EFFICIENT ALLOCATIONS AssOCIATED WITH FIXED PRICES AND ...

EFFICIENT ALLOCATIONS AssOCIATED

WITH FIXED PRICES AND INCOMES

by Yve.-6 BALASKO

November 1977

N° 7706

*

* Université Paris XII and CEPREMAP.

ALLOCATIONS EFFICACES ASSOCIEES

A DES PRIX ET REVENUS FIXES

- RESUME -

L'objet de cette note est une étude des allocations optimales

compatibles avec un système de prix rigides et des contraintes de

revenus. Les pr~priétés de ces allocations contraintes efficaces sont

déduites de propriétés semblables des équilibres walrasiens au moyen

d'une théorie de dualité qui présente un intérêt propre. On considère

aussi des processus dynamiques associés aux allocations contraintes

efficaces. On montre ainsi que les allocations contraintes efficaces

sont des attracteurs d'un certain procsssus dynamique.

EFFICIENT ALLOCATIONS ASSOCIATEO WITH FIXED

PRICES AND INCOMES

- ABSTRACT -

The main purpose of the paper is a study of optimal allo­

cations under price rigidities and incarne constraints. The properties

of these efficient constrained allocations are deduced from similar

properties of Walrasian equilibria by wey of a duality theory which

is of independent interest, Oynamic processes related with the con­

cept of efficient constrained allocations are also considered. For

one of these dynamic process, it is shown that every efficient

constrained allocation is locally stable.

C E P R E M A P

EFFICIENT ALLOCATIONS AssOCIATED

WITH FIXED PRICES AND INCOMES

by Yve~ BALASKO *

1. INTROVUCTION

Novembre 1977

The main purpose of this paper is a study of optimal alloca­

tions under pries rigidities and incarne constraints. Paying attention

to efficient allocations reflects the pervasive normative view point

which culminates in planning theory I considering fixed distribution

of incarnes and fixed price systems originates in Hicks's fixprice ana­

lysis of the short-term period [11], The above concern with efficient

constrained allocations in thus related to the recent developments of

equilibrium concepts under price rigidities that have followed Hicks's

pionneering approach, developments surveyed by Grandmont in [10],

The study of optimal allocations associated with given incarne

distribution and price system having not yet been dealt with in the

economic literature, this paper tries to provide a list of the most

important properties of efficient constrained allocations. These pro­

perties are deduced from similar proporties of Walrasian equilibria

by way of a duality theory which is of independent interest. Dynamic

processes related ta the above problem being also taken into consi­

deration, we define a dynamic process for which all efficient constrai­

ned allocations are locally stable,

* Université Paris XII and CEPREMAP.

- 2 -

2. THE MOVEL

We cohsider exchonge economies having Q, commodities and m

consumers.whose preference preorderi~gs satisfy the following stan­

dard assumptions [2] : monotonicity, smoothness, indifference hyper­

surfaces bounded from below and having everywhere non-zero Gaussian !I, curvature. We denote by ui = lR + JR a smooth surjective utili ty

function representing consumer i's preference preordering ~, every

consumption set being equal to JR9' • We choose the Q,-th commodity as

numeraire so that S = {p = (p1 , ... , p1

_1 , 1) 1 pi> O} denotes the

set of price vectors while B = S x :Rm denotes the set consisting of

vectors defined by commodity prices p ES and by individual incarnes

w. E lR i ,:: 1 , ... , m. Note that the individual demar.d mapping l

fi S x lR -+ JRQ, is then a diffeomorphism.

Let r E lR Q, denote the vector of fixed total resources. An

Jl m ' allocation is a m-tuple x = (x1 , ... , xrn) E (JR ) such that l xi = r

we denote by X= {x E (lRQ,)m I I x1

-· r} the space of allocations. We

also define the space X= {b = (p, w1 , ... , wm) E B I p.r = w1 + ••• + wm}

consisting of pries vectors and individual incarnes such that the total

incarne l wi is equal to the value p.r of the total resources. From now

on, we shall only consider prices p and incarnes w1 such that

b = (p, w1 , ••. , w) belongs to X. m

Let b = (p, w1 , ... , wm) EX be a given price-income vector.

The set of constrained allocations associated with b, set denoted A(b),

is defined by the equation system

f"xi = W. i = 1 ..... ., m ; l

r xi = r.

- 3 -

Note that A(b) is an affine space of dimension (m - 1) (t - 1) ; we

denote by Y: the set consisting of the affine subspaces of X which

can be defined by an equation system of the above form. Then, it is

easy to check that the above equation systom is unique: wo call it

the canonical equation system ass0ciated with the affine subspace

belonging to -*- , This dofinos a natural bijection between ~ and X

which enables us to identify these two spaces.

3. OPTIMA. ANV WALRASIAN EQUILIBRIA

An allocation x e Xis efficient or Pareto optimal if there

is no x' e X such that u.(x.) ~ u.(x'.) with a strict inequality for l l l l

at least one i. \.\le denotG by v. : X -+ m the function l

v.Cx1

, ... , x. ,.,., x) = u.(x.L Let T.(x) be the tangent space in 1 i m 1 1 1

x to the indiffArence hypersurface {y e X I v.(y) = v.(x)}. We denote l l

by H.(x) the open half-space delimited by T.(x) which contains 1 l

{y e X I v.(y) > v.(x)}. We also associate with every x e X the inter-1 1

section sets

D(x) =rlî.(x) 1

K(x) = f'IH.(x), 1

It is well-known (~nd also very Basy to check directly) that codim

D(x) is equel either to rn or tom - 1, that xis Pareto optimal if and

only if codirn D(x) = rn - 1 or, if and only if, the set K(x) is empty

[20]. Furthormore, when xis Pareto optimal, then D(x) belongs to .x the set of Pareto optima, denoted P, is diffeomorphic to Rm- 1 (Appendix,

Corollary 3) ; we define am - 1 parameter family (? of elernents of-*.

