A mathematical theory of pure exchange economies without the no-critical-point hypothesis

A mathematical theory of pure exchange economies without

the no-critical-point hypothesis

Citation for published version (APA):

Geldrop, van, J. H. (1980). A mathematical theory of pure exchange economies without the no-critical-point

hypothesis. Stichting Mathematisch Centrum. https://doi.org/10.6100/IR94027

DOI:

10.6100/IR94027

Document status and date:

Published: 01/01/1980

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OF PURE EXCHANGE ECONOMIES

WITHOUT THE

OF PURE EXCHANGE ECONOMIES

WITHOUT THE

NO-CRITICAL-POINT HYPOTHESIS

PROEFSCHRIFT

~ .

TER VERKRIJGING VJ\N DE GRAAD VJ\N DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR

MAGNIFICOS, PROF. IR. J. ERKELENS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBMR TE VERDEDIGEN OP

VRIJDAG 14 NOVEMBER 1980 TE 16.00 UUR

DOOR

.JOHANNES HUBERTUS VAN GELDROP

GEBOREN TE ROTTERDAM

1980

Prof.Dr. S.T.M. Ackermans

en

Aan de naqedachtenis van mijn ouders,

Chapter 1. Introduction

1.1. Commodities, prices and .preferences 1.2. The set of equilibria

1.3. The set of local Pareto optima 1.4. Disastrous allocations

1.5. Survey 1.6. Some examples

Chapter 2. Preliminaries on manifolds Introduction

2.1. Differentiable mappings and submanifolds 2.2. Sard's Theorem

2.3. The Whitney c"" topology 2. 4. Transversal! ty

2. 5. Characterization of local Pareto optima 2.6. The submanifold

r.

\

Chapter 3. The set T Introduction

3.1. A first definition ofT 3.2. An alternative definition of T 3.3. Comparison with the results of Smale 3.4. T is not dependent on scale transformations

"" 1 m 3.5. T is a dense subset of C (JR ,B)

.. 1 m

3.6. Is T open in c (JR ,:R) ?

Chapter 4. '!'he set of equilibria in a pure exchange economy Introduction 4. 1. Regular economies 4.2. An example 1 3 6 7 7 8 12 13 18 19 21 22 28 31 31 36 39 42 44 52 55 55 62

Chapter 5. The set of local strict Pareto optima in a pure exchange economy

Introduction 5.1. Regular pairs

5.2. Local structure _{of ecro:i=1} m 5.3. A local optimal part of·e

cr 5.4. Some examples

Chapter 6. Trade curves Introduction

zi, u)

6.1. Construction of a vector field, generating trade curves

6.

2. An example Epilogue References Index Samenvatting Curriculum vitae 69 70 74 80 84 94 96 103 105 106 108 109 110

CHAPTER 1

INTROVUCTION

In this monograph we consider a

pure erechange economy

without producers. There are R. durable goods and m agents. We assume R. 2: 2, m 2: 2.

' R.m

Each point x = (x₁, ••• ,xm) € E represents an

allocation,

where

xi

=

(x~,

••• ,x:) € E1 is the

commodity bundle

of

agent

i (1

s

i

s

m) • We assume that for each agent the whole of E1 is his

conswrrption set,

the set of possible commodity bundles. For a more elaborate discussion of the terms "goods", "consumption sets" and the sign convention, concerning

x~

< 0, xi = O,

x~

> O, see Debreu [1].

With each commodity, say the h-th one, is associated a real number, its

• h h .

pnce

p • The price p may be positive (scarce commodity), null (free com-modity) or negative (noxious comcom-modity). The

pl'ice system

is the R.-tuple

1 R. R.

p

=

(p '• • • ,p ) € .E •

1 R.

The

vatue

of a bundle a = (a , ••• ,a ) , relative to the price system p, is · the standard inner product

R.

p•a :=

L

h=1

h h

P a •

We assume p {: 0. Two price systems p and q are

equivalent

i f there is sane positive A € E such that q

=

Ap. Hence, if we take a price system p, we

R.-1

always choose p € S , i.e. p•p

=

1.

R.m R.-1

Each point (x,p) € E x s · defines a

state

of the economy.

Given two bundles a and a' in ER. one and only one of the following three alternatives is assumed to hold for agent i:

(1) a

is preferred to

a';

'· (2) a

is indifferent to

a';

(3) a'

is preferred to

a.

It is convenient to introduce a

preference relation

f

on E1 for agent i. Then the foregoing alternatives read as follows:

{1) _{a' .S a} and -, (a .S a') i i (2) a' :ka and a~ a' ; i ~ (3) a :;, _a' and -,(a' :S a) • i i

The binary relation :k is assumed to be

Pefle:dve

and

transitive.

i t

The preference relation :;, is said to be

continuous

if for each a' e: JR the

.

t

i

t

sets {a e: JR

I

a~ a'} and {a e: JR

I

a'~ a} are closed.·

~ . ~ t

If

f

is continuous, there is a continuous function ui: JR -+ JR satisfying for ·all a, a':

For a proof see Debreu [1], page 56-59.

Suph a function ui is called a

utiZity function,

representing the preferen-ces of agent i. In order to use the calculus of differentiable manifolds

and maps we assume utility functions to be

smooth.

See Chapter 2.

In economic literature several assumptions are proposed about preference relations and, consequently, utility functions, the relevance of each of them being a matter of taste or realism. We mention here:

(1)

nonsa 1-a 1-on,

t . t.

i.e. for each a e: :R t there is some a' e: :R t preferred to a,. (2)

convereity,

i.e. for each a e: :Rt the set {a' e: JRt

I

ui(a')

~

ui(a)} is

convex;

(3)

monotonicity,

i.e. u. (a') > u. (a) whenever a' ;. a and a'h ~ ah for allh.

~ l.

We

do

not make any of these assumptions. In

OU!'

modeZ the aZass of util.ity

functions coincides with the clAss of smooth functions

:Rt -+JR.

Given some u'!:;ility function ui and some bundle a, the set of points a' , preferred to a by agent i, can have locally different shapes depending on the

gradient

To get same insight in the possible situations we assume for the moment u

1 {a) = 0 and i = 2. Then, up to degeneracies one has the following

A} Here Du

1 (a) .; 0. Agent i has a clear idea in which directions to move in order to increase his utility.

B} 'l'he same picture as (A) but Du

1 (a} = 0. It is possible that some other utility function, representing the same preference relation, has a gradient .; 0 at a. C)

0

D)

0

E)

1 •

2. The

.6

e.t

o

6

eqlUU..bJti.a.

Here DUi (a) = 0 and the point a represents a local minimum for ui. Each direction improves the posi-tion of i.

The point a represents a local maximum for u

1• This situation is described as

Zocat satiation.

The point a represents a

point of

doubt.

If some direction improves the position of i, its opposite direction equally does.

Now we assume that aqent i is endowed with some

initiat

bundle r i e BR,.

R.-1

Then, given some price system p e S , he faces his

budget set

It will be his aim to maximize his utility function ui on this budget set. If such a maximizing bundle exists we can find it in the set Ei (p,ri) of points x

Given initial bundles r = {r₁, •.• ,rm) and utility functions u

=

{u

1, ••• ,um) the set Eex(r,u) of

e:ctendsd equil-ibrium states

is defined by:

{x,p) e E (r,u) if and only if ex 1.2.2.

