The questionnaire topology on some spaces of economic preferences

Journal

of Mathematical

Economics

22 (1993) 479494.

North-Holland

The questionnaire topology on some spaces of economic preferences James Redekop University

of Waterloo,

Submitted

February

Waterloo,

Ont., Canada

1990, accepted

September

1991

In this paper we deline and characterize two well-known topologies on spaces of preferences, namely the Kannai and closed-convergence topologies. We show that on spaces of economic preferences (e.g. continuous and monotonic preferences on RI+), both topologies coincide and have an especially simple characterization. Namely, they agree with the questionnaire topology, which takes as a base the set of questionnaire sets Q((xl,yl), ,(x,,y_)) = fly=, {p 1x$yi} where 6 denotes strict preference. The characterization permits the statement and proof of some theorems on the innocuousness of the assumption of continuity, given monotonicity or convexity; and it allows for dramatically simplified proofs of some denseness results resembling those of Kannai (1974), Mas-Colell (1974) and Araujo (1975).

1. Introduction In economic theory, the space P,(X) of continuous preferences on topological space X is usually endowed with the topology of closed convergence [Grodal (1974), Hildenbrand (1974)]. Since the definition of this topology is rather artificial, it is gratifying that in most situations in economics, this topology coincides with the one proposed by Kannai (1970), which has a natural and appealing characterization. In this paper we show that in some economic situations, these topologies coincide and are equal to a third topology, dubbed the questionnaire topology which is defined to take as a base the collection of sets of the form

Q((x,Y,),..., txn3 Yn))

=

i, i=l

where j denotes strict preference. sets. The rationale for the name ‘Which do you prefer, xi or y,?’ each question, then by posing Correspondence

Waterloo,

Ontario,

03044068/93/$06.00

J Math

E

to: James

Canada, c

Redekop, N2L 3Gl.

1993-Elsevier

(P 1 xitYi>9

(1.1)

The sets Q in (1.1) are called questionnaire is that if we ask an agent the n questions and the agent responds (truthfully) ‘xi’ to this questionnaire we will have elicited Department

of Economics,

Science Publishers

University

B.V. All rights reserved

of Waterloo,

480

J. Redekop, Questionnaire topology on spaces of economic preferences

precisely the information that the agent’s preference lies in the set Q in (1.1). Aside from the obvious simplicity of the characterization in (1.1) the utility of this topology lies in the ease with which one can prove that certain subsets of P,(X) are dense as subsets of other sets. One can prove such statements by induction on n in (1.1): and in such proofs the base case is usually trivial, while the induction step entails only a minor alteration to some preference whose existence and properties are guaranteed by the inductive hypothesis. Also, the restriction to continuous preferences is not essential to the questionnaire topology, as it is with the Kannai topology and the topology of closed convergence. Thus it can make sense to ask whether or not continuity is a regularity ocndition, given some other assumption, in the same way that concavifiability is a regularity condition given convexity of a continuous monotonic preference [see Kannai (1977)]. We include proofs of only the key results, namely Propositions 2.1 and 2.2. The remaining proofs consist of mostly straightforward but sometimes tedious (Lemma 4.2) reasoning, and therefore are collected in an appendix available from the author. 2. The Kannai and closed convergence topologies The Kannai topology on P,(X) is defined by Kannai minimal topology .a,, if it exists, such that the set S={(x,y,p)EXXXXP,(X)/X~y}

(1970)

to be the

(2.1)

is open in the product topology .F x .Y x -9, on X xXx P,(X). (Here fi denotes strict preference, and in what follows 3 denotes indifference according to p, for any preference p on X.) The question of existence of Y,, is not trivial; however, a common situation in which 3, does exist is given in Lemma 2.1 [Kannai (1970)]. Let (X,.f) be a regular, locally compact topological space. Then .& on P,(X) exists, and can take as a subbase the collection

~=:B(O,,O,)10,,0,~.~,~,~~cornpact},

(2.2)

B(O,,O,)={PEP,(X)l~,~~*}.

