excess demand functions, equilibrium prices and …
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EXCESS DEMAND FUNCTIONS, EQUILIBRIUM PRICES
AND EXISTENCE OF EQUILIBRIUM
by
Kam-Chau Wong
Discussion Paper No. 283, October 1995
Center for Economic Research Department of Economics University of Minnesota Minneapolis, MN 55455
Excess Demand Functions, Equilibrium Prices, and Existence of Equilibrium
Kam-Chau Wong t
Department of Economics Chinese University of Hong Kong
Shatin, Hong Kong E-mail: [email protected]
Phone: (852) 2609-7053 Fax: (852) 2603-5805
September 6, 1995
Abstract: For continuous excess demand functions, the existing literature
(e.g. Sonnenschein [1972, 1973], Mantel [1974], Debreu [1974], Mas-Colell
[1977], etc.) achieves a complete characterization only when the functions
are defined on special subsets of positive prices. In this paper, we allow the
functions to be defined on a larger class of price sets, (allowing, for exam
ple the closed unit simplex, including its boundary). Besides characterizing
excess demands for a larger class of economies, it is also a useful tool for
proving other results. It allows us to characterize the equilibrium price set
for a larger class of economies. It also permits extending Uzawa's observa
tion [1962]' by showing that Brouwer's Fixed-Point Theorem is implied by
the Arrow-Debreu Equilibrium Existence Theorem ([1954], Thm. I.).
JEL Classification: D51, Dl1, D50
Keywords: Excess demand, Exchange economy, Equilibrium, Fixed-point
t This paper was written while I was visiting the Department of Economics of
the University of Minnesota in the summer of 1995; the hospitality of the department
is gratefully acknowledged. I also wish to thank Professor M. Richter for his helpful
comments and suggestions on earlier versions of this paper.
1
1. Introduction
In the existing literature of characterizing continuous excess demand
functions (Sonnenschein [1972, 1973], Mantel [1974], Debreu [1974], Mas
Colell [1977], etc.), a complete characterization is achieved only for the
special case where the functions are defined on compact subsets of posi
tive prices.! By weakening the preference requirement of monotonicity to
insatiability, we are able to characterizing a larger class of excess demand
functions, even those defined at boundary points, where some prices are
zero. In particular, Theorem 1 below gives a complete characterization for
any continuous excess demand function defined on any compact subset of
nonnegative nonzero prices, including, e.g. the closed unit simplex. This
extension has applications in several areas. We give two examples.
The first application is a characterization of equilibrium price sets. In
the series of papers by Sonnenschein [1972, 1973] and Mas-Colell [1977], it
has been established that any non-empty compact set in the relative interior
of the closed unit simplex is indeed the set of equilibrium prices for some
exchange economy (with monotone preferences). By using Theorem 1, we
drop the "interior" res.triction, showing that any non-empty compact set
in the closed simplex is the set of equilibrium prices for some exchange
economy (with insatiable preferences).
The second application concerns the connection between two very fun
damental theorems - Brouwer's Fixed-Point Theorem and the Arrow
Debreu Equilibrium Existence Theorem ([1954], Thm. I). Uzawa [1962]
proved that Brouwer's Theorem is equivalent2 to the equilibrium existence
1 McFadden et aI. [1974] considered not-necessarily-continuous excess demand functions on the set of positive prices.
2 In a formal sense, all mathematical theorems are equivalent, since they can be derived from the same set of axioms for mathematics. But in the present context, we use "equivalent" in the usual informal sense, meaning that there is a fairly direct proof
2
theorem for excess demand functions defined on the closed unit simplex.
By Theorem 1, it follows that the latter is implied3 by the Arrow-Debreu
Equilibrium Existence Theorem for economies. Consequently, Brouwer's
Theorem is implied by the Arrow-Debreu Theorem. This formalizes De
breu's observation4 that "the proof of existence of a competitive equilibrium
requires mathematical tools of the same power as a fixed point theorem"
(Debreu [1982], p. 720).
The proof of Theorem 1 also has some interest of its own. It uses
modifications of the methods of both McFadden et al. [1974] (for the de
composition ofthe given function into individual excess demand functions)5
and Debreu [1974] (for the construction of preferences for the individual ex
cess demand functions). It sheds some light on the complementary nature
of the basic ideas of these two different methods.
2. Statement of Results
Let6 R+ = {p E JRI : P ~ OJ, fl = {p E JR~ : E~=l Pi = I}, o fl= {p E fl : P » O}, S = {p E JRI : p » 0 & IIpll = I}, and Sf = {p E S :
Pi ~ f for every i} for every real number f > O.
We will consider any exchange economy E = {(b,wi,JR~)~=d such
of one from the other.
3 See Footnote 3 above.
4 Cf. also Sonnenschein [1973], p. 352 and Sonnenschein [1982], p. 691.
5 I.e. we use oblique projections, as given in McFadden et al. [1974], rather than the orthogonal projection of Debreu [1974]. The reason is that the former method allows us to handle situations in which a continuous excess function is defined at boundary points.
6 For any x, Y E R,I, we write x ~ Y to mean Xi ~ Yi for every i = 1,···, /, and write x> Y to mean Xi > Yi for every i = 1"", I. Also, ". " denotes the Euclidean norm.
3
that
for all i, Wi » 0 E JR~ and ti is a continuous, insa-tiable, strictly convex preference relation on the consump- (1) tion space JR~.
Let G ~ JR~ \ {O}. A function ( : G -+ JRl is an excess demand function if
( is continuous and such that:
WL) p. «(p) = 0 for every pEG,
H) «(p) = «(..\p) for every pEG and every ..\ > 0 with ..\p E G,
BB) «( G) + q » 0 for some q » 0 E JR1•
It is well-known that for any function ( : G -+ JR1, if there exists an
exchange economy E = {(b,wi,JR~)~=l} satisfying (1) and such that
I
{«(p)} = L: {z E JR~: [p. z ~ p. Wi] i=l (2)
I
& ('v'y E JR~)[(p. y ~ p. Wi) =* (z by)]} - L: {wd i=l
for all pEG, then ( is continuous and satisfies (WL), (H) and (BB), i.e.
( is an excess demand function. A partial converse is provided by the
Sonnenschein-Mantel-Debreu Theorem (Debreu [1974]), which shows that
for any f > 0, any excess demand function ( : Sf -+ JR1, there exists an
exchange economy C = {(b,Wi, JR~)~=d satisfying (1) and
every ti is monotone, (3)
and such that (2) holds for all p E Sf.7 By dropping (3), we obtain the
following result, which considers excess demand functions defined on any
compact subset of JR~ \{O} (including e.g. the whole closure of S, ~, etc.),
rather than just Sf.
