Articles

I

Consumer Decisions andRevealed Preference

I have been doing research in economics
more than sixty long years. In hindsight, we recall that in the 1960s when we were ambitious
students in Japan, there existed two popular yet
opposing textbooks in economics. Remarkably, the were nicknamed the "Red Text" andthe "Blue text." The Red text was well-represented by the U.S.S.R. Academy Economics
Institute (1958) Economic Textbook : the thirdrevised version [ Japanese translation, 1958],
which was no doubt the most authoritarian
text of the "Red Bloc" or the Soviet socialist
bloc.1)
The issue of "socialism versus capitalism"
constituted the central theme of the red text.
The honorable coauthors of the big text took
pride in reaching the following provocative
conclusions:

Reevaluating the Foundations ofSamuelson's Economic Analysis

Yasuhiro Sakai

Shiga University / Professor Emeritus

The "Red Text"of the U.S.S.R.
Institute versus the"Blue Text"of Paul A. Samuelson

We have thus far discussed the whole processes of economic development of asociety. As a result, we have reached themost important conclusion that from ahistorical viewpoint, capitalism is destined
to collapse whereas socialism is marchingfor its final victory over socialism. There
should be no other way possible. We are so
confident of such historical inevitability.
(U.S.S.R. Econ Institute, Japan ed. (1959),
p. 1050).
In contrast to the powerful Red Text, theBlue Text, presumably being regarded as a
I am grateful to Lionel W. McKenzie (Rochester), JamesW. Friedman (Rochester), William Brock (Chicago) andHiroshi Atsumi (Toronto) for helpful suggestions, andalso to Trout Rader (Missouri), Edmond Malinvaud(Paris) and Werner Hildenbrand (Bonn) for constantencouragement. I am also thankful to Masashi Tajima(Shiga) for technical assistance.

0041) For the rivalry between capitalism and socialism, see

Tsuru (1961) and Galbraith (1977).

THE HIKONE RONSO

Summer / July 2023 / No.436

strong rival, seemed to be rather modest andeven hesitant to the general public. Indeed,Paul A. Samuelson , one of the leading economists in the "Blue Block", wrote the worldpopular textbook Economics (1955, 7th edition
1967), gently defending the American capitalist
bloc in the following way:
America leads Russia, but will the gap narrow? (Samuelson, 7th ed., 1967, p. 791.)
Comparing the "Red Text" and the "BlueText", the majority of the people seemed to be
more impressed by the powerful "Red Text"
than the moderate "Blue Text." There existed
the minority group including myself, however,
who thought that the power of the "Red Wave"
would be only temporary, and eventually overtaken by the "Blue Wave", which would act likethe "Great Wave" of Japanese ukiyo-e Hokusai.
Let us recall the street demonstrations andcampus boycotting which greatly hindered theJapanese students in their study. After somehesitation, I decided to go to the United States
so that I could continue my graduate study
without unduly political and psychological interruptions. I just wanted to get out of theJapanese university disturbance in the 1960's,
thus daring to jump into the very core of thecapitalist economy.
At appearance, it looked like an escape trip.
However, in reality, it was an immense challengeor an enormous adventure to me. As the saying
goes, god helps those who help themselves. In
order to get financial and psychological help
from the outside organizations, I had to do a lotof preparations for studies in advanced economics and higher mathematics.

Consumer Decisions and Revealed PreferenceII

Revealed Preference
Theory:Paul A. Samuelson's
Analysis asa Starting Point

Paul A. Samuelson's Foundations of Economic Analysis (1947) served as a sort of bible
for all the graduate students who already mastered his popular text Economics (1955, 7thedition 1967) and wanted to study at a moreadvanced level. The very first page per se wasdecorated with the following impressive passage:"Mathematics is a Language. "American mathematician. Willard
Gibbs (1839-1903)
I certainly guess that the young and ambitious Samuelson was quite impressed by such ashort yet powerful message by an influential
mathematician when he decided to write his
masterpiece Foundations of Economics. Not
only that, Chapter 1 of Samuelson (1947) begins with the following impressive passage:
The existence of analogies between central
features of various theories implies the existence of a general theory which underlies theparticular theories and unifies them with respect to those central features. This
fundamental principle of generalization by
abstraction was enunciated by the eminent
American mathematician E. H. Moore
more than thirty years ago. It is the purpose of the pages that follow to work out
its implications for theoretical and applied
economics. (Samuelson 1947, p. 3.)

Yasuhiro Sakai005

As was well-stated by E. H. Moore (18621932), althoug h there exist a variety ofparticular theories, there is a general theory
which underlies those particular theories andunifies them with respect to those central features. This generalization principle was clearlystated by the eminent mathematician Moore,and later confirmed by the young lion Samuelson. When I opene d the first pa g e ofSamuelson's Foundations, I myself was greatly
shocked like a thunderbolt from a clear sky,and firmly felt that this was the time to say
good-bye to the infertile and unproductive
controversy between the "Red Text" and the"Blue Text" mentioned above. Following Moor
and Samuelson, I then felt that I must do everything in my power toward a "high ivory
tower" of unifying particular theories.
Revealed preference theory was first pioneered by the young Samuelson in his shortarticle in 1938, and later fully developed in his
main work Foundations 1947. Before Samuelson came on the stage, the existing theories ofconsumer demand were based on the traditional idea that consumers made consumption
decisions to maximize their utility levels. For
details for this point, see R, G. D. Allen (1936),
J.R. Hicks (1st ed. 1930, 2nd ed. 1948), and others.
Revealed preference theory initiated bySamuelson requires us to make a drastic change
in our way of thinking. Instead of stating withutility functions or ordinal preferences to derive demand functions, we first pay ourattention to consumer's demand behavior, andlater infer consumer's preferences on the basisof a certain set of "rational behavior assumptions." The question to ask is what the "rational006