(2.3)

where

Thus under all unions (The result countability X G iw” that

the hypotheses of the lemma, Y, exists and is equal to the set of of finite intersections of sets in 58 [together with C#Jand PC(X)]. actually stated by Kannai [1970, p. 797, Remark (l)] has second in place of regularity; however, it is clear from the proof for regularity is the property required).

J. Redekop, Questionnaire topology on spaces of economicpreferences

481

The other ‘classical’ topology of interest in this section is the topology of closed convergence on P,(X) [see Hildenbrand (1974)], defined as follows: If p E P,(x) then

R(P)= 1(x,Y)E X x X 1XPY)

(2.4)

is a closed subset of Y = X x X, in the two-fold product of the given topology on X. We define a sequence of preferences {p,} to closed converge to p if the sequence of closed sets {R(pJ} closed converges to R(p); and in this case we write pn 2 p. In terms of elements of X, it is easy to show that pn 2 p if and only if: (a) if x@y, then there are open sets 0, and 0, such that XE O,,~E O,, and O&O, for all sufficiently large n; and (b) if xpy, then for all open 0, and 0, containing x and y respectively, there is a sequence of pairs (x,, y,) such that for all sufficiently large n we have

x, E ox, Yn E o,, and x,P.Y~.

(2.5)

It can happen that the sequence {R(p,)} has a closed that R does not correspond to any preference via (2.4). of defining a topology all that matters is that the truth statement ‘pn * p’ always be well-defined, for then we can

limit RcX x X but But for the purpose or falsehood of the let

&={UGP,(X)IifpEUandp,tp,thenp,EU for all sufficiently to get 4 E&, intersections.

P,(X) EA.,

and

closure

of xc

under

large n)

unions

and

(2.6) finite

Lemma 2.2. Let (X,3) be a locally compact topological space. Then the topology of closed convergence on P,(X) is at least as fine as the topology on P,(X) which takes .B in (2.2) as a sub-base. Equivalently, a~&.

Sufficient conditions for the reverse inclusion, i.e. &c&, are somewhat more specialized but they hold in every example to be studied in this paper. First, we need a consition on preferences which will allow & to be a separated topology when restricted to those preferences. For lack of better terminology, say that p is pair-wise locally nonsatiated (PLN) if: For all (x, y) such that xpy, and for every pair of open sets O,, 0, containing x and y respectively, there exist 5 E 0, and qeOy such that @q.

(2.7)

482

J. Redekop, Questionnaire topology on spaces of economic preferences

Thus a continuous p is PLN if and only if Rim, in the obvious notation. It is easy to see that if p is PLN, then (a) implies (b) in the characterization of closed convergence in (2.5). Lemma 2.3. countable

Let

(X,9)

be regular

and locally

base. Let P be any set of PLN

compact,

continuous

and let .a have a

preferences

on X. Then

,a,lP=.a,,P.

Now we are going to state some assumptions on preferences which will enable the convenient characterization promised for .a, and .$c when restricted to sets of preferences satisfying those assumptions. Loosely speaking, we need there to exist a binary relation on X which is analogous to > on R, with which preferences are consistent. Formally, let > be an asymmetric relation on X, and let 2 be the relation derived from > via

(2.8) Obviously, if PEP,(X) and x> y implies zdy, then xky suppose that there is a countable dense subset K GX, following conditions on >: (1) For any x EX and and an open set 0 (2) For any YE X and and an open set 0

every open containing x, every open containing y,

set U containing such that Ok& set U containing such that 720.

implies xpy. Now and consider the

x there

are 5 E U n K,

y there

are q E U n K,

Proposition 2.1 Let X have a countable dense subset K and let PC P,(X) be a set of continuous PLN preferences on X; and let > be a relation on X satisfying conditions (I) and (2) such that x>y implies xpy for all p E P. Then -eClP’&IP. Proposition

2.2.

