I

ArticlesRevealed Favorability andIndirect Utility
Extending the Samuelson Approach thePrice-Income Space＊

Yasuhiro Sakai

Shiga University / Professor Emeritus

Revealed Preference
and Revealed Favorability:
Duality Relations
in Consumer Demand

It is Paul A. Samuelson (1947, 1965,1967)
who newly adopted a revealed preference approach to consumer choice theory, thus bravely
departing from the ordinal utility approach,which had been traditional and dominant for along time before Samuelson appeared on theeconomics stage. Such ordinal utility is conveniently shown by a direct utility function of
goods, and has been well-developed by E.
Slutsky (1915), R. G. D. Allen (1936), J. R.
Hicks (1939, 2nd ed. 1946), and others.
The purpose of this research is to carefully
examine the axiomatic foundations of the indirect utility function of normalized-prices, which
was later developed as a dual of revealed preference approach a la P. A. Samuelson (1947, 7thed. 1967). Remarkably, there has been a strong
revival of interest in the indirect utility function since it was first studied by Hotelling(1932) and others a long time ago.
It is now well-known that there exists thebasic duality relations between the direct andindirect utility functions: namely, maximizing
the direct utility of commodity is equivalent tominimizing the indirect utility of prices andincome, with the identical budget constraint
imposed on both instances. In conjunction
with such duality relation, a number of propositions on the structure of utility functions anddemand systems have been established by many
economic theorists. It is of great interest,
therefore, to reexamine the duality relations in
the light of revealed preference theory, whichhave been long neglected so far.

*This research is a completely revised version of Sakai (1975),with many new figures added. I am grateful to Lionel W. McKenzie (Rochester), James W. Friedman (Rochester), Hiroshi Atsumi (Toronto), Koji Okuguchi (Tokyo Metropolitan), Marcel
K. Richter (Minnesota), Hugo Sonnenschein (Boston), andWerner Hildenbrand (Bonn) for valuable suggestions. I am also
thankful to Masashi Tajima (Shiga) for technical assistance.

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In order to visually understand the duality
relationship between the direct and indirect
utility approaches, let us consider the simple
world with two commodities, x1 and x2, andtheir prices, p1 and p2, for illustrative purpose.
If we divide those prices by income, then wereadily obtain normalized prices q1 and q2 inwhich q1 = p1 / m and q2 = p2 / m. In the traditional consumer choice world, a representative
consumer is supposed to maximize his (direct)
utility U(x1, x2) subject to his income budget
q1 x1 + q2 x2 ≦ 1. It follows from the resulting
consumer equilibrium that x1 and x2 are functions of the price pair (q1, q2), so that we may
duly write x1 = x1(q1, q2) and x2 = x2(q1, q2).
If we substitute x1(q1, q2) and x2 = x2(q1,q2) for the direct utility function U(x1, x2),
then we can rightly obtain the following indirect utility functions:
V(q1, q2) = U(x1 (q1, q2), x2(q1, q2)). 

(1)

(A)

The normalized price space(q 1 , q 2 )(A)

The normalized price space(q 1 , q 2 )(A) The normalized price space (q1, q2)(B) The commodity space (x1, x2)(B)

The commodity space(x 1 , x 2 )Fig. 1 The dual relationship between the indirect and di(B) between
The commodity
space(x 1 , xutility
2 )the indirect
and direct
approaches
Fig. 1 The dual relationshiprect utility approaches

It is reasonable for us to say thatFig.
the1 funcThe dual relationship between the indirect and direct utility approaches
tion V is indirect utility function because theutility is indirectly related to the normalizedprice pair (q1, q2) through the "medium" of the
maximizes the direct utility U(x1, x2) subject tothe budget set q1 x1 + q2 x2 ≦ 1.
commodity pair (x1, x2). With the preparaWe are now in a position to exactly define
tions aforementioned, we are ready to say thatthe normalized-price analog of revealed preferthe consumer is supposed to minimize his indience relation as follows. First, let us firstrect utility V(q1, q2) subject to the budget set
suppose that a commodity bundle x1 is chosen
x1 q1 + x2 q2 ≦ 1.
from the budget set b(q1) associated with a
The dual and symmetrical relationship benormalized price vector q1.
tween the indirect and direct utility function
Second, let us also assume that this bundle
approaches may vividly be visualized in Fig. 1.
1
x belongs to b(q0), namely the budget set assoIn the upper panel (A), the (normalized) price0
point q minimizes the indirect utility V(q1, q2)
ciated with a different normalized-price vector
q0. Then, we duly say that the new budget set b
subject to the budget set x1 q1 + x2 q2 ≦ 1. In
0the lower panel (B), the commodity point x(q0) is directly revealed more favorable than theold budget set b(q1). Or more simply, we cay