7 Different variants of the theorem were obtained by McFadden et a!. [1974]. MasColell [1977]. Mantel [1979] and others.
4
Theorem 1 (Characterization of Excess Demand Functions). Let
K be a compact subset of IR~ \{O}. Then for any ( : K -+ IRI to be an
excess demand function, it is necessary and sufficient that there exists an
exchange economy [ = {(ti, Wi, IR~)~=d satisfying (1) and such that (2)
holds for all p E K.8
Proof. The sufficiency is clear. And the necessity follows immediately
from Lemma 1 and Proposition 1 in Section 3. Q.E.D.
We will now discuss two applications of Theorem 1. The first one con
cerns a characterization of equilibrium price sets. For any exchange econ
omy [= {(b,Wi, lR~H=d, a competitive equilibrium is a tuple (p, (Xi)~=l)
such that p E 1R~, Xi E IR~ for all i = 1, .. " I, and
E.1) (2:!=1 Xi - 2:!=1 Wi) $ 0 and p. (2:!=1 Xi - 2:~=1 Wi) = 0,
E.2) (p. Xi $ P . Wi) & (Vy E IR~)[(p. y $ p. Wi) ~ (Xi ti y)]
for all i = 1, ... , I.
A competitive equilibrium price for [ is a vector p E IR~ such that
(p, (Xi)~=l) is a competitive equilibrium for [ for some vector (Xi)~=l' We
use E£ to denote the set of competitive equilibrium prices for the economy
[in ~.
Following the work of Sonnenschein [1972, 1973], Mas-Colell [1977]
found a necessary and sufficient condition for any set K in the relative o
interior ~ of ~ to be the set of equilibrium prices for an exchange economy
[ = {(b,Wi, IR~)~=l} satisfying (1) and (3), namely that K is a non
empty compact set. In Theorem 2 below, we relax the requirement (3)
for economies, but we obtain a more general characterization condition for
equilibrium price sets by dispensing with the "interior" restriction.
8 By (H). it is clear that Theorem 1 also covers the case where K = R~ \{o}.
5
Theorem 2 (Characterization of Equilibrium Price Sets). A set
K ~ D. is a non-empty compact set if and only if K = Ee for some
exchange economy £ = {(ti,wi, JR~)~=d satisfying (1).
Proof. The "if' part is well-known. To show the "only if" part, it is clear
that if ( : D. -t JRI is an excess demand function and £ = {(ti, Wi, JR~ )I=d is an exchange economy such that (2) holds for every p E D., then we have:
Ec = E" where E, = {p ED.: ((p) ~ O}. By Theorem 1, it suffices to
construct an excess demand function ( : D. -t JRI with K = E,. To do this,
we will modify the methods given in Uzawa [1962, p. 61], in Mas-Colell
[1977, Corollary 1] and Mas-Colell [1985, p. 195]. First, we pick any p E K.
Then we define a function e : D. -t JR! by e(p) = p - (p . p/p . p)p. It is
clear that e is an excess demand function. It is also easily verified that for
every p E D., one has: e(p) ~ 0 if and only if p = p. Finally, we define a
function ( : D. -t JRI by ((p) = Ape(p), where Ap = minpeK lip - pli. Then
clearly the function ( is an excess demand such that E, = K, and we are
done. Q.E.D.
We now discuss our second application of Theorem 1. As e?lphasized
by Sonnenschein [1972, 1973] and Debreu [1982]' characterizations of ex
cess demand functions are useful in studying relationships between general
proofs of existence of competitive equilibrium for exchange economies and
fixed-point theorems.
Uzawa [1962] proved that Brouwer's Fixed-Point Theorem is equivalent
to the following existence theorem:
Statement A (cf. Gale [1955], Nikaido [1956] & Debreu
[1956]). For every excess demand function ( : D. -t JR!, there
exists an equilibrium p E D., i.e. ((p) ~ O.
6
However, Uzawa's result only shows that the existence of fixed-points is
equivalent to the existence of equilibrium for excess demand functions de
fined on t::.. But how does equilibrium for excess demand functions defined
on t::. relate to equilibrium for economies? This has remained an open ques
tion, which the existing literature (e.g. the Sonnenschein-Mantel-Debreu
Theorem, Mas-Colell's Theorem [1977]' etc.) does not answer because its o
excess demand functions are defined only on certain subsets of t::., not on
t::.. However, our Theorem 1 does cover the situation.
Theorem 3 (Existence of Fixed-Points and Equilibrium Existence).
i) Brouwer's Fixed-Point Theorem is equivalent to Statement A.
ii) Statement A is implied by the Arrow-Debreu Equilibrium Existence
Theorem ([1954], Thm. I).
Proof. Assertion (i) was shown by Uzawa [1962]. And (ii) follows imme-
diately from Theorem 1. Q.E.D.
Therefore, by combining (i) and (ii) in Theorem 3, we see that Brouwer's
Theorem is implied by the Arrow-Debreu Theorem.
3. Technical details.
Here we will establish the two results (Lemma 1 and Proposition 1)
which we have used in the proof of Theorem 1. The following Lemma 1
extends demands from "small" to large price sets, permitting us to focus
on the special case where an excess demand function <: is defined on all of
IR~ \{o}.
7
Lemma 1 (Extension of Excess Demand Functions). Let /{ be a
non-empty compact subset of JR~ \{O}, and ( : /{ -+ JRI be an excess de
mand function. Then there exists an excess demand function ( : JR~ \ {O} -+
JRI such that (IK = (.
Proof. By (H), it clearly suffices to consider the case where /{ ~ .6.. By
Tietze's Extension Theorem (cf. Munkres [1975], p. 212), there exists a
continuous function ( : .6. -+ JRI such that (IK = (. Then we define a
function ( : .6. -+ JRI by (p) = (p) - (p . (p)lp . p)p. It is clear that (
is an excess demand function such that (IK = (. Finally, we extend the - - - I
function ( to the whole space JR~ \{O} by defining (p) = (pi Ei=l pt) for
any nonnegative nonzero p fI. .6.. This extension is clearly an excess demand
function whose restriction to /{ is (. Q.E.D.
By Lemma 1, the following proposition then establishes the necessity
part of Theorem 1.
Proposition 1 (Characterization of Excess Demand Functions de
fined on JR~ \{O}). Let ( : JR~ \{O} -+ JRI be an excess demand function.
Then there is an exchange economy C = {(ti,wi, JR~)~=d satisfying (1)
and such that (2) holds for every p E JR~ \{O}.
To characterize excess demand functions on compact subsets of posi
tive prices, Debreu [1974] used "orthogonal" projection to obtain individual
preferences ti and endowments Wi. By contrast, in order to permit bound
ary prices p (i.e. some Pi = 0), we will use the "oblique" projection as in
McFadden et al. [1974]. Thus, we will normalize the price set JR~ \{O} to
the set P, the intersection of JR~ with the sphere ofradius 2 IIqll centered
at -q, i.e.