behavior assumption" are all about. Although
there may be several assumptions conceivable,
Samuelson's choice was as simple and lucid as
what we now call "the weak axiom of revealed
preference."
The fundamental idea of Samuelson's analysis is that "consumers reveal their preferences
through their market behavior" if people's judgment is " rational and consistent." As is seenFig. 1, let us suppose that there are two goodsin the market, x1 and x2, and two bundles ofgoods, x = (x1, x2) and y = (y1, y2). Let us also
suppose that the two budget sets, B(p, m) andB(p', m'), and the two isoquants or utility
curves, u and u', are drawn as nicely as in Fig. 1.
If the commodity bundles x and y are respectively chosen at the budget sets B(p, m)
and B(p', m') by the consumer, we may conveniently obtain the following relations:x = h(p, m) ∈ B(p, m) ;

(1)

y = h(p', m') ∈ B(p', m') .

(2)

It is clear in Fig. 1 that the two bundles, x =(x1, x2) and y = (y1, x2), belong to the samebudget set, B(p, m) = { (x1, x2) : p1 x1 + p2 x2 ≦m}. Although these two bundles can bebought at (p, m), it is only x that is actuallychosen .
Under such circumstances, Samuelson
characteristically believes that x is revealed preferred to y . Someone might think that this is a
too strong statement to accept. Later developments of consumer demand theory, however,
have more or less taken Samuelson's side, with acertain qualification. In fact, we now knowthat the weak axiom of revealed preference a laSamuelson is not strong enough to guarantee

THE HIKONE RONSO

Summer / July 2023 / No.436

i = 1, ..., n. Therefore, we will work within then - dimensional vector space Rn.
The commodity space Ω , or the set of allconceivable bundles, is defined as follows:Ω = {x = (x1 , ... , xn) : x ≧ 0}.(3)

Let P and M be the price space and the income space, respectively. We make thefollowing assumptions:
P = {p = (p1 , ... , pn) : p ≧ 0},(4)

M = {m : m ≧ 0}.(5)Fig. 1 Samuelson claims that x is "revealed preferred " to y .
B( p, m) and B ( p', m') are respectively indicated by the vertiFig. 1
Samuelson claims that ｘ is " revealed preferred " to Now,
ｙ . we denote by B the family of all comcally-shaded triangle and the horizontally-shaded triangle.
B (p , m ) and B ( p ' , m ') are respectively indicated by thepetitive budgets, where for each (p, m) ∈ P ×

M, eachvertically-shaded triangle and the horizontally-shaded triangle.
. budget is defined as follows.

the derivation of utility functions. We need tob(p, m) = {x ∈ Ω : px ≦ m}.(6)have something more than the weak axiom :bundles can be bought at (p , m ) , it is only ｘ that is actually chosen .
Actually, the strong axiom of revealed preferLet h be a single-valued demand function
Under such circumstances, Samuelson characteristically believes that x is
ence that was later proposed by the coolon B, that is, a function which to each b(p, m)
revealed preferredto Houtakker
y . Someone
mightthat tothis is ∈a too
to bundle x = h(b (p,headedshouldbethink
adoptedM ,strongassignsstatement
a commodity
accept.
Laterdevelopments
consumerdemand theory, however,more hereor less
successfully
derive ofutilityfunctions..
m)). It ishave
supposed
that h(b(p, m)) = 0 for
In side,short,with
this isathebrief essenceof revealedm we= 0.now
Weknow
denoteby theX the range of the detaken Samuelson's
certainqualification.
In fact,that
preference
theory,showing
thestarting
point
mand
functionh:weak axiom of revealed preference a la Samuelson is not strong enough to guaranteeof Samuelson's unique analysis. More detailed
the derivation of utility functions. We need to have something more than the weak
and technical discussions will gradually be doX = ∪ {h(b(p ,m)) : b(p, m) ∈ M}.(7)axiom : Actually, the strong axiom of revealed preference that was later proposed by
ne in the following sections.
the cool-headed Houtakker should be adopted to successfully derive utility functions..
In what follows, we make the two assumpIn short, this Someis the brief
essence of revealed preference
tions:theory, showing theTechnical
starting point of Samuelson's
unique analysis. More detailed and technical discussions
Preparations(D1) X is a convex set in Ω .
will gradually be done in the following sections.
We are concerned with a consumer faced
withthe problem of choosing a commodity(D2) h satisfies the budget equation ph(p,III Some Technical Preparations
bundle subject to given prices and income. Letm) = m for any (p, m) ∈ P × M.
us assume that there are n commodities, labeledIII

We are concerned with a consumer faced with the problem of choosing a commodity

bundle subject to given prices and income.

Let us assume that there aren5Consumer Decisions and Revealed Preference

Yasuhiro Sakai

007

For convenience, for h(b(p, m)) and b(p, m)∈ B, we will also simply write h(p, m) and (p,m) ∈ B.
We are now in a position to define "preference relations in terms of h" as follows. To thisend, suppose the following relations hold:(SRP) There is b(p, m) ∈ B such that x= h(p, m), y ∈ b(p, m), and x ≠ y.
Then, following Samuelson (1947), we saythat x is directly revealed preferred to y, and we
simply write xSy. If we look back at Fig. 1
above, we may easily understand the graphicalmeaning of the relation (SRP). Indeed, seeing
is believing !
Fortunately or unfortunately, (SRP) is not
only one of revealed preference relations conc eivab le ; Inde e d , Houtha kker (1950)
succeeded in extending the Samuelson revealed
relation (SRP) in a very natural way. More specifically, let us consider a new preference
relation specified as a long chain of (SRP):(HRP) For some finite sequence u1, ... ,um, we have xSu1S ... umSy .
Then, in line with Houthakker (1950), we
say that x is indirectly revealed preferred to y,conveniently writing xHy. In other words, H isthe transitive closure of S on Ω. Clearly, xSy
implies xHy, but not the other way around.
S a mu e l s o n's w e a k a x i o m ( W ) a n d
Houthakker's strong axiom (S) of revealed
preference are formulated as follows:(W) For any x, y ∈ Ω, xSy implies 〜 ySx.
(S) For any x, y ∈ Ω, xHy implies 〜 yHx.

