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A remark on uniqueness of fix-price equilibria in an economy with production
Journal
of Mathematical
Economics
22 (1993) 421430.
A remark on uniqueness equilibria in an economy Hubert B.E.T.A., Submitted
North-Holland
of fix-price with production
Stahn* UniversitC Louis Pasteur, Strasbourg I, France March
1991, accepted
September
1992
Schulz’s assumptions leading to uniqueness of fix-price equilibria cannot be readily generalized to economies with production. In such a setting, the effective demand of a consumer not only reacts to changes in the perceived constraints but also to a modification of the realized profit levels
1. Introduction In General Equilibrium Theory, it is well-known that several restrictions on the demand side, even if the production side is well-behaved, are not sufficient to ensure uniqueness of an equilibrium in an economy with production. Gross substitution or more generally diagonal dominance of the Jacobian matrix of the normalized excess demand’ evaluated at an equilibrium point does not imply uniqueness in a production economy [see Kehoe (1982, 1984)]. In this paper, I want to point out that a quite similar problem arises for tix-price equilibria. In an exchange economy, following Schulz (1983), one knows that uniqueness is guaranteed due to some assumptions on the nature of the ‘spillovers’ of changes in each consumer’s perceived constraint on the non-constrained components of his effective demand. However, restricting the effective supply of each producer in the same manner is no longer sufficient to obtain uniqueness in an economy with production. Thus, the use of this result must be taken with caution unlike Benassy (1988) who seems to point out that Schulz’s (1983) work easily generalizes. Correspondence to: Hubert Stahn, Bureau d’Economie Thiorique et Appliquie, Universiti: Louis Pasteur, 38 Boulevard d’Anvers, 67070 Strasbourg Cedex, France. *I would like to thank C. d’Aspremont, J.-P. Benassy, R. DOS Santos Ferreira and an anonymous referee for a number of helpful comments. Remaining errors are only of my responsibility. I also wish to acknowledge the financial support of the Deutsche Forschungsgemeinschaft, Gottfried-Wilhelm-Leibniz Fiirderpreis. ‘By a normalized excess demand function, I mean a function in which the price of one good is set to 1 and the excess demand associated to this good is taken away. 03044068/93/$06.00
c
1993-Elsevier
Science Publishers
B.V. All rights reserved
422
H. Stahn, On uniqueness offix-price equilibria
Intuitively a fix-price equilibrium can be viewed as a fixed point of a map which associates to each agent’s effective demand a recomputation of this demand based on his perception of the constraints resulting from the choices of the other agents. Therefore to obtain uniqueness, one needs some restrictions on the manner in which each agent reacts to the choices of the others. In a pure exchange economy, the only connections between the different agents’ decision rules are indeed given by the perceived constraints. Some restrictions on the nature of the ‘spillovers’ due to changes in each agent’s constraints are therefore sufftcient to ensure uniqueness insofar as these assumptions guarantee diagonal dominance of the identity minus the Jacobian matrix of the effective demand. But in a production economy, this is no longer true because each consumer’s budget becomes endogenous. As a consequence every modification of the firms’ transactions has an effect on each consumer’s effective demand not only through changes in the perceived constraints but also through a modification of the profit distribution. Introducing some assumptions in order to satisfy diagonal dominance on the first kind of effects does no longer ensure the same property for the sum of the two effects. To make this argument more formal, let me briefly characterize in section 2 a fix-price equilibrium in a production economy. In section 3, I will show that Schulz’s uniqueness conditions cannot be readily extended to production economies. In section 4, I will exhibit an example satisfying Schulz’s extended conditions but where several equilibria may still exist. Section 5 will be devoted to some concluding remarks.
2. Fix-price equilibria in an economy with production Consider an economy with L produced goods and an additional one, called money, which serves both as a numeraire and as a medium of exchange. Assume the existence m consumers indexed by i, and n producers indexed by j. Consumer i is characterized by his initial endowments oi E rW$ and moi ~0 and by profit shares Bij~ [0, 11, j= 1,. . , n. His preference ordering over consumption bundles (x,m) E RL,” is represented by a utility function -JUi:IwL++ ’ -+R continuous, quasi-concave and strictly monotone. In assume (i) 1&iE P( rW4;‘), Mi(x, m) E Rq=’ V(x, m) E R”,Y+‘, (ii) addition, h’.(?2~~i(~~,m)~h