!

I

xi =

I

ri • i=1 i=1 m xi£ Ei(p,ri)]i=l • See also Smale [15].

The first· condition is inspired by the definition of pure exchange econo~ mies, in which allocations are admissible if and only if they can be

ob-m

tained by redistri):>ution of the total, resowoces l:i=l ri, given by the ini-tial endowments.

The second condition is the first order criterion which implies that each agent i finds himself endowed with a bundle in his budget set which is. a possible local maximum for ui. Hence the adjective "extended".

Given an economy, defined by (r,u) = {r

1, ••• ,r ,u1, ••• ,u ), we al~o

con-tm 1-1 m m

sider the set Ecr (r,u) c: lR x S of critical equilibrium states (x,p) for which the following holds:

m m

I

xi =

I

ri ,

i=1 i=l

1.2.3.

Then, as is easily seen, Eex(r,u) c: Ecr(r,u).

'

The system {1.2.3) consists of 1+m-l +m1 equations in m+m1+1-l unknowns p,,x,p). Roughly speaking· there is in general locally one solution (or none), .so the set Ecr(r,u) is in general a discrete one.

'l'o make more exact the notion of the vague statement "in general" we use methods of GLobal, JmaZysis, especially of transversaUty theorems. See for instaJ).ce Golubitsky and Guillemin [6], . Hirsch (9], or Dierker [2J.

introduced I in Chapter 3. For this set T we are able to prove the

. General :vesult I:

Tis dense and far each

u

£ T

there is a dense.set of initial endowments

r

far which

Ecr(r,u)

is

disCl'ete.

Since the set of smooth functions in its turn is dense in the set of

con-I

tinuous fuhctions it follows that the assumption of smoothness for utility functions is not too bold.

Moreover, the question whether T is also open, is in this context less in-terestinq,:since in the set of continuous functions no neighbourhood is filled up ~Y functions satisfying sane differentiability condition, whereas

o~nness

ot

T by its very nature may only be studied in the set of at least

twice differentiable functions. See Chapter 3.

Smale [15] restricts the consumption space of each agent to the closure of the positive orthant of

Il'

and, using only utility functions with non-zero gradients troves also for a certain set Y of utility tuples that i t has the desired pr~ties: there is a dense set of r for which Eex(r,u) is discrete.

Y itself is dense.

His methods are based upon the existence of the

nornred gradients

V E JaR, •

As will be shown in several examples there are many pairs (r,u) for which, according to our results, the set Ecr(r,u) is discrete, and contains states

{x,p) for whiCh some bundles xi are stationary points for the corresponding utility function ui.

Moreover, ~e shall show that our methods, applied to utility functions satisfying Smale's

no-critical-point hypothesis,

lead to the same set of economies as obtained by Smale. So~

our results generalize those of Smale.

A second topic in pure exchange econanies is the set of local

strict Pa:l'eto

optima:

Given sane w e

Jr/·,

being the total resources in the economy, and utility functions ui one considers the set 8(w,u) of admissible allocations x e Etm

for which the following holds:

There is an open neighbourhoOd

0

c JR!m of x such that for each admissible allocation y e

0,

y

F

x, there is at least one i such that ui(yi) < ui(xi}. In this context prices, and consequently budget sets, are not involved and the only. criterion for redistribution of w is non--decreasing of utility. So, given sane admissible allocation r, agents are willing to accept sane admissible allocation x if and only if ui(x

1) ~ u

1

(ri)J~.

1

• If we take into

account that such redistribution has to be realized by exchanging small amounts of goods, it seems reasonable to assume that no trade takes place from r e B.

1-1

It will be shown that x e 8 implies (x,p) e Eex (x,u) for sane p e S • Acting in the same spirit as in the definitions of equilibria we introduce sets Sex(w,u) and acr(w,u} as follows:

(1)

(2}

X € 6 (w,u) if and only if X is admissible and (x,p) E Eex(x,u} for

ex t-1

sanepeS

X € 6 (w,u) if and only if X is admissible and {x,p) € E

0r(x,u) for cr t-1 sanepes Bence 8 c 6ex c 6cr" The conditions 1. 3.1.

I

Y

xi • w , i=1 m

Dui (xi) = li p]i=1

constitute a system of 1 + tm equations in m + tm + R. - 1 unknowns (A ,x,p) , the solutions of which determine points x e acr· In general the set of solutions is parametrized by m- 1 variables.

If u

E T th~Pe

is a dense set of total reeouraes

w~

for

~hiah

e

0r(w,u)

is a

submanifoZdl of dimension

m-1.

r

For the definition of submanifold see Chapter 2.

It will be shown that eex is the intersection of ecr with a closed set. Hence the set e _ex and a fortiori the set 9 may"not be a submanifold. Moreover, i;E one restricts the consumption sets to the open or closed

posi-i !/, •

tiVe Orthant in JR 1 the preCiSe deSCrJ.ptiOn Of the StrUCture Of

a

Or e

', • , ex,

is a compli~ated affair. It turns out to be necessary that one invokes the theory of stratified manifolds with corners. See for instance Wan [23], Smale [17], or Schecter [11]. We do not enter into these problems. We only remark that, as for the structure of ecr our results generalize those of Smale [16] :in the same sense as described in the case of equilibria.

i

Omitting the no-critical-point hypothesis leads to some special effect, If all of the /!utility functions ui have som.e critical point, say zi' then

!/,-1 m

(z,p) € E~{z,u) for all p € S and z E ecr{w,u) for w

=

ti•l zi.

We may coruiider such a point z as

disastrous

for the economy. None of the agents has some specified

short run demand,

indicated by the direction of the gradients Dui (zi), being all zero.

It will be shown that for the set of pairs (r,u), (w,u) respectively, in-dicated in the general results I and II there are no admissible disastrous allocatioruk.

1.

5.

SU/I.vey o6 :the conten:t.s o6

:th.U.

rnonogJt.aph

In Chapteri 2 we give a summary of some standard. material of global analysis, contained in Sections 2.1 until 2.4. Section 2.5 is Cl.evoted to a proof of a well-known theorem on local Pareto optima. This proof is based upon methods, used by the author in his paper [5]. In Section 2,6 a s~ifold f is introduced, which plays an important role in the sequel.

Chapter 3 bontains the Cl.efinition of the set T and the proof that T is Cl.ense.

Fur~ermore,

a discussion about the openness of T is given in 3.6.

The proof of the general result I is the main topic of Chapter 4. Moreover, we introduce some subsets of T, which are open in the set of utilities u. chapter 5 is devoted to the proof of the general result II and some criteria for points in ecr on which one can decide whether they are points in

e

or not, are discussed.

Chcipter 6 contains some elements of trtade aUl'Ve8 • The introduction to this

topic is postponed to the first section of Chapter 6.

1. 6. Some.

examptu

Before we end this chapter we present some standard illustrative pictures, intended to give an impression of the relationship between equilibria and

'

Pareto optima.