P = {pE JR~: (p+q). (p+q) = 4q .q},
8
(4)
where q is given in (BB). In particular, we will decompose any given excess
demand function ( on P into individual "oblique" excess demand functions
¢1, ... , ¢I : P -t IRJ. By a function t.p : P -t IRJ being an oblique excess
demand junction,9 we mean that there exist a vector a » 0 E JRI and a
continuous function (3 : P -t JR such that:
a) (3(p) > 0, b) t.p(p) = (3(p)h(p),
(5)
for every pEP, where h(p) is the oblique projection along the direction of
p + q of a on the tangent space of p (see. Figure 1 below), i.e. 10
p·a h(p)=a- ( )(p+q).
p. p+q (6)
Then we can apply the rationalization result for oblique excess demand
functions given below in Lemma 3.
Proof of Proposition 1. Let ( : JR~ \ {O} -t IRJ be an excess demand
function. We pick any vector q » 0 given in (BB), and consider the set
P defined in (4). Then we can pick vectors a1,···,al »0 E JRI such
that {a1,···, an} is linearly independent and such that for every pEP,
((p) + p + q is contained in the interior of the convex cone spanned by
{a1,···,at} (see Figure 1 below, cf. also Mas-Colell [1977, p.125].) For
every pEP, we can write ((p) + p+ q = E~=l {3i(p)ai, where (3i(p) > 0, in
a unique fashion. It is clear that the functions (3i(p) are continuous. It is
9 This notion is a modification of the function defined in McFadden et. a!. [1974], p. 364, Equation (2). There, is defined only on the relative interior of P, {3 is a notnecessarily-continuous function, and a is only required to be nonnegative nonzero (i.e. not necessarily positive). Also, when q = 0 and a = ei the positive unit vector on the i coordinate axis of R', then h(p) given in (6) becomes Debreu's orthogonal projection function bi(p) = ei - ((p. e;)/(p. p»p.
10 Actually, h(p) is well-defined for every pER' with (p + q) . (p + q) = 4q . q, by the following fact: *) (p + q) . (p + q) = 4q . q implies p . (p + q) > o. To see (*), consider any such a p. Then we have: p. (p + q) = (q + q) . (q + q) - q . (p + q) = (q + q) . (q + q) - (q + q) . (p + q) + q . (p + q) ~ q . (p + q). Therefore, p . (p + q) ~ (1/2)(p. (p + q) + q . (p + q» = (1/2)(p + q) . (p + q) = 2q· q > o.
9
also clear that these ai can be chosen so that for every i, conditions (A-B)
in Lemma 3 below holds with a = ai.
Then consider the oblique excess demand functions tPi : P -+ IR} de
fined by tPi(p) = f3i(p)hi(p), where hi(p) is defined in (6) with a = ai (see
Figure 1 below). By Lemma 3 below, there exists an exchange economy
£ = {(~i,Wi,m~)~=d satisfying (1) and such that for every i = 1,·· ·,1
and every E P, equation (8) below holds with r.p = tPi, W = Wi and ~=~i.
Also, by an easy linear algebra argument (cf. McFadden et al. [1974], p.
365), it follows that ((p) = E~=l tPi(p) for all pEP, and so (2) holds for
very pEP. Then by (H), we have: (2) holds for every p E m~ \{o}. Q.E.D.
We now show, as claimed in our proof of Lemma 2, that an appropriate
choice of the oblique projection parameters q and a yields an endowment
and preference relation rationalizing individual oblique excess demands on
P.
Lemma 3 (Rationalization of Individual Oblique Excess Demand
Functions on P). Let q, a:» 0 Em', and define P by (4). Let f3 : P -+ m be a continuous function satisfying (5. a). Let r.p : P -+ m' be the oblique
excess demand function defined by (5.b). Assume:
A) a·a=4q·q, and
B) a-qftm~.
Then there exist a vector W :» 0 E m' and a preference relation ~ on m~
such that
~ is continuous, strictly convex and insatiable (indeed :c + Aa ~ :c for every A > 0 E m and every :c E m~),
(7)
and
{r.p(p)} = {:c E m~ : [p.:c $ p. w]
& ('v'y E R+)[(p. y $ p ·w)::} (:c ~ y)]} - {w} (8)
10
for every pEP.
Lemma 3 requires an (insatiable) rationalization ~ on all of P. Debreu
[1974] showed how to construct (monotone) preferences rationalizing his
individual "orthogonal" excess demand functions on compact subsets Sf of
S. To apply the spirit of Debreu to our context, we will prove the following
Lemma 4, where we obtain rationalizations on compact subsets, but of a
larger space
Q = {p E JR' : (p + q) . (p + q) = 4q . q and p . a ~ O}, (9)
which contains all of P (see Fig. 2 below). (Cf. Footnotes 15 and 18.)
Also, there we extend the consumption space from JR~ to JR' , and construct
(insatiable) preferences defined on JR' , not just JR~. As we will see, Lemma
3 follows as a consequence of Lemma 4.
Lemma 4 (Rationalization of Individual Oblique "Excess Demand
Functions" on Qf). Let q, a » 0 E JR', and define Q by (9). Let (} =
maXpeQ p . a. Let f3 : Q -t JR be a continuous function satisfying (5.a) on
Q. Let cp : Q -t JR be the corresponding oblique "excess demand" function,
i.e. cp is defined by (5.b).11 Let f > 0 E JR and define
Qf = {p E Q : f ~ p. a ~ (} -fl. (10)
Then there exist a preference relation ~ on JR' such that:
i) ~ is continuous and strictly convex, (11) ii) z + Aa ~ z for every A> 0 E JR and every z E JR',
and
{cp{p)} = {z E Bp : z ~ y for every y E Bp} (12)
11 By Footnote 10, h(.) is well-defined on Q, and so does rp(.).
11
for every p E Qf' where Bp = {x E JRl : p' x ~ OJ.
Before proving Lemma 4, we apply it to prove Lemma 3.
Proof of Lemma 3. Since Lemma 4 obtains a preference relation only
for sets Qf' we first ensure that P is contained in some Qf' By an easy
geometry argument, it is clear that (A) implies that the vector a maximizes
uniquely the function p ~ p . a on the sphere {p E JRl : p . p = 4q . q}, and
therefore the vector p = a - q maximizes uniquely the function p ~ p' a on
Q. By assumption (B), p ft P, and hence: p. a < p. a = ci for every pEP.
Also, since a » 0 and P ~ JR~ \{O}, we have: 0 < p. a for every pEP.
Then it follows that P ~ Qf for every sufficiently small f > O.
Now, let f3 : P -t JR and cp : P -t JRl be as given in Lemma 3. It is
clear that we can pick a continuous function P : Q -t JR such that Pip = f3
and P(p) > 0 for every p E Q. We define a function cp : Q -t JRl by
cp(p) = P(p)h(p), and we pick any f > 0 with P ~ Qf' By Lemma 4,
there exists a preference relation t on JRl satisfying (11) and such that
{cp(p)} = {x E Bp : xty for every y E Bp} for every p E Qf' and hence
{cp(p)} = {x E Bp : xty for every y E Bp} for every pEP. Then we pick
any w » 0 E JRl with cp(P) + w » 0, and it is clear that the translation !:
of t from JRl to JR~ defined by x !: y ¢::::> x - w t y - w satisfies (7)
and such that (8) holds for every pEP. Q.E.D.