H. Stahn, On uniqueness of fix-price
equilibria
423
n) and maximal sales (zi,, 50 and Zjh 20 h=l,..., L, i=l,..., m and j=l,..., m and j = 1,. . , n). If each agent chooses his optimal for h=l,..., L, i=l,..., plan by taking into account each upper and lower bound on his net-trade (in other words compute his constrained demand or supply) rationing never occurs at equilibrium. To avoid such situations, let me introduce the following effective excess demand and supply functions. Let the hth component of the solution to the program max %Jwi + z, m)
subject
to
Oi+Z~O,
p.Zsbi=m,i+
i dijnjt j=l
be Zi, = rih(fi, zi, bi) the consumer i’s effective excess demand for good h and let me define his effective excess demand 5i: rWq x lQ! x R, + lQL by Zi= obtained by solving L programs of this Si(zi,zi,bi)=(5ih(Zi,Zi,bi))h=l,...,L, kind. Let the hth component of the solution to the program
maxPY subject
to
YeYj
and
Vk=l,...,L,
k#h,
ZjkiYkIyjk
j’s effective supply for good h and define his by ~j=rj(zj,zj)=(rjh(Zj,4j))h= i,,.,,~. Because effective demands and supplies do not clear the markets, a associates to rationing scheme F: RL(“‘+“)+ lRLCm+n’is required. This function the set of all effective plans Z=(Ti, ?j)iE 1,,_,,,,j= 1,_,,, ,E IR~(~+“) the transaction fi,,= Fi,,(T) or Tjh= Fj,,(.2) realized respectively by consumer i= 1,. . . ,m or producer j = 1,. . . , n on market h= 1,. . . , L. As usually, assume that F is continuous and satisfies consistency, voluntary trade, and efficiency. But if rationing occurs for some agents, this one must be consistent with their perceived constraints. Thus, a functional relation must exists between effective plans and perceived constraints. Following Benassy, let me introduce a constraint perception rule G: RL(“‘+“)+ RZLCm+“) such that (Z,,,,z,,,)= (G,,,(Z’),G,,(q) where h= l,..., L and index a stands for i= 1,. , m and j=l ,. . .,n. Moreover assume that this rule satisfies continuity and some consistency assumptions with respect to the rationing scheme [see Schulz (1983, assumptions Gl, G2, G3)]. It immediately follows that a fix-price equilibrium corresponds to a be .5jh=qj,,(Zj,~j) CffeCtiVe
SUPPlY
the producer
qj:
R”,
X FF
+
RL
H. Stahn, On uniqueness offix-price equilibria
424
situation in which each agent’s effective demand or supply is consistent with his perceived constraints and in which the budget of each consumer is defined, at equilibrium, by the realized profits. Definition. A fix-price equilibrium in a production economy such that: vector (?*, z *,z*,z=*,(bir)i=,,,..,,)E[W4L(m+”)~[W: 2: = ti(Zz, $, b:)
(i)
Vi=l,...,m,
(ii)
(z*, Z*) = G(z”*) and
(iii)
Vi=1 ,...,m,
Vj = 1,. . , II,
by a
ZT = u]~(FT,~~*),
Z=*= F(z”*), Qijp.
b; = i j=
and
is defined
Fj(2*).
1
3. The main argument consequence of this definition, fixed point of the following map: As
a fix-price
a
Z. (~(m+n)L
G,b;
C;P(m+n)2L+m,24
equilibrium
is defined
as a
[~(m+n)L
(~H(Zi,Zi,ZJ’Zj,bi)i=l,_._,m,j=l,_..,
n++(qT
with
43=(bi(fJ)i=l,...,m=
moi+ (
i
jt 1
@ijP’Fj(fl
. i=l.....m
In this ‘smooth’ economy, it is easy to verify that ci and qj are continuous Lipschitzian functions2 for all i and j. Moreover if one assumes that G and F are Lipschitzian, then Z also satisfies this property. To obtain mathematical conditions for existence and uniqueness, one can readily generalize of every Schulz’s (1983) degree argument. 3 Hence, if on each neighborhood z”~Z~‘(0) the map Z(F)=?-Z(F) (whenever Z is differentiable) is such that all the principal minors of its Jacobian matrix are different from zero and maintain their sign, then uniqueness of a fix-price equilibrium is satisfied.
*Use for instance Mas-Cole11 (1985, Cl, p. 31). 3With continuous Lipschitzian function, the index formula is obtained in a quite similar way as in a differentiable framework because these functions are almost everywhere differentiable. One only has to take care of local uniqueness. This problem can be solved by using Kawamura’s (1979) result.
425
H. Stahn, On uniqueness offix-price equilibria
Following McKenzie (1960), if one proves that the Jacobian matrix of g(5) (whenever defined) has a positive quasi-dominant diagonal (pqdd) then uniqueness is achieved. To obtain this result Benassy (1988) recalls Schulz’s (1983) argument which intuitively states that any change in each agent’s perceived constraints ‘spills over’ to the other markets by less than 100% in value terms. In order to formalize this condition, one has to consider [see Schulz (1983, condition U, p. 57)] the following class of optimization programs with KG {l,...,L}: max u2i(z + oi) subject
to
z+oi~O,
to
YE Yj
pzsbi
and
for kEK,
z,=i,
and
maxPY subject
and
jam=y,
for kc K,
where i and E are fixed arbitrary values.4 If one denotes the associated class of solution y?(j), then the condition reads as follows: Condition
U extended.
For all K $ (1,.
