilenberg-Montgomery’s Secondly, by assuming that

Assumption PR”. $=$J, 04~~ for two u.h.c. correspondences & and cjl with non-empty, convex, compact values, respectively, from S x n;= 1 83 to X, a closed subset of a Euclidean space, and from X to S”.

The latter assumption is needed by Bonnisseau (1988) to deduce the existence result of Dierker et al. (1985) from Theorem 2.1’. We leave the reader to check that the proof given below can be simply adapted by using a product argument and an extension theorem of Cellina (1969).


3. Some consequences of Theorem 2.1

3.1. Loss-free pricing rules

In this section, we consider the case of a private ownership economy whose firms follow loss-free pricing rules, i.e., c1i = .. . = ~1, =0 in Assumption BL or, equivalently, p E ~j(yj) implies p* yjzO, for all j. Actually, we state the following result in the more general setting of a pricing rule 4 for the whole economy, as defined above.

Theorem 3.1. The private ownership economy S=((Xi St Wt), (Q 4, (6,)) has an equilibrium if Assumptions C, P and PR’ hold and i$

Assumption B,. The set of attainable allocations A(w) is bounded.

Assumption SAo (survival assumption). For all i, Oi E Xi + Rr+ + .

Assumption LF (loss-free assumption). For all (P,(Yj))ESxn~=laI;.,

(qj)E$(p,(yj)) implies qj’yj~O, for all j= l,.. .,n.

The proof is given in section 5. We point out that under the stronger Assumption B, it is a direct consequence of Theorem 2.1’.

The first example of a loss-free pricing rule is given by a profit maximizing firm whose production set I$ contains the null production, i.e., OE Yj. Indeed, let pePMj(yj) then p.yjzp.O=O. We now deduce from Theorem 3.1 the existence of a Walras equilibrium which we recall is an element ((x:),(yj*),p*) in R’” x R’” x R’ satisfying conditions (a), (c), of Definition 2.1 with (b) replaced by profit maximization, i.e., for all j, yj* E q and p* E PMj(yr).

Corollary 3.2. [Debreu (1959)]. The private ownership economy d=

((Xi, _i i, wi), ( I;), (eij)) h as a Walras equilibrium if Assumptions C and SA, hold and if:

Assumption P,. For all j, 0 E 5; Y = x3= 1 q is closed, convex and - R’+ c Y

Assumption B,. Y n - Y = (0).

Proof: We first associate to 8 the economy b=((Xi, <i,Oi)r( 57 PMj),(eij)) where yj is the closed convex hull of Yj- R’+. Clearly, in the economy 8, the firms follow loss-free pricing rules, and Assumptions C, P, PR and SAO are satisfied. Noticing that

F ?= 1 q=xr= 1 5, one deduces that the set of

attainable allocations of is bounded as in Debreu (1959). Hence, by Theorem 3.1, d has a Walras equilibrium ((xr),(jj),p*). But, there exist yj* E 5 (j=l,..., n) such that ~~=lJj=~=ly~ and p*.~jn=iyj*=p*.C;=r~j= sup p* .I;= I Fj= sup p* .I;= r Yj. Consequently, ((x:), (yi*), p*) is a Walras equilibrium of 8. 0


Another important example of a loss-free pricing rule is given by the average cost pricing rule which is defined, for yje aYj:., by

ACj(yj)={pER:)p’yj=O} if Yj#O,

AC~(O)=CICO {per: 13 {p’,y’} c R:[a~\{O}],

An average cost pricing equilibrium of the private ownership economy d = ((Xi, <i, Oi), (YJ, (eij)) is then defined as an equilibrium of the economy 6 in the sense of Definition 2.1 with ~j= AC,, for all j, i.e., each firm is instructed to follow the average cost pricing rule.

At this stage it is worth making some comments about the above definition of AC,(O). If we had taken for ACAO) the larger set {p E R’+ Ip. 0= 0} = R’+, then for every Walras equilibrium ((xr),p*) of the associated exchange economy (Xi, 4,, oi), the element ((x”),(yT), p*) with yj* =O, for all j, would have been an average cost pricing equilibrium of d. This is to limit this possibility that we have taken the above definition of AC,(O).

Corollary 3.3. The private ownership economy d =((Xi, -& wi), ( $),(eij)) has an average cost pricing equilibrium if Assumptions C, P, B,, and SA, of Theorem 3.1 hold and if, for all j, 5 n R: = (0).

Proof. It is a direct consequence of Theorem 3.1 by checking that, for all j, the average cost pricing rule satisfies Assumption PR. Indeed, the correspondence AC, is clearly upper hemi-continuous with convex, compact values, and the assumption Yj n R: = (0) implies that AC, is non-empty- valued. 0

3.2, A further existence result for general pricing rules

In this section, we give a consequence of Theorem 2.1, which is of interest for itself, and which will allow us to deduce the existence results of Beato and Mas-Cole11 (1985) and Vohra (1988). In all this section, we allow the pricing rule +j of the jth firm to depend upon the price vector p and the efficient production plans of the other firms. Equivalently, we assume that the normalized pricing rule 4 of the whole economy is the Cartesian product of correspondences 4j from S x n;=r aYj to S. We first posit the following assumption which will be made throughout this section. We let e =( 1,. . . , 1) be the vector in R’ with all its coordinates equal to 1.


Assumption B (boundedness assumption). The set A(o) is bounded and, for all real numbers k, the set 5 n [{ - ke) +R:] is bounded (j= 1,. . . , n).