In other words, the weak axiom (W) argues that the relation S is not symmetric,
whereas the strong axiom (S) states that the relation H is not symmetric. Each axiom clearlyshows the consistency of consumer's choice behavior in a direct or indirect way.
Let E(p) be the well-known Engel curve associated with a price vector p:E(p) = ∪ {h(p, m) : m ∈ M}, p ∈ P.

(8)

Denoting by Cls(X ) the closure of X, let us
define the following sets of commodity bundles:
A(x0) = {x ∈ Cls(X) : 〜 xHx0}, x0 ∈ X;
(9)Ā(x0) = {x ∈ Cls(X) : 〜 x0Hx}, x0 ∈ X.
(10)
On the one hand, the under-bar set A(x0)stands for the set of commodity bundles which
are not indirectly revealed preferred to a given
commodity bundle x0. On the other hand, theupper-bar set Ā(x0) denotes the set of commodity bundles to which a given commodity
bundle x0 is not indirectly revealed preferred.
Now, in order to postulate a "regularity condition" under which the weak and strong axioms
are to be established, it is quite useful for us to
introduce two different kinds of expenditurecompensation functions as follows : 2)ã(p, x 0) = sup {px : x ∈ A(x 0) ∩ E(p)},(p, x0) ∈ P × X,(11)a(p, x 0) = inf {px : x ∈ Ā(x 0) ∩ E(p)},(p, x0) ∈ P × X.(12)2) The expenditure compensation function was first in-troduced and effectively employed by McKenzie (1957).

008

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Summer / July 2023 / No.436

On the one hand, the newly defined function ã(p, x0) stands for the supremum of those
expenditures px ( = m ) at price p such that thecommodity bundle x = h(p, m) is not indirectly
revealed preferred to a given commodity bundle x0. On the other hand, the function a(p,x0) indicates the infimum of those expenditurespx(= m) at price p such that a given commodity bundle x0 is not indirectly revealed preferredto the commodity bundle x = h(p, m). It will
be seen that under assumptions (D1), (D2),and (S), the functions ã(p, x0) and a(p, x0) are
well-defined for all (p, x0) ∈ P × X.
At first glance, the meanings of those funcFig. 2 The upper-shaded set {x : xHx0} and the lower-shaded
x : ~ xHx0}. set { x : x H x 0 } and the lower-shaded set { x : ~ x H
set {upper-shaded
tions may appear to be complicated andFig. 2 The
E p
TheEngel
Engelcurve
curve
or may not, be continuous. It
difficult for the reader to comprehend. As the The
E ( (p )) may,
may, or may not, be continuous. It breaks twbreaks twice here.
saying goes, however, seeing is really .believing.
Graphical representations will help us to easily
understand what is going on here. For instance, in Fig. 2, the whole commodity We
set areX isnow in a position(R) For(p, x0) introduce
ã(p,regularityx0)∈ P × X,theto allcarefully
condition00
the union of the upper-shaded set {xplay
: xHx
}
).
a(p
,
x
≧
a very important role as a bridge between the weak and strong axiom
0and the lower-shaded set {x : ~ xHxpreference.
}, where Specifically, such a regularity condition is formulated as follo
the base commodity x0 belongs to the lowerIt would not be an easy job to describe theshaded set, but not the upper-shadedof XtheBroadly(R) set.
For allexact( p , meaningx 0 ) ∈P ×
, conditionã ( p , x 0) (R).
≧ a
( p , x 0) .
Depending on the condition of the demand
speaking, the condition (R) says that for a givcurve h(p, m), the Engel curve E(p) may,or notencommodity
vectorand a given
It would
be price
an easy
job to pdescribe
the exact
meaning of the con
0
may not, be continuous. If it happens
tobebundlex,themaximumof"worse-than-or-asBroadly speaking, the condition (R) says that for a given price vect
continuous, it can be drawn as a continuous
good-as- x 0" expenditures is greater that or
given commodity bundle x 0, the maximum of " worse-than-or-as-g
curve like E(p) in Fig. 2. So, the expenditureequal to the minimum of "better-than-or-asexpenditures is greater that or equal to the minimum of " better-than-or-a0compensation function ã(p, x ) can roughly
g ood-as x 0 " expenditures. Once ag ain,be
expenditures. Once again, graphical illustrations would help us to under
depicted as a downward-sloped line there.
graphical illustrations would help us to undergoing on here. Fig. 3 shows the situation under which the condition holds
We are now in a position to carefully
introstand what is going on here. Fig. 3 shows theshows the situation under which it is violated. As is clear from a compar
situation under which the condition holds,
duce the regularity condition which has
to play
those two figures, the regularity condition (R) requires that there be no
while Fig. 4 shows the situation under which ita very important role as a bridge between thethe Engel curve E ( p ) for any p , yet allowing for " ~H overlap
is violated. As is clear from a comparison beweak and strong axioms of revealed preference.
should be noted that there may exist " jumps" on the Engel curve since theSpecifically, such a regularity condition is fortween those two fig ures, the reg ularity
the demand function is not guaranteed here.
condition (R) requires that there be no "~Hmulated as follows:gap" on the Engel curve E(p) for any p, yet allowing for "~H overlap" on it. It should be

Consumer Decisions and Revealed Preference

Yasuhiro Sakai009noted that there may exist "jumps" on the Engel curve since the continuity of the demand
function is not guaranteed here.