(9 (ii)
respectively
. , L} assume
by z?(2) and
that:
’
Vj=l,...,n,
To have a better understanding of this condition, note that the effective demand (or supply) of an agent is differentiable as long as the relevant constraints to his optimization problem are strictly binding. If this is the case, this assumption implies that any change in one of his binding constraints leads to a change in absolute value of his optimal expenditure which is smaller than IpkdSkl where i, stands for one of his binding perceived constraints. Thus this assumption only induces some restrictions on LJ&, a,,&, a,rIj and a,,rlj. But these restrictions are no longer sufficient to ensure that a?_$? has a pqdd. In fact, by computation of ai2 (whenever defined) one obtains:
4To be more precise, these values must be chosen to these optimization programs.
in way to ensure
the existence
of a solution
H. Stahn.
426
On uniqueness
oJ fix-price
equilibria
where ‘4 =
a&&
B=
i
oijp
?,,,
Fj
jst
[
c?&.
i j'E 1
1 i-1
and ,,._, m.i’=l,._..
m
O,j,p+,,F,i, 1 i=l.__..
m, j=l,...,
n
In the lines of Schulz’s proof it is easy (but tedious5) to show that I-L&, ($.3G has a pqdd. But this is no longer sufficient to assert that ?,p satisfies the same property. Consequently, the natural extension of condition U, pointed out by Benassy (1988) tells us nothing about the third term of Zi.Z? which describes the effect of changes in the profit distribution on each consumer’s effect demand. Moreover it seems quite hopeless to find interpretable assumptions on [$I i] in order to guarantee that ai2 has a pqdd because nothing can be said, in general, on the properties of the sum of a diagonal dominant matrix and another one. Therefore, a direct study of ai.? must be performed in order to obtain a correct extension to production economies. In other words, one has to take into account that any change in a consumer-shareholder effective demand may be simultaneously due to a modification of his perceived constraints or of the realized profits of the firms. But in this case, if one looks for pqdd of a$, the required assumptions involve some restriction mixing up effective demands or supplies, rationing schemes and perceived constraints in a manner which is quite difficult to justify from an economic viewpoint [see Stahn (1991)]. Therefore Schulz’s assumptions cannot be generalized in a simple way to economies with production because, in this case, not only the changes in each consumer’s perceived constraints ‘spill over’ to the other markets but also the moditications of each firm’s realized transactions due to wealth effects.
4. An example To illustrate this argument consider the following simple macroeconomic example of a three-goods (labor, product, and money), three-agents (one producer, two consumers) model. The firm is characterized by a neoclassical production function y=f(L). Each consumer is defined by a utility function @Jci,mi/p), i= 1,2, by initial endowments in money M,,~,i= 1,2 and by a 5An explicit
analysis
of Cii: can be found in Stahn (1991).
H. Stahn, On uniqueness ofjx-price
equilibria
421
profit share 8, = 0 and 8, = 1 - 0. Assume that w = p = 1. Because there is no labor disutility assume that the two consumers respectively supply LOi and L,, units of labor. For simplicity, let me choose L,, and Lo2 such that f’(L,, + LO,)< 1. In this case, only unemployment equilibria occur and, as a matter of consequence, no rationing scheme is required for the producer on the labor market. Let L, = kz,, and L, = (1 - k)Z, be the rationing schemes of respectively consumers 1 and 2 with z, the effective labor demand. With regard to the product market, either an excess demand or an excess supply may occur. Thus, let jj=min {c^i+c”*, j) be the firm rationing scheme where demand and jj for the effective 2, +c’, stands for the effective aggregated supply. Finally, note that no rationing schemes must be explicitly set on this market for the two consumers because their labor supply is exogenous and, accordingly, is never modified by the realized transactions on the product market. Let me now define more precisely each agent’s effective demand or supply. In order to simplify this example, I will concentrate on rational K-equilibria, i.e. on situations in which the relevant perceived constraints coincide with the rationing schemes. Because each consumer is rationed on the labor market, his effective consumption plan is given by c”i=~i(L,, ~) where r? is the realized profit level. However, without labor disutility, consumer i’s (i= 1,2) effective demand is only subject to a wealth constraint. So let me redefine this function by Fi=ci(Ri) for i= 1,2, and let Ri=mio+Zi+8i(y-_L, -L2) be his wealth. Concerning the firm, labor excess demand cannot occur; its effective product supply is therefore given by j=f((f’)‘( 1)). But its effective labor demand can be constrained by the effective product demand. thus Z,=min{f-‘(I,+?,),)‘}. As a consequence of this setting, following equation system: E, =c,(m,,
a fix-price
+(k-O)2,+O.min
equilibrium
{c”i+c”,, jj}),
is a solution
to the
(1)
i (I)
! I
c”,=c,(m,,+(8-k)E,+(1-8).min{~,+c”,,~}),
(2)
9=f((f’)_
(3)
t,=min
l(l)), {f-‘(C”i SF,),
(f’)-‘(l)}.
In other words, a fix-price equilibrium is, as usually, a fixed point of Z(T) where z” stands for (?,,F,, jj, z,) and Z(Y) is given by the right-hand-side terms of (I). In order to obtain uniqueness, one has to study the Jacobian matrix of Z(Y) = Z- Z(?). Following Schulz, if this matrix is such that all of its principal
H. Stahn, On uniqueness offix-price equilibria
428
minors are different from zero and maintain their sign for each 5, uniqueness is achieved. Yet, the natural extension of Schulz’s assumptions is not able to guarantee this property. In order to illustrate this fact let me concentrate on the Keynesian case characterized by i3i +E, < j. In a first step, I will show that this property fails under condition U extended, and in a second one, I will point out that several equilibria may occur. Let me first compute this Jacobian matrix. One immediately obtains
1- @J;(R,),
&2=
1- (1 - &-$(R,),
0, -(.f’o”r’(Z, +FJ-l,
0,
For a Keynesian equilibrium condition U implies that -Vi=1,2,
- Qc;(R,),
-(I -@$(R,),
-(f’of-‘(c”l of this model,
0, -to-k)c;W,)
+2,))_‘, 0, the natural
extension
of Schulz’s
is rationed on the b ecause each consumer labor market and non-rationed on the commodity market, on the commodity market b ecause the firm is rationed and non-rationed on the labour market.