In the following, we let

Ajk=aY$n [{ -ke}+R$], (j=l , . . . ,4,

and we denote by $ the attainable set of the jth firm, i.e., the projection of A(o) onto I$

Theorem 3.4. The economy d = ((Xi, li, ri), (rj, +j), w) has an equilibrium if Assumptions C, P, PR’, B and WR hold, if for some real number k, cc{-ke}+R:+, (j=2,...,n) and if

Assumption BL,. There exists a real number a1 such that, for all

(P, Y 1 ,..., y,)~SxdY~ xfl~=2djk,qE$1(p,yl ,..., y,) implies q’y,Za,.

Assumption WBMC. For all jE (2,. . . , n}, for all (p,(y,))~S x JY, x nyz2 AT, there exists q E ~j(p, y,, . . . , y,) such that q,,=O for all good h such that yj,,= -k.

Assumption SAI, (survival assumption). For all (p, (yj)) E S x 8Y, x nT= z Ai, (p,(yj)) E PE implies p.(Cy, 1 Yj+W) >infp.xy= 1 Xi.

In the above theorem, Assumptions BL, on the first firm, and WBMC on the (n- 1) other ones replace Assumption BL of Theorem 2.1. In fact Assumption WBMC will allow us to modify the production sets 5 and the pricing rules $j( j = 2,. . . , n), for Yj outside { - ke} + Rt+, in such a way that Assumption BL will also be satisfied by the firms j = 2,. . . , n. We prepare the proof of the theorem by a lemma, the proof of which is left to the reader. In the following, we denote by n,:R’-+{ -ke} + R’+ the projection mapping onto { - ke} + Rt+, i.e.,

n,(y)=(max{-k,y,},... ,max{-k,y,}) for y=(yJ in R’.

Lemma 3.1. Let k be a real number and let Y be a closed subset of R’ such that Y-R’+cY and Yn{-ke)+R’+ is non-empty and bounded. We let y= Yn[{-ke)+R:]-Rt+. Then ? is a non-empty, closed subset of Y such that F-R’+ c t i3yn[{-ke}+Rt+]=aYn[{-ke}+R$], and n,(y)~aY for all yEaF

Proof of Theorem 3.4. We first associate to d the economy b=((Xi, Ai, Fi), (~,~j),O), where ?i=Y,, for j=2 ,..., n, s=qn[{-ke}+R’+]-R’+, and ~j, pi are defined, for (p,(yj)) in R’+ x nyzl al;., by


j=2 ,..*, n,

?i(P, (Yj)) = ri(P9 Y 19 xk(YA,. . . > nk(Yn)) +(11m)(P’~j”=~(Yj-~k(Yj)))7

i=l,...,m.

Clearly, the assumption that 8~ 5 n [{ - ke} + R’+ +] and the definitions of $j and pi, imply that every equilibrium of d is in fact an equilibrium of 6’. The proof will thus be complete if we show that $ satisfies the assumptions of Corollary 2.2. It is clearly the case for Assumptions C, WR, P, B [since, for all co’ zco, the set A”(o’) is bounded] and also for Assumption PR’, noticing that Assumption WBMC implies that, for j = 2,. . . , n, the normalized pricing rule of ~j is non-empty-valued. Furthermore, each pricing rule has bounded losses; this is clearly the case for the first firm from Assumption BL, and also for j=2,. . . , n, since, for all (p,(yj)) in S x l-I;= i 85, qEqj(p,(yj))nS implies q.y.=q.Kk(yj)zq.(-ke)=-k.

Finally, we check that $ satisfies Assumption SA. Indeed, let (p,(yj)) E S x l’$= 1 85 such that PE n;=i Fj(P,(Yj))* Clearly, P.Yj=P.%(Yj), for j=2 , . . . , n, and p E f-)3= I $j(p, y,, nk(y2), . . . , rcJyJ), hence, from Assumption SA, one deduces that p.(Cy=i yj+W)=p.yi+p.(~~=z nk(yj))+P*O> infp .cF! i Xi, which ends the proof of Theorem 3.4.

We now give several consequences of Theorem 3.4.

Corollary 3.5 [Beato and Mas-Cole11 (1985)]. The economy d =((Xi, ii, ri), (yj, 9j),O) has an equilibrium if, for all i, OEX, and XicR$, if Assumptions C, P, PR’, B, WR hold, if for some real number k, q c ( - ke} + R: +( j = 1,. . . , n) and if:

Assumption BMC. For j= 1,. . .,n, for all (p,(yj))eS x nyzI A:, for all 4E+j(PvY19.*. , y,) one has qh = 0, for all good h such that Yjh = - k;

Assumption SAL. For all (p, (Yj)) E: S x n$= 1 A!, (P, (Yj)) E PE implies p.(C;=iyj+O)>O.

We notice that Assumption BMC will be shown to be satisfied by a firm following the marginal (cost) pricing rule under the additional assumption that its production set Yj satisfies q= K,-R:, for some non-empty compact subset Kj of R’ [see Lemma 4.2(c) below].

Proof Let 8 be the economy ((Xi, <i, ?i), (q, $j), o), where Fl=


[Yin{-ke}+R:]-R:, for j=2,..., n;= 1 a&

n, q= 5 and for all (p,(yj)) E R’+ x

~1(P,(Yj))=~l(P,~k(Yl),YZ,..‘, ~,)n{q~R’Iq.(~l-71k(~1))=0),

4j(P9 (Yj))=4j(P9 Tc(Y~)~YZ,. . .T YA j=T 0.. 9 %

~i(P,(Yj))=ri(P,71k(Y1),Y2,...,Yn)+(1lm)(P.(Yl--~(Yl))), i=l,...,m.