Revealed Favorability and Indirect Utility

Yasuhiro Sakai

047

vealed preference relation on the commodity
space, the indirect revealed favorability relation
may be defined as what we can call "transitive
closure" of the direct revealed favorability relation on the price-income space. The strong
axiom of revealed favorability requires that theindirect revealed favorability relation be asymmetric.
In the traditional revealed preference approach, it is always assumed that the range ofthe demand function has the convex property.
In case the new revealed favorability approachis adopted, however, it will be seen that such astringent assumption can be discarded; thereFig. 2 Revealed favorability relation.
b(q0) is directly revealed more favorable than b(q1) ; or simfore, the results obtained in this paper help to1
is directly
morefavorability
favorable than qrelation.
.
ply, q0Fig.
2. revealed
Revealed
clarify the role by the convexity condition on
the rangeb (q 0 ) is directly revealed more favorable thanb (q 1 of) ; the demand function in the theoryof consumer's demand. It should also be noor simply, q 0 is directly revealed more favorable than q 1 .
say that the normalized-price vector q0 is directly
ticed that, in contrast to the previous results ofrevealed more favorable than the normalizedKuga (1969), Weddepohl (1970), and others,
price vector q1. This is presumably because there
the continuity or Lipschitz condition is not
0
imposed at all on the demand function here. 2)
is some commodity bundle, say x , in the new0budget set b(q ) which is better than all comThe contents of the remaining sections
1
may be outlined in the following way. In Secmodity bundles in the old budget set b(q ).
Such revealed favorability relation may be welltion 2, a set of exact definitions and detailed
4 this paper isillustrated in Fig. 2. 1)
assumptions used throughout
carefully introduced. Section 3 thoroughly disIn the traditional revealed preference theory a la Samuelson (1947) and Houthakker
cusses the question whether there exists the(1950), there exist the famous two axioms.
duality relation between revealed favorability
They are the weak and strong axioms of rerelations on the normalized-price space andreveled preference relations on the commodivealed preference. In a very similar way, we canrightly define the normalized-price analogs ofties. In Section 4, we establish an important
those axioms. More specifically, the weak axitheorem concerning the relationship between
the weak and strong axioms of revealed favorom of revealed favorability requires that thedirect revealed favorability relation thus deability. As will be seen below, the strong axiom
holds if and only if the weak axiom plus a cerfined be asymmetric. Needless to say, this isnothing but the normalized-price counterpart
tain "regularity condition" both hold. Interestinglyof revealed preference on the commodity
enough, this regularity condition is a relationshipspace. Likewise, analogous to the indirect rebetween two definitions of income compensated
1) The term "more favorable" was first used by Wed-depohl (1970)0482) For example, see Hurwicz & Richter (1971), Richter(1966), Richter (1971), Uzawa (1960, revised 1971), andmany others. Strangely enough, it is rather common toassume that the demand function has the convex property. One of main purposes in this paper is to do awaywith such deep-rooted tradition.

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Autumn / Oct. 2023 / No.437

functions that must hold if the strong axiom
holds.
Section 5 deals with the representation
problem of revealed favorability relations. It isshown that, given the strong axiom of revealed
favorability together with some other usual assumptions on the demand function, there
exists a real-valued function (namely, an indirect utility function) of normalized-pricevectors, with the desired properties of minimality, lower semi-continuity, strict quasiconvexity, and monotonicity. A nice bridge
between the indirect and direct utility functions will also be built in this research
whenever the inverse of the demand function
exists. Final several remarks will be made in
Section 6.