I(p/w).(Xi/aLi)l
-~WP)~(&/~~)~<~
Due to my peculiar setting, these conditions can be rewritten as lc;(~i)l< 1, ’ < 1. But these restrictions are no longer [c;(~,)IO, and D, = D,= 1 -f?.c’,(R,)-([email protected];(R,)>O but D,= l-Q.c;(R,)-(1 -U).c;(R,)+ (f’ofP’(cl +c~))~-’ .(Q-k).(c;(R,)-c;(R,)) is unsigned. Thus nothing can be said about uniqueness. Let me now point out that because D, is unsigned, several equilibria may occur. In order to illustrate this argument graphically, I will, as usually, work in the production/employment space. By summing up eqs. (5) and (6) and by setting down y=c”, +E2, system (I) can be rewritten as follows:
(I’)
y=c,(mo,+(k-H)~~,+e~y)+c,(m,,+(8-k)~L,+(1-e).y),
(5)
t,=f-l(y).
(6)
Furthermore it is easy to verify that any solution (y, z,) of (I’) in the production/employment space induces a unique solution (c”i, C,, j, z,) of (I). Thus, one only needs to concentrate on (I’).
H. Stahn.
On uniqueness
of fix-price
429
equilibria
Kd
Xd Fig. 2
Fig. 1
Moreover, if Schulz’s uniqueness assumptions are satisfied, the following is the implicit function true: 1 - 8. c;(R 1) - (1 - 0) . c)2(R2) > 0. By applying theorem to (5) there exists a continuous function Y: [O,L,, +L,,] + [w, + such that
Y(L,)=c,(m,,
+(k-epZ,+e.
+c,(m,,
+(e-k)&+(l
Y(L,))
-e). Y(Z,)),
(7)
and dY
(k-Q.(c;(R,)+c’,(R,)) dZ,==c;(R,)-(1 -B)c;(R,)’ Thus (I’) can be rewritten
(8)
as (9)
y = Y(L), (I”)
(10) 1 Y =f(L). Accordingly, a Keynesian equilibrium can be viewed as an intersection of Y(L,) and f(z,) in a production/employment space. Assume, without loss of generality, that m,,, and mo2 are chosen in order to ensure existence. Insofar as the slope of Y(L,) is smaller than the slope of the production every equilibrium point, a Keynesian equilibrium is unique. But now remark that
oD,
2 0 because
at equilibrium
Therefore a unique equilibrium may occur (for tion for instance) (as depicted in fig. 1) as well as in fig. 2), because nothing can be said about the to the slope of f(Z,) at a equilibrium point or of D,.
function
at
L, = f _ ‘( j). Cobb-Douglas utility funcseveral equilibria (as shown slope of Y(E,) with respect equivalently about the sign
430
H. Stahn, On uniqueness
offix-price
equilibria
5. Conclusion In this paper, 1 have shown that as in standard Walrasian analysis, some conditions for uniqueness of a fix-price equilibrium in exchange economies do not necessarily extend in a simple way to production economies. Schulz’s assumptions which require that changes in quantity constraints spill over onto the other markets by less than 100% in value terms [Benassy (1988, p. 43)] are not sufficient to ensure uniqueness unlike Benassy seems to point out6 Furthermore, in a simple macroeconomic model, restrictions on the marginal propensity to consume are not sufficient to ensure uniqueness if one assumes profit distribution. If this is not the case, uniqueness can be established [see for instance Hildenbrand and Hildenbrand (1978)].’ ‘Accordingly. the existence of the objective demand concept which is introduced in his paper must be arbitrarily assumed. His general equilibrium concept with imperfect competition suffers from the same restriction as the CournottWalras concept. ‘Hildenbrand and Hildenbrand (1978) were interested in Malinvaud’s (1977) approach in which profit distributions are typically not taken into account
References Benassy, J.-P., 1988, The objective demand curve in general equilibrium with price makers, Economic Journal 98 (conference 1988), 3749. Hildenbrand, K. and W. Hildenbrand, 1978, On Keynesian equilibrium with unemployment and auantitv rationing, Journal of Economic Theory 18, 255-277. Kawamura, Y., 1979, Invertibility of Lipschitz continuous mappings and its application to electrical network equations. SIAM Journal of Mathematical Analysis 10, 253-265. Journal of Mathematical Economics 10, Kehoe, T., 1982, Regular production economies, 1599165. Kehoe. T., 1985, Multiplicity of equilibria and comparative statics, Quarterly Journal of Economics 100, 119-147. McKenzie, L., 1960, Matrices with dominant diagonals and economic theory, in: K. Arrow, S. Karlin and P. Suppes, eds., Mathematical method in the social sciences (Stanford University Press, Stanford, CA) 47-62. Mas-Colell, A., 1985, The theory of general economic equilibrium: A differentiable approach (Cambridge University Press, Cambridge). Malinvaud, E., 1977, The theory of unemployment reconsidered (Blackwell, Oxford). Schulz, N., 1983, On the global uniqueness of fix-prix equilibria, Econometrica 51, 47768. Stahn, H., 1991, Strategies en prix et equilibre: Une approche en equilibre general, Doctoral Dissertation (University Louis Pasteur, Strasbourg).
of Mathematical
Economics
22 (1993) 421430.