- t;. -

by associatir,g v1ith every Pa:;.~eto optimum x E P the affine spnce

D(x) E _3E • ïhis enr:,bJ.Gs w: te forrnulats tv10 types of problems occu-

ring in the spaco X. One, callsd problcm (/), is concerned with pro­

pe:~ti.es of the el8ï:1onts of the fé1mi1y fP pass:Li1g th· -.::,_:gh a given x E X

this co:cresponds to \;Jalrasi6n equilibrium thcory, the vector x E X

representing initial endowmgnt3 while the price vector p e S associated

by the canonical equation system \tJith every e:;.ement of JE.. passing through

x is an squil:i.brium price vector for U:s i:1i tial endm.msnts x e X. The

nther problem, callsd (I), is concer~ed with prcperties of the inter-

section of the S8t o+ Pareto optima p \,/Jth a g::_ve1î elP~,-3n-c of ·x ; this correEponds to efficient constrained allocation th~ory, the elemen~

of 3( beinG the sst of conJtrainod allocation~ associated

with a given price-income vcctor be X

EXAMPLE. Teke (2. mJ = ~, 2) so that one cen use tMs 2dgeworth box

representation; Xis , . C,18 plena of Edgm·Drth box, )E is the set of

l:!.nss of X whi i.e P deno ces the "contract c;~!"VE". 1-i' x be longs to P,

then D(x) is the com~oG t~ngGnt to thB two irdifforencs curves passing

through x 1~ P. The far;,i.ly cP :Ls then obtainsd t1v rnuk-i_1-;i~ :' vary in the

set P.

4. THE Vl'AL-:'.T\/ Tf itCR\!

main purpose of this section is to show thet (W) and (I) (respectively

CI) and (\IJ)J are cct:_11J)ly s:·Ji.,a:ent, · t1:1:~;;l1 dsfi1~:-,r; ë: r:iualHy theory

betvieen (W} and ( I J, L ,3. behJDe11 \1,,aL.'üsiar1 equ:U ibriurn theory and ef-

ficient constrained allocat~on theory.

- 5 -

The first step in establishing this duality theory begins by

defining preferences or utilities on the dual space X, The individual

Q. demand mapping f. = S x JR ->- 1R being cJ di ffeomorphism, we define a . J.

preordering by taking the inversa image by_ fi of the

preordering ~ on JR 9', Thon, the fodirnct utility function

ui = ui O fi : S x IR + 1R associated wi th ½_ represants the preorde-

ring ~-· There is nn need recalling here the well-known properties of J.

~i (see [15] and [16]). The construction associating ui with ui, when

repeatod, yields ui = ui. hence the ~dual" framework.

Let v i : X -i- JR be defined by the formula

We denote

,j • J.

by

hypersurf,Jce

(p

T. (b) l

{b'

W,, ,,,,,W.,,,,, W) 1--+ ~1.(p

1 J_ m

the tangont hyperplans in b to the ~ ~

E X V. (b') = v.(b)}. Let D(b) = J. l

n

w.). l

indifference

T. (b) l

; ltJ8

also introduce H.(b),the halfspace of X dolimited by T.(b) containing l l

fo• EX I v.(b') < :.(b)} and we ckifine K(b) = n H.(b). J. l l

THEOREM 1. The following properties are equivalent

(1) b EX is a Pareto minimum for the v. J.

i=1, ••• ,m

(2) codim O(b) = m - 1

By a Pareto minimum, ws meon that there is no b' EX such

that v.(b') s v.(b) with at least one strict inequality. Note that a l J.

Pareto minimum for the vi becomos a Pareto optimum for - vi. i = 1, .•.• m.

- 6 -

Proof. (1) => (2) : If b c Xis a Pareto minimum, the condition

codim O(b) = m - 1 results from the first-order necessary condition.

(2) <=> (J) : Taking into t3ccount the equ,::1lity grad ui (fi (p , \,\)) = Ài P ,

an easy calculation shows that the equation of the tangent hyperplan in

b = (p, w1

•••. , w) to the indiffer8nce hypersurface m

{b' E 8 1 v.(U) = v.(b)} takes the form 1 l J..

Therefore, D(b) is defined by the equation of X

p' ,r

and by the m equations

\' l w'.

l

p'.f.(p, w.) = w'. l l l

i = 1 , . , , , m.

Codim O(b) is equal to the r~nk of the matrix defined by the coefficients

of these equations

f1 m

t-1 r

t-1 f1

K-1 f'7 ,_

f t--1 m

1 1

1 0

0 1

0 0 1 1

-' (Note that individuals are denoted by indices and commodities by exponents

therefore, f~(p , w1 ) is consumer i's demand of commodity k).

- 7 -

This rnatrix can have rank m - 1 only if the dependance relation

k r =

rn I

i=1

holds. Walras law irnplies the equality

rn r = I

i=1 f.(p, w.L

l l

!<. = 1 9, - 1

(3) => (1 l : Instead of shm·Jing that the second--order sufficient condi­

tions are satisfied, we shall use an alternative approach based on proper-

ties of independant interest.