We assume

t

= m

=

2 and use the so-called

Edge~Portth-bo:r:

in :rt2• See also Debreu [1], Hildenbrand-Kix:man [8], Dierker [2], Smale [16], and many other authors.

The horizontal axis represents quantities of commodity I and the vertical 2

quantities of commodity II. Let w e :R be the total resources in the eco-nomy. We measure quantities for consumer 1 from the origin and quantities for consumer 2 from w. Then each point in :rt2 represents an admissible location. For instance, the origin corresponds with the allocation

4 4

(O,w) e JR , whereas w represents the allocation (w,O) e :R • Points within the open rectangle through 0 and w and sides parallel to the axe.s corre-spond with allocations (x

1,x2) ,in the positive orthant. The closure of this rectangle is generally denoted as the Edgeworth-box. We do. not confine our~ selves to these allocations, so our set of allocations is the whole of JR2• Given utility functions u₁ and u

2, throuqh each point in JR 2

there pass two curves, the indiffertenae curvee for u₁ and u

2, indicated by a solid curve for u

1 and a dotted curve for u2• If we consider equilibria we indicate the initial bundles (r₁ ,r₂) by the initial bundle r₁•

Each allocation (x

1,x2) is indicated by the bundle x1 • . '(1) If u

1 and u2 both satisfy strong convexity assumptions we have pic-tures as in Fiqure 1 •

---+---tr

_{Figure 1.}

Edge-worth box for strongly conv~x utility functions ..

Obviously, each point where the indifference curves are tangent is a point; of

eex

and even a point of 8.

The set 8 is the curve, denoted by Edgeworth's cont:r>aat curve (o-o-o) t passihg throur;rh.all these tangent points.

Given r₁, one finds the equilibria (x,p) by selecting those points x 1 on 9 where the common tangent passes through r ₁, as is the case .for x₁

and y₁, but not for z₁•

(2) In Figure 2 the utility functions satisfy weak convexity conditions. The s$t of equilibria contains a one-dimensional set. Intuitively one sees that a slight perturbation of the utility functions breaks down the whole structure, in accordance with general result I.

I I

.

"

) I

Figure 2~

(3) In Figure 3 the utility functions do not satisfy convexity conditions •

• w

~I

Figure 3.

Edgeworth box for concave utility functions.

The indifference curves u

1 == 2 and u2

=

1 are tangent at x1 and x1 is a point of ecr' Clearly

x

₁ is not a point of 9 since each point in the

shaded region is better for both of the agents. Equally x

1 does not correspond with an equilibrium since u

1 and u2 both increase along the line x

1r1•

(4) In Figure 4 the function u

1 satisfies the convexity condition but u2 does not.

I I

Fiqure 4.

Edgeworth box for u

(5)

As on~ sees the situations in x

1 and y 1, both being points of diffetent from each other.

x1 is!not a point of 9, since all points in the shaded region

eex,

are are better for both of the agents. The concavity of u

2 overrules the con-vexity of u₁ at x₁•

y₁ is a point of 8, due to the fact that the convexity of u₁ is

stron~er than the concavity of u₂ at y₁•

In FiJure 5 we consider utility functions defined by:

where we assume a 1

+

a2 tl w. I I U2'--;#' ; u₁::.-J_..,₁-'~.__ I I I I I I I I l l l Figure S. •w

Edgeworth box for utility functions wtth satiation points.

6 = 9 ex is the closed segment between a₁ and w - a₂• If r l lies between th~ verticals through a₁ and w -a₂, the set E ex (r, u) consists of only one p¢int. Otherwise E (r,u) is empty.

1

ex

In cake w = a₁ + a

2 the set

e

ex consists of only one point, namely a1, but now

e

is the whole of :at2. Moreover 1 for every r

1 the point cr

(a₁ ,p) where p .1. a

1- r1 represents an equilibrium. Clearly the point a₁ is disastrous and we do not have the general situation as stated i~ general results I and II.

CHAPTER 2

PRELIMINARIES ON MANI'FOLVS

The first four sections of this chapter contain a summary of standard topics from global analysis, in a form adapted to the context they will be used in. For instance, all of,the manifolds considered here are submani-folds of some Euclidean space.

IJ) Section 2.5 we prove a well-known theorem on local Pareto optima. This theorem has been proved formerly by Smale [20], and Wan [22], but the proof given here is basically different from theirs. See also [5].

In Section 2.6 we introduce a subset

r

of :Rtm and show that

r

is a sub-manifold. This set

r

plays an important role in the sequel.

OUr main references for this chapter are [6] and [9] •. We do not, at least not before Section 2.5, present proofs of the statements we make. They can be found in [6] or [9].

· n 1 n .1 nT

Points in E. are given as a row x

=

(x , ••• ,x ) or as a column (x , ••• ,x ) ,

~ context making clear which form is chosen.

The topology in En will always be the metric topology, induced by the ·standard inner product, defined by

n X. y :=

I

h=l h h X y ' llxll := (x • x)! • n-1 n n

The unit sphereS c E. is the set of points x € E. satisfying llxll • 1. Furthermore, given a subset

u

of En its interior is denoted by

u

or int

u,

and its closure by

u.

2.

7.

V.£6 nte.n.t<.a.ble. mapp.in.g.6

and

.6ubma.n.i..6o£.cl6

~ n

Let 0 be an open subset of :R , and k a nonnegative integer.

2 .1.1. DEFINITION. ck (U, lRm)

is the set of az:t maps

f: u + E.m

being

k

times

diffePentiable UJith alZ. del'ivatives

up

to ord.eP

k

continuous on

o .

..

2.1.2. DEFINITION. c"' (U,JRm) :=

n

Ck(O,JRm)

is the sat of smooth maps

We shall extend the notion of differentiability up to order k to maps X + Y, where X and Y are submanifolds, and we shall define later on the sets

k ..

C,(X,Y) aqd C (X,Y).

2 m

Let f E: C i(U,.JR) and x

0 E:

u.

Then by Taylor's theorem there exists a unique

1 n m .

linear m~ JR + JR , denoted by Df (x

0) , a un1que symmetric bilinear form D2f (x

0) : ~n x JRn + JRm, and a map p: u +

lffl

such that for all x ;;: JRn suffi-ciently

c~ose

to x

0 the following holds:

i

where

1 n n 1 m m

With respect to the coordinates (x , ••• ,x ) on :R and (y , ••• ,y ) on JR the

del'ivative

Df(x₀) has the matrix

[

()fi

l

i=l , ••• ,m

- j

<xo> .

ax

J=l, •••

,n

Equally, tlb.e

second del'ivative

D2f (x

0) for f: JRn + lR is denoted by 2.1.3.

~[ 32f Ji=1, ...

,n i i j . . !

ax ax

J=l, ••• ,n • 1 m 1 k m h

CHAIN RULE,

Let

f ;;: C (U,.JR )~ q £ C (V,.JR ) and f(U) <= V c: JR •

Ten

the composition

q • f E c1 (U, lRk) and D (q o f) (x

0) .. Dq (f cx0> ) o Df (x0)

fOP, each x 0 e

o.