The economy 2 satisfies all the assumptions of Theorem 3.4, noticing that Assumption BMC is stronger than Assumption WBMC. Thus, by Theorem 3.4, there exists an equilibrium of b, which is clearly an equilibrium of

8. cl

Corollary 3.6. [Vohra ( 1988)].4 The economy d = ((Xi, _i i, ri), (5, 4j), o) has an equilibrium if, for all i, OeXi and XiC R’+, for all j, OE I;, if Assumptions C, P, PR, BL,, B hold, iffor some real number k, 8 c { - ke} + R: +( j= 2,. . . , n), and ifi

Assumption VSA. For all (p,(yj))ESxaK xn;=~A: for which

PEh(P,Yl,...? y,) and for some i~{l,..., m}, ri(p, (yj)) 5 0, then the following assertion is true: for some jE (2,. . . , n}, for all q E $j(p,(yj)), there exists a good h such that [Yjh > - k and q,, > ph], or [yj, = -k and qh < p,].

Proof. We first associate to d the economy d’ = ((Xi, li, ri), (5, vi), 0)) where vi = 4i and the normalized pricing rules fj( j = 2,. . . , ) are defined, for

(PT Y 1,. . . , y,) in S x n;= I a?, as follows:

if yj${-ke}+aR’,, ~j(P,y,,...,y”)=~ji(P,y,,...,~,);

if yj E ( - ke} + aR:, vi(p, y,, . . . , y,) consists of all q E S

such that, for some q’ E ~j(p, y1 , . . . , y,), one has:

qhsq;, if -k=Yjh,

qh=& if -k<y,,<O and

qh 2 6 if Osyjh.

Since $c I;.n{-ke}+R$+,forj=2,..., n, every equilibrium of I’ is in fact an equilibrium of &‘. The proof will thus be complete if we show that b’ satisfies the assumptions of Theorem 3.4. It is clearly the case for Assump- tions, C, P, BL1, 8, PR’ and WBMC. We now show that Assumption VSA implies Assumptions SA, and WR. Indeed, if Assumption SAI, is

4Actually, in the last version of his paper, Vohra is not making Assumption BLI, i.e., that the first firm has bounded losses.

J.Math-B


not satisfied by b’, then for some (p,(yJ) E S x aY, x nyE2 Af one has

P E n;= 1 +ji(P, (Yj)) and p.(C;=1yj+o)~infp’C~=“=Xi~O, since OEXi for all i. Then, clearly, for some i E (1,. . . , m>, one has ri(p,(yJ) SO, hence, Assump- tion VSA implies that, for some j E (2,. . . , n}, for all q E $j(p, (yj)) there exists a good h such that [Yjh > - k and qh> p,J or [Yjh= - k and qh < p,,]. But p E ~j(p,(Yj)) and YjE A: imply that, for some 4 E $,(p, (Yj)), one has phz qh for all good h such that yjh> -k, and p,,sq,, for all good h such that yjh = -k, which contradicts the above inequalities. Let us now suppose that Assump- tion WR is not satisfied by d’, then for some (p,(yJ) E APE and for some i, ri(p,(yj)) ginfp. Xi SO. The end of the proof is the same as above, noticing that (y2,. . . , y,) E n:=2 A!, since yj~ %, for all j. 0

We end this section by a result which considers the case of a single firm in the economy. This assumption is implicitly made in the work of Beato (1982), Brown and Heal (1982), Cornet (1982) and Mantel (1979) who also assume that the firm follows the marginal pricing rule (cf. the next section). Here we assume that the firm follows a general pricing rule +i, which is, as in section 2.1, a correspondence from d Y, to R:,

Corollary 3.7. The economy ~=((Xi, li, ri),(Y~, 91),O) has an equilibrium if, for all i, 0 E Xi, Xi c R’+, if 0 E Y,, if o E R’+ +, if Assumptions C, R, P, PR hold and if

Assumption B,. The set A(w) is bounded.

Assumption SA,. For all (p,y)~SxaY~, pe4,(y) and y+ozO imply p.(y+w)>O.

It is worth noticing that, in the above theorem, the losses of the (single) firm are not assumed to be bounded and that the survival assumption is only made on the set of attainable production equilibria.

Proof. We first claim that there exists a correspondence $r from aY, to S satisfying the following conditions, for all (p, Y)ES x aY,: (i) $i(y) = $l(y) if YE~Y~~{-w}+R$; (ii) $1 is u.h.c. with non-empty, convex, compact values; (iii) pi Gl(y) implies p.(y+o) >O. Then Corollary 3.7 is a direct consequence of Corollary 2.2 and the above claim, noticing that 6, has bounded losses since the above condition (iii) implies that p. yl c1 dAf inf{-q.wlqES}, for all pEtJI(y).

We now prove the above claim and we let, for E > 0, and y E 8 Y,,

where t: YI + [0, l] is a continuous function such that t(y) = 0, if y E ?I dAf Y, n [{ -w} + R:], t(y) = 1, if y E Y, and y# B( Yr;, E), which clearly exists by Tietze-Urysohn’s theorem.