Let p(x) be the unique price vector in S associated with the

Pareto optimum x e P. i.e. p[x) e Sis colinear with grad u. (x.l for any i. l l

The mapping

g p -+ B

defined by the formula

x 1-:---> (p(x) , pCx).x1 , ... , p(x).xm)

is smooth, takes its values in B[r), satisfies the equalities

vi(x) = v1 (g(x)) with i = 1 •... , m. and finally gis a diffeomorphism

between P and B(r) (Appendix, Corollary 4).

Assume that there exists be B(r) nota Pareto minimum for the

V." There exists b' E- B(r) such that V. ( b') :Ç V. (b) with at least one l l l

strict inequality. Th1:m -1

g ( b') is a Pareto optimum for the V. since it l

belongs to P but is also Pareto dominated by -1

g (b) : a contradiction.

Q.E.O.

- 8 -

Let* be the set consisting of the codimension m - 1

affine subspaces of X that are not perpendicular to S. It results A

from this definition that X consists of the affine subspaces of

X which can be defined by equations of the type

W. ::: p.X. J. J.

1=1, ... ,m

where p describes S, Recall that, by definition, we have Lx ~ r. i

,.. Such an equation system being unique for any element of X this

enables us to identify X with X. Let /:i consist of the spaces

O(b) where b varies in B(r), the-set of Pareto minima for the

V, i=1, ... ,m. 1

" We denote by a X -+ X and by A X-+ Y.. the identification

mappings

a : X 1---> a(x) {(p ' \;J1 , • Cl • ., w ) E X W, = p.xi i = 1 .J ' •• , m l

A b 1--> A(b) r (x1 X ) E X 1 i 1 = "\.X = ;, Q ,. r. :, W. = p.x. = ) ••• .P m 1 1

·" = {), THEOREM 2, \Ale have a(P) tO A(B(r)) = , and

_F'roof. Let x E P; then y= (g(x), g(x).x1

, ... , g(x).xm) belongs

to a(x). We deduce from f.(g(xl • g(x).x.) = x. that 1 . 1 1

'f.(g(x) , 0çr(x).x,) = \" x. = r L. 1 1 L 1

m}

mL

so that, by theorem 1, y is a Pareto minimum. Therefore, O(y) belongs

to :f> . From the procf of theorern 1. vJe have

D(y) = {(p, w1 , ... , wm) EX W.

l i=1,.,,,m}

hence O(y) = a(x). Therefore, a(x) belongs to ~~

- 9 -

A similar proof works for the second relation&hip. Let

b E f-J(r) ; clerH'ly, f(bJ = (f1

(p .. 1,11

) ,.,., fm(p, ~\lm)) belongs

to P, and D ( f ( b l), \"1hich be longs to 11.), is oasily seen to bo equ,31

() () 1-f.) to Ab . Thersfore, we have Ab, c 0

Q.E.O.

The theory (~,J) deals v.Ji th the elements of the -Family f> passing through a ~ivsn x EX while (I) deals with the intersection

of P with a given element of X. Using the transformations a and A,

we see that (W) becomes the study of the intersection of B(r) with

a gj_vr3n element of X, which is theory (I), Simil,:irly, (I) becomes

the study of the elements of the family ,Ppassing through a given

b E X, Le. theory (W). Theories (\AJJ and (W) (respectively (ï) r:ind

(I))are in a very natural sense dual theories, Therefore, the identity

between (I) and (W) (respectively [W) and (I)) provides a duality

theory between (W) and (I). As a consoquence, a property established

for (W) yields a dual property for (Il and vice-versa. This view-

point, which is going to bs devsloped on a systsmntic basis, has

been already encountered in equilibrium analysis where problems

dealing with the number of equilibria or with singular economies,

i.e, problems concerning (W), were solved with the help of their

dual formulation through (Il [3]. The duAl view-point will be used

in this paper to establish a list.of properties of efficient cons­

trained allocations (theory (Ill deduced from Walrasian equilibrium

theory (theory (W)).

- 10 -

5. EFFICIENT COMSTR\IMEV ALLOCATTO:\IS

Let b = (p, w1

, .•.• w) e X be o given price-income vector. m

The first question coming to minci is whether there always 8Xists an

efficient allocation consistent with these given prices and incarnes.

Linder the assumptions of section 2 ~bnut consumers' preferences. we

have the theorem :

THEOREM 3, There exists at least one. efficient constrained allocation

x e X associated with any given prioe-in~ome vector

b = (p, w1

, ..• , w ) m

EX,

The strong assumptions on smoothness and strict convexity

of prsferences can be weakened while theorem 3 remains true. A way

of proving such generalizations is to approximate prsferences by

preferences satisfying the strong assumptions of section 2, a stan­

dard method in differential topology {see e.g. Hirsch's proof of

Brouwer's fixed~point theorem [14]. p. 14). Alternative proofs of

theorem 3 based on convex analysis are.left as an exercise.

Theorem 3 is an existence statement. Therefore, the next

natural quastion to ask is whether there cre one or several (in that

case, how many) efficiBnt constrained allocations. A subsidiary ques-

tian is how these efficient constrained allocations do depend on the

parameters, especially on the price-income vector be X. To deal with

these problems, we introduce the concept of a regular price-income

vector which we deduce from the concept of a regular economy defined

in Walrasian equilibrium analysis ([ô], [2] and [3]), Thus, we say

- 11 -

that b EX is regular (resp, singular) if the affine space A(b) is

transverse (resp. not transverse) to the manifold P of P3reto optima.

