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General equilibrium with producers and brokers
Economics Letters 0165-1765/93/$06.00
41 (1993) 257-263 0 1993 Elsevier
General brokers Existence
257 Science
Publishers
equilibrium
B.V. All rights
reserved
with producers
and
and regularity
Elyits Jouini * ’ INSEE-ENSAE, 14 Boulevard Adolphe Pinard, 75 014 Paris, France Laboratoire d’Economt?rie de 1’Ecole Polytechnique, Paris, France
Hkdi Kallal Department Received Accepted
of Finance, Stern School of Business, New York University, New York, USA
20 January 1993 9 March 1993
Abstract In this paper we prove the existence of general equilibrium with transaction costs generalizing Hahn’s (Review of Economic Studies, 1973, 40, 449-461) model by introducing producers and nonconvexities (in particular we allow for increasing returns in transaction sets). We also recover any exchange economy as a special case and this allows us to analyze the effects of small frictions on bid-ask prices, consumption vectors and utilities. We prove that, generically, the induced perturbations are of the same order as the frictions.
1. Introduction Several authors have introduced transaction costs in a general equilibrium framework [in particular Hahn (1973) and Starrett (1973)]. In Hahn’s model there are, besides consumers and producers, a special kind of producer: brokers. Brokers purchase at the ask price the goods supplied by consumers and producers, transform them according to a feasible transaction set and sell them back, at the bid price. In these papers, however, transaction sets are assumed to be convex cones. Hahn (1973) points out that this assumption is ‘pretty terrible. [It] rules out increasing returns when causal observation suggests that set up costs are an important feature in transaction technologies’. In this paper we are not making any convexity assumption on the production and transaction sets. This is, we believe, the typical case for transaction sets, as is pointed out in Hahn (1973) and
*We are indebted to Michel Balinski for his encouragement and to Jean-Marc Bonnisseau and Jose Scheinkman for helpful discussions. We are also grateful to Andreu Mas-Cole11 and the other participants of the University of Paris I Equilibrium Theory workshop. The financial support of the Laboratoire d’Econom&rie de 1’Ecole Polytechnique, of the Department of Economics of the University of Chicago and of NSF grant SES 9022643 are gratefully acknowledged. * Correspondence to: E. Jouini, INSEE-ENSAE, 14 Boulevard Adolphe Pinard, 75 014 Paris, France.
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E. Jouini and H. Kallal I Economics
Letters 41 (1993) 257-263
Starrett (1973). Indeed, there are important fixed costs involved in transactions. For instance, in markets requiring immediacy such a fixed cost is time: the dealer must be present on the market at all times. The main purpose of this paper is to study the effect of small frictions (in the transaction possibilities) on equilibrium. We show that, generically, at equilibrium the perturbations in prices, in the consumption vectors, and in the utility of the consumers is of the same order as the frictions introduced in the transaction possibilities. In section 2 we introduce the general framework of this paper. In section 3 we establish an existence result of an equilibrium. In section 4 we study the effect of small frictions.
2. The model ’ Let $5’ be an economy with e goods, h = 1,. . . , t, m consumers, i = 1,. . . , m, n producers, j=l,..., n, and r brokers, k = 1,. . . , r. The ith consumer is described by a continuous convex and locally non-satiated preferences preorder 5 ; on a consumption set Xi = R?+ and initial endowments OJT. We denote by o* the vector of total initial endowments, i.e. w* = Cy=, WT. The jth producer is able to transform an input vector y, E R: into an output vector yj E R: , as long as ( y,, y; ) belongs to the feasible production set Y, C RF. Transactions are carried out by the brokers. The kth broker buys zk E Rt and sells z; E R: constrained by (zk, 2;) E Z,, where Z, is his feasible transaction set. We assume that for all j (resp. all k) the production set YJ (resp. transaction set Z,) is non-empty, closed and (q + R: x R<) fl Ry C Yj [resp. (Z, + R: x R?) fl Ry C Z,], i.e. less can be produced with more. We also assume that for every ‘efficient’ production (resp. transaction) plan (y,, y;) E 8Y [resp. (z,, 2;) E aZ,], the firm (resp. the broker) sets the price p in (P,(Yj, yJ) (resp. IJJ~: aZk-+ S,,) is an upper semicontinuous ’ [resp. t&(zk, z;)], where ‘p,: aYj+$, convex compact value correspondence called the pricing rule. For instance, this formalization takes into account profit maximization in the convex case and marginal pricing or cost-average pricing in general. For a price system (p, q), production plans (y,, y;) and transaction plans (zk, z;) the revenue of the ith consumer is equal to wi(p, q, (yj, yi), (zk, z;)), where (w;) is a collection of positively homogeneous [with respect to (p, q)] mappings from Ry x R2e” x R2er into R satisfying Cz”=, Zk, z;>> = CT=, (-p . y, + 4. y;> + c;=, (p . z; q * z/J. Wi(P, 97 (Y,, Y;), ( The ith consumer purchases a vector xi at the price p and sells a vector .x: at the price q constrained by (xi, xi) E B(p, q, wi), where B(p, q, w,) = {wi +x, -x1: (xj,x:)ERy andp.x; q . xi 5 w;}. This merely says that his net consumption w, + xi - xl must be in his budget set.