Clearly, for all s>O, $i satisfies conditions (i) and (ii) and we shall show that (iii) holds for E> 0 small enough. Indeed, we choose E>O so that

O<p(y)~finf{p,(y+o)IpE~,(y)}, f or all YE B( $, E) n aY,. This is clearly possible since the function p:aY, + R is lower semi-continuous [by the upper hemi-continuity of $1; see Berge (1966)] and since O<inf {p(y))y o aY, n R}, by Assumption SAL. Then condition (iii) is clearly satisfied from above if YE B( PI,&). We now let YE aY, such that y$ B( fl,~), then PE $r(y) implies p. yZ0 which, together with w E R I+ + and PES, implies that p’(y+o) >

0. cl

4. Marginal (cost) pricing rule

4.1. The marginal pricing rule and the bounded losses assumption

We first recall some definitions. If q is a closed subset of R’ and y is an element in I$, then the tangent cone Trj(y), in the sense of Clarke (1975), to I$ at yj consists of all vectors u in R’ such that, for all sequences (y’} c I$ and {t’} c (0, + co) converging, respectively, to y and 0, there exists a sequence {u’> converging to v such that, for v large enough, y’+ t”v”E 5. Clarke’s normal cone N,,(yj) is then defined by polarity as follows:

Nyj(y)= TY,(y)o={p~R’[p.vSO, for all veTYj(y)}.

We then define the marginal pricing rule MR., of the jth firm by

MRj(yj) = NYj(yj) for yj E d I;..

The above definition coincides with the standard one when I$ has a ‘smooth boundary, that is, if y,~aYj, then N,,(yj) is the half line of outward normal vectors to Yj at yj. Furthermore, if Yj is convex, the marginal rule coincides with profit maximization, i.e., N,,(yj)=PMj(yj) for yj~a~, and we refer to Clarke (1983) for the proof of these results. We also refer to Bonnisseau and Cornet (1985) for the link between the above marginal pricing rule and the standard marginal cost pricing rule for which the firm is instructed (i) to choose its inputs by cost minimization and (ii) to set the prices of its outputs according to its marginal costs.

We have chosen above the definition of normal cone in the sense of Clarke (1975) as in Cornet (1982) instead of the one of Dubovickii and Miljutin (1965) as in Guesnerie (1975) for, under Assumption P, the marginal pricing rule as defined above, will satisfy Assumption PR. Formally we have the following result [cf. Cornet (1982) Rockafellar (1979), for the proofl.

Lemma 4.1. Let 3 be a non-empty closed subset of R’ such that q-R: c 5. Then, for all yj~aI$ NYj(yj) is a closed, convex cone of vertex 0, included in

136 J.M. Bonnisseau and B. Cornet, Equifibria and bounded losses pricing rufes

R’+ and not reduced to (0). Furthermore the correspondence yj + N,,(yj) n S, from aYj to S, is u.h.c. with non-empty, convex, compact values.

We point out that Lemma 4.1 is not true, in general, for the normal cone of Dubovickii-Miljutin. Actually, when it is assumed, as in Beato (1982), that the normal cone of Dubovickii-Miljutin satisfies Assumption PR, it coincides with Clarke’s normal cone [see Cornet (1987)].

The next lemma gives a geometric characterization of Assumption BL for the marginal pricing rule, under Assumption P. Formally it says that Assumption BL holds if and only if the production set rj is star-shaped.

Lemma 4.2. Let 5 be a non-empty, closed subset of R’ such that 5-R: c I;-.

(a) Let aj be a real number. Then the two following conditions are equivalent:

(i) for all y~aI$for all p~N,~(y) nS, then p’yzaj; (ii) for all YE 5, then [aje, y] c 6 (star-shaped).

(b) Conditions (i) and (ii) are satisfied g there exists a non-empty, compact subset Kj of R’, if aj=inf,,{y,Iy=(yh)EKj} and if one of the following conditions holds: C.l Yj= Kj- R’+ or C.2 Yj~Kj is convex.

(c) Furthermore, if & satisfies C.l, then, for every real number If such that Kjc(-Ee)+R$+, for all yj~ aYj and all p E Nyj(yj), yjh s - E implies ph=o.

The condition Cl is assumed by Beato and Mas-Cole11 (1985) in their existence theorem of marginal (cost) pricing equilibria together with Assump- tion BMC. Condition C.2 has been formulated to take into account, when 1=2, the standard example of a production set with a fixed cost and then a convex one. Both conditions, together with the star-shaped one, are illustrated in fig. 1.

Proof of Lemma 4.2, For the sake of simpler notations we omit the index j in the following.

Part (a). (i)+(ii). Since Y is closed, it suffices to show that, for all a’ <a and all y E x [a’e, y] c X Suppose it is not true, then there exists a’ <a, y’ in Y and t’ in [0, l] such that (1 - t’)a’e + fy’ # I: Let to = sup {t E [6, l] I(1 - t)a’e + ty’ 4 Y} and let y0 =( 1 - t,)a’e+ toy’. Then *yO E aY and clearly, there exists a sequence {t’} + 0 such that for all v, t”>O and y0 + t’(a’e- y,,) 4 Y and we shall contradict this last assertion. We first show that ae - y, E T,(yJ. Indeed, since y,-,~aY; condition (i) implies that, for all p~N,(y,)\{0}, (p/p*e)y,Za, hence p. (ae - yO) 5 0. This implies that ae - y, E N,(y,)’ = Ty(yJ”‘, from the definition of the normal cone. But T,(y,) = T,(y,)” from the bipolar theorem [cf. Rockafellar (1970)] since T,(y,) is a closed convex cone of vertex 0 [cf.

J.M. Bonnisseau and B. Cornet, Equilibria and bounded losses pricing rules

(a) Y star-shaped (b) Y=K-Rf (clY\K=C is convex

Fig. 1

Clarke (1983)]. This shows that cle-y,E 7”(y,), which thus implies that there exists a sequence {u’} + cre-y, such that y,, + tYuY~ Y for all v. But ~1’ < c1 implies that, for v large enough, u” 2 cr’e-y,, hence y, + t”(a’e - y,) E {y, + t’~“j -R’+ c Y-R’+ c X which contradicts the fact that y,+t”(cr’e-yy,)$ I:

(ii)*(i). It suffices to show that, for all y in aY ae-ye T,(y). Indeed, let {(Y’, f’,} = Y x (0, + ~0) b e a sequence converging to (y, 0) and let u” = ue - y'. Then {u’} -+ u, for v large enough, t” E [O, l] and, by (ii), y’+ Pu” = y'+ t’( ae - y’) E [ae, y’] c Y!