A remark on uniqueness equilibria in an economy Hubert B.E.T.A., Submitted
North-Holland
of fix-price with production
Stahn* UniversitC Louis Pasteur, Strasbourg I, France March
1991, accepted
September
1992
Schulz’s assumptions leading to uniqueness of fix-price equilibria cannot be readily generalized to economies with production. In such a setting, the effective demand of a consumer not only reacts to changes in the perceived constraints but also to a modification of the realized profit levels
1. Introduction In General Equilibrium Theory, it is well-known that several restrictions on the demand side, even if the production side is well-behaved, are not sufficient to ensure uniqueness of an equilibrium in an economy with production. Gross substitution or more generally diagonal dominance of the Jacobian matrix of the normalized excess demand’ evaluated at an equilibrium point does not imply uniqueness in a production economy [see Kehoe (1982, 1984)]. In this paper, I want to point out that a quite similar problem arises for tix-price equilibria. In an exchange economy, following Schulz (1983), one knows that uniqueness is guaranteed due to some assumptions on the nature of the ‘spillovers’ of changes in each consumer’s perceived constraint on the non-constrained components of his effective demand. However, restricting the effective supply of each producer in the same manner is no longer sufficient to obtain uniqueness in an economy with production. Thus, the use of this result must be taken with caution unlike Benassy (1988) who seems to point out that Schulz’s (1983) work easily generalizes. Correspondence to: Hubert Stahn, Bureau d’Economie Thiorique et Appliquie, Universiti: Louis Pasteur, 38 Boulevard d’Anvers, 67070 Strasbourg Cedex, France. *I would like to thank C. d’Aspremont, J.-P. Benassy, R. DOS Santos Ferreira and an anonymous referee for a number of helpful comments. Remaining errors are only of my responsibility. I also wish to acknowledge the financial support of the Deutsche Forschungsgemeinschaft, Gottfried-Wilhelm-Leibniz Fiirderpreis. ‘By a normalized excess demand function, I mean a function in which the price of one good is set to 1 and the excess demand associated to this good is taken away. 03044068/93/$06.00
c
1993-Elsevier
Science Publishers
B.V. All rights reserved
422
H. Stahn, On uniqueness offix-price equilibria
Intuitively a fix-price equilibrium can be viewed as a fixed point of a map which associates to each agent’s effective demand a recomputation of this demand based on his perception of the constraints resulting from the choices of the other agents. Therefore to obtain uniqueness, one needs some restrictions on the manner in which each agent reacts to the choices of the others. In a pure exchange economy, the only connections between the different agents’ decision rules are indeed given by the perceived constraints. Some restrictions on the nature of the ‘spillovers’ due to changes in each agent’s constraints are therefore sufftcient to ensure uniqueness insofar as these assumptions guarantee diagonal dominance of the identity minus the Jacobian matrix of the effective demand. But in a production economy, this is no longer true because each consumer’s budget becomes endogenous. As a consequence every modification of the firms’ transactions has an effect on each consumer’s effective demand not only through changes in the perceived constraints but also through a modification of the profit distribution. Introducing some assumptions in order to satisfy diagonal dominance on the first kind of effects does no longer ensure the same property for the sum of the two effects. To make this argument more formal, let me briefly characterize in section 2 a fix-price equilibrium in a production economy. In section 3, I will show that Schulz’s uniqueness conditions cannot be readily extended to production economies. In section 4, I will exhibit an example satisfying Schulz’s extended conditions but where several equilibria may still exist. Section 5 will be devoted to some concluding remarks.