It remains to prove Lemma 4. As mentioned earlier, our proof follows
the spirit of Debreu's proof [1974]: our steps (stages (3.1-7) below) parallel
his. However, we require modifications so that we can rationalize oblique
excess demand functions (rather than orthogonal ones) (cf. Footnote 14)
and allow boundary prices or even negative prices (cf. Footnote 19).
As in Debreu [1974], we endow the set C of non-empty, closed subsets
12
of JRI with the topology of closed convergence (see Hildenbrand [1974],
p.15). From here on, continuity of a function from any subset of JR into C
is always understood with respect to that topology.12
To illustrate the ideas of our proof of Lemma 4, consider the following
facts 13:
Facts (Indirect Rationalization). For every p, r E Q:
a) p. h(p) = O. b) iEr -I p and 0 < r· a ~ p. a, then r· h(p) > O.
(13)
Proof. Fact (13.a), which states that h(p) is in the tangent space of p,
holds as h being a projection function. To prove Fact (13.b), we apply the
arguments in McFadden et al. [1974], p.364, as follows. Consider any pair
of distinct r,p E Q. By definition (6), we have:
r·h(p) = r.a-p.a(r.(p+q)/p·(p+q)). By (*) in Footnote 10, we have:
p.(p+q) > O. Also, r.(p+q)-p·(p+q) = (r+q)·(p+q)-(p+q).(p+q) < O.
Therefore, we have: r· h(p) > 0, and so (13.b) holds.
Q.E.D.
Fact (13.b) suggests that the function p I-t -p . a plays a role similar
to an indirect utility function for the oblique projection h.14 For example,
by viewing -r' a and -p' a as the indirect utilities of rand p, and h(p) as
the bundle purchased under p, one can interpret (13.b) as follows: if p-lr
and the indirect utility at r is less than or equal to that at p, then what
12 For any X ~ n ', the interior of X is denoted by IntX, and the boundary of X is denoted by ax.
13 Fact (13.b) has its antecedent in McFadden et al. [1974], p. 364. There the price set is the relative interior of P, rather than Q, and a is any nonnegative nonzero vector (i.e. not necessarily positive).
14 By contrast, Debreu [1974] used p .... -p . e, as the indirect utility function for his orthogonal projection function b'(p) = e, - «p. e,)/(p. p»p, where e, is the i-unit vector.
13
is purchased under p is not affordable at r.15 Similar intuition can also be
applied to the function <.p = f3h as given in the Lemma 4, where f3 > 0 (as
in (5.a)).
Corresponding to the "indirect utility" function p I-t -p. a, we will
define the indirect weakly-preferred sets Ut = {p E Q : p . a :::; t}, the
indirect indifference sets Vt = {p E Q : p. a = t}, and the indirect weakly
worsen sets L t = {p E Q : p . a ~ t}, where t E [0, a]. (See Figure 2
below.) Actually, in our proof of Lemma 4, we will transform the sets Ut
into a profile of (direct) weakly-preferred sets M t which constitutes to a
preference on JR' satisfying (11) and rationalizing <.p (i.e. (12) holds) on Q£.
Each (direct) weakly-preferred set M t we will construct is correspond
ing to the indirect weakly-preferred set Ut and rationalizes <.p, therefore it
must satisfy: i) Mt is weakly above the budget set Bp for every pELt, and
ii) for every p E Vt, the set Mt intersects with Bp only at <.p(p). This moti
vates our use of the convex16 cone L; = {x E JR' : p . x ~ 0 for every p E
L t }, which is the set of all commodity bundles which are either unaffordable
or just-affordable at every pELt. Thus (i) simply states: i') M t ~ L;.
Also, it can be verified from (14) below and that for every t E (0, a) and
p E Vt, the convex cone L; (which by definition is weakly above Bp) in
tersects with Bp only at the ray {Ah(p) : A ~ a}. Therefore, (ii) simply
means: ii') Mt n In; = <.p(Vt). (See Figure 2 below; ref. (15.a) below; cf.
Debreu [1974], p.18, paragraph 3.)
15 This intuition fails at anypairofboundarypointsp,r ofQ, wherep·a = r·a = O. In particular, for any such a pair of p, r, we have: h(p) = h(r) = a, and r.h(p) = p.h(r) = O. By this fact, and by picking {3 with {3(p) #: {3(r), one can construct an oblique "excess demand" function iP violating the Weak Axiom of Revealed Preference at P. r, and so it is impossible to rationalize iP on the whole set Q by any preference t. This explains why in Lemma 4, we can obtain rationalization only on Q., rather than the whole set Q.
16 By (13) and (14) below, it can be verified that L: is indeed a strictly convex cone (i.e. any ray in 8L: passes through 0) for every t e (0,&). This observation, however, will not be used in our proof of Lemma 4. because we will use (13) and (14) directly.
14
We now characterize the interior and boundary of L;. This result is
useful for constructing the sets M t .
Facts (Characterization of IntL; and (n;). a) For every t E [0, a],
IntL; = {x E JRI : p . x > 0 for all PELt}.
b) For every t E (0, &) and every x E JRI,
i) if x = 0, then x E 8L;, and
(14.a)
ii) if x :j:. 0, then one has: x E 8L; if and only if (14.b) x = )"h(p) for some).. > 0 and some p E vt.
Proof. To show part (a), consider any t E [0, &] and any x E R. It is easily
verified that if p . x > 0 for every pELt, then x E IntL;. Conversely, let
x E IntL;. Suppose p. x = 0 for some pELt; then for any sufficiently small
u> 0, we have: x - up E IntL; ~ L; and p. (x - up) = -up· p < 0; and
a contradiction is derived. Thus we must have: p. x> 0 for every PELt.
And (14.a) holds.
To show part (b), it is clear that (14.b.i) is immediate from (14.a). It
remains to show (14.b.ii) for any t E (0, &) and any nonzero x E JRI . First,
suppose x = )"h(p) for some).. > 0 E JR and some p E vt ~ Lt. Notice that
for every r E Lt , we have: r· a? t = p. a, and so r· x = )..r . h(p) ? 0 by
(13). Therefore, x E L;. Then by (13.a) and (14.a), we have: x E 8L;.
Now, suppose x E 8L;. By (14.a), there exists apE Lt with p. x = O.
We will show that: A) p E vt and B) there exists).. > 0 with x = )"h(p).
To show (A), we suppose by contradiction p rt vt, i.e. p. a > t. Then
consider the function F(-y, u) = (-yp - ux + q) . (-yp - ux + q) - 4q· q, which
has F(l,O) = 0, and 8F~~,O) = 2p· (p+ q) > 0 by fact (*) in Footnote 10.