’ If x = (x,) and y = ( y,,) are vectors in R’, we let x. y = Ci=, xhy, be the scalar product of R’, and IJxI( = (x .x)112 be the Euclidean norm. The notation x 2 y (resp. x + y) means x,, 2 y, (resp. xh > yh) for all h; we let R: = {x E R’: x 2 0) and R:, = {x~R’:x>O}.Ifx~R’wedenotebyx’ and n- the vectors, respectively, with coordinates xi = max(O, x,) and equal to 0 except the hth xi = max(O, -x,,). For h = 1,. , e we denote by eh the vector in R’ with all coordinates equal to 1 and by S, = {p E R:: C:=, coordinate which is equal to 1. We denote by e the vector in R ’ with all coordinates ph = 1). For A c R’, we denote by CIA, 8A and ri(A), respectively, the closure, the boundary and the relative interior of A. 2 Given two topological spaces X and Y, a correspondence r$ is said to be upper semi-continuous if 4 is locally bounded and if the graph of 4, i.e. {(x, y) E X x Y: y E 4(x)}, is closed.
E. Jouini and H. Kallal I Economics
259
Letters 41 (1993) 257-263
Definition 2.1. An equilibrium of 8 = ((Xi 5 ;, w,, w,), (Y), (2,)) is a list (p, (zk, z;)) in Ry x Ry”’ X Ry” X RF’ such that: (i) for all i, y + xi -x: is a best element for _< 1 in the budget set
q, (x,,xi),
(yj, Yi),
B(P, 4, Wi(P, 47 (Y,, Y’i), (Zk> Z’k))) ; (ii)
for all i, (Y,, yj) E dq and (P, 4) E Vj(yj, ~j);
for all k, (zk, z;)EU, and (p, q)E c~,x,+c;~,y,~c;=,z;, (iv) { c;, x; + c;=, yj 2 c;=, Zk ;
(iii)
slr,(zk,zL);
p . (CE, x, + ql y, - c;=, z;) = 0, 09 ( q . (CE”,, x;. + c;=, y; - CL=, Zk) = 0. For w E Rt let A(W) = {((y,, yj), (z,, z;)) EIIr=, aY, X II;,, a.Z,: 2; - Cl,=, zk + w 2 0} be the set of attainable production and transaction at this stage the following boundedness and survival assumptions. Assumption
(B).
Assumption
(WSA).
For every
(P,
cP,(Yj, y;))wX=,
For every
o z w*, the attainable
CT=r Yi - Cy=r Yj + CL=, plans. We can introduce
set A(w) is bounded.
(SA(w)) holds: w 2 w*, the following assumption and such that (p,q)E(n;=, E R? x A( w > such thatprq (cI,(zk, 2:)) we have for all i, wi( p, q, (y,, yl), (zk, z;)) + q . wi > 0. For every
4, (Y,,
y’j),
(zk, z;>)
Assumption (WSA) means that for an economy with initial endowments o at least equal to w*, the ith consumer can generate a positive total revenue by selling his endowment (at the price q) for all production-transaction marginal pricing equilibria that are in the attainable set.
3. The existence
result
In this section we state and prove the existence of an equilibrium as defined fact, using the results of Bonnisseau and Corne (1991) we prove the existence pricing equilibrium. Also, as we shall underline in the proof, we can show the equilibrium for general pricing rules with bounded losses using Bonnisseau and We recall that, for a closed set YE R”, a perpendicular vector to Y at y E Y the set &(y)={p~R’? Then, Clarke’s hull of the set
normal
{P ER’:g(y,)
3p>O,
VY’EY,
p.yzp.y’-plly-y’ll’}.
cone to Y at y [see Clarke
(1983)],
C Y, (Y,)+Y,
(P,)+P
3(p,)
in section 2. In of a marginal existence of an Cornet (1988). is an element in
CRY,
d enoted
N,.(y),
is the closed convex
and vq, p4 E ly(y)>
.
As in Guesnerie (1975) and Bonnisseau and Cornet (1991) the marginal pricing rule consists in fulfilling the first-order necessary condition of profit maximization. Formally, this can be stated: (P? 4) E U;(Y,* Yl) if (-P, 4) ENy(Y,, Yi) and (P, q) E (Clk(zk, 2;) pricing rule is of interest because the second welfare theorem production economy. In our case this result continues to hold,
if
C-q,
P> E &,(zk,
holds for a marginal in spite of transaction
2;).
This
pricing costs,
260
E. Jouini
and H. Kallal
because the definition of Pareto optimality as well as the production possibilities.