Part(b). Let cl=inf(y,IhE(l,..., /),y=(y,,)~K). We first suppose that Y= K-R’+. Let YE k: then y=y’-u for some (y’,u)~K x R$. But y’-aez0 and for all tin [O,l] one has ty+(l-t)ae=y’-[(l-t)(y’-cre)+tu]EK-R$=Y, hence [tLe, y] c Y and (ii) holds. We now consider the case where Y\K = C is convex. Let y E I: then either YE K and [ae, y] c {y} - R$ c E: or YE C and [ae, y] ccl Cc Y since cre E cl C and Y is closed.

Part (c). We first choose k <f; such that Kc { - ke} + R’+ + and we notice that Y=[Yn{-ke}+R$]-R$. As in section 3, we denote by nk the projection mapping from R’ onto { - ke} + R’+. We first show that, for all YE al: zJy)-y belongs to T,(y). Indeed let YE aY let {(t’. y')} c R, x Y be a sequence converging to (0,~) and let vy = n,(y’) -y’. Then {v’} + u and, for v large enough, t”~(0, 1) and y’+ tVuY E [y’, am] c Y since rck(yv) E Y and y’Snk(y’). Hence n,(y)-YE T’(y), and for all PENN, OLR.(n,(y)-y)= 1; = 1 pAmax { - k Y,J - YJ- C onsequently, R,,[max { - k, y,,) - y,,] = 0, for all h and if yh 6 -R< -k, one deduces that ph = 0. 0

4.2. Existence of marginal pricing equilibria

If each firm in the economy S=((Xi, si,ri),( Q),o) follows the marginal


pricing rule, an equilibrium of d is simply called a marginal pricing equilibrium. As already noticed, if each firm has a convex production set, the marginal pricing rule coincides with profit maximization, hence the notions of marginal pricing equilibria and of Walras equilibria are identical.

Theorem 4.1. The economy S=((Xi, <i, ri),(YJ,o) has a marginal pricing equilibrium if Assumptions C, P, B, R and SA hold and if, for all j, Yj is star-shaped.

Proof It is a direct consequence of Theorem 2.1, Lemmas 4.1 and 4.2. 0

We now deduce from Theorem 4.1 a result of Brown et al. (1986) on the existence of marginal pricing equilibria when all the firms in the economy are convex but one.

Corollary 4.2 [Brown et al. (1986)]. The economy d =((Xi, ii, rJ,( YJ, co) has a marginal pricing equilibrium if Assumptions C and P hold, if, for j=l ,...,n-l,~isconuex,OE~,OEY,,ifA(C~=l q)n -A(x3=1 Y,={O}andif:

Assumption R*. For all i, OEX~, XiC R’+ and ri(p,(yj)) =6ip*(Cj”= 1 yj+O), for some fixed ai > 0 (i = 1,. . . , m) such that cr= 1 6, = 1.

Assumption SA*. For all y, E 8Y,, p E N,“(yJ n S implies sup p. cj”S: 8 + p. y,+p. o >O, where $ denotes the attainable set of the jth firm.

Proof. It is a consequence of Theorem 4.1 and we only check that Assumptions SA and BL (or equivalently the star-shapedness of Y,) are satisfied, since all the other assumptions of Theorem 4.1 are clearly satisfied. For j=l,..., n- 1, the firms have convex production sets Yj which contain 0 and they are profit maximizing, hence p E N,,(yj) = PMj(yj)_ implies p. yj >= p. 0 =0 = ~j. From the boundedness assumption, the sets I$ are compact, hence, from Assumption SA*, one deduces that the losses of the nth firm are also bounded since p E Nrn(y,) n S implies that p. y, 2 a,, d&f inf { -4. co - sup q .cJZ: ?j)q E S}. Finally, Assumption SA is a consequence of Assump- tion SA* since, for all yje8Yj (j=l,...,n) and all pin;=, NYj(yj) nS one has

(j:l ’ ) :I: II=1

P’ c y.+w =supp. 1 yj+p*(y,+o)~supp. c $+p.(y,+o)>o. j=l

Remark 4.1. It is worth pointing out that all the existence results of marginal pricing equilibria in the literature allowing more than a single firm,


i.e., n> 1 [Theorem 4.1, Corollary 4.2 and also the result of Beato and Mas- Cole11 (1985) stated in Corollary 3.5 for general pricing rules] do assume implicitly or explicitly that the losses of each firm are bounded below.

In a previous paper [Bonnisseau and Cornet (1985)], we reported an existence result of marginal cost pricing equilibrium in which the production sets are not assumed to be star-shaped (which allowed us to consider the production set Y={(y’,y2)~R21y’~0 and y2s(y’)2}) but we made the additional assumption that the boundary aYj is smooth, to be able to use techniques from differential topology, more precisely Morse’s theory. So neither Theorem 4.1 implies our previous existence result nor the converse is true. We further point out that the proof of Theorem 2.1 (hence of Theorem 4.1) relies essentially on Kakutani’s theorem, whereas more elaborate arguments are used in the other paper of ours.