II

Definitions andAssumptions

In this section, we will introduce certain
definitions and basic assumptions to be usedthroughout this research. In particular, revealed favorability relations on the set of priceincome vectors will rigorously be defined interms of a system of axioms.
We are concerned with the behavior of arational consumer, who chooses a set of commodity bundles subject to market prices andincomes. The commodity space Y is the set of allconceivable commodity bundles. For convenience, we assume that it is the non-negative
quadrant of the n-dimensional vector space Rn :Y = { y : y = (y1, ..., yn) & 0 ≦ y ∈ Rn },  (2)where yi denotes the quantity of commodity i for i = 1, ..., n. It should not be necessarythat every y ∈ Y is chosen subject to some normalized-price configuration.
The price-income space P × M is the positive orthant of the (n + 1)-dimensional vector
space:
P × M = { (p, m) : (p, m) = (p1, ..., pn, m)& 0 ＜ (p, m) ∈ Rn+1 }, 
(3)where pi denotes the price of commodityi(i = 1, ..., n) and m represents the consumer's
income.
Henceforth, we will also use another priceconcept by means of "normalization." To thisend, let us define the normalized-price space Q
is the positive orthant of the n - dimensional
vector space:
Q = { q : q = (q1, ..., qn) & 0 ＜ q ∈ Rn },(4)where qi = (1 / m) / pi any i = 1, ..., n.
As can easily be seen, the normalized-pricespace Q is actually the set of all conceivablecombinations of "normalized-prices," in thesense that "the sum of all xi-weighted qi's over i"is always unity: namely, ∑i xi qi = 1. Because ofsuch normalization process, the income component is safely dropped out of our discussion.
It is noted that both the commodity space Y
and the normalized- price space Q have thesame n-dimension. 3)
For a given price vector q ∈ Q, the budget
set b(q) is defined as follows.
b(q) = { y ∈ Y : q y ≦ 1 }. 

(5)Let B be the family of all budget sets:

3) It is noted that in the present case, Ω is nothing but

the positive orthant of R n and hence happens to be equal
to P.

Revealed Favorability and Indirect Utility

Yasuhiro Sakai049

B = ∪ { b(q) : q ∈ Q}. (6)modity bundles in b(q 1). We can formallyb ( q 0 ) F 1 b ( writeq 1 ) such relationship as follows.
Now, let h be a non-empty demand correspondence (function) on B, that is, a function 0b(q0)F 1b(q1)⇔b ( q ) ⊃ h ( q 1 0) and 1 q 0 0≠ q1 1 ,which to each b(q) assigns a non-empty subset.
⇔ b(q ) ⊃ h(q ) and q ≠ q ,Such a subset may be called a choice set, being⇔ q0x0 = 1 ≧ q0x1for x0 ∈ h(q0), x1 ∈0
1 q
q 0willx 0 = 1 ≧h(qq 10),xandforh ( q 1 ) , and q 0⇔ wewritten as h(b(q)). In what follows,≠xq10. ∈ h ( q 0 ) , x 1 ∈(8)
obey the following conventions. That is to say,for h(b(q)) and b(q) ∈ B, we will also simply
In order to avoid the lengthy expression. In order to avoid
the lengthy expression b ( q 0 )F 1 b ( q 1 ) , it iswrite h(q) and q ∈ B, respectively. Further, inb(q0)F 1b(q1), it is more convenient for us tofor usto desimplysimply
writewrite
q 0q0FF 11q
believethe light of (5.5), we let X be the rangeof theq1.1 .WeWe
believethat that
Fig. 3 Fig.
is 3 is helpful fo
helpfulunderstandingtrue meaningofmand correspondence h :the true meaningof forEq.
(8).
It theis noted
that although
the ess
Eq.
(8).
It
is
noted
that
althoughtheessenceof0
1
1
favorability relation q F q
was already pointed out in the last Fi
revealed favorability relation q0F 1q1 was alX = b(B) = ∪ { b(q) : q ∈ Q}
it wasoutshownthecommodity
space ra= { x ∈ Y : x ∈ h(q) for some unfortunate
ready pointed
in the laston
Fig. 2,
it wasquite
q ∈ Q}.  (7) that
normalized-pricespace. that
To itcorrect
suchonathe
mismatch,
unfortunate
was showncommod- we can draw t
X needthe
not revealed
ity space favorability
rather than therelation
normalized-priceIn general, we find X ⊂ Y, butwhichq 0 space.
F 1 q 1 is now r
be identical to Y ; indeed, X may be a proper
To correct such a mismatch, we can draw thecorresponding normalized-price space. For this point, see Kuga (1subset of Y.
new Fig. 3, in which the revealed favorability(1970,
andSakai(1977).
Besides, we make the following usual asrelation
q0F 1q1 is now rightfully on the corresponding normalized-price space. For thissumption:
point, see Kuga (1969), Weddepohl (1970),
and Sakai (1977).
(H) For all q ∈ Q and all x ∈ h(q), we
have q x = 1.
The assumption (H) means that h is (positively) homogeneous of degree zero withrespect to q and that the whole budget is spent.
Now, we are in a position to exactly define
revealed favorability relation on Q in terms of hand b as follows. Suppose that the budget setb(q0) associated with a price vector q0 contains
the choice set h(q1) associated with a distinct
price vector q1. Then, we say that b(q0) is directly revealed more favorable than b(q1). For
this relation, we will simply write b(q0)F 1b(q1),since there is presumably some commodity
bundle in b(q0) which is better than all com-