Under the same hypotheses

as in Proposition

2.1, Yqlp is the

minimal topology on P for which

is open in the product topology on P.

topology

on X x X x P. That

is, &,r

is the Kannai

Notice that in Proposition 2.2 we do not write So,, =Yk ,p, because the proposition asserts both the existence of the Kannai topology on P and its

J. Redekop, Questionnaire topoiogy on spaces

ofeconomicpreferences

483

equality to .a,,,. The notation Y,,, would make sense only if Yk were already known to exist as defined, on some set of preferences containing P. 3. Examples In this section we present economic examples of topological spaces (X,4 and sets of preference P on X which satisfy the hypotheses of Propositions 2.1 and 2.2. We also present some examples which do not satisfy all those hypotheses. Let X,=R$ with the usual 3.1. for all i, and let P be any subset of

Example

xi>yi

topology.

Let x>y

denote

thenxiy},

P,,(X,)={p~P,(X,)lifx~y,

that

(3.1)

the weakly monotonic continuous preferences on Xi; and let K be the set of points in X, with rational coordinates. It is not hard to check that the triple (Xi, P, K) satisfies all the requirements of Proposition 2.1. Example

bound

3.2.

Let X, be the set of bounded

sequences

of real numbers,

with

b;

x, E R, and Ix,1 5 b for all rr}.

x’={{x”Inll}[ Endow

X2 with the product Q,={rE[WIItl=b

topology,

or r is rational

let x>y

denote

that x,>y,

for all i, let

and Irl
(3.2)

K:={xEXb/xiEQbfor

lzisn;xi=bfori>n},

(3.3)

K’Y={xEXbIxiEQbfor

lzisn;xi=-bfori>n},

(3.4)

and finally let K_=

K,=&C!+, It=1

(j K”,

K=K+

UK-.

(3.5)

n=l

If P is any subset of P,,(X2) then (X2, P,K) Example

= {PE P,(X,)

I if

x>y, then

satisfies all the assumptions

@Y>,

of Proposition

(3.6) 2.1 and 2.2.

3.3. Let J= [a, b] and let X,=D(J), the set of distribution functions on J; endow X, with the topology of weak convergence; define C-G to indicate first-order stochastic dominance, and let P be any subset of

J. Redekop, Questionnaire

484

topology

P,,(x,)={p~P,(x,)lifF>G

on spaces of economic preferences

then F@G},

(3.7)

the continuous preferences on X, which agree with first-order stochastic dominance. Let K be the set of rationally weighted convex combinations of point masses at rational points in J. Then one can show that (X,, P, K) satisties all the hypotheses of Propositions 2.1 and 2.2. Example 3.4. Let X4= RN, the set of all sequences of real numbers, and endow X, with the product topology as in Example 3.2. Again let x>y mean that xi>yi for all i. This time the hypotheses of section 2 fail, because for any open set 0 and sequence 5 it is not true that 025. Also, given the uniform topology on X, the hypotheses fail because of the lack of any countable dense subset. Example

3.5. Let J= [a, a) or (- m,b] or (- cx;, as), and proceed as in Example 3.3. The triple (X,, Pdc(X5), K) does not satisfy the hypotheses of Propositions 2.1 and 2.2, since any spheroid B,(F) either contains a sequence {F.lnzl} w h’K h IS undominated by any GED(J) (if J=[a, m)), or contains a sequence which does not entirely dominate any GE D(J) (if J =( - a, h]), or contains sequences of both types (if J =( - ‘x), GO)).

In Examples 3.1-3 the assertions are true for any subset P of preferences described therein but this is not necessarily true when consider proper subsets of the sets X involved. That is, it is possible (X, P, K) satisfies the hypotheses of section 2, but that (X’, P’, K’) does K’ is a countable dense subset of X’, and where X’cX,P’=PI,,, topology is -al*.. For instance, if in Example 3.1 we let X’={x&+

Icxi=l}Sx,

the we that not, the

(3.12)

and 122, then no open subset of X’ dominates any point in X’ according 2. Therefore conditions (1) and (2) do not hold in this environment.

to

4. Properties

In this section we state some of the basic topological properties of spaces of preferences such as in Examples 3.1-3 when they are endowed with the questionnaire topology. We begin by positing some additional structure on X and the preferences in the set PS P,(X) of interest which will allow Yo,, to be metrizable. These extra conditions are satisfied in Examples 3.2 and 3.3 but not in Example 3.1; however the metrizability result holds for those preferences, as shown by Kannai (1970, Theorem 2). Lemma 4.1.