- -Let I denote the set of singular price-income vectors and~-= X\ I the set of regular price-income vectors.

-THEOREM 4, The set Î of regular price-income vectors is open dense - ~

in X. Any b ~ {R. has an odd number of Rfficient constrained alloca-

-"" tians tissocié}tod wi th, Eech such ,~llocntion depends smoothly __ on b E tK•

Actually, theorem 4 can be improved to show that ,.'\ is

not only open dense, but that its complemant I is closed of Lebesgue

measure 0, But saying that an efficient constrained allocation

x E A(b) depends smoothly on b E fl, 1,<Je mean that there is a smooth

mapping x

(Notation

LJ + X where LI is a neighbourhood of b such that x = x(b)

x denotes both a mapping and an allocation l). A straight-

forward consequence of theorem 4 is the constancy of the number of

efficient constrainGd allocEJtions over every connected compcnent of (K,.

The role played by the set of Pareto minima B(r) in the

study of the number of efficient constrained allocations is illustrated

by the next theorem,

THEOREM S. The set B(r) of Pareto minima in X belongs to a connected

component of k. There is only.one efficient constrained allocation

associated with every b in that component.

We deduce from these three theorems the following picture

of the dependance of efficient constrained allocations on price-income

vectors : the set of regular price-income vectors ;~ is partitionned

- 12 -

into arcconnected componenté ; the number of efficient constrained

allocations is a constant odd number over every arcconnected compo-

nent ; the set B(r) is contained in.one of thsse components, component

denoted LJ; thers is only one efficient constrained allocation asso­

ciated with any b E LJ; furthermoro, b E Rbalongs to U if the

(Euclidean) distance from b to B(r) is small enough or, by theorem 2,

if the difference vector I f.(p, w.) - ris not too large, l l

Proof of theorem 3 to 5. These theorems deal with properties of

theory (I) ; their dual statements are properties of (W). Since

they are formally identical to well-known properties of theory (W),

we just have to adapt the proofs established in the case of (W) to

the case of space X, For the reader's convenience, however, we keep

to a minimum the necessary background of theory (W) so that most proofs

are repeated,

In the Cartesian product Px X. we define E as the set

consisting of the pairs (x, b) such that x c A(b). It is easy to

check that Eisa closed subset and a smooth submnnifold of Px X.

Cloarly, E is the analogue in (W) of the equilibrium manifold E of

Walrasian equilibrium theory (W) [2]. The "dual" Oebreu mapping

rr E + X

is obtained by restricting the natural projection (x, b) t---> x to E.

The mapping rr is smooth since È is a submanifold of Px X, By defini-

tion, we have

;-1(b) = {(x, b) e Px X I x F A(bl} ~ (F n A(b)) x {b},

- 13 -

We see from this equality that the study of efficient constrained

allocations amounts to the study of the dual Oebreu mapping w.

This equivalence butweEJn w rrnd P n A(b) can be given a formal basis

following the lines of section G of [3]. Skipping the details, it

can be shown that bis a regular (resp. singul~r) price-income

vector if and only if bis a regular (resp. singular) value of rr.

Basides smcothness, another important fenture of the

Debreu m~pping is its properness, i.e. the inverse image of every

"' compact set is compact. Let us show that w is also proper. Let

K c X be compact. Consider

;-1 (K) = { (x , b) E E I b E K rmd X E A(b) n P}.

Clearly, ;-1 (K) is closed in Px X. From

:;-· 1 (K) c K x (A(l<'J n P)

where A(K) • u A(b), we deduc~ that ;-1 (K) is bcunded if A(K) n P bEK

is bounded. From Appendix, Corollary 6, we just need to show that

m the image of A(K) n P by the imbedding u : P + ~ whero

u x ,___> (u1

(x1

) •... , u (x )) m m

is bounded. Oefine u. l

x E A(b), we have

~- Sup u . ( f. ( p , ~,. ) ) bE:K l l l

since p.x. ~ w. for l l

u.(x.):::: u,(f.(p, w.)) ~ u. l l l l l l

i = 1 ,.,,, m

So that u[A(K) n Pl is bounded from above.

- 14 -

Define x(i) the unique Pareto optimum such that uk(x(i)) = uk, k ;,! i

(Appendix. Corollary 5). Any x e A(K) n P satis~ies the inequalities

k ;,! i

by definition of x(i), Since xis a Pareto optimum, we have

u.(x.) ~ u.(x(i)) otherwise x(i)) would Pareto dominate x. Thsrefore, 1 1 1

u(A(K) n P) is bnunded from below by (u1

(x(1)) , ... , um(x(m))).

l•/e are nm,J reedy to apply to ;' standard resul ts of elemen­

tary differential topology Ci being proper, sections 1 and 2 of [14] -

are still valid in our context). Tho set I of singular values is

-closed of measure 0 ' its complement J~ = X / l is open dense. The -

mapping 1T defining a finite covering of /r<, M :;- 1 (bl is finite for

b e /,\. and constant over every connect,3d component of .). This proves

the first part of theorem 4.