I Economics
ought
Letters
41 (1993) 257-263
to take the transaction
possibilities
into account
Theorem 3.1. Under Assumptions (B) and (WSA), the transaction cost economy 25 = ((X,, w,), (Yi), (2,)) admits a marginal pricing equilibrium and we have p 2 q. Proof.
Forj=l,..., q =
n,k=l,...,
randi-l,...,
{(-y,, y;) ER=: (y,, Y;> E
2, = {(z;,
-zk)
2, = {(xi, -xi)
_i , , wi,
m,let
r,, -
E R2”: (zk, 2;) E Z,)
(0) x R: ,
- R: x 101 >
E R*‘: xi 2 0, xi 2 0 and x, -x:
+ wT 2 0} ,
and let 3 i be the natural preorder induced by 5 1 [i.e. (x,, -xi + wT)a i(Xi, -Xi + wT> if (Xi-X:_+WT)~,(~,-XI+OT)]. Let $5 be the economy, with 2& goods, defined by ((X,, 3,, W,, GT), (q), (Z,)), where for sets of this economy and where 6, i=l,... , m, 6: = (0, O), where <(q>, (Z,)) are the production is defined by the formula: Gi(p, q, (-y,, yj), (z;, -zk)) = wi(p, q, (yj> Y;>, (z,, z;)). In this transformed economy, producers and brokers are indistinguishable in their behavior: they take (p, q) for the price vector. All the assumptions of Bonnisseau and Cornet (1991) are satisfied and gu_arantee the existence of a marginal pricing equilibrium (p”, 4, (X,, -x”:.), (-Y;,y”J), (2”:, -,SYk)) for 8. We can then easily 0 verify that (p”, @, (x”,, x”:), (Y;, (j?)‘), (Fk, (FL)+)) is an %-equilibrium.
4. Small frictions
in an exchange
economy
In this section we analyze the effect of small frictions on an exchange economy. The definitions of the previous section extend straightforwardly to this case. The results that we obtain could easily be generalized to production economies. Let % = (X,, 5 ;, wi) be an exchange economy with (w,) % 0. We shall see that its equilibria can be recovered as equilibria of a transaction economy % obtained from % by adjunction of a transaction set Z,, = ((2, z’) E Ry: z’ 5 z and z’ 5 2~). Note that since Z, is convex the first-order condition for profit maximization is also sufficient and the profits of the broker are always non-negative. We assume that these profits are evenly distributed across consumers. 3 The economy 8 satisfies all the assumptions of our existence result. Proposition
economy % = (X,, _< i, wi), then 1s an equilibrium of the transaction economy @ = ((Xi, _< i, a.+), Z,). C onversely, if (p, q, (x,, xl), (z, z’)) is an g-equilibrium, then p = q and (p, (x, -xl + w,)) is an g-equilibrium. (P,
p,
((x,
4.1.
-w;>+,
Let (p, (x,)) be an equilibrium (Xi - qn,
v;“,,
(x, - qr,
of the exchange
qI,
(Xi - q)+N
The proof of this proposition is left to the reader. From now on we shall identify an exchange economy ’ This assumption
is in fact innocuous
since at the ‘S-equilibria
with its associated
the broker
always
makes
transaction
zero profits.
economy
E. Jouini and H. Kallal I Economics
obtained by adjunction of the transaction without producers and with a transaction
Letters 41 (1993) 257-263
set 2,. Let us consider now a transaction set Za sufficiently ‘close’ to Z,,.
261
economy
Let Z, be a sequence of convex subsets of R2’ and let rr, be the projection mapping on Z,. The sequence (Z,) is said to converge to a convex subset Z of R2” if the sequence r,, converges to the projection mapping 7~ on Z for the Ce2 norm on every compact subset of RZe\A, where A is a null sets is the same as in Mas-Cole11 measure subset of R2’. This topology on transaction-production (1985, section 3.8). For our purpose we shall strengthen the assumption on the preferences assuming that, as in Balasko (1988, section 2.3), the preordering _< i is representable by a quasi-concave smooth utility function ui for which the first-order sufficient condition for strict monotonicity and the secondorder sufficient condition for strict quasi-concavity are satisfied. What follows can, however, be generalized to non-convex transaction sets noting that, under the free disposal assumption, aZ’, appears as the graph of a function A,,: el+ R and hence n,, can be replaced by A,, in the definition of convergence. In the following theorem, we only consider convex perturbations of the set Z, in order to apply Mas-Colell’s (1985) regularity results. There is no real difficulty in extending our result to non-convex perturbations using regularity results as in Jouini (1992a,b). Theorem 4.2. For almost every initial endowment o E R’JJ’, and for every exchange equilibrium of the economy $? = ((X,, _< i, wi), Z,) if we perturb slightly Z, we can find near this equilibrium an equilibrium of the perturbed economy. This means, in particular, that by introducing small frictions into an exchange economy we obtain consumption vectors and bid-ask prices in a neighborhood of the exchange equilibrium. Consequently the bid-ask