Fig. 3 Revealed favorability relation on the normalized-pricespace : q0F 1q1 ⇔ b(q0)F 1b(q1) ⇔ b(q0) ⊃ h(q1) & q0 ≠ q1.

Fig. 3

Revealed favorability relation on the normalized-price space

q 0 F 1q 1 ⇔ b ( q 0 ) F 1b ( q 1 ) ⇔ b ( q 0 ) ⊃ h ( q 1 ) & q 0 ≠ q

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Autumn / Oct. 2023 / No.437

Now, suppose that for a finite sequence r 1,..., r k ∈ Q, we find the following sequence ofrelations:
q0F 1r 1F 1... F 1r kF 1q1 (9)Then, we can say that q0 is revealed more favorable in k steps than q1, and we write q0F kq1.
Moreover, if there exists a certain finite integer
k for which the relation q0F kq1 holds, then weshould say that q0 is indirectly revealed more favorable than q1, and we simply write q0F q1. Inother words, F is the "transitive closure" of F 1,
or the smallest transitive relation including
F 1on Q. It is easy to see that q0 ≤ q1implies
q0F 1q1, and hence q0F q1.
Corresponding to these two relations F 1and F on Q, let us introduce the two axioms of
revealed favorability, namely, the weak axiom(WF ) and the strong axiom (SF ) in the following fashion:(WF) For q0, q1 ∈ Q, q0F1q1 implies ~q1F1q0,0q,(SF) For q0, q1 ∈ Q, q0F q1 implies ~ q1F

where, in general, the symbol "~ABC"
stands for the "negation of ABC." These two
axioms asserts that the consumer's behavior
should be "directly" or "indirectly" consistent;
therefore, they are quite analogous to the traditional revealed preference axioms of Samuelson(1947) and Houthakker (1950) on Y, the commodity space. The duality relationshipbetween revealed favorability relations on Q
and revealed preference relations on Y will be
investigated in more details in the next section.