\,Je knmv by theorem 1 that B(r) is diffeomorphic to IRm- 1,

hence cnnnected, To show that B(r) belongs to one connected component A

of~. we just need to prove that B(r) is included in~J i.e. that A(b)

is transverse to Pif be B(r) which is the same thing as showing that

A(p(x) , p(x).x1

,,,,, p(x).x) associated with the Pareto optimum m

X= (f1(ri, w

1)

proved in [20~.

f (p, w )) is transverse to P, a property m m

To complets theorem 5, taks any h e B(rL We have

f(b) = ( f Co 1 . w1J , ••• , fm(p, wm)) c P n A(b). Take now x e P n A(b) ;

from p.x. = w. i"' 1 ,, .. , m, 1,ie deduc8 that u.(x,) ~ u.(f,(p, w,)), 1 1 1 1 l l 1

hence xis Pareto dominated by f(b). so that x = f(b). We have alrsady

--1 shown that#=(P n A(b)) = 1 ; this is e~uivalent to 73/-=1T (b) = 1.

- 15 -

The other statem8nts are straightforward consequences of

the degree concept. Take b E 6(r) ; since f/= rv-î 1T (b) = 1, the (topo-

"' --1 r logicnl) degree of 1T is odrj, hence =/1 'IT (b) is odd for every b E I'.!\,

Furthermore, 'IT is surjective otheruise its dagree wou1d IJ8 equal to O.