spread is small. Since our argument relies on an implicit function theorem, frictions induce exactly first-order perturbations in prices and allocations. Furthermore, the difference in utility for the ith consumer is of second or higher order only if consumers have colinear net trade vectors, which is easily shown to be nongeneric. Note that, in general, this result does not hold for all initial endowments o since if 8 is a singular exchange economy [see Dierker (1982) and Mas-Cole11 (1985)] it is then easy to see that slight perturbations of the parameters of the economy can induce jumps in the equilibrium prices. Hence, using Balasko’s lemma (1988, lemma 5.2.1) we can show that these jumps in prices induce jumps in utilities and consequently in allocations. If the economy 8 = (X,, si, w,) is a regular exchange economy [see Dierker (1982), Mas-Cole11 (1985) and Balasko (1988)], then it admits a finite number of equilibria. If for all equilibria (p, (xi)) of 8 and h = 1, . . . ,8 we have xh # of, we say that the economy 8 is transactive. Lemma 1. If m 2 2, for almost every w E RTe the economy K = (X,, 5 ,, CO,)is a regular transactive economy. The proof of this lemma is obtained by classical arguments of differential topology. Let us now define the demand correspondence Bj of the ith consumer when facing the price vector (p, q), with p %-q, as the set of solutions (xi, xl) of the following program:
262
E. Jouini
and H. Kallal
(max ui(wi + xi -xi)
(9)
1
I Economics
Letters
41 (1993) 257-263
,
X,20, Xj20,
lp.xi-9.x:
10.
It is easy to show that in this case the solutions (xi, xi) must satisfy xi *xi = 0. We extend the previous definition to the case where p 2 q by adding the condition x, *x: = 0. The only effect of this constrain is to get rid of the trivial indeterminacy of solutions when ph = qh for some h. In fact, strict quasi-concavity of ui implies that bj is a well-defined function for p 2 q. Lemma 2. Let (p*, p*)Eri(S,,) a neighborhood of (p*, p*). This lemma
is a consequence
such that bi(p*,
of a classical
implicit
p*) = (-lzi,ai) with ii +a: 90.
function
Then B, is %’ in
theorem.
Proof of Theorem 4.2. The result is obvious for m = 1. Assume now that m 2 2 and let w E RY$ such that 8 = (X,, _< i, w,) is a regular transactive economy (by Lemma 1 this is true for almost every w). In fact, in what follows we shall assume that the consumers are directly characterized by their demand functions and we shall define 8 by (X,, D,, wi). Let G?= ((X,, 8,) w,), Z,) and g be the economy obtained from $. Since Z0 is a polyhedral cone of the activity matrix A = ’ (-I,, Z,) and the ith demand function fii in the transformed economy 2 is defined by fii(p, q) = (xi, -xi), where (xi, x:) = bi(p, q), we have, following Mas-Cole11 (1985, Proposition 6.4.1) that % is regular if for all equilibrium of g the matrix
WP,
-a&(~, -‘A
9) = (
-(P,
q) 4)
A 0
‘(~7 q) 0
0
0
1
has full rank, which is easily verified. We then obtain consequence of Mas-Colell’s results (1985, section 6.4).
the result Cl
of Theorem
4.2 as a direct
References Balasko, Y., 1988, Foundations of the theory of general economic equilibrium (Academic Press, New York). Bonnisseau, J.M. and B. Cornet, 1988, Existence of equilibria when firms follow bounded losses pricing rules, Journal of Mathematical Economics 17, 119-147. Bonnisseau, J.M. and B. Cornet, 1991, General equilibrium theory with increasing returns: the existence problem, to appear in Equilibrium theory and applications: Proceedings of the 6th international symposium (Cambridge University Press, Cambridge). Clarke, F.H., 1983, Optimization and nonsmooth analysis (John Wiley, New York). Dierker, E., 1982, Regular economies, in: K. Arrow and M. Intriligator, eds. Handbook of mathematical economics, vol. II (North-Holland, Amsterdam) 795-830, ch. 17. Guesnerie, R. 1975, Pareto optimality in nonconvex economies, Econometrica 43, no. 1. l-29. Hahn, F.H., 1973, On transaction costs, inessential sequence economies and money, Review of Economic Studies 40, no. 4, 449-461. Jouini, E., 1992a, An index theorem for nonconvex production economies, Journal of Economic Theory 57, no. 1, 176-196.
E. Jouini and H. Kallal I Economics
Letters 41 (1993) 257-263
263
Jouini, E., 1992b, Structure de I’ensemble des Cquilibres dune Cconomie productive, Annales de I’Institut Henri Poincare: Analyse non-lineaire 9, no. 3, 321-336. Mas-Colell, A., 1985, The theory of general economic equilibrium: A differentiable approach (Cambridge University Press, Cambridge). Starrett, D., 1973, Inefficiency and the demand for ‘money’ in a sequence economy, Review of Economic Studies 40, no. 4. 437-448.