5. Proofs

5.1. Preliminary lemmas

We prepare the proof of Theorem 2.1’ by two lemmas. The first one states that the boundary 85 of each production set r;. is homeomorphic to a Euclidean space of dimension 1- 1. In the following, we let e’ ef {y E R’( y. e = 01, be the orthogonal space to the vector e=(l,. . . , 1).

Lemma 5.1. Let 5 be a non-empty, closed subset of R’ such that T-- R$ c 5 and I;# R’. Then, for all s E e’, there exists a unique real number, denoted by Aj(s), such that s-lj(s)e E aYj. Furthermore, the function Aj: s E e’ --f R is Lipschitzian and the mapping Aj:s +s--lj(s)e from e’ to a? is a homeomor- phism (with inverse proj,l,ay,, the restriction to aY, of the projection mapping onto e’).

The mappings Aj and nj are illustrated in fig. 2.

Proof of Lemma 5.1. We first establish that, for every s be’, there exists a real number lj such that s-lje E Y> Indeed, let y’, y” be elements in R’ such that y’~ I; and y” $5, then the set {A E R 1 s-Ae E I$} is non-empty (since it contains sup,, s,,-inf,,$,) and is bounded below by inf,s,,- sup,, yi. We let ~~j=inf{~ERIs-~eE I$}, then one clearly sees that s-n,e~aI;.

We now show simultaneously the uniqueness part and the fact that the function Aj( .) is Lipschitzian. Let sl, s2 be elements in e’ and let 2: and Ajz such that yld~fsl-A~e~a~ and yZdzfs2-I:e~arj. Without any loss of generality, we can assume that A’ zA2 and we show that y’ -y2# R$ +; indeed, if y’ -y2 E -R’+ +, then y’ E Yj-int Rt+ tint Y., a contradiction with y’ ~85. Thus, there exists he (1,. . . , l} i such that y,,=s~-A.~j2yyhl=s~-A~,


Fig. 2

hence lnj’ -$I =$ -A: 5s: -si 5 (Is1 -s’I\. This shows firstly that, for every SET’, the element Aj(s) is uniquely defined since s1 = s2 implies that Aj = A:. Secondly, this shows that the function lj:e’ + R is Lipschitzian since Inj(S')-nj(S2)(~IIS'-S211.

We end the proof by showing that the mapping nj:S -+ s - lj(s)e has proj,l18Yj for inverse. One clearly has proj,,(s-IJs)e) = s for every s E e’ and we now show that proj,,(y) - ;Ij(proj,,(y))e = y, for every y E 83. Indeed, one clearly has y = proj,ly - lie with Aj= - y * e/l and the above uniqueness property and the fact that y E arl;. imply that Aj(proj,lyj) = lj. l-J

In the following, for every integer k, we let

Xf=Xi n [{ke} - R:], i=l,...,m.

We now state a lemma which is a direct consequence of Assumptions R and SA under the additional assumption that, for all i, X,C R’+ and OE Xi since in this case, inf p * Xl = inf p * Xi( = 0).

Lemma 5.2. Under Assumptions PR’, SA and R, if K is a compact subset of R’(“’ I), then, for k large enough

(p, (yj)) E PE n K implies ri(p, (Yj)) > inf p* X: for all i = 1,. . . , m.

Proof of Lemma 5.2. If it is not true, there exists a sequence {pk,(yfi), i”} in


[PEnK] x(1 , . . . ,m} such that Ik = rc(p”, (y;)) 5 infp’ * Xf. Since K is compact, without any loss of generality, we can suppose that the sequence (p”,($} converges to some element (@,(jj)) E K and that the sequence {ik} is constant, say ik= 1. Since 6 is u.h.c. from Assumption PR’, (fi,(jj)) EPE; hence, from Assumptions SA and R, r,(j$ (jj)) > inf pa XI. This implies that there exists x,EX, such that r,(p,(jjj))>P.xl. Then, for k large enough, x,EX:, hence rksinfpk.X:spk.xl. At the limit when k+ CO, one gets limkrk=rl(jk(jj))s p.x,, a contradiction with the above inequality. 0

5.2. Proof of Theorem 2.1’

We fix .s>05 arbitrary and we let

k=(p~R’I~~=~p~=l and ~,,z--.s,h=l,...,I}.

We choose xiEXi(i=l,..., m), arbitrary and we choose r large enough so that p.Zi<r (i=l,...,

(P, (Yj)lES x ny=l t,

m), pews& for all PES,, and ri(p,(yj)) SF, for all i, all where q is the attainable set of the jth firm. Such a

choice is clearly possible since the functions li are continuous and the attainable set A(w) is compact. We now let E =(e’)n and yj(s) = sj - lj(sj)e, for all j, all s =(sJ E E, where Aj:e’ + R is the function associated by Lemma 5.1 with the production set 5 (noticing that Assumption B implies that q#R’, for all j). Then from Assumption B and Lemma 5.1, there exists a closed ball B with center 0, in the Euclidean space E so that

(5.1)

From Lemma 5.2, we can choose k large enough so that

ri(p, (yj(s))) >infp.X: (i = 1,. . . , m) for all (p, (yj(s))) E PE

such that SE B, (5.2)

ZieXf and Ric{ke}-R$+ (i=l,...,m), (5.3)

where 8, denotes the attainable set of the ith consumer. We introduce the standard definitions of the (quasi-)demand correspon-

dences of the bounded economy. For all i and all (p, w) E R’ x R, we let

sThe reader who is interested in the existence of a free-disposal equilibrium, may take E=O and S,=S in the following.