2. Fix-price equilibria in an economy with production Consider an economy with L produced goods and an additional one, called money, which serves both as a numeraire and as a medium of exchange. Assume the existence m consumers indexed by i, and n producers indexed by j. Consumer i is characterized by his initial endowments oi E rW$ and moi ~0 and by profit shares Bij~ [0, 11, j= 1,. . , n. His preference ordering over consumption bundles (x,m) E RL,” is represented by a utility function -JUi:IwL++ ’ -+R continuous, quasi-concave and strictly monotone. In assume (i) 1&iE P( rW4;‘), Mi(x, m) E Rq=’ V(x, m) E R”,Y+‘, (ii) addition, h’.(?2~~i(~~,m)~h
H. Stahn, On uniqueness of fix-price
equilibria
423
n) and maximal sales (zi,, 50 and Zjh 20 h=l,..., L, i=l,..., m and j=l,..., m and j = 1,. . , n). If each agent chooses his optimal for h=l,..., L, i=l,..., plan by taking into account each upper and lower bound on his net-trade (in other words compute his constrained demand or supply) rationing never occurs at equilibrium. To avoid such situations, let me introduce the following effective excess demand and supply functions. Let the hth component of the solution to the program max %Jwi + z, m)
subject
to
Oi+Z~O,
p.Zsbi=m,i+
i dijnjt j=l
be Zi, = rih(fi, zi, bi) the consumer i’s effective excess demand for good h and let me define his effective excess demand 5i: rWq x lQ! x R, + lQL by Zi= obtained by solving L programs of this Si(zi,zi,bi)=(5ih(Zi,Zi,bi))h=l,...,L, kind. Let the hth component of the solution to the program
maxPY subject
to
YeYj
and
Vk=l,...,L,
k#h,
ZjkiYkIyjk
j’s effective supply for good h and define his by ~j=rj(zj,zj)=(rjh(Zj,4j))h= i,,.,,~. Because effective demands and supplies do not clear the markets, a associates to rationing scheme F: RL(“‘+“)+ lRLCm+n’is required. This function the set of all effective plans Z=(Ti, ?j)iE 1,,_,,,,j= 1,_,,, ,E IR~(~+“) the transaction fi,,= Fi,,(T) or Tjh= Fj,,(.2) realized respectively by consumer i= 1,. . . ,m or producer j = 1,. . . , n on market h= 1,. . . , L. As usually, assume that F is continuous and satisfies consistency, voluntary trade, and efficiency. But if rationing occurs for some agents, this one must be consistent with their perceived constraints. Thus, a functional relation must exists between effective plans and perceived constraints. Following Benassy, let me introduce a constraint perception rule G: RL(“‘+“)+ RZLCm+“) such that (Z,,,,z,,,)= (G,,,(Z’),G,,(q) where h= l,..., L and index a stands for i= 1,. , m and j=l ,. . .,n. Moreover assume that this rule satisfies continuity and some consistency assumptions with respect to the rationing scheme [see Schulz (1983, assumptions Gl, G2, G3)]. It immediately follows that a fix-price equilibrium corresponds to a be .5jh=qj,,(Zj,~j) CffeCtiVe
SUPPlY
the producer
qj:
R”,
X FF
+
RL
H. Stahn, On uniqueness offix-price equilibria
424
situation in which each agent’s effective demand or supply is consistent with his perceived constraints and in which the budget of each consumer is defined, at equilibrium, by the realized profits. Definition. A fix-price equilibrium in a production economy such that: vector (?*, z *,z*,z=*,(bir)i=,,,..,,)E[W4L(m+”)~[W: 2: = ti(Zz, $, b:)
(i)
Vi=l,...,m,
(ii)
(z*, Z*) = G(z”*) and
(iii)
Vi=1 ,...,m,
Vj = 1,. . , II,
by a
ZT = u]~(FT,~~*),
Z=*= F(z”*), Qijp.
b; = i j=
and
is defined
Fj(2*).
1
3. The main argument consequence of this definition, fixed point of the following map: As
a fix-price
a
Z. (~(m+n)L
G,b;
C;P(m+n)2L+m,24
equilibrium
is defined
as a
[~(m+n)L
(~H(Zi,Zi,ZJ’Zj,bi)i=l,_._,m,j=l,_..,
n++(qT
with
43=(bi(fJ)i=l,...,m=
moi+ (
i
jt 1
@ijP’Fj(fl
. i=l.....m
In this ‘smooth’ economy, it is easy to verify that ci and qj are continuous Lipschitzian functions2 for all i and j. Moreover if one assumes that G and F are Lipschitzian, then Z also satisfies this property. To obtain mathematical conditions for existence and uniqueness, one can readily generalize of every Schulz’s (1983) degree argument. 3 Hence, if on each neighborhood z”~Z~‘(0) the map Z(F)=?-Z(F) (whenever Z is differentiable) is such that all the principal minors of its Jacobian matrix are different from zero and maintain their sign, then uniqueness of a fix-price equilibrium is satisfied.
*Use for instance Mas-Cole11 (1985, Cl, p. 31). 3With continuous Lipschitzian function, the index formula is obtained in a quite similar way as in a differentiable framework because these functions are almost everywhere differentiable. One only has to take care of local uniqueness. This problem can be solved by using Kawamura’s (1979) result.
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H. Stahn, On uniqueness offix-price equilibria
Following McKenzie (1960), if one proves that the Jacobian matrix of g(5) (whenever defined) has a positive quasi-dominant diagonal (pqdd) then uniqueness is achieved. To obtain this result Benassy (1988) recalls Schulz’s (1983) argument which intuitively states that any change in each agent’s perceived constraints ‘spills over’ to the other markets by less than 100% in value terms. In order to formalize this condition, one has to consider [see Schulz (1983, condition U, p. 57)] the following class of optimization programs with KG {l,...,L}: max u2i(z + oi) subject
to
z+oi~O,
to
YE Yj
pzsbi
and
for kEK,
z,=i,
and
maxPY subject
and
jam=y,
for kc K,
where i and E are fixed arbitrary values.4 If one denotes the associated class of solution y?(j), then the condition reads as follows: Condition
U extended.
For all K $ (1,.