41 (1993) 257-263 0 1993 Elsevier
General brokers Existence
257 Science
Publishers
equilibrium
B.V. All rights
reserved
with producers
and
and regularity
Elyits Jouini * ’ INSEE-ENSAE, 14 Boulevard Adolphe Pinard, 75 014 Paris, France Laboratoire d’Economt?rie de 1’Ecole Polytechnique, Paris, France
Hkdi Kallal Department Received Accepted
of Finance, Stern School of Business, New York University, New York, USA
20 January 1993 9 March 1993
Abstract In this paper we prove the existence of general equilibrium with transaction costs generalizing Hahn’s (Review of Economic Studies, 1973, 40, 449-461) model by introducing producers and nonconvexities (in particular we allow for increasing returns in transaction sets). We also recover any exchange economy as a special case and this allows us to analyze the effects of small frictions on bid-ask prices, consumption vectors and utilities. We prove that, generically, the induced perturbations are of the same order as the frictions.
1. Introduction Several authors have introduced transaction costs in a general equilibrium framework [in particular Hahn (1973) and Starrett (1973)]. In Hahn’s model there are, besides consumers and producers, a special kind of producer: brokers. Brokers purchase at the ask price the goods supplied by consumers and producers, transform them according to a feasible transaction set and sell them back, at the bid price. In these papers, however, transaction sets are assumed to be convex cones. Hahn (1973) points out that this assumption is ‘pretty terrible. [It] rules out increasing returns when causal observation suggests that set up costs are an important feature in transaction technologies’. In this paper we are not making any convexity assumption on the production and transaction sets. This is, we believe, the typical case for transaction sets, as is pointed out in Hahn (1973) and
*We are indebted to Michel Balinski for his encouragement and to Jean-Marc Bonnisseau and Jose Scheinkman for helpful discussions. We are also grateful to Andreu Mas-Cole11 and the other participants of the University of Paris I Equilibrium Theory workshop. The financial support of the Laboratoire d’Econom&rie de 1’Ecole Polytechnique, of the Department of Economics of the University of Chicago and of NSF grant SES 9022643 are gratefully acknowledged. * Correspondence to: E. Jouini, INSEE-ENSAE, 14 Boulevard Adolphe Pinard, 75 014 Paris, France.
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E. Jouini and H. Kallal I Economics
Letters 41 (1993) 257-263
Starrett (1973). Indeed, there are important fixed costs involved in transactions. For instance, in markets requiring immediacy such a fixed cost is time: the dealer must be present on the market at all times. The main purpose of this paper is to study the effect of small frictions (in the transaction possibilities) on equilibrium. We show that, generically, at equilibrium the perturbations in prices, in the consumption vectors, and in the utility of the consumers is of the same order as the frictions introduced in the transaction possibilities. In section 2 we introduce the general framework of this paper. In section 3 we establish an existence result of an equilibrium. In section 4 we study the effect of small frictions.
2. The model ’ Let $5’ be an economy with e goods, h = 1,. . . , t, m consumers, i = 1,. . . , m, n producers, j=l,..., n, and r brokers, k = 1,. . . , r. The ith consumer is described by a continuous convex and locally non-satiated preferences preorder 5 ; on a consumption set Xi = R?+ and initial endowments OJT. We denote by o* the vector of total initial endowments, i.e. w* = Cy=, WT. The jth producer is able to transform an input vector y, E R: into an output vector yj E R: , as long as ( y,, y; ) belongs to the feasible production set Y, C RF. Transactions are carried out by the brokers. The kth broker buys zk E Rt and sells z; E R: constrained by (zk, 2;) E Z,, where Z, is his feasible transaction set. We assume that for all j (resp. all k) the production set YJ (resp. transaction set Z,) is non-empty, closed and (q + R: x R<) fl Ry C Yj [resp. (Z, + R: x R?) fl Ry C Z,], i.e. less can be produced with more. We also assume that for every ‘efficient’ production (resp. transaction) plan (y,, y;) E 8Y [resp. (z,, 2;) E aZ,], the firm (resp. the broker) sets the price p in (P,(Yj, yJ) (resp. IJJ~: aZk-+ S,,) is an upper semicontinuous ’ [resp. t&(zk, z;)], where ‘p,: aYj+$, convex compact value correspondence called the pricing rule. For instance, this formalization takes into account profit maximization in the convex case and marginal pricing or cost-average pricing in general. For a price system (p, q), production plans (y,, y;) and transaction plans (zk, z;) the revenue of the ith consumer is equal to wi(p, q, (yj, yi), (zk, z;)), where (w;) is a collection of positively homogeneous [with respect to (p, q)] mappings from Ry x R2e” x R2er into R satisfying Cz”=, Zk, z;>> = CT=, (-p . y, + 4. y;> + c;=, (p . z; q * z/J. Wi(P, 97 (Y,, Y;), ( The ith consumer purchases a vector xi at the price p and sells a vector .x: at the price q constrained by (xi, xi) E B(p, q, wi), where B(p, q, w,) = {wi +x, -x1: (xj,x:)ERy andp.x; q . xi 5 w;}. This merely says that his net consumption w, + xi - xl must be in his budget set.