Q.E.O.

~~~~~~- Thoorom 3 to 5 are adaptations tn tha theory of efficient

constrained allocations of well-known properties of Walrasian equi-

libria. As an exercise, we suggest a study of ti1e following topics

deduced by duality of miscellaneous properties of Walrasian equilibria

the dependance of e-ff~cient constrained allocations upon regular

bcf.R.C4J; the measure of the set of pride-income vectors

having mnre than a given number of effisient constrained alloca-

tians associated with [4] 1 the relationship existing bstwsen the

number of efficient constrained allocations(a qualitative informa­

tion) and the determinateness of these efficient constrained allo­

cations (a quantitative propert0 [5] ; the local bshavior of

efficient constrained allocations when be~ crosses the set!

of singular price-income vectors [3].

6. NOM-TATONNEMENT VVNAMICS ANV 1JUAL1TY THEORY

By non-tatonnement dynamics in the spacs X. ws mean dynamic

processes which, when restricted to a purs sxchange framswork, repre-

sent the way exchanges that increase everyone's utility can proceed ([1]

chapter 12 and [10]).A trade (or admissible) path represents the

evolution through time of the commodity bundles owned uy every economic

agent. We say a path ~ [0,1] +Xis smooth by pieces if it is a fini-

te union of smooth paths ; note that ~ has a right d9rivati,1e ~; every-

where on [0,1).

by pieces paths.

For the sake of sim~licity, we consider only smooth

- ·rn -

DEFrnITION 1. _0_pnth cp [0,1] + X smooth by pieces is admissible if

rpr (t) e K(cj>(t)) t E [0,1),

An admissible path is complcite if i ts end point cp ( 1) is

Pareto optimal, i.e. K(cp(1)) = 0, Note that cp[1) can be the only

Pareto optimum of an admissible path.

The definition of an admissible path is actually given in

term of the cone field K. It has a straightforward extension ta any

dynamics defined by a cone field. Thus. a dual admissible path is a l

srnooth by pieces path cp : [0,1] + X such that cpr(t) e K(cp(t)) for

te [0,1). Completeness occurs when cp(1) is a Paretti minimum, i.e.

Kfcp(1)) ~ ~ which also means that cp(1) belongs to B(r).

Let b = (p. w1 , ...• w) e X be considered as a proposal rn

for prices and individual incarnes in a planning process. The pro-

posal is feasibJe if and only if I f. (p '

ltJ. ) is :<; r ;• using Walras 1 1

la1t1, the inequality I f. (p , W,) ::;; r is easily seen to be equivalont l 1

to I fi(p, wi) = r. In other words, if b does not belong ta B(r), i.e.

is nota Pareto minimum, then bis net feosible. This remark leads

ta the economic interpretation of a dual admissible path already

pointed out by Malinvaud in a larger framework [13]. Consider a pro­

posal that is not feasible. The planning board revises the proposal

in a way that leads to a decrease of everyone's utility level compa­

red to the initial proposal, this process being illustrated by way

of an admissible path. The properties of monotone processes

leading to feasible proposals starting from a given initial proposal

have been extensively studied in the recent years in a framework inclu-

ding production and public goods (see [9], [13], and Tulkens's survey

- 17 -

article [22] 1 for recent mathsmatical developments, ses [6], [7] and

[18]). The next theorem,dual version of theorem 2.1 in [18], is inclu-

ded here for the sake of completeness.

THEOREM 6. Let b EX and let b' e D(r) be a faasible price-incoms

vector such that u.(f.(p' , w'.)) < u.(-f.(o, w.JL There exists a l l l ]_ ]_ ' l

smooth complets admissible path starting from band ending in b',

Proof. Apply the same proof as in [1BJ. Q.E.D.

Another problem related with monotone processes is the

stability of final outcomes, a question considered by Smale in the

case of pure exchange economies [19],

DEFINITION 2. The allocation x e Pis stable for the non-tatonnement

dynamics_ K _if,, given a neighborhood U(x) _9f x J~ P, there exists a

neighborhood V(x) _in X such that any complets admissible path star­

tinf from V(x) ends in U(x).

This is a property of local stability which can be defined

for any dynamics generated by a cone field. With our assumptions con-

cerning preferences, Smale has shown that every Pareto optimum is

stable for the non-tatonnement dynamics K. Considering the dual

statement, we have :

THEOREM 7. Every feasible proposa! be B(r) is stable for the dual

non-tatonnement dynamics K.

Proof. Apply the same proof as in [191. Q"E.O,

- 18 -

In cther words. a feasible proposal be B(r) bscomes, when slightly

perturbed, a proposal b' close to b which, though not feasible any­

more, is the starting point of complets admissible paths that all

end at feasible proposals close to b.

1. COM8TR/l.HJEV VYf,JAMICS

This section is devoted to the study of stability proper­

ties of efficient constrainod allocations which raises the prelimi­

nary question of defining dynamic processes that either reflect

actual dynamics or could be used to reach such efficient constrained

allocations.

We bogin by mantionning a special class of dynamic processes,

namely the algorithms capable of cornputing efficient constrained allo­

cations. From Scarf's algorithm which computes the Walrasian equilibria

associated with given initial endowments x e X [~7], we obtain by

duality a new algorithm that computes the intersection of the set

B ( r) of Pareto minima in X v.Ji th the tif fine space a ( x) E f ; El

straightforward adaptation of this new olgorithm to the same inter­

section problem formulated this time in the space X provides an al­

gorithm computing the intersection of the set P of Pareto optima in

X with the affine spaco A(b) e *· We add a few words for mentionning

the possibility of dofinins dual versions of Smale's smooth formulation

of Scarf's algorithm [21] and dual versions of the Walras tatonnement.

From now on, we restrict our attention to the formulation

of dynamic processes that could present a real practical intsrest

for reaching in a somewhat decentralized way efficient constrained

allocations associated with a given pries-incarne vector be X.

- i g -

r Let X = (x1 , ... ~ X ) E X ·' W8 recall that X. E JR'" is ,3