By the Implicit Function Theorem (cf. Rudin [1976], p. 224), we can pick
a iT > 0 small enough and a l' close enough to 1 so that F(1', iT) = 0 and
15
(.:yp - ux) . a > t. Then we have the contradiction .:yp - UX E Lt and
(.:yp - ux) . x = -ux . x < O. Thus have shown (A).
To show (B), observe that p . x = 0 :::; p . x for every pELt. Then p
minimizes p. x subject to: i) p. a = t and ii) (p+q). (p+q) = 4q· q. The first
order condition (cf. EI-Hodiri [1970], p.27) of this minimization problem
yields A, p E IR with x = Aa+2p(p+q). Since 0 = p·x = Ap·a+2pp·(p+q) =
0, we have: x = A(a - (p. alp· (p + q))(p + q)) = Ah(p). It remains to
show that A > O. First, since x :I 0, we have A :I o. Second, we suppose
by contradiction A < 0, then we can pick any r :I pin L t , and so by (13.b)
we have: r· x = Ar· h(p) < 0, contradicting to the fact that x E L;. Thus
we have A > 0, and hence (B) holds. This completes our proof for (14.b).
Q.E.D.
We now give several useful topological properties for the cones L;. First, for every t E [0, a], since L t is compact, it follows that L; is closed.
Then by the Krein-Milman (cf. Royden [1988], p. 242), it follows from (14)
that L; is indeed the convex cone spanned by h(Vi). Also, it is clear that
each Vi is compact; and it is easily verified (e.g. using the Implicited Func
tion Theorem) that the function t ~ Vi is continuous on [0, a]. Therefore,
the function t ~ L; continuous on (0, &) into £.17
We now present our proof of Lemma 4.
Proof of Lemma 4. Let a, q, Q, &, f3 and <p be as given in Lemma 4.
Consider any (: > O. Notice that the claim of the lemma is trivial when
Qf is empty. We now consider the non-trivial case, i.e. Qf is non-empty.
We will construct a family {Mt : t E (-oo,oo)} of (direct) weakly-preferred
sets (defined in (25) and (27) below) which satisfies the following properties
17 Also, by using (13) and (14), it can be shown the following hereditary property: if t' < t, then L;, \{O} C IntL;. This property, however, is not required in our proof of
Lemma 4.
16
(see Fig. 2 below):
a) (Rationality) for every t E [f, Q - f]: i) M t ~ L;,
ii) M t n 8L; = <p(lIt), b) (Closedness) every M t is closed, c) (Graduality) the function t t-+ M t is continuous, d) (Open Hereditary) ift' < t, then Mt' ~ IntMt, (15) e) (Strict Convexity) every M t is strictly convex, f) (Insatiability) IntMt 3 x + Aa for every A > 0 E JR,
every x E M t , and every M t ,
g) (Progressiveness) for every x E JRI , there exist t,t' E (-00,00) with x E Mt and x ¢ Mt,.
Before constructing these M t , we define a preference relation t on JRI by
x t y ¢::::::> (Vi E (-oo,oo))[y E Mt => x E Mt]. By (15.b-c) and (15.g), it
is easily verified that the preference t has {Mt : t E (-00, oo)} as its family
of weakly-preferred sets, {8Mt : t E (-oo,oo)} as its family of indifferent
sets, and {IntMt : t E (-oo,oo)} as its family of strictly-preferred sets.
Then it follows clearly from (15.b-g) that t satisfies (11).
We now show that t rationalizes <p on QE' i.e. (12) holds for every
p E QE' Consider any p E QE' and let t = p. a. We need to show that:
A) Mt' n Bp = 0 for every t' < t, and B) Mt n Bp = {<p(p)}. To see (A),
consider any t' < t. By (15.a.i) and (15.d), we have: Mt' S; IntMt S; IntL;,
and so by (14.a), we have: p.y > 0 (hence y ¢ Bp) for every y E Mt,. Hence
(A) holds. To see (B), by (15.a.ii), (15.d) and (14.a) we have: p . y > 0 for
every y E M t \<p(lIt). Also, by (13.b) and (5.b), we have: p. y > 0 for every
y = <p(p') with p =P p' E lit. Then we have: M t n Bp S; {<p(p)}. Also, since
p E lit, by (15.a.ii) we have <p(p) S; M t . Notice that <p(p) E Bp. Thus we
have: M t n Bp ;2 {<p(p)}. Hence (B) follows. And so (12) holds for every
pE QE'
As mentioned earlier, we will obtain the profile of weakly-preferred
sets M t from the profile of indirect weakly-preferred sets Ut by making use
17
of the sets L;. This will be done along the lines of Debreu [1974], with
appropriate modifications (e.g. see Footnote 19).
(3.1 Indirect Stage: picking Ut , where t E [E, a- E] ) We begin by listing the
following properties for the family {Ut : t E [E, a - En of indirect weakly
preferred sets:
a) (Compactness) every Ut is compact, b) (Graduality) the function t t-t Ut is continuous, (16) c) (Hereditary) if t' < t, then Uti CUt,
Facts (16.a) and (16.c) are obvious. And (16.b) is easily verified (e.g. by
using the Implicited Function Theorem).18
(3.2 Direct Stage: transforming Ut to Dt , where t E [E, a-E]) To transform
the indirect preference to a direct preference, as the first step we define
Dt = tp(Ut) for every t E [E, a - E] (see Figure 3 below). Then the sets Dt
satisfy the following properties:
a) (Rationality) for every t, i) D t ~ L;,
ii) Dt n En; = tp(Vt), b) (Compactness) every Dt is compact, c) (Graduality) the function t t-t Dt is continuous, d) (Hereditary) if t' < t, then:
i) Dt l ~ Dt , and ii) Dt' ~ IntL;.
(17)
Facts (17.b-c) are clear from (16.a-b). And (17.d.i) holds by (16.c). To
prove the rest of them, consider any t E [E, a - E] and any x E Dt = tp(Ut}.
Then x = {3(p)h(p) for some p E Ut . To see (17.a.i), for every r E Lt , since
r·a ~ t ~ p·a, it follows from (13) that r·x = {3(p)r.h(p) ~ O. Thus, x E L;,
and so (17.a.i) holds. We now show (17.d.ii). Suppose x E Dtl for some
t' < t. By repicking p, we can assume p E Uti, and so p. a = t' < t :$ r· a
18 By contrast, the graduality property is not necessary for the profile of sets Ut nP. This explains why we have to extend our price set from P to Q.
18
for every r E Lt. By (14.b), we have: r· x = f3(p)r . h(p) > 0 for every
r E L t , and so x E IntL; by (14.a). Thus (17.d.ii) holds. It remains to show
(17.a.ii). Suppose p E Vt. Then p. x = f3(p)p. h(p) = 0, and so: x E (n; by (14.b). Thus, we have: Dt n 8L; ;2 tp(Vt). Also it follows from (17.d.ii)
that D t n8L; ~ tp(Vt). Therefore (17.a.ii) follows.