Revealed Favorability and Indirect Utility

III

Revealed Favorability
versus Revealed Preference

In this section, we will examine whether
there exists the duality relationship between
revealed favorability on the price space and revealed preference on the commodity space. We
are also concerned with the question of how
this duality relationship corresponds to themore familiar one between the indirect and direct utility functions.
As mentioned above, revealed preference
relations on Y are defined in terms of h and b as
follows. If there is q ∈ Q such that x ∈ h(q), y∈ b(q), and x ≠ y, then we say that x is directly
revealed preferred to y, and we write xSy. If
there is some finite sequence u1, ..., uk of elements of Y such that x Su1S ... SukSy, then we
say that x is indirectly revealed preferred to y,
and we write xHy .
On the one hand, the weak axiom (WP) of
revealed preference requires that the direct revealed preference relation a la Samuelson S be
non-symmetric:(WP) For x, y ∈ Y, xSy implies ~ ySx .
On the other hand, the strong axiom (SP)of revealed preference requires that the indirect
revealed preference relation H be non-symmetric:(SP) For x, y ∈ Y, xHy implies ~ yHx .
It is easy to show that if h satisfies (WF) on
Q, then for any x ∈ X, the inverse image of x
by h contains a single element; therefore, h isuniquely invertible. In contrast, it is also easily
seen that if h satisfies (WP) on X, then for any

Yasuhiro Sakai051

q ∈ Q, the image of q by h is a singleton; therefore, h is single-valued. Hence, (WF) does not
imply (WP), and vice versa. It is equally clearthat (SF) does not imply (WF), and vice versa.
This point will more sharply be understood bymeans of graphical illustrations.
On the one hand, let us consider simple
examples of indifference curves on Q with"pointed" portions, or equivalently, those of indifference curves on X with "flat" portions. As
is seen in Fig. 4 (A) & (B), they indicate theconsumer satisfying (WF), but not (WP). On(A)

Indifference curves have(B)" pointed " sections on X .

.

Fig. 4.

the other hand, as is clear in Fig. 5 (A) & (B),symmetrical examples of indifference curves on
Q with "flat" portions (and hence those of indifference curves on X with "pointed" portions)indicate the consumer satisfying, (WP) but not(WF). If h happens to be a one-to-one correspondence between Q and X, however, then it
is readily seen that (WF) holds on Q if and only if (WP) holds on X, and also that (SP) holds
on Q if and only if (SP) holds on X. 4)

Indifference curves have
" flat " sections on Q .

The consumer satisfies (WF), but not (WP).

(A) Indifference curves have(A) Indifference curves have"pointed" sections on X.

" pointed " sections on X .

(B) Indifference curves have "flat"(B) Indifference curves havesections on Q.
" flat " sections on Q .

Fig. 4 The consumer satisfies (WF), but not (WP).
Fig. 4The consumer satisfies (WF), but not (WP).

(A) Indifference curves have "flat"
sections on Q.

(A)

Indifference curves have(B) Indifference curves have"pointed" sections on X.
(B)

Indifference curves have

Fig. 5 The consumer satisfies (WP), but not (WF).
" flat " sections on Q .
" pointed " sections on X .
Fig. 5(A)

to Richter (1966) for clarifying this line4) I ,am
The consumer satisfies (WP)
butindebted
not (WF).

Indifference curves have
" flat " sections on
Fig. 5

052

Q.

(B)

ofIndifference
argument. curves have

" pointed " sections on

X.

The consumer satisfies (WP), but not (WF).

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Autumn / Oct. 2023 / No.437

IV Relationship between
the Weak and Strong
Axioms of Revealed
Favorability

In this section, we will study the relationship between the weak and strong axioms of
revealed favorability. Let us suppose that thedemand function h satisfies the homogeneity
assumption (H). Then, it will be seen that thestrong axiom (SF) of revealed favorability
holds if and only if the weak axiom (WF) of
revealed favorability together with a certain"regularity condition" holds. As will be seen,such a regularity condition is a relationship between two kinds of "income compensatedfunctions" that must hold if the strong axiom
holds.
To this end, we first establish the following
important lemma.
LEMMA 1 (the closeness of the set { q ∈Q : ~ qFq0 }).
Suppose that the demand function h satisfies (H) and (WF). Then, for anyq ∈ Q, the set { q ∈ Q : ~ qFq0 } is closed
in Q.
The proof is very technical and omitted
here. For details, see Sakai (2023).
Now, for a given p ∈ Q and a given q0 ∈ Q,
let us define the "M-plus and M- minus sets" ofincomes as follows :M + (p, q0) = { m ∈ M : ~ (1/m) pFq0 } ;
(10)



M - (p, q0) = { m ∈ M : ~ q0F (1/m)p}.
(11)