’ If x = (x,) and y = ( y,,) are vectors in R’, we let x. y = Ci=, xhy, be the scalar product of R’, and IJxI( = (x .x)112 be the Euclidean norm. The notation x 2 y (resp. x + y) means x,, 2 y, (resp. xh > yh) for all h; we let R: = {x E R’: x 2 0) and R:, = {x~R’:x>O}.Ifx~R’wedenotebyx’ and n- the vectors, respectively, with coordinates xi = max(O, x,) and equal to 0 except the hth xi = max(O, -x,,). For h = 1,. , e we denote by eh the vector in R’ with all coordinates equal to 1 and by S, = {p E R:: C:=, coordinate which is equal to 1. We denote by e the vector in R ’ with all coordinates ph = 1). For A c R’, we denote by CIA, 8A and ri(A), respectively, the closure, the boundary and the relative interior of A. 2 Given two topological spaces X and Y, a correspondence r$ is said to be upper semi-continuous if 4 is locally bounded and if the graph of 4, i.e. {(x, y) E X x Y: y E 4(x)}, is closed.
E. Jouini and H. Kallal I Economics
259
Letters 41 (1993) 257-263
Definition 2.1. An equilibrium of 8 = ((Xi 5 ;, w,, w,), (Y), (2,)) is a list (p, (zk, z;)) in Ry x Ry”’ X Ry” X RF’ such that: (i) for all i, y + xi -x: is a best element for _< 1 in the budget set
q, (x,,xi),
(yj, Yi),
B(P, 4, Wi(P, 47 (Y,, Y’i), (Zk> Z’k))) ; (ii)
for all i, (Y,, yj) E dq and (P, 4) E Vj(yj, ~j);
for all k, (zk, z;)EU, and (p, q)E c~,x,+c;~,y,~c;=,z;, (iv) { c;, x; + c;=, yj 2 c;=, Zk ;
(iii)
slr,(zk,zL);
p . (CE, x, + ql y, - c;=, z;) = 0, 09 ( q . (CE”,, x;. + c;=, y; - CL=, Zk) = 0. For w E Rt let A(W) = {((y,, yj), (z,, z;)) EIIr=, aY, X II;,, a.Z,: 2; - Cl,=, zk + w 2 0} be the set of attainable production and transaction at this stage the following boundedness and survival assumptions. Assumption
(B).
Assumption
(WSA).
For every
(P,
cP,(Yj, y;))wX=,
For every
o z w*, the attainable
CT=r Yi - Cy=r Yj + CL=, plans. We can introduce
set A(w) is bounded.
(SA(w)) holds: w 2 w*, the following assumption and such that (p,q)E(n;=, E R? x A( w > such thatprq (cI,(zk, 2:)) we have for all i, wi( p, q, (y,, yl), (zk, z;)) + q . wi > 0. For every
4, (Y,,
y’j),
(zk, z;>)
Assumption (WSA) means that for an economy with initial endowments o at least equal to w*, the ith consumer can generate a positive total revenue by selling his endowment (at the price q) for all production-transaction marginal pricing equilibria that are in the attainable set.
3. The existence
result
In this section we state and prove the existence of an equilibrium as defined fact, using the results of Bonnisseau and Corne (1991) we prove the existence pricing equilibrium. Also, as we shall underline in the proof, we can show the equilibrium for general pricing rules with bounded losses using Bonnisseau and We recall that, for a closed set YE R”, a perpendicular vector to Y at y E Y the set &(y)={p~R’? Then, Clarke’s hull of the set
normal
{P ER’:g(y,)
3p>O,
VY’EY,
p.yzp.y’-plly-y’ll’}.
cone to Y at y [see Clarke
(1983)],
C Y, (Y,)+Y,
(P,)+P
3(p,)
in section 2. In of a marginal existence of an Cornet (1988). is an element in
CRY,
d enoted
N,.(y),
is the closed convex
and vq, p4 E ly(y)>
.
As in Guesnerie (1975) and Bonnisseau and Cornet (1991) the marginal pricing rule consists in fulfilling the first-order necessary condition of profit maximization. Formally, this can be stated: (P? 4) E U;(Y,* Yl) if (-P, 4) ENy(Y,, Yi) and (P, q) E (Clk(zk, 2;) pricing rule is of interest because the second welfare theorem production economy. In our case this result continues to hold,
if
C-q,
P> E &,(zk,
holds for a marginal in spite of transaction
2;).
This
pricing costs,
260
E. Jouini
and H. Kallal
because the definition of Pareto optimality as well as the production possibilities.