(3.3 Open Hereditary Stage: transforming Dt to E t where t E [f, a-f].) The
sets Dt satisfy only the property of hereditary, but not the open hereditary.
To obtain this property, we will expand each Dt to a union of closed balls
Nt•x centered at x EDt, while maintaining rationality, graduality and
compactness. (See Figure 4 below.)
In defining the radii of the closed balls Nt •x , we will make use of the
function d : [E, a - f] X JRI --t JR defined by d(t, x) = min{llx - yll : y E
UPEL,Bp}. Since the sets L t are compact, it follows that the sets UPEL,Bp
are closed, and so the function d is well-defined. Also, it is easy to see that
d(t, x) is non-increasing in t. Moreover, observe that each UPEL,Bp is the
closure of the complement of L;, then by the continuity of the mapping
t I-t L;, it follows that the function d is continuous.
We now define the closed ball Nt •x = {y E JRI : IIy-xll::; (tja)d(t,x)}
for every x E D t and every t E [f, a- fl. We define the set Et = UXED,Nt.x
for every t E [f, a - fl. Then the sets E t satisfy:
a) (Rationality) for every t, i) E t ~ L;,
ii) E t n 8L; = tp(Vt), b) (Compactness) every E t is compact, c) (Graduality) the function t I-t E t is continuous, d) (Open Hereditary) if t' < t', then EtA ~ IntEt .
(18)
Facts (18.b-c) follows clearly from (17.b-c) and the continuity of the func
tion d. To show (18.d), let t' < t. For any x EDt', we have: x E Dt
by (17.d.i). Then it clearly suffices to show that Nt ,.x ~ IntNt •x . By
19
(17.d.ii) we have 0 < d(t, z). Since d(t', x) ~ d(t, x) and t' < t, we have: t' t --{;d(t', x) < {;d(t, x). Therefore, we have: Nt"x ~ IntNt,x; and so (18.d)
holds. To show (18.a), consider any t and x EDt. It is clear that Nt,x ~ L~,
and so E t = UxeD,Nt,x ~ Lt, i.e. (18.a.i) holds. It remains to show
(18.a.ii). Since Et ;2 Dt , by (17.a.ii) we have: Et n 8L; ;2 tp(Vt). We now
show that Et n 8L; ~ tp(Vt). To see this, consider any y E Et n 8L;. Then
y E Nt,x for some x EDt. Suppose by contradiction that x E IntL;. Then
we clearly would have: Nt,x ~ IntL;, and so: IntL; ;2 Nt,x :3 Y E 8L;, a
contradiction. Therefore, we must have: x E 8L;; so d(t, x) = 0, and hence:
Nt,x = {x}. Therefore, we have: 8L; :3 Y = x EDt; and so y E tp(Vt) by
(17.a.ii). Thus, we have: E t n 8L; ~ tp(Vt). Hence (18.a.ii) holds.
(3.4 Convexity Stage: transforming Et to Ft , where t E [f,o- - f].) Since
Et may not be convex, we let Ft be the convex hull of Et . (See Figure 5
below.) Then the sets Ft satisfy:
a) (Rationality) for every t, (i) Ft ~ L;,
(ii) Ft n 8L; = tp(Vt), b) (Compactness) every Ft is compact, c) (Graduality) the function t t-t Ft is continuous, d) (Open hereditary) if t' < t, then Ft , ~ IntFt, e) (Convexity) every Ft is convex.
(19)
Facts (19.b-d) follow clearly from (18.b-d). And (19.a.i) follows from (18.a.i)
and the fact that the cones L; are convex. To see (19.a.ii), consider any
t E [f,o- - fl· Since Ft ;2 E t , by (18.a.ii) we have: Ft n 8L; ;2 tp{Vt).
To show that Ft n 8L; ~ tp{Vt), consider any x E Ft n 8L;. By the
definition of Ft , we have: x = L:~=l AiXi for some n, some real numbers
AI, ... ,An > 0 with L:~=l Ai, and some vectors Xl, ... Xn E E t ~ L;. Since
x E 8L;, we have: Xl,"', Xn E 8L;. By (18.a.ii), there exist PI,'" ,Pn E Vt
such that Xi = tp{Pi) for all i. We claim that PI = ... = Pn. Suppose
not, then it follows from (13.b) that for every r E L;, we have: 0 <
20
f3i(p;)r . h(Pi) = r . IP(Pi) = r . Xi for some i. Also, for every i, since
Xi E L~, we have: r· Xi 2:': O. Therefore, we have: r· X = 2:7=1 Air· Xi > 0
for every r E L t ; and so X E IntL~ by (14.a). Thus we have derived the
contradiction: IntL~ 3 X E 8Lt. Therefore, we must have: P1 = ... = Pn;
and so x = Xl = '" = Xn E IP(Vt). Hence Ft n 8L; 2 IP(Vt). And (19.a.ii)
holds.
(9.5 Insatiability Stage: transforming Ft to Gt , where t E [e, it - e].) Since
the sets Ft does not satisfy the instability property. To obtain this prop
erty, we will adding them with a convex cone A spanned by a closed ball
centered at a, while maintaining the properties of rationality, graduality
and convexity.19 (See Figure 6 below.) Of course, this leads us to weaken
the property of compactness to closedness.
The closed ball we pick can be any No = {x E JRI : IIx - all < c5} with
c5 > 0 E JR and such that p. x > 0 for every pELf and every x E No. We
then define the cone A = {AX: A 2:': 0 E JR, x E No}, which is clearly closed
and convex, and has non-empty interior. Also, for any z E A, any pELt
and any t E [e, it - e], we have: i) p. z 2:': 0, and ii) p. z > 0 if z # O. Then
by (i) we have: A ~ L;; and by (ii) we have: z E IntL; for every non-zero
z E A. Also, it is easy to see that z + Aa E IntA for every z E A and every
A> 0 E JR.
For every t E [C, it - e], we define Gt = Ft + A. Then the sets Gt satisfy
19 Debreu [1974] made use of n~ to obtain monotone weak-preferred sets. Our
convex cone A shall be a proper subset of n' in order to maintain rationality property for allowing "non-positive" prices P (i.e. Pi < 0 for some i) in L 1• (Cf. the proof of (20.a.i) given below.)
21
the following properties:
a) (Rationality) for every t, i) Gt ~ L;,
ii) G t n 8L; = <p(Vt), b) (Closedness) every Gt is closed, c) (Graduality) the function t t-+ Gt is continuous, (20) d) (Open Hereditary) if t' < t, then G t , ~ IntGt ,
e) (Convexity) every Gt is convex, f) (Instability) x + Aa E IntGt for every A > 0,
every x E Gt and every Gt .