Fig. 3(R)

Fig. 3(R)

is satisfied here. E (p ) is not continuous, jumping once.

Fig. 3 (R) is satisfied here. E(p) is not continuous, jumping
is satisfied here. E (p ) is not continuous, jumping once.
once.

IV Equivalence of the Weakand Strong Axioms

In this section, we would like to establish
an important equivalence theorem of the weak
and strong axioms of revealed preference onCls(X), the closure of the range of the demand
function. More exactly, we wish to demonstrate that, for the demand function satisfying(D1) and (D2), the strong axiom (S) holds on
Cls(X) if and only if the weak axiom (W) together with the regularity condition (R) holds
on Cls(X). This is no doubt the core of our research, requiring a careful, step-by-step inquiry.
We need several preliminary results in
preparation for the equivalence theorem. To
this end, we first note the following lemma.
LEMMA 4.1 (Ā as a closed set).
Let demand function h satisfy (D1, 2) andthe weak axiom (W). Then, for any x0 ∈ X,
the upper-bar set Ā(x0) is a closed set in Cls(X).

is violated here. E (p ) is not continuous, jumping once.

Fig. 4(R)

Fig. 4p ) is notjumping once.
(R) 4is violated
here. E (here.
E( pcontinuous,
Fig.
(R) is violated) is not continuous,
jump-

ing once.

Equivalence of the Weak and Strong Axioms

quivalence of the Weak and Strong Axioms

Proof. It suffices to show that the set {x ∈ Cls(X) : x0Hx}, x0 ∈ X, is an open set in Cls(X).
First, if x0 = 0, then this set is evidently
empty, so that it is open. Next, if x0 ≠ 0, thenlet us choose any x 1 ∈ X such that x 0Hx 1.
Then, by the definition of H, there exists an x2∈ X such thatx0 = x2 or x0Hx2 

his section, we would like to establish an important equivalence theorem of the

(13)

and
strong axiomsof revealed
preferenceCls (X )equivalence, the closure theoremof the rangeof
his section,
we would
like to establish
anon
importantof theand p2 x2 ≧ p2 x1, x2 ≠ x1.
t
1
2
Let
on satisfying(D1) andthe strong
axiom
holds on Cls
) if
and
only
if x = (1 – t)x + tx , 0 ≦ t ≦ 1. Then,emand
function.
More(D2),
exactly,
we wishto (S)
demonstrate
that,(Xforthedemandt
wefindregularity
conditioneakaxiom (W)) (holds
onand
Cls
(X if
).. x ∈ X since X is convex by (D1). So,on satisfying(D1) togetherand (D2),withthe thestrongaxiom (S)holds on (RCls
X ) ifonly
have the following:
eak axiom (W) together with the regularity condition (R ) holds on we
Clsshould(X )..

emand
function.
exactly,
we wish on
to demonstrate
that, fordemandand strong
axiomsMoreof revealed
preferenceCls (X ), the closureof the range
of

10

2 2
x ≧ p2xt, xt ≠ x1 .
p10

010

THE HIKONE RONSO(14)

Summer / July 2023 / No.436

This relation together with (W ) mustyields the following results:
ptx2 ＞ ptxt.

(15)

Because of definition of xt, we must havethe following equation:(1 – t) (ptx1 – ptxt) + t(ptx2 – ptxt) = 0,
whence we find pt xt ＞ ptx1 by (15.). Here,
we can choose a neighborhood V(x1) of x1 suchthat
ptxt ＞ pty for any y ∈ V(x1) ∩ Cls(X).(16)As is seen above, the proof of Lemma 4.3 istedious but straightforward. Its graphical andintuitive outline, showing that the set {x :x0Hx} is open, may be given in Fig. 5. Thislemma will effectively be utilized in the equivalence theorem.
To reach the final goal of the equivalence
theorem, we need to pass through the expenditure compensation functions to be statedbelow.
LEMMA 2 (ã and a are well-defined )Let demand function h satisfy (D1, D2)
and the strong axiom (S). Then, for any (p, x0)∈ P × X, the expenditure compensation functions ã (p, x0) and a (p, x0) are well-defined.

If we combine Eqs. (13), (14) and (16), weThis ensures the desired result that the set 3)y ∈ V(x1) ∩ Cls(X). Thisobtain x Hy for any
Proof. To see that the ã(p, x0) is well-defined,00{ x ∈ Cls (X ) : x H x } , x ∈X , is an open set in Cls ( X ) .
Q.E.D.
it suffices to seethat the following set is nonensures the desired result that the set {x ∈ Cls
00empty and bounded above (see Fig. 6 for an(X) : x Hx}, x ∈ X, is an open set in Cls(X).
As is seen above, the proof of Lemma 4.3 is tedious but straightforward.
Itsillustration) :
Q.E.D.

V ( x 1 ) ∩ Cls0 (X) .

graphical and intuitive outline, showing that the set
given in Fig. 5.

{x :x 0 H x } is open, may beThis lemma will effectively utilized in the equivalenceT(p, x0)theorem{px : x≡

∈ A(x0) ∩ E(p)}.(17)To this end, let us choose any (p, x0) ∈ P ×
X, and note x0 = h(p0, m0) ∈ B. There are two
possibilities for the value of m0 : namely, m0 =0 or m0 ＞ 0. If m0 = 0, then it is clear that T(p,x0) = {0}. If m0 ＞ 0, then the set {x : 0 ≦ x ∈Cls(X) & p0x ＜ m0} is nonempty. It is possible
to find an m1 ∈ M such that, in Cls(X), the hyperplane px = m1 lies below the hyperplane p0x= m0. Therefore, we have p0h(p, m1) ＜ p0x0.
This implies x0 S h(p, m1), so that x0H h(p, m1).
It follows from (S) that we find ~ h(p, m1)Hx0,
implying that p h(p, m1) ∈ T (p, x0).
This ensures that for any (p, x0) ∈ P × X,
Fig. 55 The
Thesetset: }xis0 open.
H x } isAopen.
A graphicalisthe
illustrated
set T(phere.
, x0) is nonempty.
Fig.
{x{:xx0Hxgraphical
outline isoutline
illustrated here.