Let X be any topological

space, let I c [w be an interval, let P be

J. Redekop, Questionnaire topology on spaces of economic preferences

a set of continuous preferences function such that: ifs>

t then g(s)pg(t)for

on X, and suppose that g:I+X

485

is a continuous

all PEP;

(4.1)

and for all x E X there exist s, t E I such that for all p E P, we haue g(s)pxpg(t).

(4.2)

Then for all p E P and x E X there is a unique t E I satisfying g(t)px; p E P the equation U,(x) = (t E u 1g(t)jx}

(4.3)

defines U,:X+R to be the unique continuous utility representation also satisfies the normalization U&g(t)) = t for all t E I. Proposition 4.1. Let X be compact, let on X satisfying conditions (I) and (2) I = [a, b] and let g:I-+X be a continuous g(s)>g(t) for s> t. Define U, as in (4.3).

d(p,d = max1u,(x) XSX

and for all

u,(x)l

for p which

p a set of contnuous PLN preferences for some partial order > on X; let function satisfying g(b)kX>g(a) and Then the metric (4.4)

metrizes .Yo, p,

These metrization results show that the questionnaire topology, on the given sets of preferences P, satisfies the strongest of the four separation axioms, i.e. normality. The restriction to the right set of preferences is crucial, for if we allow somewhat larger sets of preferences then even the weakest of the separation axioms may fail. For example, if p and q have the same indifference map, except that q has a thick indifference class corresponding to some range of preference according to p, then we will have pfq while xQy implies xjy. In this case any questionnaire containing q also contains p, so the singleton {p} is not a closed set in the questionnaire topology on the set of nondecreasing preferences (or any set containing both p and q). In the terminology of Royden (1988) [but not Kolmogorov and Fomin (1970), for example] $Q is not a Tychonoff topology in this case. Since _aOIP is a metric topology in the sets P of preferences studied in this paper, one cas ask whether or not P is a complete metric space. The answer appears to be ‘no’, in general. For example if U represents some p E P,,(R’+) = P and I/: rW’+ + [w is nondecreasing but not monotonic, then the

486

J. Redekop,

topologyon spaces

Questionnaire

of economic

preferences

preference represented by I/ + (U - V)/ n will be in P for all n. This sequence will be Cauchy according to Kannai’s metric (1970, eq. 3.1), but cannot converge to any preference in P. Similar sequences can be constructed in the settings of Examples 3.2 and 3.3; thus in general our sets of preference P are not complete metric spaces in the topology $FQelp.They are therefore not compact either, and as the next result shows they are nor even locally compact, in general. Let X = IF?: and let P = P,,(X), as in Example 3.1. Let Q be any Lemma 4.2. nonempty questionnaire subset of’ P. Then jbr any x E X, there exists an open set 0 containing x such that for all PEP there exists a preference qE Q such that plo=qlo; or more compactly

Plo=Qlo.

(4.6)

Thus we can say that every open subset of P is locally universal. As in the previous discussion, we can construct Cauchy sequences in PI, which have no limits in P/o, and we can pick such a sequence {p,> so that pnlx_o is constant. Thus such a {p,} can be chosen to be a Cauchy sequence in Q with no limit in (2, so this topology is in general not locally compact, since any compact subset of a metric space must be complete as a metric subspace. We now consider the non-hereditary properties of connectedness and local connectedness for the special cases of the following subsets of P, = P,,(X), which was defined in Example 3.1, with X = rW$: P,=P,,(X)={p~P,,(X)~ifx~yandx#y,thenx~y},

(4.7)

P, = P,,,(X)

(4.8)

P, = P,,,(X)

= {p E P,,(X) = {p E P,,(X)

1Vx E X, {y 1ypx} is convex}, 1if x@y, x # y, 0 < t < 1, then

Itx + (1 - t)ylBx} G P,,(x), P,=P,

(4.9) (4.10)

n P,.