While the "M—plus set" M + (p, q0) represents the set of incomes m such that (1/m) p is
not indirectly revealed more favorable than q0,
the "M—minus set" M-(p, q0) represents theset of incomes m such that q0 is not indirectly
revealed more favorable than (1/m)p. Then,the "m - plus and m -minus income compensation functions", m + and m -, can respectively
be defined in terms of M + and M - in the following way : 5)
m + (p, q0) = sup { m : m ∈ M + (p, q0) } ;
(12)



m - (p, q0) = inf { m : m ∈ M - (p, q0) }.
(13)

We can give economic interpretations tothose two functions m + and m - in the following fashion. Let p and q 0 respectively be agiven "absolute" price vector and a given "relative" price vector q0. Then, whereas the function
m + (p, q0) stands for the supremum (or the
least upper bound) of incomes m such that (1/m) p is not indiretly revealed more favorable
than q0, the function m - (p, q0) stands for the
infimum (or the greatest lower bound) of incomes m such that q0 is not indirectly revealed
more favorable than (1/m) p. It is clear by definition that M + (p, q0) ⊃ (0, m + (p, q0)) andM - (p, q0) ⊃ (m - (p, q0), + ∞).
Now, we are ready to establish the following useful lemma.
LEMMA 2 (the closeness of the M-plus set
M + (p, q0)).
Let us suppose that the demand function hsatisfies the homogeneity assumption (H) and5) Income compensation functions were first introducedby McKenzie (1957) and Yokoyama (1953), independently,
in terms of preference orderings on the commodity
space.

Revealed Favorability and Indirect Utility

Yasuhiro Sakai053

incomesm such that(1/m) p is "no better than "to the infimum of incomes msuch thatq 0 is greater than or eq

q 0 is " no better than "(1/m) p .

m ∈ M } (R)for any
given pricevectorp, yetthe weak axiom (WF). Then, forTheany (p,q0) ∈ condition
regularity
requiresthat that
there
bealno " ~F - gaps " in0
P × Q, the M-plus set M + (p, q ) is closed in M.
lowing for the existence of "~ F- overlaps" inray { (1/m ) p : m ∈ M } for any given price vector p , yet allowing for thethe relevant ray. Comparison of Fig. 6 and Fig.
existence of " ~ F - overlaps " in the relevant ray. Comparison of Fig. 6 and7 may graphically show what the regularity
The proof is omitted here. For details, see
may graphicallywhat thecondition (R) really means.
condition(R)regularity
really means.
Sakai (2023). According to 7Lemma
2, we find show
00Now, it is high time for us to establish a
M + (p, q ) = (0, m + (p, q )) whenever h satisvery important theorem concerning the relafies (H) and (WF). We are now approaching
to the final goal of the equivalence theorem.
tion the weak and strong axioms of revealed
To reach it safely, we need to pass through thefavorability.
following midpoint.
LEMMA 3 (the existence of m + (p, q0)
and m - (p, q0))
Suppose that the demand function h satisfies the homogeneity assumption (H) and thestrong axiom (SF). Then, for any (p, q0) ∈ P ×
Q, the income compensated functions m + (p,q0) and m - (p, q0) exist and are finite.
The proof of lemma 3 is omitted here. For
details, see Sakai (2023).
Now, we are ready to formulate a regularity
condition in terms of income compensation
functions m + (p, q0) and m - (p, q0) :Fig. 6 The regularity condition (R) is satisfied : m + (p, q0)≧ m - (p, q0).
The
m + (p , q 0 ) ≧ m - ( p , q 0 ) .
0 regularity condition (R) is satisfied :Fig. 6+ (p, q ) ≧(R) For any (p, q0) ∈ P × Q, m
0m - (p, q ).

As will be seen, the regularity condition(R) plays a critical role in making a bridge between the weak and strong axioms of revealed
favorability, (WF) and (SF). In plain English,
it states that for any given price vector (p, q0),
the supremum of incomes m such that (1/m) p
is "no better than" q0 is greater than or equal tothe infimum of incomes m such that q0 is "nobetter than" (1/m) p.
The regularity condition (R) requires thatm+
Fig.
7 The (R)regularity(p,, q 0 ) .
0 )Fig. 7 The regularity
condition
is not condition
satisfied: (R)m +is(pnot, qsatisfied:＜ m- (pthat there be no "~F- gaps" in the ray { (1/m) p:q0) ＜ m - (p, q0).