To reach the final goal of the equivalence theorem, we need to pass through the3) I am grateful to Uzawa (1960) for providing a guide to
expenditure compensation functions to be stated below.
the proof of this lemma. Its completely revised version
was contained in Chipman & Hurwicz & R ichter &Sonnenshein (1971).
LEMMA 2 ( ã and a are well-defined )Let demand function
any ( p , x 0 )a( p , x 0)

h satisfy (D1, D2) and the strong axiom ( S ).

∈ P × X , the expenditure compensation functions ã ( p , x 0)

Proof .

andare well-defined. .

Consumer Decisions and Revealed Preference
3)

Then, forTo see that theã ( p , x 0)

Yasuhiro Sakai

is well-defined, it suffices to see that the011

implies

x 0 S h (p , m 1 ) , so that x 0 H h (p , m 1 ) .

It follows from (S ) that we

find ~ h (p , m 1 ) H x 0 , implying that p h ( p , m 1 ) ∈ T ( p , x 0) .

0This ensures that for any ( p , x 0 ) ∈0P × X , the set T ( p , x ) is nonempty.

Next,
for any (p, x ) ∈ P × X, let us choose
shown in Lemma 4.2, the expenditure com( p , x 0 ) ∈P × X , let us choose an m 2. ∈ M such that202
pensationfunctions ã(p, x0) and a(p, x0) are. ∈ Mthatsuch
It follows
2. m
p x 0 ＜ manIt follows
p hthat(p , pxm 2 )＜= mm.2 ＞
p x 0 , implying
that
2
2
0
for any (p, x0) ∈ P × X. In order) = mthath(p,＞ ppxh (, pimplyingh ( p , m 2 )that
H x 0 ph(p,. Thismensures, m 2 ) is athat
upperbound towell-defined
the set202
to derive the regularity condition (R), we rely
T ( p , x 0) .m )Hx . This ensures that ph(p, m ) is a uppera the( p ,setx 0)T isfor any ( p , x 0 ) ∈P ×on
X proofproceeds
The proof
thattoby contradiction. To this end, let us
bound(p,well-definedx0).
in a similar way.The proof that a(p, x0) is well-defined for
Q.E.D.
dare to suppose to the contrary that ã(p, x0) ＜0
any (p, x ) ∈ P × X proceeds in a similar way.
a(p, x0) for some (p, x0) ∈ P × X. Then, choose
an m ∈ M so that ã(p, x0) ＜ m ＜ a(p, x0), and
Q.E.D.
let x = h(p, m). Then, we immediately obtainthe following:
Next, for any

ã(p, x0) ＜ px ＜ a(p, x0).

(18)

By the definition of ã and a, we obtain thefollowing:
xHx0 and x0Hx,

Fig. 6 The proof of LEMMA 2 : a graphical outline
Fig. 6

The proof of LEMMA 2 : a graphical outline

With all the preparations aforementioned,
we are now in a position to state and prove theequivalence theorem, which certainly represents the most important result of this chapter.
THEOREM 3 (EQUIVALENCE THEOREM)
Let the demand function h satisfy (D1,
D2). Then, the strong axiom (S) holds if andonly if the weak axiom (W) and the regularity
condition (R) both hold.
Proof . (i) (S) ⇒ (W) & (R) : 4)
Let h(p, m) satisfy the strong axiom (S).
Then, by definition per se, the weak axiom(W)is obviously implied by (S). Note that, as was(19)which clearly contradicts the strong axiom(S). So, in order to get rid of the contradiction,
we should have ã(p, x0) ≧ a(p, x0), which is
nothing but the regularity condition (R).
(ii) (W) & (R) ⇒ (S) :Next, we will prove the weak axiom (W)together with the regularity condition (R) im13
plies the strong
axiom (S) . Once again, we relyon the proof by contradiction. So, let us assume otherwise. Then, there exist somecommodities x, x0 ∈ X such that the following
relations hold:
xHx0 and x0Hx.

(20)

Let x = h(p, m) for some (p, m) ∈ B. Then,we will show that Eq. (20) implies the following:ã(p, x0) ≦ px ≦ a(p, x0) . (21)4) I am indebted to Lionel W. McKenzie for suggesting
the present proof, which was better than my previous
proof.

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Summer / July 2023 / No.436

Indeed, if we assume on the contrary thatã(p, x0) ＞ px, we could take an x* ∈ A(x0) ∩ E(p) for which ã(p, x0) ≧ px* ＞ px. The last inequality yields x* Sx, which together with xHx0
entails x* Hx0. This contradicts x* ∈ A(x0). In
order to get rid of the contradiction, we should
originally have ã(p, x0) ≦ px.
The proof that x0 Hx, x ∈ E(p), implies px≦ a(p, x0) proceeds in an analogous way.
Now, let us specifically assume px = m =a(p, x0). Then, take a sequence {pxi} converging to px such that xi ∈ Ā(x0) ∩ E(p). This ispossible by the definition of a. Define the new
set Y(p0, x0) as follows (see Fig. 7 for an illustration) :Hence, in the light of Eq. (21), we musthave the following;ã(p, x0) ≦ px ＜ a(p, x0).(23)This clearly contradicts the regularity condition (R). And now, we can happily declarethat the proof is complete.
Q.E.D.