142 J.M. Bonnisseau and 13. Cornet, Equilibria and bounded losses pricing rules

5~(p,w)=(xi~B:(~,w)Ixlixi, for all xE/$(p,w)} if w>infp.X:;

fF(P, w) = 51(P, w) if w>infp.Xf,

={xiEXl(p*xi=infp.X:} if w5infp.X:.

We let rc be the projection mapping from R’ onto S and we define the correspondence F from (I:= I Xi) x B x S, x S” to itself as follows:

4

F(x, S, P, (Pj)) = n F,(x, S, P, (pj)), where

Fl(xv S, P, (PJ) = f fb, min {C rA4P), (Yj(s)))); i=l

i (p-pj).(~j-e>)~O, for all (QEB j=l

F3(X,S,p,(pj))= qES,I(q-q’). X- i yj(S)-W 20, for all q’ES, j=l

FAX, S, PT (Pj)) = ‘$(x(P), (Yj(S))).

The correspondence F is clearly upper hemi-continuous with non-empty, convex, compact values by Assumption PR’, Lemma 1 in Debreu (1962), Lemma 5.1 and the maximum theorem [in Berge (1966)]. Consequently, by Kakutani’s theorem, there exists a fixed-point (x*,s*,p*,(pf)) of the correspondence F. But x*=x7= 1 x:, for some x: ~Xf(i= 1,. . . ,m) and if we let yf=yj(s*)(j=l,..., n) and s* =(sT) one has

x: E ff(p*, min {rr, Fj) with rl = ri(n(p*),(yr)) (i=l,...,m); (5.4)

j$I(p*-pf).~~~ i (p*-P?)*s~ for all(sJEB; j= 1

(5.5)

p*.(~lxt-~lY~-o)~P~(~lx~-~lY~-~) for all PEG

(5.6)

(Pj*) E $(4P*), (Yf)). (5.7)


The following claims will show that ((xr),(yy),p*) is an equilibrium of d.

Claim I. (p*,,,., p*) E $(p*, (y,tN, i.e., (P*, (yj*N E P-E.

Proof: We first establish that

i~lx:-j~iyfog*et -Ri++ u (0) with

y*Ep*. ( ii,x;- i y?-w . j=l >

(Q-9

Indeed, one deduces from (5.6) that O~p*.(~~=“=, ~t--~j”=~ yj*-o--y*e) for all PES, or, equivalently, ~~=i x:-c;=, y?-o-y*e~S,Oc {v~R*]p.uS 0, for all PE SE>, the negative polar cone of S,, which is clearly a subset of

-R:+ u (0) and the result is established. We now claim that y* srnF+f--cc (the proof of which will be given below)

and we deduce from it the fact that (p*, . . . , p*) E $(p*, (~7)). Indeed, from (5.8) and our choice of the ball B in (5.1) one deduces that s* =($) E int B. Consequently, by (5.5) the gradient at s*=($) of the linear function

tsj) + Cj”= 1 (P* -Pi*) ’ j s is equal to 0, hence p* =pT, for all j. But, for all j, pj*

belongs to S, hence p* = pi* belongs also to S, and 7c(p*) = p*. Consequently, by (5.7), (P*, . . . , P*) E &P*, (yj*,,.

To show that y*=p* .(x7= 1 xr - Cj”= 1 y: - 0) 5 mf- u + f, we first notice that p* .(-co) 5 f, from our choice of r? Furthermore, for all i,

xf E ff(p*, min (r:, fj) by (5.4); then either xf E <f(p*, min (P:, fj>, and p* . xr S min { rl, f} 5 f, or p* . xi* = infp * .Xf and infp* .XfzLp*.XisI;, from our choice of k, Xi and r; hence, for all i, p* . XT 5 F. Thus, it only remains to show that p* . x7= i yj* 2 CI. Indeed, taking (sj) =0 in (5.5) one gets x7= 1 (p* -pj*) . sj* 20 which implies that p* . xi”= r y; 2 xi”= 1 pr. yJ since y; *Zf yj(s*) = ST - nj(sT)e by Lemma 5.1 and noticing that p* . e =py. e. But Assumption BL and the fact that (p?) E $(n(p*), yy)) [by (5.6)] imply that CT= I pj* .yr Za, hence p*.=&yj*&Y.. 0

Claim 2. C~=n=l.:-Cjn=lyj*-o~-R:+u{O}.

Proof: In view of (5.Q it suffices to show that y*sO. But y*=p*. (C~=“=xxT-CJ=lyj*-O)=C~=“=(p* . x: -rF) and we shall show that p* . xi* 5 r:, for all i. Indeed for all i, XT ~ff(p*, min {rf, f}), by (5.4), rf =ri(p*,(yT)) > infp*.X: by (5.2) and Claim 1, and f>p*. Xi 2 infp* .X:, from our choice of f and Xi. Consequently, for all i, min (r:, F] > inf p* . X: and X:E &p*, min {r: 3) which implies that p* . xi*=< min { r$ f} 5 rr. 0


Claim 3. For all i, xi* E &I*, rr).

Proof. We have shown previously that x: ~&*,min {rT,F}), for all i, and from Claim 2, ((x:),(yf)) is an attainable allocation, hence yj* E pj, for all j. Therefore, for all i, r: =rJp*,(yj*)) SF, from our choice of ?, hence xi* E t:(p*, rl). 0

Claim 4. Cy= 1 x: = ES= 1 yj* + w.