I Economics
ought
Letters
41 (1993) 257-263
to take the transaction
possibilities
into account
Theorem 3.1. Under Assumptions (B) and (WSA), the transaction cost economy 25 = ((X,, w,), (Yi), (2,)) admits a marginal pricing equilibrium and we have p 2 q. Proof.
Forj=l,..., q =
n,k=l,...,
randi-l,...,
{(-y,, y;) ER=: (y,, Y;> E
2, = {(z;,
-zk)
2, = {(xi, -xi)
_i , , wi,
m,let
r,, -
E R2”: (zk, 2;) E Z,)
(0) x R: ,
- R: x 101 >
E R*‘: xi 2 0, xi 2 0 and x, -x:
+ wT 2 0} ,
and let 3 i be the natural preorder induced by 5 1 [i.e. (x,, -xi + wT)a i(Xi, -Xi + wT> if (Xi-X:_+WT)~,(~,-XI+OT)]. Let $5 be the economy, with 2& goods, defined by ((X,, 3,, W,, GT), (q), (Z,)), where for sets of this economy and where 6, i=l,... , m, 6: = (0, O), where <(q>, (Z,)) are the production is defined by the formula: Gi(p, q, (-y,, yj), (z;, -zk)) = wi(p, q, (yj> Y;>, (z,, z;)). In this transformed economy, producers and brokers are indistinguishable in their behavior: they take (p, q) for the price vector. All the assumptions of Bonnisseau and Cornet (1991) are satisfied and gu_arantee the existence of a marginal pricing equilibrium (p”, 4, (X,, -x”:.), (-Y;,y”J), (2”:, -,SYk)) for 8. We can then easily 0 verify that (p”, @, (x”,, x”:), (Y;, (j?)‘), (Fk, (FL)+)) is an %-equilibrium.
4. Small frictions
in an exchange
economy
In this section we analyze the effect of small frictions on an exchange economy. The definitions of the previous section extend straightforwardly to this case. The results that we obtain could easily be generalized to production economies. Let % = (X,, 5 ;, wi) be an exchange economy with (w,) % 0. We shall see that its equilibria can be recovered as equilibria of a transaction economy % obtained from % by adjunction of a transaction set Z,, = ((2, z’) E Ry: z’ 5 z and z’ 5 2~). Note that since Z, is convex the first-order condition for profit maximization is also sufficient and the profits of the broker are always non-negative. We assume that these profits are evenly distributed across consumers. 3 The economy 8 satisfies all the assumptions of our existence result. Proposition
economy % = (X,, _< i, wi), then 1s an equilibrium of the transaction economy @ = ((Xi, _< i, a.+), Z,). C onversely, if (p, q, (x,, xl), (z, z’)) is an g-equilibrium, then p = q and (p, (x, -xl + w,)) is an g-equilibrium. (P,
p,
((x,
4.1.
-w;>+,
Let (p, (x,)) be an equilibrium (Xi - qn,
v;“,,
(x, - qr,
of the exchange
qI,
(Xi - q)+N
The proof of this proposition is left to the reader. From now on we shall identify an exchange economy ’ This assumption
is in fact innocuous
since at the ‘S-equilibria
with its associated
the broker
always
makes
transaction
zero profits.
economy
E. Jouini and H. Kallal I Economics
obtained by adjunction of the transaction without producers and with a transaction
Letters 41 (1993) 257-263
set 2,. Let us consider now a transaction set Za sufficiently ‘close’ to Z,,.
261
economy
Let Z, be a sequence of convex subsets of R2’ and let rr, be the projection mapping on Z,. The sequence (Z,) is said to converge to a convex subset Z of R2” if the sequence r,, converges to the projection mapping 7~ on Z for the Ce2 norm on every compact subset of RZe\A, where A is a null sets is the same as in Mas-Cole11 measure subset of R2’. This topology on transaction-production (1985, section 3.8). For our purpose we shall strengthen the assumption on the preferences assuming that, as in Balasko (1988, section 2.3), the preordering _< i is representable by a quasi-concave smooth utility function ui for which the first-order sufficient condition for strict monotonicity and the secondorder sufficient condition for strict quasi-concavity are satisfied. What follows can, however, be generalized to non-convex transaction sets noting that, under the free disposal assumption, aZ’, appears as the graph of a function A,,: el+ R and hence n,, can be replaced by A,, in the definition of convergence. In the following theorem, we only consider convex perturbations of the set Z, in order to apply Mas-Colell’s (1985) regularity results. There is no real difficulty in extending our result to non-convex perturbations using regularity results as in Jouini (1992a,b). Theorem 4.2. For almost every initial endowment o E R’JJ’, and for every exchange equilibrium of the economy $? = ((X,, _< i, wi), Z,) if we perturb slightly Z, we can find near this equilibrium an equilibrium of the perturbed economy. This means, in particular, that by introducing small frictions into an exchange economy we obtain consumption vectors and bid-ask prices in a neighborhood of the exchange equilibrium. Consequently the bid-ask spread is