Y(p0, x0) = Ā(x0) ∩ {x ∈ Cls(X): a(p, x0)(22)≦ px ＜ a(p, x0) + 1}.Fig. 7

Fig. 7 The set Y(p0, x0) in the proof of THEOREM 4.3. is0,x 0 ) in the proof of THEOREM 4.3. is illustra
set Yof( pTHEOREM
The setillustrated.
Y ( p 0,x 0 )Fig.
in 7theThe
proof
4.3. is illustrated.

Let us focus on the sequence {xi} aforeNeedless to say, THEOREM 3 (the equivmentioned. Since xi ∈ Y(p0, x0) for sufficiently
Needless to say, THEOREM 3 (the equivalence theorem) represents thlarge i, we may without loss of generality asalence theorem) represents the most important
important3resultin thischapter. As istheorem)
easily seen in the detailed proof
aforemeNeedless to say,
THEOREM(theequivalencein thischapter. As is easilyrepresents
seen in thethe mosis aLEMMAresult
sume that the original
sequence {xi} itself
1 , demonstrating that the set Ā ( x 0 ) is closed under (W) , plays a
0 0
0 this chapter. As is easily seen in the detailed proof aforementioned
resultdetailed
proof
aforementioned,
1, may havesequence in Y(pimportant, x ). SinceĀ(xin) is a closedrole in the
proof part.
Indeed,if the set Ā ( x 0 ) is LEMMANOT closed, we0 (W)0 ) xisthe0 ) ( and
0 Ha
x set
∉ Ā (x Ā∉ closedA set(x 0)Ā(x, meaningthatxunx criticaand x
1 , demonstrating
that
the
xthat
under
, playsset in Cls(X) byLEMMA
means of LEMMA
1, Y(p0, x0in
) whichdemonstrating) is closed
and ã ( p , x 0) = a (p , x 0)0= p x , x ∈ E ( p ) . For a graphical outline of this indeed a compact
in proofCls(X).
Therefore,
criticalin thepart. the casrole inset
the
part.
Indeed, ifder
the(W),
set Āplays( x a) is
NOTroleclosed,weproofmay havesatisfies the regularity condition ( R ),see Fig. 8 . In such a case, h surelyk0
we can take a convergent
subsequence{x
Indeed,
if
the
set
Ā(x
}
)
is
NOT
closed,wemay x H x 0⊂the xstrong∉ axiomA (x 0()S ) ,. meaning that x 0 H x andin which x ∉ Ā (x 0 ) and
i
0 0
1
k
0 0
0
{x } in Y(p , x ).and
have the case in which x ∉ Ā(x outline
Let xã (=plim
x a Y(p,x )) and xof∉this poin
, x 0) =∈(p , x 0) =⊂p x , x ∈ E ( p ) . For a graphicalĀ(x0). Then, we obtain px1 = px. Here, notesee Fig. 8 . In such a case, h surely satisfies the regularity condition ( R ), but nothat either x1 = x or x1 ≠ x since h may, or may
the strong1 axiom ( S ) .
not, be continuous.
If x = x then, from Eq.
(4.20), we have x0Hx1, contradicting that x1∈ Ā(x0). If x1 ≠ x then the equality px = px1
yields xSx1, which together with Eq. (20) implies x0Hx1. This also contradicts that xi ∈Ā(x0).
To get out of the contradiction, we can not
specifically assume px = a(p, x0).
Fig.
8 8 Theset setY ( pY (,xp0,)xis0)illustrated
here forthe proofof Theorem
3.
Fig.
The
is illustrated
here
for theproof of0

0

Theorem 3.

It is worthwhile to discuss how our equivalence result distinguishes itself froprevious results in the related literature.

Fig. 8First of all, we can say that the hyp

The set Y ( p 0,x 0 ) is illustrated here for the proof of Theorem 3.

It is worthwhile to discuss how our equivalence result distinguishes itself from th
Yasuhiro Sakai
013
previous results in the related literature. First of all, we can say that the hypothesiConsumer Decisions and Revealed PreferenceA(x0), meaning that x0Hx and xHx0, and ã(p,x0) = a(p, x0) = px, x ∈ E(p). For a graphicaloutline of this point, see Fig. 8. In such a case,h surely satisfies the regularity condition (R),but not the strong axiom (S).
It is worthwhile to discuss how our equivalence result distinguishes itself from theprevious results in the related literature. Firstof all, we can say that the hypothesis of THEREM 4.3 is definitely weaker than that of vanMoeseke's Theorem 2.5 [4, p. 99 ] since : (i)neither Lipschitz nor continuity conditions are
imposed on the demand function h, (ii) X neednot be the positive or nonnegative orthant of
Rn ; it can be any convex subset.
Let us recall that Sonnenschein's Example
2 ( 1971, p.274) shows that (D1, D2) and ( S )do not imply the continuity of h. In view of
THEOREM 4.3, it is easily seen that the regularity condition (R) [even together with (D1,
D2) and (S)] does not suffice to guarantee thecontinuity of h. Besides, based upon the strong
axiom of revealed preference (S), Hurwicz andRichter (1971) carefully studied revealed preference theory without demand continuity
assumptions.5)
In the light of these works mentioned
above, the significance of our equivalence theorem without demand continuity assumptions isquite evident. Generally speaking, continuity
assumptions may be useful yet strong in any
subject of research. They may unfortunately be
too strong in some research areas including thepresent consumer demand theory.