That is, we are going tions of monotonicity 01-. j14 we have: -

to consider all possible combinations of the assump(weak or strict) and convexity (weak or strict). For

Proposition (4.3+ j). Pj is a connected set, and any questionnaire is connected. Thus P, - P, are connected and locally connected.

subset of Pj

5. Some denseness results The questionnaire

topology

enables

the statement

and proof

of denseness

J. Redekop, Questionnaire topology on spaces of economic preferences

487

results which do not assume continuity of preferences, but rather assert that continuity itself is an innocuous assumption, given some other assumption. Such results do not make sense in either classical topology because the Kannai topology does not exist on any set of preferences which includes at least one discontinuous preference, and the closed convergence topology prohibits convergence to any discontinuous preference. So in our next result we use the questionnaire topology on P(X) as a topology of interest in its own right, rather than justifying its existence by showing its equivalence to some classical topology. Similarly to the last section, we concentrate on the special case Xs R’ with its usual topology. We let P, = P,(X)

= {p E P(X)

1vx, y E x, x > y*xiy},

(5.1)

P,=P,(X)={pEP(X)IVx,yEX,[xZyandx#y]*xfiy}, P,=

P(X) jVx~X,{yl

P,,(x)={p~

P, = PC2(X)

= {p E P(X)

ypx} is convex},

(5.3)

IVx E X, {y 1yix} is convex},

(5.4)

and assume

in (5.3) and (5.4) that X is convex.

Proposition

5

( j - 4).

(5.2)

Pj n P,(X)

For 5 5 jsS

we have:

is dense in Pj.

Thus the assumption of continuity is innocuous given either form of monotonicity (5.1H5.2) or either form of convexity (5.3H5.4). Our next results are similar to those in Kannai (1974) Mas-Cole11 (1974) and Araujo (1985). These results are intended to illustrate clearly the utility of adopting our characterization of the usual topology on certain spaces of preferences as the questionnaire topology on these spaces. Therefore we include complete proofs of these results in the body of the paper, since the aforementioned utility arises from a great reduction in their length.

Proposition

5.5.

Recall

P, = (weakly monotonic, weakly convex continuous preferences on X = ET+}.

(5.5)

P, = {pi P, I p has a utility representation which is strictly concave and C” >.

(5.6)

Let

Then

P,

is dense relative to P,.

488

Proof:

J. Redekop,

Questionnaire

topology on spaces of

We have to show that any nonempty

4

economic

questionnaire

preferences

subset

Q(xi,Yi)cP,

(5.7)

contains an element of P,, which we prove by induction on n. It permissible to begin the inductive proof with the base case n=O, which trivial since it amounts to the assertion that P, # 0. For n > 1 we let

is is

(5.8) show

that the set in (5.7) is 9GP* satisfies -uiQyi for 1 5 assume that yipy, for 15 i 5 n the worst of the xi, according

not empty, and assume inductively that some is n - 1. Without loss of generality we may - 1, and hence x$y, for 15 i 5 n. Let x* denote to p, so that

Yn41+Px*i=~(x*). Now B(x,) is a closed convex set, so there is c E R’ such that for all ZE B(x,); and necessarily CE 5X:. Now let @( .) denote normal cumulative distribution function, and let

(5.9) cy, < cx* < cz the standard

F,(~‘)=i[l-@(t(s-k))]ds

(5.10)

0

for t > 0 and y > 0. Clearly F, is a strictly increasing, strictly concave valued function of y, and as t-+ a, F, approaches the function F,(y) = y,

0<

1’ <

C” real

k, (5.11)

= 1,

g>k.