small. Since our argument relies on an implicit function theorem, frictions induce exactly first-order perturbations in prices and allocations. Furthermore, the difference in utility for the ith consumer is of second or higher order only if consumers have colinear net trade vectors, which is easily shown to be nongeneric. Note that, in general, this result does not hold for all initial endowments o since if 8 is a singular exchange economy [see Dierker (1982) and Mas-Cole11 (1985)] it is then easy to see that slight perturbations of the parameters of the economy can induce jumps in the equilibrium prices. Hence, using Balasko’s lemma (1988, lemma 5.2.1) we can show that these jumps in prices induce jumps in utilities and consequently in allocations. If the economy 8 = (X,, si, w,) is a regular exchange economy [see Dierker (1982), Mas-Cole11 (1985) and Balasko (1988)], then it admits a finite number of equilibria. If for all equilibria (p, (xi)) of 8 and h = 1, . . . ,8 we have xh # of, we say that the economy 8 is transactive. Lemma 1. If m 2 2, for almost every w E RTe the economy K = (X,, 5 ,, CO,)is a regular transactive economy. The proof of this lemma is obtained by classical arguments of differential topology. Let us now define the demand correspondence Bj of the ith consumer when facing the price vector (p, q), with p %-q, as the set of solutions (xi, xl) of the following program:
262
E. Jouini
and H. Kallal
(max ui(wi + xi -xi)
(9)
1
I Economics
Letters
41 (1993) 257-263
,
X,20, Xj20,
lp.xi-9.x:
10.
It is easy to show that in this case the solutions (xi, xi) must satisfy xi *xi = 0. We extend the previous definition to the case where p 2 q by adding the condition x, *x: = 0. The only effect of this constrain is to get rid of the trivial indeterminacy of solutions when ph = qh for some h. In fact, strict quasi-concavity of ui implies that bj is a well-defined function for p 2 q. Lemma 2. Let (p*, p*)Eri(S,,) a neighborhood of (p*, p*). This lemma
is a consequence
such that bi(p*,
of a classical
implicit
p*) = (-lzi,ai) with ii +a: 90.
function
Then B, is %’ in
theorem.
Proof of Theorem 4.2. The result is obvious for m = 1. Assume now that m 2 2 and let w E RY$ such that 8 = (X,, _< i, w,) is a regular transactive economy (by Lemma 1 this is true for almost every w). In fact, in what follows we shall assume that the consumers are directly characterized by their demand functions and we shall define 8 by (X,, D,, wi). Let G?= ((X,, 8,) w,), Z,) and g be the economy obtained from $. Since Z0 is a polyhedral cone of the activity matrix A = ’ (-I,, Z,) and the ith demand function fii in the transformed economy 2 is defined by fii(p, q) = (xi, -xi), where (xi, x:) = bi(p, q), we have, following Mas-Cole11 (1985, Proposition 6.4.1) that % is regular if for all equilibrium of g the matrix
WP,
-a&(~, -‘A
9) = (
-(P,
q) 4)
A 0
‘(~7 q) 0
0
0
1
has full rank, which is easily verified. We then obtain consequence of Mas-Colell’s results (1985, section 6.4).
the result Cl
of Theorem
4.2 as a direct
References Balasko, Y., 1988, Foundations of the theory of general economic equilibrium (Academic Press, New York). Bonnisseau, J.M. and B. Cornet, 1988, Existence of equilibria when firms follow bounded losses pricing rules, Journal of Mathematical Economics 17, 119-147. Bonnisseau, J.M. and B. Cornet, 1991, General equilibrium theory with increasing returns: the existence problem, to appear in Equilibrium theory and applications: Proceedings of the 6th international symposium (Cambridge University Press, Cambridge). Clarke, F.H., 1983, Optimization and nonsmooth analysis (John Wiley, New York). Dierker, E., 1982, Regular economies, in: K. Arrow and M. Intriligator, eds. Handbook of mathematical economics, vol. II (North-Holland, Amsterdam) 795-830, ch. 17. Guesnerie, R. 1975, Pareto optimality in nonconvex economies, Econometrica 43, no. 1. l-29. Hahn, F.H., 1973, On transaction costs, inessential sequence economies and money, Review of Economic Studies 40, no. 4, 449-461. Jouini, E., 1992a, An index theorem for nonconvex production economies, Journal of Economic Theory 57, no. 1, 176-196.
E. Jouini and H. Kallal I Economics
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Jouini, E., 1992b, Structure de I’ensemble des Cquilibres dune Cconomie productive, Annales de I’Institut Henri Poincare: Analyse non-lineaire 9, no. 3, 321-336. Mas-Colell, A., 1985, The theory of general economic equilibrium: A differentiable approach (Cambridge University Press, Cambridge). Starrett, D., 1973, Inefficiency and the demand for ‘money’ in a sequence economy, Review of Economic Studies 40, no. 4. 437-448.