V

The Effectiveness andLimitations ofRational Decisions

In his first printing of Foundations of Economic Analysis (1947), Paul A. Samuelson
remarked as follows;
I was aware that each field involved interdependent unknowns determined bypresumably efficacious, interdependentequilibrium conditions — a fact which has
always been generally realized. But, andthis leads me to the second fundamental
purpose of this work, it had not been
pointed out to my knowledge that there
exist formally identical meaning ful theorems in these fields, each derived by anessentially analogous method. This is not
surprising since only the smallest fraction of
economic writings, theoretical and applied,
has been concerned with the derivation ofoperationally meaningful theorems.
(Samuelson 1947, p. 3)
The main purpose of Samuelson's Foundations (1947) was to obtain a series of operationally
meaningful theorems, which had unfortunately
been very rare in the 1940s. In fact, when it
published at long last after an unavoidable delay due to the Second World War, it received asort of mixed blessing in the then dominanteconomics profession. For instance, Kenneth
Boulding (1948), one of the finest contempo-5) When I established the equivalence theorem with nocontinuity assumptions, I received a very positive response from Edmond Malinvaud, one of the leading theoretical economists. In fact, in his famous advanced
text, Malinvaud (1972, revised 1985) praised my early paper (1973) by saying that Sakai finished the equivalence
theorem in consumer demand theory. It was certainly anice surprise to me. Note that besides my regularity
condition ( R ), there are several other regularity conditions conceivable ( see Sakai (1973) ).

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Summer / July 2023 / No.436

raries at that time, wrote the following rather
critical and cynical review :The Foundations is an important book. It
should be studied not only by the mathematically baptized but also by those who,
like myself, hang on to n-dimensions by
the skin of their teeth. No economist who
studies it can fail to profit by it. Nevertheless, the present reviewer cannot help
feeling a certain sense of rapidly diminishi n g m a r g i n a l p r o d u c t i v i t y. i n t h eapplication of mathematics to economics.
There is an elusive flavor of John StuartMill about the Foundations which makes it
seem less like a foundation than a coping
stone, finishing an edifice which does nothave much further to go. It may well bethat the slovenly literary borderland between economics and sociology will be themost fruitful building during the years tocome and that mathematical economics
will remain too flawless in its perfection to
be very fruitful. ( Boulding 1948, p. 199.)
As Boulding eloquently tells us, Samuelson's Foundations is an important book. Everyeconomist would agree with this. Indeed, bymaking use of the weak axiom of revealed preference with help of his friend Houthakker,
Samuelson succeeded in deriving many meaningful theorems in consumer choice theory. So
far so good. There exist a certain number of
economists, however, who might rather put6) Before Boulding's review appeared, the ordinal utili-

ty approach represented by Allen (1936) and Hicks (1939,
revised 1946) dominated consumer demand theory for along time. Although Hicks was influenced by the rise of
revealed preference approach, he took an independent
middle path between the two approaches. For such a
subtle position, see Hicks (1956).

7) It was in 1969 that noted theoretician John Hicks

published a very interesting book A Theory of Economic
History, in which two apparently different subjects –Theory and History – were nicely combined into one. In

Consumer Decisions and Revealed Preference

themselves in Boulding's position. We shouldbe very careful of Boulding's remark on Samuelson's Foundations, which never admires that
Foundations is a very important book, only saying that it is just an important book. This
descriptive difference looks small, but may
sometimes be larger than we imagine.6)
As far as I am concerned, my own position
on Foundations is rather delicate. When I wasyoung and gay, I found much interest in mathematical economics, and read Samuelson's
many books and papers repeatedly and energetically. To tell the truth, this paper is no
more than a completely revised version ofChapter One of my Ph.D. thesis Foundations ofConsumption and Production Theories submitte d to the Un iversit y of R o chester in
September 1972. Professors Lionel W. McKenzie and James W. Friedman were my thesis
advisers, thus giving me kind suggestions andconstant encouragement. When I was later
teaching economic theory at the University ofPittsburgh, my attitude on the application ofmathematics to economics gradually changed
into the direction of more economic reality
rather than pure theory per se, with growinginterest in building a bridge between Theory
and History. Since John R. Hicks happens tobe McKenzie's teacher at Oxford, I am a sort ofHicks' grand child, and then have followed the
new Hicks who has found growing interest in
Theory of Economic History.7)
In hindsight, although Boulding's prediction that "the slovenly literary borderland

Japan also, well-known economist Takashi Negishi(1989) wrote an outstanding book History of Economic
Theory a la John Hicks. In postscript of the book, Negishi eloquently remarked. " The currently dominating
mainstream theory is not the only possible theory. The
study of the history of economic theory is important,
not only from the heretical point of view, to show how
and why the current mainstream theory is wrong, butalso from the mainstream point of view, to spur on thedevelopment of its own thinking." (Negishi 1989, p.
385.)

Yasuhiro Sakai

015between economics and sociology will be themost fruitful building during the years tocome" may not be totally right, I do think thatit has been fairly correct. In fact, from the
1980s to the present 2020s, there have gradually yet surely emerged new waves of social
sciences toward "more psychology, more sociology and still more history". Amartia Sen (1977,
1987) is one of those economists, which hasconstantly issued a strong warning against excessively rational and unduly consistent humanbehavior, for it would possibly lead us to the
non-realistic world where all men and womenas "rational fools" make neither mistakes nor
contradictory behaviors. As the saying goes,
extremely spicy things are not good for thestomach. Very noted Japanese essayist Shichihei Yamamoto (1983) wrote a very popular yet
sarcastic essay A study of "koo-ki" or social pressure, arguing that the heart of ordinary people
with cool head sometimes tend to get hot andexcited under impersonal and social pressure.
For instance, at the end of the Second World
War, many young and talented Japanese pilots
dared to take on zero fighter planes and carryout "kamikaze" or suicidal attack missions withno hope of victory. And now, in parallel withthe emergence of the Economics of Uncertainty, we can see the coming and growing wave of
Behavioral Economics, with Akerlof & Schiller(2009) , Kahneman (2003) and Thaler (2015)
being recent leaders, in which many biases andirregular human behavior cannot be neglected
and should be taken care of in standard social
sciences.
In conclusion, we can argue that revealed
preferenc e the or y a la Samuelson and
Houthaker is no longer perfect, having many
weaknesses and limitations. Not only that, alltheories in economics and related fields should
not be free of shortcomings and defects. I do
declare, however, that Samuelson's reveal preference theory is still alive today and can befurther developed.8)
We can still learn new lessons from oldteachings.
References
⦿Akerlof, GA, Schiller, RJ (2009 ) Animal spirits: howhuman psychology drives the economy, and why it matters for global capitalism. Princeton: Princeton University Press.