Before stating the implicit function theorem we need some notations.

n m 1 k

Let U c lR and V c lR be open and F E

c

(U xv, lR ) • Then, given x

0 E u, Yo € v, we define (1) F E cl (U,JRk) by F (x) := F(x,y 0) , Yo Yo (2) F E cl (V ,JRk) by F (y) := F(x 0,y) • xo xo

2.1.4. IMPLICIT FUNCTION THEOREM. Let U c lRn and V c lRm be open sets. Let

oo m

(x₀,y₀) E u x v and FE

c

(uxv, lR) be such that rank DFx

0<y0) =.m.

Then there is an open neighbourhood

u•

c

u

of x

0 and a map

cp E C00 (U' ,JRm) satisfying

(2) F(x,cp(x)) = F(x

0,y0) for aZZ x E u•

See [9]~ page 214.

As a consequence of the implicit function theorem (and vice versa) one has

, n oo , n

" 2.1.5. INVERSE FUNCTION THEOREM. Let U c lR be open and f E C (U,JR ) • Let

x₀ E

u and

rank Df(x₀) = n. Then there are open neighbourhoods

u

₁ of

and oo n • •

x

₀

~ u₂ of _{Yo := f(x0}> a map g E

c

_{<u 2,JR )} satisfy1-ng

(2) f(g(y)) = y for aZZ y E u

2, and g(f(x)) = x for aZZ x E u1•

See [9]~ page 214.

Now we come to the definition of a submanifold of dimension k as a subset of some Euclidean n-space where n ~ k. Intuitively a k-dimensional

submani-k

fold has locally the structure of an open subset of E .

2.1.6. DEFINITION. A subset y c lRn is a submanifoZd of JRn of dimension k

there e:dsts an open neighbOU'I'hood

u

of

y

0 and

a

q~ E:

c""

(U, lRn)

such

: +

thqt

q~(y

₀

) = 0~ rank Dql(x) = n

for all

x E: u~

an4

q~ (V)

=

Y

n u,

: { 1 n n

I

k+l n }

zvh~re V :• (z , ••• ,z ) E: E z = ... • z "' 0 •

Moreover~

the pair

(U,cp)

is called a submanifotd chart for

Y

at

y

0•

So the submanifold chart (U,cp) provides a local parametrization of Y by means of the first k coordinates of cp(x), in a neighbourhood of y

0• For example, each open subset of JRn is an n-dimensional submanifold of lRn, and a Q-dimensional submanifold of En is a discrete set.

It shouU

be

emphasised that the topology on a submanifotd of

lRn

is the

one~

induced by the topol-ogy on

En.

J:f not nec;:essary we do not specify in the future the Euclidean space in w~ich a sqbmanifold is contained, nor the dimension of the submanifold. The definition of submanifold is not always as manageable as desired in order to find out whether a subset 'of JRn is a submanifold or not. The followingitheorem will be useful in the sequel.

2.1.7.

~OREM.

A subset

Y c:

x(l

is a submanifotd of

JRn

of dimension kif

ani/. only if for every point Yo

E: Y

there e:dsts

an

open

neighbour-hood 0 of

y

0 and

a

ljl E:

c ..

(O,mn-k)

such that

rank Dljl(x)

=

n- k

for

aZZ

x E:

0

and

Tji+(O) = Y n

0.

See [6]~ page 9.

As one sees the submanifold Y is locally defined as the solution set of the equation 1jl (x)

=

0, constituting n - k equations in n unknowns. Due to the implicit function theorem the set Y is locally parametrized in a smooth way

by sane k""tuple fran the n coordinates in En. +

The kernel Dljl (y

0) (0) of Dljl(y0) has dimension k.

on

the other hand, if (U,cp) is a submanifold chart for Y at y

0 one has the k-dimensional subspace

-1 , n

Dq~ (()) (V) of JR • Fran the definitions i t follows

foljl.(y

_{0> <O>}

= Dql-

1co> <v> .

i

2.1. 8.

~INITION. Let

Y c ».n

be a k-dimeneiona't submanifo'td. Let

Yo

be a ,,

POfnt in

Y and (U,cp)

a submanifotd chart for

Y

at

Yo·

Then

Ty

0Y

is

t~ set of pairs (y

₀

,~y)~ zvhere

ByE: Dcp-l(O) (V).

The set

Ty

aarted the tangent spaae to

Y

at

y

0•

It

has

the stz>Uetu:l'e of a

k-dimensional veetor

~>paae~

isomorphie to

Dip -1 (0) (V).

Gecmetrically Ty Y is the set of all tangents to smooth curves on Y 1

pass-O .

ing through y₀, considered as a Euclidean space. In general, speaking of Ty

0Y, we only give the second ccmponent of the pair (y0,6y), i.e. oy e ~n. Now we introduce differentiability and derivatives for maps, defined on submanifolds.

2 .1. 9. DEFINITION.

Let

xi c: E. ni

be

ki

-dimensional submanifo

Zd.8

for

1 "' 1, 2.

Let

xi e xi and (Ui ,cp₁)

be submanifold charts for

xi

at

xi (i = 1,2) and f:

x

₁ -+ x₂

be a

map aueh

that

_{f <x1} l = x₂• Furthermore~

v

₁ and

v

₂

are defined

as

in

2.1.6.

Let

k

be a nonneqative inteqer.

(1) The

map

f: x

1 -+ x2

is said to be of

class

d"

at

x1

if the

map

-1 . . • k

cp

2 o f o cp 1 : V 1 + V 2 1.8

of

a wasB C

at

0 e V 1 •

(2) ck<x₁ ,x₂)

is the set of maps

f: x₁ -+ x₂

being of elaas

ck

at

every point

x₁ e x₁• 00 .. k (3) C (X 1

,x

2) :=

n

C (X1

,x

2) k=O

(4)

Given

f e

c

1

cx

1

,x

2

>~

x₁ e

x

₁ and (x_{1,ox1) e}

Tx

1

x

1

~ the map

Tx

1 f: Tx1 x1 + Tx2 x2

is defined

as

fo

l

'!A::Ms:

or shortly:

(5) Tx

1f<x1,6x1> := (f(x

1

),Df(x

1

)6x

1

>~ ~here

the definition of

DfCx₁> foll~s

from

(4).

In general, speaking of the derivative Tx/• we only give the second part, i.e. Df(x₁)ox₁, or shortly Df(x₁).

2.1 !10. LEMMA. The

Cartesian product

x₁ x x₂

of wo submanifolds is a

sub-manifold

and dim{X

I

'1'(1 ) (Xl X X2) = { (Xl ,x2,ox1 ,ox2)

j

oxl € 'l'x1X1' OX2 € 'l'x2X2} • f'l'x2

See [ 6]~ page 5.

. 0 1..- •

2.1.11. ~INI'l'ION.

Let

f €

c

(X,lR)

Ulru:re

X

-z.s some submanifoZd. Then the

support

Supp f

is the a losure of

f + ( lR \ { 0}) •

2.1.12. DEFINITION.

Let

{ua}ae:A

be a family of subsets of a submanifoZd

x~

suah that

UA ua •

x.

In other

U~ord.s~ {Ua}a£A

is a aover of x.

ae:

Then

{Ua}ae:A

is said

to

be loaa'Lly finite if for every

x € X

there

is

an

open neighbourhood

0 c:

x

of x

suah that

0 n ua =

rl

for aU

but a finite number of

a's

in

A.