Fact (20.b) holds because of (19.b) and the fact that A is closed. Facts
(20.c-d) follow clearly from (19.c-d). Fact (20.e) follows clearly from (19.e)
and the fact that A is convex. Fact (20.a.i) follows from the fact that
each L; is a convex cone containing Et and A. It remains to show (20.f)
and (20.a.ii). Consider any t and any x E Gt . Then x = y + z for some
y E Ft and some z E A. To see (20.f), consider any A > O. Then we
have: z + Aa E IntA, and so x + Aa = y + z + Aa E Ft + IntA ~ IntGt .
Thus (20.f) holds. To see (20.a.ii), since Gt ;2 Ft , by (19.a.ii) we have:
G t n 8L; ;2 <p(Vt).We now show that G t n 8L; ~ <p(Vt). Suppose x E 8L;.
Then we have: z = 0, because otherwise, we would have z E IntL;, and
so x = y + z E IntL;. Therefore, we have: Ft 3 y = x E 8L;. Then by
(19.a.ii), we have: x E <p(Vt). Thus, we have shown that G t n 8L; ~ <p(Vt).
Hence (20.a.ii) holds.
(3.6 Strict Convexity Stage: trons/orming Gt to Mt , where t E [f,o.
f].) The sets Gt are not strictly convex. To obtain this property, we will
transform them to the sets M t as given in the beginning of this proof of
the lemma, where t E [f, 0.- fl. To do this, we pick any vector p E Q with
p·a = a and let Tp be the tangent space ofp, i.e. Tp = {y E JRI : p.y = OJ.
We will define strictly concave functions JJt : Tp -t JR, and will define
Mt = {y + sa : y E Tp and JJt(Y) $ s E JR}. (See Figure 7 below, cf. also
(26) below.)
22
To define the functions f..lt, we first define two functions as follows. For
every t E [e, 0- eJ and every y E Tp, let It(Y) be the least s such that
y + sa E Ct , and let At (y) be the least s such that y + sa E L;. (See Figure
7 below.) We claim: i) 0 ~ At{y), ii) At(Y) ~ It{y), and iii) It{y) < 00. To
see (iii), it suffices to find any s with y + sa E Ct. To do this, we can pick
any x E Ft and pick any s E JR with y - x E s(Na - a), where Na is given
in the second paragraph in Stage 3.5. Then we have: y + sa = x + sz for
some z E Na , and so y+sa E Ft+A = Ct. Hen<;e (iii) holds. To see (i), by
(iii) we have: y + sa E Ct ~ L; for some s E JR. Also, notice that for every
s < 0 we have: p. (y + sa) = sp· a = so < 0, and so: y + sa f/. L; because
pELt. Therefore (i) follows. Finally, (ii) is clear from (20.a.i). Thus the
claim is established.
We now list four useful properties for the functions (t, y) ~ At(Y) and
(t, y) ~ It(Y). First, they are continuos, by (20.c) and by the continuity
of the mapping t ~ L;. Second, each At (.) and It (.) is a convex function
on Tp for every t, because each Ct and L; is a convex set. Third, for every
t and every y E Tp, since the sets L; and C t are closed, it follows that
y + At (y) E en; and y + It (y) E aCt. Fourth, for every t and every y E Tp,
by the previous property and (20.a.ii), we have: At(Y) = IdY) if and only
if Y + At(Y) E <p(Vt).
We pick a continuous convex function p defined on {(s, s) E JR~ : s ~
s} (e.g. the one given in the appendix of Debreu [1974]) such that:
(a) p is strictly increasing in each variable, (b) p(s, s) = s for every s E JR+, (21) (c) if s -:/= sand/or s' -:/= s', then p is strictly convex on
the segment [(s, s), (s', s')].
Then we define JJt (y) = p( At (y), It (y)) for every t E [e, a] and every Y E Tp.
Clearly, the function (t, y) ~ JJt(Y) is continuous.
We now show that JJt(-) is a strictly convex function for every t E [£, a].
23
To see this, consider any t E [f, a - f], any pair of distinct vectors y, y' E Tp,
and any r E (0,1). Then:
Jlt(ry+ (1- r)y')
= p(>'dry + (1- r)y'), 'Yt(ry + (1- r)y'»
~ p(r>.t{y) + (1- r)>'t(Y'), T'Yt(Y) + (1 - rht(Y')) (22)
~ rp(>.t{y), 'Yt (y» + (1- r)p(>'dY'),'Yt(y'» (23)
= rJ-lt(y) + (1- r)J-lt(Y')·
Therefore, it suffices to show that either (22) or (23) is a strict inequality.
There are three possible cases, as follows.
First, suppose >'t(Y) = >'t(y'). Notice that y + >'t(y)a, y' + >'t(y')a E
8L;. By (14.b), there exist real numbers 8,8' ~ 0 and vectors p,p' E \It
such that y + >'t(z)a = 8h(p) and y' + >'t(z')a = 8'h(p'). We claim that
p :/; p'. Suppose not. Then we have: (8 - 8')h(p) = (y - y'), and hence:
(8 - 8')p . h(p) = p. (y - y') = O. Notice that 8 :/; 8' because y :/; y'.
Therefore we have: p. h(p) = 0, contradicting to (13.b) because a . p =
a > t = p. a. Thus we must have: p :/; p'. Similarly, it can be shown
that 8,8' > O. Then it follows from (13.b) that for every pELt, we have:
o < p·(r8h{p)+(1-r)8' h(p'» = p.{ry+(1-r)y' +(r>'t(z)+(1-r)>'t(y'))a).
Therefore, by (14.a), we have: ry+ {1- r)y' + (r>'t(z) + {1- r)>'t(y'))a E
IntL;. Then it follows from the definition of >'t(ry + (I - r)y') that we
have: >'t(ry + (I - r)y') < r>'t(Y) + (1 - r)>'t(y'). Then by (21.a) and the
convexity of the function 'Yt(-), it follows that the inequality (22) is strict.
Second, suppose >'t(Y) :/; >'t(y'), and suppose >'t{Y) = 'Yt(Y) and >'t(y') =
'Yt(y'). Then there exist p,p' E \It such that: y + >'t(y)a = <p(p) and
y' + >'t(y')a = <p(p'). We claim that p:/; p'. Suppose not. Then we have:
y+>'t{y)a = y' +>'t{y')a; and hence have the contradiction: 0 = p{y-y') =
(>'t(Y) - >'t{y'))p· a = (>'t(Y) - >'t(y'))a :/; O. Thus, we must have: p:/; p'.
24
By applying the arguments used in the last paragraph, it follows that the
inequality (22) is strict.
Third, suppose AdY) i= At(y') and suppose At(Y) i= It{y) and/or
At(y') i= Idy')· Then (At(y)"t{y)) i= (At(Y'), Idy')), and so (21.c) im
plies that the inequality (23) is strict. Thus we have shown that J.ld·) is a
strictly convex function for every t E [f, Q - f).