q, and Now let U,(z) be a strictly concave C” utility function representing for t, T >O consider the preference q*(T, t) whose strictly concave C” utility is

u*(z) = U,(z) + TF,(cz). The remainder of the proof, which we omit, consists of showing sufficiently large T and t we will have q*( 7: t) E Q as required. 0

(5.12) that

for

For many purposes the preferences in P, are not the most convenient ones. For example, in general equilibrium theory it is nice to know that demand functions are smooth. If we insist that utility functions be C” on all

489

J. Redekop, Questionnaire topology on spaces of economic preferences

of RI+, then we must allow for the possibility that corner solutions will exist for some admissible preferences and strictly positive price vectors. In general this means that demand functions will not be smooth. Therefore let rcE rW$+ and w > 0 denote prices and income, respectively, and consider P,, = {p E P, 1p yields demand

functions

fi(rr, w) which are I@++

valued, C”, and satisfy (A) limminn,_+Ollf(rt, w)ll= co}.

(5.13)

There are about the nicest preferences a general equilibrium theorist could ask for. For p E P,, it is sufficient that p be represented by a utility function U, which is C” on FL!‘+ +, strictly concave and monotonic, and which satisfies 0

00 vi(x)

+

o

*

for all non-negative values of xj, conditions, in the following sense: Proposition Proof.

P,,

5.6.

(5.14)

xi+m j#i.

But

these

are

merely

regularity

is dense relative to P,.

Again, we have to show that if

PEQ=i)Q(xi>Yi)cPZj

(5.15)

1

then there is some q E Q n P,,. For n=O we can let q have the utility xx,!” and given the proposition for n- 1, we can add some multiple of F,(cz) to the utility for q E ()l- ’ Q(xi, y,), as in the proof of Proposition 0 5.5. We conclude this section by proving that finiteness of the set of exchange equilibria is a regularity condition on preference profiles, in the same sense that positive valued C” demand functions are a regularity assumption on preferences, as shown in Proposition 5.6. The result here resembles the main theorem of Araujo (1985), who showed that within an appropriate set of profiles of smooth utility functions, the set of profiles with infinitely many equilibria is contained in a closed set of measure zero. The concept of ‘measure zero’ applies to separable Banach spaces of real valued functions, and therefore not directly to the spaces of preferences analyzed here. In fact, since the class of measures assumed in the result necessarily gives zero measure to a dense set of utilities (e.g. the ones which are somewhere not twice differentiable), it is difficult to attach an interpretation of probabilistic genericity to the theorem. But the theorem does imply that the nice

490

J. Redekop,

Questionnaire

topology

on spaces of

economicpreferences

profiles of preferences are dense, because these profiles have utility profiles which are dense in the set of smooth utility profiles, which itself represents a dense set of preference protiles. As for the complement of this set of profiles, it is natural to conjecture that the set of pathological economies is also dense, because infinite numbers of equilibria can occur in arbitrarily small regions of the allocation space, and basic open sets in our topology determine preferences only on finite sets. Thus our next result is an approximation theorem, not a genericity theorem. See Mas-Cole11 (1985) for a discussion of the ‘generic point of view’. In what follows we presume an exchange economy of N agents characterized by endowment vectors oi E rW$ and preferences pin P,. Given these endowments and preferences, an exchange equilibrium consisting of allocations (x1.. . xN) E rW:” and a price vector n E S: = {n E @+ 1c$ nj= l} occurs if each agent optimizes given rt and markets clear, i.e. (1) Vi E N, yiixi implies (2) C~~i=C~Wi.

rcy > rrxi = rcai, and

We will write (p, o) to denote the profile (pr.. . pN, ml.. co,), which will be assumed to be in P: x Rtf, and we will denote allocation-price pairs by (X,X) E iwy x S’. The pair (X,X) is called a regular equilibrium given (p,o) if there is a neighbourhood U containing w and a smooth function g: U +rW:” x S’+ such that g(w)=(x, rr) and such that g(o’)=(x(o’),rc(o’)) is an equilibrium given (p, 0’) for all o’ E U. Proposition

5.12.