2/1.13. TBEOREM (Existence of a partition of unity, subordinate to an open cqver of X.)

L~t

X

be a submanifoZd

and {ua}a€A

an open aover of

x~

i.e. all of

t~e ua

are open

and U u = X.

There is a family

{f } A

of smooth

! aEA B· a ae:

mdpe

x + lR

eatis fyi7lf!

I

q>

fa(x) e: [0,1]

for aU

a € A and

all

x e: x;

(2) Supp fa c: u<X

for aZZ

<X € A;

(3) {Supp f } A

is a loaaZly finite aover of x;

a <X€

(4)

L

fa(x)

=

1

for all

x €

x.

(Thie is a finite

sum, due to

aE:A

locally finiteness. Moreover, this shOUis that the interiors of

Supp fa

form

an

open, locally finite aover of

x.)

see [9]~ page 43.

2.1.14. COROLLARY.

Let

x

be a eubmanifoZd. Let

u

and

v

be open subsets of

X

U~ith

u

c:

v.

Then there is

an f € c~(X,lR)

suah that

ffx) • {

~

• f(x) •

1 i

sre

[6]~ page 17.

if

X € U ,

if

X

f

V ~

otheruise.

2 .1 .15. COROLLARY.

Let

c

be a a tosed subset of

:m.n.

Then there ewists a

81T/Ooth function

f: JRn + E

suah that

f (x) ~ 0

everyuJhere and

....

c

= f (0) •

See [6], page 17.

2.2.

Sand'~

Theo4em

Let a € JRn and b E: En and bk. > ak, k = 1, ... ,n. '!'hen C (a,b) is the closed

n k k k

block consisting of all points x E: E , satisfying a s x s b , k = 1, ••• ,n. '!'he volume of C(a,b) is

n~=l

(bk- ak).

2.2.1. DEFINITION.

A subset

s

c 'lil.n

is thin in

JR0

if fo'l' evecy

£ > 0

the'l'e

is a aountabte aove'l'i'Yifl of

s

with btoaks in

JR03

the sum of whose

volumes is less than

E.

2.2.2. DEFINITION.

Let

x

be an n-atmensionat submanifold.

and s

be a subset

of

x.

Then

s

is said to be thin in

X

if ther>e e:cists a aountab le

open aove'l'i'Yifl

u₁ ,u₂, •••

of

s and

ahart-ma:ps

!fl₁ ,!f1₂, ...

so that

!fl₁(u

1 n S)

is thin in

En,

fo'l' atl i.

As a consequence of 2.2.2 one has the following: if

s

is thin in

x,

then s

does not contain an open subset of x, so its complement is dense in X• See 'lso 2.2.7.

2.2.3. LEMMA.

Let

m < n and Y c Em

be a submanifold. Then

f(Y)

is thin in

R0,

for> aU

f E: c"" (Y, JRn).

See [6], pa_ge 31.

2.2.4. DEFINITION.

Let

X c: JRn

be a k-dimensionat submanifold.

andY c 'lil.m

be a p-dimensional submanifold.. Let

f €

c

1 (X,Y). (1) corank Df (x

0) := m.i.n (dim

x,

dim Y) - rank Df (x0) ,

for>

x0 E: X. (2) x

0

is said to be a a'l'itiaal point of

f

if

corank Df Cx0) >

o.

Othel'IJise

x₀

is aatled a

Nguta'l'

point of

f.

The set of a'l'itiaal

points of

f

is denoted by

C[f].

(3

Yo

€ Y

is said to be a ariticaZ vaZue off if

yo

€ f(C[f]).

0ther'1Pi8e

y

0

iB said

to be a roeguta:z. vaZue of

f.

2.2.5. REMARK.

(1)

Foro the definition of

Df(x

0) ..

Bee

2.1.9.

(2)

Sinae ..

a8

stated in

2.1.9 ..

one may inte:r!proet

Df(x

0)

aa a. Unearo

map

E.k -+-

~.. a ari tiaaZ point

x

0

is a point rvheroe

Df

haa

not

fuU

roank.

{3}

FX'om the thirod parot of

2.2.4

it fotZows that evexy pointy

€ Y

not being in

f(X}

iB a roeguta:z. value of

f.

2.2~6. THEOREM (Sa.rd).

Let

X and Y

be submanifolds

and f €

c""

(X,Y).

Then

the set of aritiaal values of

f

is thin in

Y.

Se' [6] .. page 34.

2,2.7. cOROLLARY (Brown).

Let

x andY

be submanifolds

and f " c""(x,Y).

Then

th~

set of roeguta:z. values of

f

is dense in

Y.

se~ [6] .. page 36.

2.3. The

Whitney

c""

Topotogy

Let

A~

be the vector space of polynomials in n variables of degree

s

k,

which have their constant term equal to zero.

A~

is isomorphic to some Euclidean space.

Given f € Ck(JR.n,E.), we define.the continuous map

as follows:

k

• j f(x) := (x,f(x), Dk f(x)) ,

where Dk

~

(x) is the polynomial of degree

s

k given by the Taylor expansion

I '

of f at

xi

_: up to order k after the first term. Since E.n _, x E. K Ak _n is iso-morphic to some Euclidean space, i t has the metric of that space, denoted

k

2.3.1. DEFINITION,

Let

f

e

C00(lRn,JR)

and

o

e

c0(JRn,JR+),

ltJhere

:R+

is the.

set ofpositive

rea~s.

Then

It can be shown that these sets, given n and k, form a base for a topology

co n k

on c (JR ,:R), called the Whitney c top<)logy, or shortly

wk.

So

0

e ~ i f

and only if for every f e

0

there is a o e c0(JRn,JR+) such that

~~(f~o}

c

0.

Then w~ c

wk

for

t

~

k •

..

• co co n

2.3.2. DEFINITION. w := U wk

is the Whz.tney

c

topology on

c

(JR ,:R). k=O

,

0

""

n ""

Hence a subset c C (JR ,JR) is open in the Whitney c topology, or shortly c""-open, if i t is open in the Whitney ck-topology, or shortly

~-open,

for some k <: 0.

Within the same terminology: F c C00(lRn,JR) is C00-dense, if and only if i t is ck-dense for each k <: 0.

2.3.3. LEMMA.

Let

{fm}meJN

be a sequence of functions in

C00(lRn,:R).

Then

fm + f

in the Whitney

ck

topol-ogy~ if

and

on~y if there is, a compact

subset

K c lRn and

an

m

0 e N

suah that

f (xl = f (xl

for all.

x (. K,

k k m

m <: m~ and j fm + j f

unifoi'ITily on

K.

See [6), page 43.

2.3.4. THEOREM. C00(lRn,JR)

is a Bcdre apace in the Whitney

c""

topology.

(So, the intei'Bection of a countab

~e co~ ~ction

of

c ..

-open

and

-dense

subsets of

c'"'

(lRn ,JR)

is

c""

-dense.)

See [6), page 44,

2.3.5. THEOREM.

Given

n,m e: E.

the set

c""(lRn,:R)m

is a Bcdre apace in

the

product

topology~

induced by the Whitney

c"" topo~y

on the factors

.., n

c

(JR ,:R) • See [6], page 47.