⦿Allen, RGD (1936) Professor's Slutsky's theory of con-sumer's choice. Review of Economic Studies, 1936.

⦿Boulding, K (1948) Samuelson's foundations of eco-nomic analysis: a book review. Journal of Political
Economy.

⦿Chipman JS, Hurwicz L, Richter MK, Sonnenschein

HF (eds.) (1971) Preferences, utility, and demand. New
York: Harcourt.

⦿Chambers, E and Cristopher, F (2016) Revealed prefer-ence theory. San Diego: Cambridge University Press.

⦿Galbraith, JK (1977) The age of uncertainty. Boston:

Houghton Mifflin.

⦿Hicks, JR (1st ed. 1939, 2nd ed. 1946) Value and Capital.

Oxford: Oxford University Press.

⦿Hicks, JR (1956) A revision of demand theory. Oxford:

Oxford University Press.

⦿Hicks, JR (1969) A theory of economic theory. Oxford:

Oxford University Press.

⦿Houthakker, H (1950) Revealed preference and theutility function. Economica 17, pp. 159-170.

⦿Kahneman, D (2003) Maps of bounded rationality:

psychology for behavioral economics. American Economic Review, 93-5, pp. 1445-1475.

⦿Malinvaud, E (1972, revised ed. 1985) Lectures on mi-croeconomic theory. Amsterdam: Elsevier Science
Publishers.

⦿McKenzie LW (1957) Demand theory without a utility

index. Review of Economic Studies, 24, pp.185-189.

⦿Nicholson, W (2005) Microeconomic theory: basic

principles and extensions. Mason. Ohio: Thomson.

⦿Negishi, T (1989) History of economic theory. Amster-

dam: North-Holland.

8) For the reevaluation of revealed preference in thetwenty-first century, see Niholson (2005), Szenberg &Ramrattan & Gottsman (2006), Varian (2006), andChambers & Cristopher (2016) .

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⦿Sakai, Y (1973) Equivalence of the weak and strong axi-

oms of revealed preference without demand continuity
assumptions: a "regularity condition" approach. Journalof Economic Theory, 8-3, pp. 292 - 304.

⦿Sen A (1987) On ethics and economics. Oxford: Black-well.

⦿Samuelson, PA (1938) A note on the pure theory of con-sumers' behaviour. Economica, New Series, 5(17), pp.
61-71.

⦿Samuelson, PA (1947, 8th ed. with a new intro. 1976)

Foundations of economic analysis. Cambridge, U.S.A. :Harvard University Press.

⦿Samuelson, PS (1955, 7th ed. 1967) Economics: introduc-

tory analysis. New York: McGraw Hill.

⦿Sen, A (1977) Rational fools: a critique of the behav-

ioural foundations of economic theory. Philosophy andPublic affairs, Vol. 6 (Summer), pp. 317-344.

⦿Szenberg, M, Ramrattan, L , Gottesman, A A (eds.)(2006) Samuelsonian economics and the twenty-first
century. Oxford: Oxford University Press.

⦿Thaler, R (20**) Misbehaving: the making of behavioral

economics. New York: Norton.

⦿Uzawa, H (1960) Preference and rational choice in thetheory of consumption Contained in Arrow, K.J., Karlin, S, Suppes, P (eds) Mathematical methods in the social Sciences, Chapter 9, pp. 129 -148. Stanford: Stanford University Press. Also contained in Newman, P(ed.) (1968) Readings in mathematical economics, Volume 1: Value theory, Chapter 27, pp. 297 - 316,

⦿Tsuru, S (ed,) (1961) Has capitalism changed ? Tokyo:

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⦿Varian, HR (2006) Intermediate microeconomics: amodern approach, 7th ed. New Delhi: Affiliated EastWest Press.

⦿Yamamoto, S (1983 ) A study of ku-u-ki or social pres-sure (in Japanese). Tokyo: Yamamoto Publishers.

Consumer Decisions and Revealed Preference

Yasuhiro Sakai

017

Consumer Decisions and Revealed Preference
Reevaluating the Foundations of Samuelson's Economic Analysis

Yasuhiro Sakai

This paper discusses consumer decisionand revealed preference from many new angles,
thus carefully reexamining the foundations of
economic analysis. It is Paul A. Samuelson'scontribution who boldly introduced to microeconomic theory the brand-new concept of
revealed preference against the then standard
doctrine of ordinal utility. The main result of
this chapter is that Houthakker's strong axiomof revealed preference holds if and only if Samuelson's weak axiom of revealed preference and acertain "regularity condition" newly introduced
here both hold. This equivalence result distinguishes itself from all the previous works in thesense that no continuity assumptions are imposed on the demand function. In the light of
uncertainty and behavioral economics in thepresent times, the significance and limitationsof Samuelson-type economics are also discussed. Throughout this research, manygraphical illustrations are attempted for givinga helping hand to understand mathematicallemmas and theorems.
Keywords:revealed preference, Samuelson, theweak axiom, Houthakker, the strong
axiom, regularity conditions, behavioral economics

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Consumer Decisions and Revealed Preference

Yasuhiro Sakai019