ProoJ: Indeed, from Claim 2, I?! i xr - cj”= 1 yf -w E - R: + u (0) and, from Claim 1, p* ~ScR’+\{of. The proof will thus be complete if we show that p* * (IF= lx:-~j”=lyj*-co)=o. But p*.(c~=“=~xi*-cjn_~yj*-o)=

CF= 1 (P* . x:-r:) and x: E t$p*, r:) (Claim 3) implies that p* . x: S r:, for all i. We end the proof by showing that p* + XT = r:, for all i. Suppose, on the contrary, that, for some i, p* . x: < rr. Let Ni = {x E Xi 1 p* . x < rr and x E {ke} - R’+ +}. Then, Ni is an open subset of Xi for its relative topology, and Ni contains xz [since ((xl), (yj*)) is an attainable allocation by Claim 21 which implies that x: E Xi c { ke} -R’+ + . Hence, from the local non-satiation assumption in Assumption C, there exists XiE Ni such that Xi>X:, which contradicts the fact that XT belongs to <:(p*,r:). 0

Claim 5. For all i, X: >i Xi for all Xi E Xi such that p*. xi s r:.

Proof It is a consequence of Assumption C and the facts that xr E gf(p*, r:) (Claim 3) and Sic {ke} -RI+ +, for all i. We refer to Debreu (1959, p. 87) for the proof. 0

5.3. Proof of Theorem 3.1

The proof of Theorem 3.1 is similar to the one of Theorem 2.1’ and we let as above Xf=X;n [{ke}-R:] (i=l,..., m), yj(s)=sj--nj(sj)e (j= l,..., n) and S,={p~R’\~~=~p,,=l and p,,z-~,h=l,....,l}. From Assumptions B, and SA, we can choose k, E>O and a closed ball B with center 0 in the Euclidean space Ed&’ (e*)” satisfying the following assertions:

~EX:+R:+ and X,c {ke} -R’+ + (i=l,...,m), (5.9)

p.Ui>infp.Xf for all pES, (i=l,...,m), (5.10)

I I s’E i _Vj(S)+COE i Xi+R$ c int B. j=l i=l

(5.11)


We let F’ be the correspondence from (cyS1 Xf) x B x S, x S” to itself defined by

Flfx~ % P, (Pj>> = f f: Ps P. wi + i @ijP- YjCs) 3

i=l ( j=l > F:lx2 % P, (Pj)) = Fv(x~ ST P9 (Pj)) as defined above for v=2,3,4

Then, the correspondence F’ admits a fixed-point (x*, s*,p*,(pT)). As above, x*=~~= 1 x: for some xr E X: (i = 1,. . . , m), and we let s* =(sr) and yj*=yj(s*)(j=l,..., n). Then ((xr), (y;), p*, (pi*)) satisfies the fixed-point properties (5.5) to (5.7) together with

X” E fF(p*, ~7) with $ = p* ’ I + i Bijp*. y? (i=l,...,m). j=l

(5.12)

We now show that ((xr),(yT),p*) is an equilibrium of 8. The proof is similar to the one of Theorem 2.1’ and it consists to show successively that ((xF),(yj*),p*) satisfies the above Claims 1 to 5 with only a change in the order between Claims 1, 2 and 3 and their proofs. The proofs of Claims 4 and 5 are unchanged.

Claim I. For all i, xi* E <f(p*, rr).

Proof From (5.12), for all i, xT~ff(p*,r:). Hence from the definition of f:, it suffices to show that r~=p*.wi+~~=, 19~~p*.yj*>infp*.X:. But from (5.10) for all i, p* .o,>infp* 1 Xf, hence the proof will be complete if we show that p* . yr 2 0, for all j. Indeed, from (5.5), taking sj= 0 and si’ = sf, for j’ #j one gets (p* - ~7). SF 2 0, hence p* . yj* 2 pr. yr since yr dSf yj(s*) = sj* - lj(sf)e, by Lemma 5.1 and noticing that p* . e = pi* . e. Consequently, p* . y; 2 pi* . y? 2 0 by Assumption LF and the fact that (~7) ~$(n(p*),(yj*)) by (5.7). 0

Claim II. ~~Ix,i*-~~=l yT--OE -R’++ u (0).

Proof: From Claim I, for all i, XT E sf(p*, rr), hence p* . XT s rr. Conse- quently, p* .(Cr_ 1 XT - xi”= 1 yj” - o) = cF= 1 (p* . xi* - rr) 5 0. Hence, from (5.6),


0 2 p* * CC?= r x: -cj”= r yr 1 w), for all p E S,, or equivalently Cy= 1 XT - Til!;-;S:, tE negative polar cone of S, which is clearly a subset of

Claim III. (p*, . . . ,p*) E $(p*,(yr)), i.e., (p*,(yf)) E PE.

Proof. From Claim II, I;= 1 yj(s*) + CO EC:= I Xi+ R$, hence from our choice of B in (5.1 l), s* = (~7) E int B. Consequently, by (5.5), the gradient at s* =(sj*) of the linear function (sj) + c;= 1 (p* -pT) .sj is equal to 0, hence p* = pf, for all j But, for all j, pj* belongs to S, hence p* (= pr) belongs also to S and x(p*) =p*. Consequently, from (5.7) (p*, . . . ,p*) E $(p*, (yf)). 0

The end of the proof consists in the above Claims 4 and 5 (the proofs of which are unchanged).

Remark 5.1. One cannot weaken, in general, Assumption B in Theorem 2.1’ by only making Assumption BO, that is, the set of attainable allocations is bounded. Indeed, let us consider an economy with two goods, a total initial endowment o= (1, l), 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, and normalized pricing rules, $r and 4, defined for (yi, yZ) E 13Yr x aY, by

Y,= -R:; d2(y2)={(+3. A)}.

We first notice that the set PE is empty, 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 B [taking o’ = o + (1, l)] but Assumption B0 is satisfied.

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