2. 4. T

l!.aYl4:veJ!..4a.U;ty

I I

I

2.4.1.

DEF~NITION.

Let

X, Y and Z

be submanifolde

and f

~ c

1(X,Y).

Let

z

c

Y and x

a point in

x.

Then

f

is said to

inte~sect

z

t~aneve~s~l~

at

x

(denoted by

f

m

z

at x) if

eithe~ (1) f(x)

I

z

o~

As an interpretation of this definition,2.4.1 we give the following. Since Z c Y, the tangent space to Z at z ~ Z is a linear subspace of the tangent space to Y at z. Now, given x €

x,

we have:

f

m

z at~

is equivalent to: If

f(x) € z~

then

f~ eve~

dy

€ Tf(x)Y the~e

a:rte

oz

~ Tf(x)z and cSx € Tx X

such that

!

2.4.2.

J

1c5y

=

dz + Df(x)c5x •

,INITION.

Let

x~ Y and

z be subma.nifolds

and

z

c Y.

L(Jt

f €

c

1(X,Y), and B

a subset of

z.

Then

f

is said to

inte~seat

z

t~anave~saZly

on

B iff~

every

x € X eithe~

(1) f(X)

I

B 0~

(2) f(x) € B and Tf(x)y = Tf(x)z + Df(x.) (TXX) •

2.4.3. DEFINITION.

Let

x ..

Y and Z

be submanifolde

and Z c Y.

Let

f € c1 (X, Y) •

Then

f

is said to intersect

z

trunsve~saZZy

(denoted

by

f At

zJ

if

f

m z

at

eve~y

point

x €

x.

2.4.4. TBEOREM.

Let

x ..

Y and

z

be submanifolde ..

whe~e

z

c Y.

Let

f € C01(X,Y) and f

m z.

Then

f+(Z) c

x

is a submanifold

and

dub f+(Z)

=

dim X- dim Y +dim Z •

! • + .. _

FurothemoN .. g1-ven

x .:: f {Z)

tr"" tangent epaae to

f+(Z)

consists of

al~ cSx € T X

satisfying

Of (x) cSx € Tf (x) Z.

Theorem 2.4.4 provides a powerful tool in order to construct submanifolds. We shall use it frequently in the sequel.

Now we give, in a very specialized setting, a theorem concerning transver-sality of a parametrized family of maps.

(.t+1)m .. Let R, ~ 1, m ~ 1 be integers and Ban open subset of E • Let

be a smooth map, and W a sublilanifold of E2.tm+m. We define for each b E B

the smooth map

by:

In this context we have:

2.4.5. THEOREM. If ~

.n

w the set {b E B

I

~b

.n

w} is dense in B.

See [6], page 53.

Before ending this introduction to some topics of global analysis we give a theorem in which second derivatives are involved.

2. 4. 6. DEFINITION. Let f E c"" (:IRn, E) • Then f is said to be a Morse function

if Df(x) = 0 implies D2f(x) is nonsinguZar, for aU x E En.

2.4.7. THEOREM (Morse). The set of Morse functions is open and dense in

C00 (:IRn ,E). See [6], page 63.

2.5.

Ch~ete4lzation

o6

local Pa4eto optima

This section is devoted to a well-known theorem on local Pareto optima. The proof is essentially the same as in the author's paper [5], where a slight-ly generalized form has been presented.

00 _{R, . R, id}

Let R, ~ 2, m ~ 2 and ui E C (JR ,E), i = 1, ••• ,m. Given w. E:R we 'Cons er

the set Aw consisting of the points x E E.tm satisfying

E{=1

xi = w. So Aw is the set of admissible allocations in a pure exchange economy with total resources w.

As pointed out in 1.31 a first order condition for a point z e ~ to be a local Pareto optimum is that there are nonnegative reals /... 1 and some

R.-1 J.

peS such that Dui(zi) = !.

1p 1 i = l1 ••• ,m. (See also Chapter 5.) We restrict ourselves here to the case that Dui(z

1) p 01 i = 1, ••• ,m.

R.m

2.5.1. OEE'INITION. Let z e :R 1 and Dui (zi) p 0, i = 1, ••• ,m.

'

I

I { •,tm m

N : = <v₁ , ••• 1 v ) e :It

I

L

z

m

i=1

2.5.2. DE]!'INI'l'ION.

Let

z

e

lR R.m , w

e

lR , R. z € Aw' p .

e

S R.-1 , Dui (zi) • AiP1

wiirh

Ai > 0, i = 1, •••

,m.

Then

Hz: Nz -+ :R

is defined

as fo~lA::Ms: I

This map Hz acts as a generalized second derivative we use in order to estabiishlwhether a point z, satisfying the first order condition for a

I

local str~ct Pareto optimum, is optimal or not. (In Chapter 5 we extend the notions Nz and Hz to the case that Dui (ail

=

0 for some i.)

We prove the following theorem, using properties of implicit functions:

2.5.3. TBEOREM. Let we JR1 and z e A

satisfy

Dui (zi) =

!.

1

pJ~.

1

~

where

R.-1 w

p ~

s

and

/..i > 0

for

a~l i.

(1)

If

Bz(v) < 0

for aU

v

e

Nz' v p 0~

then

z

ie a

loca~

strict

Pa:t'eto

optimum.

(2)

If

Bz (v) > 0

for some

v € Nz,

then

z

is not a local strict

Pa:peto

optimum.

Before giving the proof of 2.5.3 we need some properties of the first and second derivatives of implicit functions.

CONVENTIONS •

(2) Given • f E:

c

.. (lR , JR) , Jl. ~ E: JR , Jl. t.Je def't,ne •

t

h e

n

2

_f(~)

1 1 x (JI. -1) matrix

~

1 (JI. -1)

ax" ax 2

2...Lm

- .t

ax ax

(JI.-1) x (JI.-1) matri:c

a

2 f

x 1 matri:c _--.:--[ by:

ax ax

2.5.4. LEMMA. Let f € C 00

(lRR..,JRl and!;; E: JRJI. such that

a~

(!;;)

#-

o.

ax

Then there is a neighboU!'hood 0 of ~ := <~1, ••• ,~!1.-1) and a smooth

function g: 0 + lR satisfying: (1) g(~)

=

!;;JI. ; (2l f(x,g(x)) = f(t;;) for a7:l x € 0 • - af - - -1 - - -(3) Dg(x)

= -

----r

(x,g(x)) Df(x,g(x)) for aU x € 0 •

ax

- - -

a

2 _{- - T}

_a

2 f (4)

n

2f(x) + _ _ f_ (x) Dg(x) + Dg(x)

t-=-

(x) + ax

ax!l.

ax ax

+ 32 £

<x> ng<x>

T

ng<i>

+

~

<x>

n2

g<x>

=

o

ax1 ax1 ax!l. for aU

i

€ 0 •

PROOF. (1), (2) and (3) follow"directly from 2.1.4. Writing out (3) in components we get:

for i = 1, ••• , Jl.-1. Differentiating with respect to xj, j = 11 •••

~·JI.-1,

leads to

a£

a

2

+ g ""0

1

j i •

ax ax ax

Now we come to the proof of 2. 5. 3.