As mentioned earlier, we define:
Mt = {y + sa : y E Tp, s ~ J.lt(Y)} for every t E [f, Q - f). (24)
We now show that:
{Mt : t E [f, Q - f]} defined by (24) satisfies (15.a-g). (25)
Properties (15.b-c) are satisfied because Tp is closed and the function (t, y) ...-+
I-'dY) is continuous. Property (15.e) will follow immediately from the fact
that each I-'t(-) is strictly convex, once we have shown (15.f). Our prooffor
(15.f) requires the following decomposition fact:
each x E JRI has uniquely a vector 7r:c E JRI and a s E JR such that: x = 7r:c + s:ca; in particular, one has: S:c = r.; (26) and 7r:c = X - s:ca.
Thus 7r:c is the projection of x along the direction of a on Tp. To see
(26), since p . a = Q > 0, and so 7r:c and S:c are well-defined. By algebra
calculation, it is easy to see that x = 7r:c + s:ca. To see the uniqueness of
the decomposition, let x = Y + sa, where y E Tp and s E JR. Then have:
p. x = p. y + sp· a = sp· a. Therefore, s = S:c and so y = x - s:ca = 7r:c.
We now show that {Mt : t E [f, Q - f]} satisfies (15.f). Consider any
t E [f, Q - f) and any x E Mt . Let x = y + sa, where y E T£ and s ~ I-'t(Y).
Consider any A> O. We need to show that the vector x, = x + Aa E IntMt .
By (26), x' has a unique decomposition: x' = 7r:c,+s:c,a, where 7r:c, E Tp and
25
sx' E JR are as defined in (26). Since y+(s+>.)a = x+>.a = x' = 1rx,+sx,a,
we have: 1rz' = Y and Sx' = s+>. > I-'t{y) = 1-'t{1r~), and so x' E Mt . We now
show that x' is indeed an interior point of M t . Notice that the mappings
x I-t Si and x I-t I-'t{ 1ri) are continuous on JRI; therefore, we can pick any
open ball N 3 x' such that Si > I-'t (1ri) for every x EN. Then we have:
x' E N = {1ri + Sia : x E N} C {1ri + sa : s ;::: 1-'t{1ri) , and x E N} C
{y + sa : s ;::: I-'t(y), and y E Tp} = Mt . Thus, x' E IntMt . And we have
shown that {Mt : t E [t,o- - t]} satisfies (15.f)
We now show that {Mt : t E [t,o- - t]} satisfies (15.d). Consider any
t,t' E [t,o-- t] with t' < t. Then we have: >'t'(');::: >'to, and "}'t'O > "YtO
by (20.d). By (21.a), we have: I-'t'(·) > I-'t(·). Now consider any x' E Mt"
and let x' = y + sa, where Y E Tp and s ;::: I-'t'(Y)' Then we have: x' = x+(s-I-'t{y))a, where x = Y+I-'t(y)a E Mt· Since S-I-'t{y) > S-l-'t'(Y) ;::: 0,
by (15.f) we have: x' E IntMt . Therefore, {Mt : t E [t,o- - t]} satisfies
(15.d).
It remains to show (15.a). Consider any t E [t,o- - t] and any x E M t .
Let x = y + sa, where Y E Tp and S ;::: I-'t(Y). Since "Yt(Y) ;::: >'t(Y), by
(21.a) we have: I-'t(Y) ;::: >'t(Y); and so S ;::: >'t(Y). Therefore, we have:
p. x = p. (Y+ >'t(y)a) + (s - >'t(Y))P' a;::: 0 for every pELt; and so x E L~.
Thus (15.a.i) holds.
To show (15.a.ii), consider any t E [t,o- - t) and any x E JR1. Let 1rx
and Sx be as given in (26). Then x = 1rx + sxa. It is clear that x E aL; if
and only if Sx = >'t(1rx). We claim that: I) x E cp(vt) implies x E Mt naL~,
and II) x E MtnaL~ implies x E cp(vt). To show (I), suppose x E cp(vt). By
(20.a), we have 1rx + sxa = x E L~ n Gt and so: Sx ;::: >'t(1rx), "Yt(1rx). Then
we have: >'t(1rx) = Sx ;::: "Yt(1rx) ;::: >'t(1rz ), and so: Sz = >'t(1rz ) = "Yt(1rz ). By
(21.b), we have: Sz = I-'t(1rz); and so x = 1rz + sza E Mt . Thus, we have:
x E M t n a L~. Hence (I) holds. To show (II), suppose x E M t n a L~. Since
26
x E Mt , we have: x = y + sa for some y E Tp and s E JR with s 2:: J-!t(Y).
By (26), we have: trx = Y and Sx = s; and so Sx 2:: J-!t(7rx ). Since x E L;, we
have: At(7rx ) = Sx. We assert that id7rx ) = Sx. Suppose not. Then we have
id7rx ) > Sx = At(7rx ); and so we have: J-!t(7rx ) > Ad7rx ) = Sx by (21.a-b).
Therefore, we have the contradiction: Sx 2:: J-!t(7rx ) > Ad7rx ) = Sx. Thus
we must have: it(7rx ) = Sx. Therefore, x = trx + sx a = trx + id7r)a E Ct.
Then we have: x E L; n C t , and so x E tp(Vt) by (20.aii). Thus (II) holds.
Then by (I) and (II), we have: Mt n In; = tp(Vt), and therefore (15.a.ii)
holds.
(3.7 Progressiveness Stage: defining Mt for everyt ¢ [f, a-f].) To complete
the construction of a profile {Mt : t E (-oo,oo)} as given in the beginning
of this proof for Lemma 4, we define
M _ { Mf + (f - t)a for every t E (-00, f), t - M t = M&-f + (a - f - t)a for every t E (a - f, 00), (27).
where Me and M&-f are defined in (24). It is easy to see that the family
{Mt : t E (-oo,oo)} of sets defined by (24) and (27) satisfies (15.a-f).
We now show that it also satisfies (15.g). Consider any Z E JR1, which
by (26) has a unique decomposition z = trx + sxa. By the definitions
(24) and (27), it is easily verified that: I) for every t 2:: a - f, if s 2::
J-!&-f(7rX ) + (a - f) - t, then trx + sa E M t ; and II) for every t' S f, if
s < J-!f(7rX ) + (f - t), then trx + sa ¢ Mt,. Therefore, we have: x E Mt
for every t 2:: max{a - f,J-!&-f(7rX ) + (a - f) - sx}; and z ¢ Mt' for every
t' < min{f,J-!f(7rx ) + f - sx}. Hence (15.g) holds.
Thus we have constructed a family {Mt : t E (-oo,oo)} as claimed in
the beginning of this proof for Lemma 4, and we have shown that it satisfies
(15.a-g).
Q.E.D.
27
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29
I I I I I I I I I I I I
+a1 I I I t I I I I I I
I f3 f.'11 •.. •. " J fJ://aJ. .. · .
. I ..
Figure 1: Oblique Decomposition.
30
.. · .. t;w)+p+q