(9 (ii)
respectively
. , L} assume
by z?(2) and
that:
’
Vj=l,...,n,
To have a better understanding of this condition, note that the effective demand (or supply) of an agent is differentiable as long as the relevant constraints to his optimization problem are strictly binding. If this is the case, this assumption implies that any change in one of his binding constraints leads to a change in absolute value of his optimal expenditure which is smaller than IpkdSkl where i, stands for one of his binding perceived constraints. Thus this assumption only induces some restrictions on LJ&, a,,&, a,rIj and a,,rlj. But these restrictions are no longer sufficient to ensure that a?_$? has a pqdd. In fact, by computation of ai2 (whenever defined) one obtains:
4To be more precise, these values must be chosen to these optimization programs.
in way to ensure
the existence
of a solution
H. Stahn.
426
On uniqueness
oJ fix-price
equilibria
where ‘4 =
a&&
B=
i
oijp
?,,,
Fj
jst
[
c?&.
i j'E 1
1 i-1
and ,,._, m.i’=l,._..
m
O,j,p+,,F,i, 1 i=l.__..
m, j=l,...,
n
In the lines of Schulz’s proof it is easy (but tedious5) to show that I-L&, ($.3G has a pqdd. But this is no longer sufficient to assert that ?,p satisfies the same property. Consequently, the natural extension of condition U, pointed out by Benassy (1988) tells us nothing about the third term of Zi.Z? which describes the effect of changes in the profit distribution on each consumer’s effect demand. Moreover it seems quite hopeless to find interpretable assumptions on [$I i] in order to guarantee that ai2 has a pqdd because nothing can be said, in general, on the properties of the sum of a diagonal dominant matrix and another one. Therefore, a direct study of ai.? must be performed in order to obtain a correct extension to production economies. In other words, one has to take into account that any change in a consumer-shareholder effective demand may be simultaneously due to a modification of his perceived constraints or of the realized profits of the firms. But in this case, if one looks for pqdd of a$, the required assumptions involve some restriction mixing up effective demands or supplies, rationing schemes and perceived constraints in a manner which is quite difficult to justify from an economic viewpoint [see Stahn (1991)]. Therefore Schulz’s assumptions cannot be generalized in a simple way to economies with production because, in this case, not only the changes in each consumer’s perceived constraints ‘spill over’ to the other markets but also the moditications of each firm’s realized transactions due to wealth effects.
4. An example To illustrate this argument consider the following simple macroeconomic example of a three-goods (labor, product, and money), three-agents (one producer, two consumers) model. The firm is characterized by a neoclassical production function y=f(L). Each consumer is defined by a utility function @Jci,mi/p), i= 1,2, by initial endowments in money M,,~,i= 1,2 and by a 5An explicit
analysis
of Cii: can be found in Stahn (1991).
H. Stahn, On uniqueness ofjx-price
equilibria
421
profit share 8, = 0 and 8, = 1 - 0. Assume that w = p = 1. Because there is no labor disutility assume that the two consumers respectively supply LOi and L,, units of labor. For simplicity, let me choose L,, and Lo2 such that f’(L,, + LO,)< 1. In this case, only unemployment equilibria occur and, as a matter of consequence, no rationing scheme is required for the producer on the labor market. Let L, = kz,, and L, = (1 - k)Z, be the rationing schemes of respectively consumers 1 and 2 with z, the effective labor demand. With regard to the product market, either an excess demand or an excess supply may occur. Thus, let jj=min {c^i+c”*, j) be the firm rationing scheme where demand and jj for the effective 2, +c’, stands for the effective aggregated supply. Finally, note that no rationing schemes must be explicitly set on this market for the two consumers because their labor supply is exogenous and, accordingly, is never modified by the realized transactions on the product market. Let me now define more precisely each agent’s effective demand or supply. In order to simplify this example, I will concentrate on rational K-equilibria, i.e. on situations in which the relevant perceived constraints coincide with the rationing schemes. Because each consumer is rationed on the labor market, his effective consumption plan is given by c”i=~i(L,, ~) where r? is the realized profit level. However, without labor disutility, consumer i’s (i= 1,2) effective demand is only subject to a wealth constraint. So let me redefine this function by Fi=ci(Ri) for i= 1,2, and let Ri=mio+Zi+8i(y-_L, -L2) be his wealth. Concerning the firm, labor excess demand cannot occur; its effective product supply is therefore given by j=f((f’)‘( 1)). But its effective labor demand can be constrained by the effective product demand. thus Z,=min{f-‘(I,+?,),)‘}. As a consequence of this setting, following equation system: E, =c,(m,,
a fix-price
+(k-O)2,+O.min
equilibrium
{c”i+c”,, jj}),
is a solution
to the
(1)
i (I)
! I
c”,=c,(m,,+(8-k)E,+(1-8).min{~,+c”,,~}),
(2)
9=f((f’)_
(3)
t,=min
l(l)), {f-‘(C”i SF,),
(f’)-‘(l)}.
In other words, a fix-price equilibrium is, as usually, a fixed point of Z(T) where z” stands for (?,,F,, jj, z,) and Z(Y) is given by the right-hand-side terms of (I). In order to obtain uniqueness, one has to study the Jacobian matrix of Z(Y) = Z- Z(?). Following Schulz, if this matrix is such that all of its principal
H. Stahn, On uniqueness offix-price equilibria
428
minors are different from zero and maintain their sign for each 5, uniqueness is achieved. Yet, the natural extension of Schulz’s assumptions is not able to guarantee this property. In order to illustrate this fact let me concentrate on the Keynesian case characterized by i3i +E, < j. In a first step, I will show that this property fails under condition U extended, and in a second one, I will point out that several equilibria may occur. Let me first compute this Jacobian matrix. One immediately obtains
1- @J;(R,),
&2=
1- (1 - &-$(R,),
0, -(.f’o”r’(Z, +FJ-l,
0,
For a Keynesian equilibrium condition U implies that -Vi=1,2,
- Qc;(R,),
-(I -@$(R,),
-(f’of-‘(c”l of this model,
0, -to-k)c;W,)
+2,))_‘, 0, the natural
extension
of Schulz’s
is rationed on the b ecause each consumer labor market and non-rationed on the commodity market, on the commodity market b ecause the firm is rationed and non-rationed on the labour market.