Since p €

F~-l,

at least one of the coordinates of pis not equal to zero. We assume p > 0. Other cases can be treated in a similar way. Since

au

₁ ~

--~ (z

1l = Ai p > 0, and according to the implicit function theorem there

a

xi

are SIIIOOth functiOnS ip i 1 defined On a neighbourhood

0

i Of

z

1 SUCh that

(1) i = 1, ••• ,m ;

(2) Ui (X:I.,!pi (Xi)) "'Ui (z₁) 1 i"' 1, ... ,m 1

(3) ui(xi)

~

ui(zi) is equivalent to

X~~

ipi(Xi) for all xi with xi € oi I

i

=

1, ••• ,m •

L~t 0~

:= oi X lit and

O'

:=

Oi

by X • • . X

0'

_m'

Let l E c""

(0'

n A , _w :R) be defined m

t

<x> :•

l:

lj) i

<xi>

i=l t -w

Since Oi is an open neighbourhOod hood of

z~,

the set

O'

n

Aw

is an We observe that

i

it, (z) - w ~

- !-1

of zi in :R and ll. an open neighbour-open neighbourhood of z in Aw.

0 •

If ui(xi)

~

u₁(zi) for all i, and x E

0'

n Aw, then

x~ ~

ipi(xi) for all i and, consl!!quently

so

m

w~ ~

l:

IPi(xi)

or l(x) $ 0 . i=l

For the ploof of part 1 of the theorem we assume Hz (v) < 0 for all v E: Nz,

v .;. 01 and claim that l has a local strict minimum 0 at z. If so, then z is

To prove our claim, we take oz

=

(oz₁, ••• ,&zm) where oz

1 is sufficiently m

,small and Ei=l oz₁

=

0.

£ (z + oz) - w t =

m

i!t c,i czi>

+Dq~i

czi>Eii +

i

o2~~'i

czi> (Eii,&zi> +

where and - R. +pi{ozi)} -w = + p (oz) - w .. R. lim p(oz) =

o

llozii+O ll&zll2

,In order to derive this Taylor expansion of£ in z we have used the deri-vatives of implicit functions as given in 2.5.4, together with the fact that p•p

=

1.

, m m - - p

Since Ei=l ozi = 0, i t follows Ei=l ozi

=

0 and v € Nz' So the function;,.

has the following properties:

(1) ..C (z) = 0 ;

m

(2) Df(z)oz

=

0 for all

oz,

satisfying

L

ozi

=

0; i=l

(3) the second derivative o2! (z) is definite positive as a quadratic fo:rm on the set T A • _{z w}

This proves our claim and the first part of the theorem. Turning to the second part, let v " Nz and Hz (v) > 0. Since

thete are ~reals a

1, ••• ,am such ,that

m and

L

ai

i=l

0 •

Let b := _{(b1, ••• ,bm)' where bi aip. We consider the curve x(t) in Aw}

through z:

Since uli is smooth, Dui (zi) Taylor's theorem: where l

um

t -2 p. (t) =

o ,

t+O ~ I since vi € 1 Dui(zi)J.. and p•p

So there ~s an E > 0 such that

I

~ip' ~i > 0, and p•p

=

1, we find using

2 2

+

l

t D u. (zi) (v. ,vi) + p, (t)

~ ~ ~

1. Hence it follows

for all t € (O,e), i = l, ••• ,m.

Obviously this implies that z is not a local strict Pareto optimum.

0

I

If all of lthe functions ui are strictly convex at zi' then each second derivative

o

2ui(zi) is definite negative on the kernel of the first deri-vative. In that case Hz(v) < 0 for all v # 0, v € Nz' so z € 9.

If each ui is strictly concave at zi' then Hz(v) > 0 for all v # 0, and

z ' 9.

If some oi' the ui are convex, others concave, then convexity may dominate and Hz is definite negative, or not. See also the examples in 1.6.

2.6. The .t.u.bma.n.i.&o!d

r

In 1.2 first order necessary conditions for an equilibrium (x,p) are formu-lated, one of them being Dui{xi) = Aip]~=

1

• Moreover, in 1.4 we mentioned that disastrous allocations are not welcome in our model. So we are looking

.

m

for allocations x, where DUi(xi)

=

Aip]i=l' and where at least on~ Ai is not equal to zero. This leads to the definition of the set

r

c m ,

con-R.m

sisting of those points v Em , for which there are reals A., not all

R.-1 • m . ~

zero, and some pES , satisfying vi= Aip]i=₁• See 2.6.2.

2.6.1. DEFINITION.

T:tie set

r

is parametrized by m + R. - 1 parameters and hence the following is not unexpected.

2. 6. 2. LEMMA.

r

is a submanifoZd of dimension

m + R.-1

and for eaah point

x

=

_(A1p, ••• ,AmP) E

r

the tangent spaae

Txr

tor

consists

of all

those vectors

(ox

1, ••• ,oxm} E mR.m

for whiah there are some

opE pL

and

reals

oA_{1 , •••} ,oAm

satisfYing

oxi = oAip +

Aiop]~=l'

PROOF. The proof is based upon 2.1.7.

Let z

=

(z₁, ••• ,z) be a point of

r.

At least one of the zi is not O, so we

R. m R.

assume zm .;. 0. Then ~ ;. 0 on a neighbourhood U of z in mR.m. We consider the map~:

U

+B(m-l)(R.-1 ) given by

where

i = 1, ••• ,m-1 •

Here, as in 2.5, the bar denotes that the last coordinate has been skipped. Obviously ~+(0)

=

r

o

U.

our claim is that rank D1)J(x) (m- 1) (R.-1) for all x E U. Since R.m - (m + R.-1)

=

{m -1) (R. -1), application of 2.1. 7 settles the proof.

R.m . Given ox

=

(ox

1, ••• , oxm) E m we find ~ (x) (ox} from

R. 1 - 'l":':" R. 1 -

-r-'!jli (Xi + OXi t X _m + OX ) _m

=

(X _mIll + OX ) (Xi + QX, ) - (Xi + OXi) (X + QX ) =

where

So

and

;= $i(x< ,xm) + oxm" x< + xR. ox, - x: 6x - ox:

x

+ p< (ox) ,

~ ~ m ~ ~ m ~ m ~

1

llo~~o

lloxll llpi(ox) II o •

D$ , (x. ,x ) (ox. , ox ) : ~ ~ m ~ m

Dljl(x) (ox) (D$

1(x1,x )(oxm 1,ox ), ••• ,D$ m m-1cx m-1,x )(ox m m-1,ox) m

29

With respect to cartesian coordinates, D$(x) is represented by the follow-ing matrix, also denoted by D$ (x) :

Dljl(x)

/).

_m

0

0

/).

0

m

0

where {).i ils the (R.- 1) x R. matrix:

0 -x. 1

~

0

i 1, •.•

,m,

and

0

is the (R.- 1) x R. matrix whose entries are all zero.