Let ORE lR’++ for all i. Then

R(w) = {p E Py 1there are onlyfinitely many equilibria given (p, CO), all of which are regular; every equilibrium price vector is strictly positive; in each equilibrium every agent’s allocation is strictly positive; and every agent’s (5.16) prejerence is in P,,} is dense relative to Pt (in the product Proof

topology on PT).

We have to show that if (5.17)

are N nonempty questionnaire subsets of P,, then there is a profile qER(o) such that qj E Qj for 15 j 5 N. Proposition 5.6 already shows that Pz* is dense in the product topology, so we may begin by letting

J. Redekop,

Questionnaire

topology

491

on spaces of economic preferences

(pl...p,)EP~*nCQ,x...xQ,l. We will now perturb p1 only the new preference in Qi. monotonic and concave and (5.14); and let a** denote rW:+=Y++ and all E> 0 the

(5.18)

so as to obtain a profile in R(o), while keeping Let U 1.. . U,, representing pl.. . pN, be strictly C” on lF!‘++, and satisfy the Inada conditions the set of such utilities. Then for all UE Sr+ n utility given by

V(z) = U i(z) - Eexp ( - az)

(5.19)

is also in a*.+; and the derived preference E. For 2 5 i 5 N the excess demands

p,,

is in Q1 for all sufficiently

J(n) = argmax {U,(z) 1x2 5 TCCI+> - oi are C” on S’+ + , and so therefore

small

(5.20)

is (5.21)

x,(~)=w1-&); 2

moreover (p,,,

pz..

rrxl(rc) =rcwl. Hence . p,,,, CO) if and only if

rc is

an

equilibrium

price

vector

for

x1(n) E R: f, FU,(x,(n))+~aexp( So let M denote

-uz)=)x,

(5.22)

for some J,>O.

the set of pairs (rr,A) such that

x1(n) E @+ + > kc-

VU,(X,(7c))E lw+ +;

define the normalization

(5.23)

function

v(z) = z/c zj; and consider where a=v(ln-

the

C”

(5.24) function

G: M+S’+ + x R + + given

by

G(rc, 3,)=(a, E)

VU,(x,(n))),

E=[i~rc-BU,(x,(n))],/[a,cxP(--Xl(n))].

(5.25)

[In (5.25) the third and fourth ones denote first coordinates and the others refer to individual 11. By construction, G(rc,A) =(a, E) if and only if n is an equilibrium price vector for the economy (p,,, pz.. . pN, w). By Sard’s Theorem the set of regular values of G is dense, so it includes pairs (a,~) for which E is small enought to have p,, E Q1; let (a,~) be such a pair. The remainder of the proof consists of standard arguments [see Debreu (1970, pp. 589-590)]: Each

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J. Redekop, Questionnaire topology on spaces of economic preferences

pair in G ‘(a, C) determines a locally unique equilibrium price vector, hence every point in S+ is contained in an open set with either zero equilibria or one equilibrium; a finite subcovering determines a finite set of equilibria; - these equilibria are smooth functions of (a,~) near (a,~) by the Inverse Function Theorem; and since the functions whose zeros determine equilibrium are smooth in CO,the equilibria themselves are locally smooth in o by the Implicit Function Theorem. This concludes the proof. 0

6. Conclusion The usual topologies on spaces of continuous preferences, called the ‘classical’ topologies in section 2, are the Kannai and closed-convergence topologies. These often coincide on spaces of economic preferences. And subject to some further restrictions on preferences, which are often satisfied in economic environments, these classical topologies have the characterization that they both equal the questionnaire topology, which takes as a base the family of questionnaire sets

Q((xl, Y 1). . . (Xn,Yn)) =

4{P 1X$YiI.