I(p/w).(Xi/aLi)l
-~WP)~(&/~~)~<~
Due to my peculiar setting, these conditions can be rewritten as lc;(~i)l< 1, ’ < 1. But these restrictions are no longer [c;(~,)I
(I’)
y=c,(mo,+(k-H)~~,+e~y)+c,(m,,+(8-k)~L,+(1-e).y),
(5)
t,=f-l(y).
(6)
Furthermore it is easy to verify that any solution (y, z,) of (I’) in the production/employment space induces a unique solution (c”i, C,, j, z,) of (I). Thus, one only needs to concentrate on (I’).
H. Stahn.
On uniqueness
of fix-price
429
equilibria
Kd
Xd Fig. 2
Fig. 1
Moreover, if Schulz’s uniqueness assumptions are satisfied, the following is the implicit function true: 1 - 8. c;(R 1) - (1 - 0) . c)2(R2) > 0. By applying theorem to (5) there exists a continuous function Y: [O,L,, +L,,] + [w, + such that
Y(L,)=c,(m,,
+(k-epZ,+e.
+c,(m,,
+(e-k)&+(l
Y(L,))
-e). Y(Z,)),
(7)
and dY
(k-Q.(c;(R,)+c’,(R,)) dZ,==c;(R,)-(1 -B)c;(R,)’ Thus (I’) can be rewritten
(8)
as (9)
y = Y(L), (I”)
(10) 1 Y =f(L). Accordingly, a Keynesian equilibrium can be viewed as an intersection of Y(L,) and f(z,) in a production/employment space. Assume, without loss of generality, that m,,, and mo2 are chosen in order to ensure existence. Insofar as the slope of Y(L,) is smaller than the slope of the production every equilibrium point, a Keynesian equilibrium is unique. But now remark that
oD,
2 0 because
at equilibrium
Therefore a unique equilibrium may occur (for tion for instance) (as depicted in fig. 1) as well as in fig. 2), because nothing can be said about the to the slope of f(Z,) at a equilibrium point or of D,.
function
at
L, = f _ ‘( j). Cobb-Douglas utility funcseveral equilibria (as shown slope of Y(E,) with respect equivalently about the sign
430
H. Stahn, On uniqueness
offix-price
equilibria
5. Conclusion In this paper, 1 have shown that as in standard Walrasian analysis, some conditions for uniqueness of a fix-price equilibrium in exchange economies do not necessarily extend in a simple way to production economies. Schulz’s assumptions which require that changes in quantity constraints spill over onto the other markets by less than 100% in value terms [Benassy (1988, p. 43)] are not sufficient to ensure uniqueness unlike Benassy seems to point out6 Furthermore, in a simple macroeconomic model, restrictions on the marginal propensity to consume are not sufficient to ensure uniqueness if one assumes profit distribution. If this is not the case, uniqueness can be established [see for instance Hildenbrand and Hildenbrand (1978)].’ ‘Accordingly. the existence of the objective demand concept which is introduced in his paper must be arbitrarily assumed. His general equilibrium concept with imperfect competition suffers from the same restriction as the CournottWalras concept. ‘Hildenbrand and Hildenbrand (1978) were interested in Malinvaud’s (1977) approach in which profit distributions are typically not taken into account
References Benassy, J.-P., 1988, The objective demand curve in general equilibrium with price makers, Economic Journal 98 (conference 1988), 3749. Hildenbrand, K. and W. Hildenbrand, 1978, On Keynesian equilibrium with unemployment and auantitv rationing, Journal of Economic Theory 18, 255-277. Kawamura, Y., 1979, Invertibility of Lipschitz continuous mappings and its application to electrical network equations. SIAM Journal of Mathematical Analysis 10, 253-265. Journal of Mathematical Economics 10, Kehoe, T., 1982, Regular production economies, 1599165. Kehoe. T., 1985, Multiplicity of equilibria and comparative statics, Quarterly Journal of Economics 100, 119-147. McKenzie, L., 1960, Matrices with dominant diagonals and economic theory, in: K. Arrow, S. Karlin and P. Suppes, eds., Mathematical method in the social sciences (Stanford University Press, Stanford, CA) 47-62. Mas-Colell, A., 1985, The theory of general economic equilibrium: A differentiable approach (Cambridge University Press, Cambridge). Malinvaud, E., 1977, The theory of unemployment reconsidered (Blackwell, Oxford). Schulz, N., 1983, On the global uniqueness of fix-prix equilibria, Econometrica 51, 47768. Stahn, H., 1991, Strategies en prix et equilibre: Une approche en equilibre general, Doctoral Dissertation (University Louis Pasteur, Strasbourg).