Since xR. :f 0 in U the matrix t:. has rank R. - 1 and consequently the

m m

(m-1)(R.-1) x mR. matrix Dljl(x) has rank (m-1)(R.-1) for all x E U. So r is a submanif:old of dimension m + R. - 1 •

Since Txr lis the kernel of Dljl(x), i t consists of all those ox satisfying

t:.m oxi = {).i oxm , i = 1, ••• ,m-1

p := R. p 0 0 0 -p 1 R.-1 -p

is an {!-1) x R. matrix, the kernel of which is spanned by p.

R.m m-1

Hence Txr consists of all those ox E JR for which there are reals

a.J.

₁ ~ ~= such that

Given ox E Txr there is a unique decomposition

Then

where

J.

On the other hand, given op E p and OA1 I • • • ,o>.ml the vector

ox= (6x11•••1oxm), with oxi

=

Ai op + 6;\PI satisfies ox E Txr' as is

easily seen.

D

The proof of 2. 6. 2 is based upon the assumption that at least one of the A i is nonzero.

I

m R.-1

The set {(>.

1p, ••• ,A.mp) A E JR 1 pES } is not a submanifold for R. ~ 2,

m 2: 2. For, assume that there is a smooth map F: 1lm-+ P.n, n ::; R.m, defined on a neighbourhood 0 of 0 E P.R.m such that F(A.

1p, ••• ,AmP) = 0 for all p E

st-

1 and A in a neighbourhood of 0 E

~

and rank DF(x} = n on

0

(see

2.1.7). Then, differentiating the coordinates Fj];=

1 with respect, to

Ai]~=l

we obtain

R.-1

In x E r the vector p E

s

is, up to its sign, uniquely determined, where~ R.-1

as ~ch p E S satisfies (A1p, ••• ,Amp) = 0 for (A1, ••• ,Am) = 0.

aFj t-1

Hence it follows that ---- (0) •p = 0 for all pES and all i,j. So

axi

rank DF(O) = 0 and consequently F is constant on a neighbourhood of 0, in contradiction with 2.6.2.

CHAPrER 3

THE SET

T

The equations

in the un~nowns (A,x,p) may have several: kinds of solution sets, depending

.I ., R. m

Ort the m-1tUple U E C (:m ,JR) • In general one expects that mR. equations in m + mR. + R. -, 1 unknowns have an m + R. :... 1 dimensional solution set, according to the implicit function theorem. In this chapter we define a subset T of

oo R. m

C (JR ,JR) having the property that the equations above establish an

m + R. - 1 dimensional set, for each u E T.

Section 3J1 contains the definition of T and a manageable second order criterionlfor tuples u to be an element ofT.

'

In Section 3.2 the number of unknowns is reduced in replacing the equation Dui(xi)

=

Aip by Dui(xi)

=

(Dui(xi)•p)p. This leads to a redefinition ofT. Section 3,3 is devoted to the proof that our results generalize those of Smale.

In Sectiot 3.4 we justify the use of utility functions instead of prefer-ence relations by proving that.two utility tuples, representing the same preferences, are either both in T or both not in T.

Section 3.5 contains the proof that. T is dense, and in Section 3.6 we discuss the question whether T is open or not.

!

3.1.

A

6~t de6~on

o6

T

co R. m

Let R. ~ ~' m ~ 2 and u E C (lR ,JR) • Then u induces a smooth map - m mR. R. R.m

i gU: JR X ::R X :R -+ :R X :R

I as followf!~:

3.1.1. DEFINITION

I n o er wor : T th ds ~s · th e se o m-tup es u or w c gu as t f 1 f hi h h 0 £ ~ ,.,tm+l as a ...

regular value.

Given

(~>..~x.~p)

E: Em x Etm x

E~,

the derivative ogu(>.,x,p) (o>.,ox,op)

equals

tm

Rearrangement of coordinates in E x E gives us the following matrix re-presentation of ngu(>.,x,p):

oq

_u p,,x,p) (o>.,ox,op) 2 D u₁ (x₁) 0 0

•

0 oT 1 0 -p T 01 D2um(xm) 01 oT 1 0

0

is the ~ x II. matrix with all entries zero.

0₁ is the II. x 1 matrix with all entries zero. I is the ~ x ~ identity matrix.

01 01 ->.1I

.

0~ o>.₁ T _{-1. I} -p _m 0 2p o>.m op

From the definition ofT it follows that u € T if and only if gu(l.,x,p) • 0

implies rank

oq

(>.,x,p) • tm + 1. u ~m

2 _{0 0} T 01 01 -!. I D u 1<x1) -p

...

1 (a1, ••• ,a.m,a> 0 D um(xm) 2 ₀₁ -p T -A. I m oT 1 oT 1 0 0 2p

So, the {mi'. + 1) x (m +mR. + .9.) matrix, representing ~u (!.,x,p) has not full

cank if and only if there is some {a

1, ••• ,a.m,B) '~ 0 such that

m

..

0 for all i, p•ai .. 0 for all i,

l

_A.1ai "' 2Sp • i=1

3 .1. 2. DEFINITION.

(S is

a compact

submanifoZd of dimension 1m+ 1-m-2.)

1-1 1 1

Given p E

s

we denote by

n :

lR + lR the orthogonal projection on the

. .l p

orthoplement p of p.

, "" 1 m

3.1.3. ~INITION. Let u e

c

(lR ,lR) • Then

I I G . € C«> (JRi',m X $ t JRR,m X lRR,m X JRR,) u is defined as foUoos: ( m 2 m m )

Gufx,p,a) := ITP Dui(xi)]i=1 ,D ui(xi)ai]i=1 '1!1 (Dui(xi) •p)ai •

'

Using the map Gu we give the following characterization of the set T.

3.1.4. ~OREM. u € T i f and onl;y if Gu (x,p,a) rf: 0 for aZ"l

1m (x(p,a) E lR x S.

J!.-1 PROOF. Let Gu(x,p,u)

Do.

Since rrp Dui(xi)

=

0 for all i, and p €

s

there are reals A.1 such that Dui(x1) = A.ip, i = l, ••• ,m. So gu(A.,x,p) = 0. srnce A..= Du.(xi) •p, it follows that the nonzero tuple (a

1, ••• ,a ,0)

1 1 m

belongs to the kernel of the transposed of the matrix

nq

(A.,x,p). Hence

u

u

t

T. On the other hand, we assume u

t

T. Then there is a point

~+ ~

(A,x,p) E g (0) where Dg (A.,x,p) is not of full rank. Hence there is a

u J!.mu

nonzero pair (a,8) € :R x :R satisfying:

It follows that

m

26p • p =

l

A.i ai • p

=

0 •

1=1

So 6

=

0 since p•p

=

1 and consequently a ~ 0. Since

we have

26p •

For;maps between Euclidean spaces the matrix representation provides a

0

1 straightforward tool in determination of the rank of the derivative,

where-as in general for maps between manifolds rank determination can be compli-cated (see definition 2.1.9). This was the reason why the map

gu

was intro-duced, making possible the straightforward proof of 3.1.4 •

.t-1

Now we return to the convention p € S •

"" J!. m

3.1.5. DEFINITION. Let u E C (JR ,JR) • Then the smooth map

is

defined as foll~e:

m g {A,X1p) := (Du₁(x.) -A.₁pJ.

1) •

u 1 1=

3.1.6. LEMMA.

co R. m

I