m 1 n n 1 Q,-tuple denoted (x. , .. :, , x~). \.Je associrrte with X. E JR'' the vector

Q. y. E JR such that

l

l

=

Foom r y. = \ x. = r l 1 L 1

f\ X.

l

V\J. l

1

k

1 - P1 xi

= 1

-

= itJ. l

1

, lt I! I!., Q·-1

Q,-1 -p Q-1 X.

l

i = 1 , it results

that y= (y1 , ... , ym) belongs to A(b). Tho mapping pr: X+ A(b)

is defined by the rolation y= pr(x). It is easily seen to be a

projection parallel to the vector subspace of CIR1 )m generated by

every a~m1t' s Q,

2-th basis vector (0 , • , , , 0 , 1) E IR •

DEFINITION 3. Constrained dynamics Con A(b) is the image by the

projoction pr: X+ A(b) of the non-tatonnement dynamics K.

In other words. constrainad dynamics C is generated by the

cono field

C{x) pr(K(x)) XE A(b).

The idea of constrained dynamics is that a Pareto irnproving move

starting from x E A(b) violates the incarne constraints represonted by

the price-income vector b EX; hcnco~ it is oorrected by changing

ev~ry agent's numeraire sndowments so thot the income constraints

become satisfied.

The economic moaning of the constrained dynamics is not

difficult to work out. A planning board fixes commodity prices and

incarne levels, Furthermore, the planning bDard has an accurate

knowledge of all individual incarnes. New, all agents are free to

increase their utility levels through trade. Neverthelsss, the plan-

ning board, in a second stcp corrects by a set of taxes the observed

- 20 -

discrepancies betwesn actual and planned incarnes. The econornic rele-

vance of constrained dynamics depends on two importa~t assurnptions :

a myopie behavior of the sconomic agents ; an accurate knowledge of

individual incomes. Note that such myopie behavior is cbssrved during

implernentation periods of price and wage controls when the sfficiency

of the redistribution system may be more or lsss unpredictable from

the consumers' point of view, Second, accu rate knm'llledge of indivi­

dual incarnes can be biased only by owni~ such valuable commodities

as gold; it would be sufficient to rul9 our privat8 ownership of such

cornmodities.

An equilibrium of constrained dynamics C bsing an allocation

x e A(b) such that C(x) = ~. which happens if and only if K(x) = ~.

the equilibria of constrained dynamics Cars precisèly the efficient

constrained allocations. This leads us to study the stability of ef­

ficient constrained allocations with respect to constrained dynamics.

\,~e recall that if be B(r) is reguli:ir (i.e, be fr:), then

A(b) n Pis discrste. This anables us to reformulate in a sornewhat

stronp,er form stability propertiss for constrained dynamics C.

DEFINITION 4. An efficient constrained allocation x e A(b) is stable

for constrainad dynamics C if theœ exists a noighborhood W of x in A(b)

such that, for any y e W, any smooth complets admissible path starting

at y e W ends at x.

This is just a· reformulation of dafinition 2 in the particular case

of a discrets set of equilibria.

The next theorem provides an important property of cons-

trained dynamics C.

- 21 -

THEOREM 7. If b E X is a regular price-income vector (Le. b E fU,

then ev~ry efficient constrained allocation is stable for constrai-

ned dynamics C.

Theorem 7 will result from the next lemma.

LEMMA. Let <ii : [O, 1 J -,- f\Cb) be a smooth path sta!'ting from y E A(b)

admissible for constrain8d dynamic~ C, Thero exists a smooth path

~ [0,1] + X starting from y admissible for non-tatonnement dyna-

mics K and there exists an E > 0 such that fort E [D,E), we have

qi(t) = pr o ~,(t).

Proof, If y is Pareto optimal, thcn the lemma is trivial. Assume

y i B(r). Let Y: pr- 1 (~([0,1])) be the inverse image of ~([0,1])

by pr; Y is a manifold with boundary. Let y= qi(O) and let T = ~'(O)E C(y)

be the tangent vector to <ii at t = 0 (<ii' denotes the derivative of qi).

Since TE C(y). there exists a vector T' E K(y) such that T = pr(T').

We define a smooth vector field on Y by associating with z E Y the vector

T = (T' - T) + ~'(pr(z)). z

Clearly, wo have T = T', By a straightforward application of the flow y

theorem (ses e.g. [12], chapter 4), we can construct a smooth curve

on Y starting at y and having T' as t3ngent vector

can be parametrized to satisfy the relation

~(t) = pr O ~(t) t E [0,1].

this smooth curve

Sincey does not belong to B(r), we have K(y) ~~and ~(O) c K(y). It

results from ths smoothness of the cons field K(x), x EX that ~(t)

belongs ta K(~(t)) fort small enough.

Q.E.D.

- 22 -

Proof of theorem 7. Let x c A(b) n P. We defins u. - u.(x.) i = 1 , ... , m. :L l l

We associate with any u = Cu1 ~ ... , um) ~uch that u1

< u1

i = 1 •..•• rn

the open set

V(u) - {x e X I u.Cx.) > u. · 1 l l

i = 1 , ... , ' nu.

Clearly,.V(uJ is an open nsighborhood of x in X. Furthermore, any

smooth admissible path for non-tatonnernent dynamics K starting from

sorne y e V(u) stays in V(u) sinca every agsnt's utility increases

along such a path.

The price-income vector t1 being regular, the set P n A(b) is discrets.

- -We can choose u close snough to u = Cu1

pr(V(u)) n P - {xL Take u' = Cu1 ·, .. ,. ' •••• u ) so thùt rn

u') such that u. < u' < u m 1 i i

and define W = pr(V(u')). Considsr a smooth admissible path

t : [0,1] ~ A(b) starting from y= ~CO) e W. Let us show that ~[[0,1])

is containsd in pr(V(u)). Actually, we will provo inclusion

1'([0,1]) c pr(V(u7 )) ,vhere pr(V(u')) c prCV(u)), Assume the contrary

and let t0 = inf {t ~(t) i pr(V(u'))}, Since ~(0) e W = or(V(u')),

we have Clearly, qi(tn) e pr(VCÜ')). by the lemma, u

consider ~ such that ~(t] =pro ~(t) in a neighborhood of t0 ; for

t > t 0 in that neighborhood, ~(t) belongs to V(u'J. Tnerefore. t(t)

belongs ta pr(V(u')) forte (t0 • t

0 + s]. a contradiction.

If ths admissible path ~ [0,1] + A(b) is complets, then ~(1) is

a Pareto optimum, i.e. ~(1) e P. Thersfors, ~(1) belongs te

P n pr(V(u]) hence ~(1) = {x}.

Q.E.O.

- 23 -

8. CONCLUSION

The main purpose of this article has been bath a study of

efficient constrainod allocations in pure exchange economies whose

prefsrences satisfy strong assumptions like smoothness, strict con-

vexity, etc

for extension