(6.1)

Thus a sequence {p,} converges to p in this topology if for all x,y such that xfiy, we have x@,y for all sufficiently large n. To appreciate the simplicity of this characterization, the reader should compare this simple condition to (a) and (b) in (section 2), which define closed convergence, or perhaps write out the convergence notion implicit in the Kannai topology as characterized in (2.2) and (2.3). The great advantage of the characterization is in the simplicity of proofs of denseness of certain sets. These theorems can often be proved by an induction argument in which the base case is trivial and in which the induction step can be accomplished by making minor modifications to some preference. Lastly, the topology admits a natural extension to sets which include discontinuous preferences; this is not true of the classical topologies. Hence, using the questionnaire topology one can state and prove theorems on the innocuousness of the assumption of continuity, given other assumptions.

Appendix Proof of Proposition

2.1

It is easy to show that &,c&,, preferences. For the reverse inclusion

for any set of P of continuous we begin by noting that for sequences

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of PLN preferences, (a) implies (b) in the definition of closed convergence (2.5). If E is not open in Yolp, then there is PEE such that no questionnaire contains p and stays in E. We can write R(j) n (K x K) = {(X,>Y,),. . . ,(X,,Y,). as a countable

Qn=

set of ordered

..)

(A.11

pairs; then let

f)Q(xi,Yi);

64.2)

i=l

and linally pick p, E Q, set containing p. To show If x$y then U,jU,, for on K and > we can find y, such that

E for all n, because for all n, Q, is a questionnaire that E is not open in 9clp, we show that pn t p.

some open sets U, and U,. So by our assumptions i: and ye, and open sets 0, and 0, containing x and

v]EK~U,,

(EK~U,,

(A.3)

and 0,259

qto,.

(A.4)

We have (5, q) =(x,, y,J for some k in (A.l) so for n 2 k we have O,@,O,. Thus (a) in (2.5) is satisfied and hence (b), from the assumption of PLN preferences. This concludes the proof.

Proqf of Proposition

2.2

We have to show that under

the hypothesis

of the proposition,

the set

~~~~={~~,Y,P~~~~~~~~~~Y}

(A.51

is open in the product topology .a x 9 x Yo,,+ and that if S is also open in 4 x 4 xY*I,, then Yolp~S*lp. These steps will establish existence of the Kannai topology on P and its equality with Yelp simultaneously. If xjy then the same 0X.5,0,, and v of the last paragraph in the proof of Proposition 2.1 will suffice to yield (x, Y,P)EO,

x 0,

x

Q(Lr)n PsS(P);

so S(P) is therefore a union of the open products YxsxYo,p. For minimality,

suppose

(x,Y,P)EO,XO,.XOpCS(P)

(A.61 in (A.6) and hence open in

that S(P) is open in .a x 9 x Y*Ip. If xjy, then (A.7)

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for some open product with O,EY*[~. In this case, O,b,O,, so PE 0,~ Q(x, y). Thus Q(x, y) is a union of sets Opt Y*lP and hence open in X*lP; and the basic open sets Q((xl, y,) . . .(x,, y,)),being finite intersections of Q(x, y)‘s, are also open in .a* IF. So .Fo, p _ c Y* I,,, establishing minimality. References Araujo, A., 1985, Regular economies and sets of measure zero in Banach spaces, Journal of Mathematical Economics 14, 61. Debreu, G., 1970, Economies with a finite set of equilibria, Econometrica 38, 387. Grodal, B., 1974, A note on the space of preference relations, Journal of Mathematical Economics 1, 279. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, NJ). Kannai, Y., 1970, Continuity properties of the core of a market, Econometrica 38, 791. Kannai, Y., 1974, Approximation of convex preferences, Journal of Mathematical Economics I, 101. Kannai, Y., 1977, Concavifiability and construction of concave utility functions, Journal of Mathematical Economics 4, 1. Kolmogorov, A. and S. Fomin, 1970, Introductory real analysis (translated by R. Silverman) (Dover, New York). Mas-Colell, A., 1974, Continuous and smooth Consumers: Approximation theorems, Journal of Economic Theory 8, 305. Mas-Colell, A., 1985, The theory of general economic equilibrium: A differentiable approach (Cambridge University Press, Cambridge). Royden, H.L., 1988, Real analysis, 3rd ed (Macmillan, New York).