and a study of related dynamic~ These results call

ta production, ta public goods and for weaker

assumptions concerning preferences, Unfortunately, this is not going

to be an easy task since the simplicity of the duality theory is

likely to be lost in such extensions. Another promising direction

for further work seems to be the application of constrained dynarnics

to equilibrium analysis. Thus, dual constrained dynamics can be for­

mulated from the constrained dynamics of section 7 (exercise : what

could be its economic interpretation ?), The dual constrained dynamics

defines an algorithm that converges to Walrasian equilibria; a compa­

rison of its performancES with those of Scarf's algorithm should be

interesting.

- 24 -

APPENDIX

We provide with this ôppondix a unified treatment of some

properties of Pareto optima for which convenient references are lacking.

We recall first that the Slutsky matrices

( af·~ c p ' \tJ.)

k af~(p

' M. (p w.) l 1·-

\,1/. ) l

' = \ + f. (p '

wi) l l

\ apk l l

3w. k,j l

are symmetric and negative definite.

Let Q be the subset of (JRi)m consisting of Pareto optima :

there is no x' = (X' Ji1 Il G • .1 X') 1 m 1

such l that l x'. = l X, and u. (x '.) ~ u.(x.J r (JRQ,)m 1 l l l l l Q = l __ x

= (x1 ., Il •• ., X ) E m with at least strict inequality one

Let f B + OR Jl.)m be the mapping

PROPOSITION 1. The mapping f 6 + (JRi)m is an imbedding having

Q êls image.

Clearly, f takes its values in Q, Let p(x) be the unique

price vector associated with the Pareto optimum x E Q, The mapping

g : X ~--> (p(x) , p(x).x1

,, .. , p(x).x) is a continuous inverse m

off : 6 + Q. Therefore, fis a smo~th mapping which defines a ho­

meomorphism bet~een Band f(B), hence an imbedding.

- 25 -

COROLLARY 1" The set of Pareto optima Q is a smooth submanifold of

i rn to JR2+m-1, ( :IR ) di ffeomorphic

COROLLARY 2. The mapping g : xi--> (p(xJ , p(x) .x1

, ... , p(x) .xm) is a

diffeomorphism betwee~ Q end B.

Let t •, n --' JR-- 9. x :IR m- 1 1 th t · t · t n f th · ~ ~ oe e res rie ion o ~ o Je mapping

PROPOSITION 2. The mapping tisa diffeomorphism.

L t Z = 9, IR m- 1 Th · t . ' t ' d b . e = = x . e mapping is oo aine y compos1ng

with the natural projection

b : Z x :IR+ Z,

We will show in a first step that tisa continuous bijection and in

a second step that its Jacobian determinant (for a suitable coordinats

system) is 7! 0.

Step 1,

ais an injection. Assume a(x) = A(y) , i.e. we have l xi= l Yi and

~ u.(y.). Let z = x + y/2; if x. ~ y. for some i, i i i l then

ui.(z1.) > u.(x.) = u,(y.) by strict quasi-concavity of u .. This con-1 l i i i

tradicts the Pareto optimality of x and y.

b restricted to a(O) is injective. Let x and y E O be such that

b(à(x)) = b(a(y)) ; i.e. l xi= L Yi and u.(x.) = u1.(yi.) for i ~ m,

l l

If u (x) ~=u (y), for example u (x) > u (y), then Pareto optima-m m m m m m m m lity of y is contradicted; hence u (x) = u (y), i,e, a(x) = a(y). m m m m

tisa bijection. Ws already know that t =boa is an injection,

Let us show that t is surjective. Let z = (r, u1 , ••• , u

11 E Z m-

be fixed and consider the optimization problem

Find x = (x1

the constraints

x) which maximizes u (x) subject to m m m

u.(x.)::;; u. i = 1 , ... , m-1 1. 1. 1.

l X, S: r 1.

Given the assumptions on preferences, proving existence of a solution

is straightforward I by the monotonicity property of preferences,

the constraints are saturated, hence u.(x.) = u. and lx. = r. Let 1. l 1. 1.

us show that a solution xis a Pareto optimum. Assume that there

exists y such that I y1 = I xi and u

1(yi) ~ u

1 Cx1

l i = 1 , ... , m

with at least one strict inequality, Choose for i = 1 , ••.• , m-1 m-1

a y'. 1.

= vm + I cy. - y:1. i=1 1 1

Clearly, y' ~ y and an easy calculation shows that u (y') > u (x ), m m m m m m a contradiction with the definition of x as a solution to a maximi-

zation problem,

The mapping t is smooth as the restriction of a smooth

mapping to a submanifold. Being a bijection, t is going to be a

diffeomorphism if it has a smooth inverse,

Step 2.

Using the diffeomorp~ism f : 8 + Q, we just have to show that

t of : 8 + Z has everywhere a non--zero Jacobian determinant, From

the relationships grad u.(f.(p, w.)) 1. 1. 1.

âf. k A. p : p 1

- p f~ and 1. ,. • êlp - - k l k

- 27 -

élf. 1 p -- = 1 (the last two squalities are deduced from Walras law élw. . 1

p.f.(p, w.l = w~ by teking derivatives), we have to show that the 1 1 ..,_

follm-Jing determinant is -"' 0 :

élf. l 1 l élp1

1 - f1

cH'. 1

-"--~ op,ii,-1

Q-1 - f1

0-1 - f"'

m-1

a-P 1

c)1,,J1

1

0

&f m-1 3f m -·-élw

m-1 ôvJ

m

0 0

1 0

Multiply line k by pk, where k = 1 •... , Q,. Add up, substituts the

resul t to line fl and chanse i t to the last line. ~Je get the determinant

élf: ôf: af1 af1 élf 1

I a/ I l 1 m-1 m ap 1 clltJ 1 d\,'1 aw 1 9a- m-1 m

élf~·-1 élf~- 1 élfQ,-i clfQ,-1 3f.Q,-1

I l l . 1 m-1 rn --- -- ---é)p apfl-1 clW élw aw 1 . 1 m-1 m

f1 ,Q-" - - f' 1 1 0 0 1 1

.

f1 Q,-1 0 1 0 - - f m-1 m-1

f1 Q-1 - - f' 0 0 1 m m

3f~ Multiply line (,Q, - 1 ) + j Lly J substract from line k, and perform

aw. ,

J these operations for ail k and j .. We get the determinant

- 28 -

M 0

= M * Icl

where M = Y M,(p, w.) is the sum cf negativs definite Slutsky w l l

matrices ; hence det M ~ O. Q.E.D.

Lst r = I x. be fixsd. Recall that l

X= {x = (X,!,:. ... , X) 1 m

and P = Q n X denotes

the set of Pareto optima in X.

COROLLARY 3. The set P of Pareto optima in Xis a submanifold of X

diffeomorphic oy x -> (u1

Cx-1) , ... , u -1(x 1n to :ITTm-

1 • 1 m-1 m- -

Note that P = t - 1 ({r} x JRm- 1 ).

COROLLARY 4. The set P of Pareto optima in_ X _is diffeomorphic by

g : x -·> (p(x) , p(x 1 .x1

COROLLARY 5. Utility level~ u1

p(x) .x ) to E:l(r). m --

•...• u 1 being given, there is a unique m-

Pareto optimum x e P such that u.(x.) = u. l l l

i = 1 •... , m 1,

Note that corollary 5 t1as an ubvious extension to the case where the

utility levels of (m-1) consumers (not necessarily the first (m-1)

ones) are given.

COROLLARY 6. A c:losed subset K ~:f_ P js compact H and only if its

- 29 -

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