Now, it is high time for us to establish a very important theorem concerning therelation the weak and strong axioms of revealed favorability.
THEOREM

4(the main equivalence theorem)

Suppose that the demand function hsatisfies (H) .

Then,

the strong axiom (SF)of revealed favorability holds if and only if both the weak axiom (WF)054

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RONSO
favorability and the regularity
condition(R) hold.

of revealed

Autumn / Oct. 2023 / No.437

THEOREM 4 (the main equivalence theorem)
Suppose that the demand function h satisfies (H) . Then, the strong axiom (SF) of
revealed favorability holds if and only if both
the weak axiom (WF) of revealed favorability
and the regularity condition (R) hold.
Proof. (a) (Necessity) Suppose that (SF)holds. Then, (WF) is obviously implied. Besides, by virtue of LEMMA 3, the two incomecompensation functions m + (p, q0) and m - (p,q0) surely exist and are finite.
Now, assume by way of contradiction thatthe regularity condition (R) does NOT hold,so we should find m + (p, q0) ＜ m - (p, q0) forsome (p, q 0) ∈ P × Q . Choose a "middle
point" m ∈ M so that the following inequalities hold:m + (p, q0) ＜ m ＜ m - (p, q0). (14)

By the definitions of m + (p, q0) and m - (p,q0), Eq. (14) implies that (1/m) pFq0 and q0F(1/m)p. This clearly contradicts (SF). So, to
get rid of a contradiction, we must conclude
that the regularity condition (RF) must hold.
(b) (Sufficiency) Suppose that (WF) and(R) both hold. Let q1, q0 ∈ Q be such that q1F
q0 Then, we will show the following relation.
m + (q1, q0) ＜ 1. 

(15)

To this end, let us dare to assume otherwise: namely, m + (q1, q0) ≧ 1. Since (WF)
yields M + (q1, q0) = (0, m + (q1, q0)) by meansof Lemma 5.2, it would follow that 1 ∈ M +(q1, q0), meaning that ~ (1/1) q1F q0, or simply~ q1F q0. Clearly, this contradict our initial as-

sumption. To get rid of a contradiction, wemust conclude that Eq. (15) must hold.
Now, let us recall the regularity condition(R) which says that m + (p, q0) ≧ m - (p, q0).
If we combine this inequality and Eq. (14),then we immediately find the following relation.
m - (q1, q0) ＜ 1. 

(16)

Let us recall that M - (q1, q0) ⊃ (m - (q1,q ), + ∞). Then, this clearly implies 1 ∈ M (q1, q0), meaning that ~ q0F(1/1)q1, or simply~ q0F q1.
To conclude, we have thus seen that under(WF) and (R) , the revealed favorability relation F is non-symmetric, meaning that q1F q0
implies ~ q0F q1. The proof is now complete.
Q.E.D.
0

As far as we know, the relation between the
weak and strong axioms of revealed favorabilityon the normalized-price space has hardly been
investigated in the economics literature. Needless to say, this is exactly the dual to the more
popular relation between the weak and strong
axioms of revealed preference on the commodity space, which has been so intensively exploredin the literature. Such non-symmetric treatment of the two approaches ―― one almost
neglected and another extensively explored ―― seems to be very strange from common
sense 6).
We should point it out that the hypothesisof Theorem 5.4 is not so strong, and indeed
amazingly rather weak. This is because (i) no
continuity condition is imposed on the demand function h, and (ii) X, the range of h,need NOT be convex in Y, the whole com6) Even in the twenty-first century, the relationship be-

tween the weak and strong axioms of revealed favorability on the normalized-price space has only occasionallybeen mentioned in the micro-economics literature. Hereagain, we see extreme difficulty to break through thehard crust of convention.

Revealed Favorability and Indirect Utility

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055

modity space. It is noted that the above
conditions (i) and (ii) are not implied by thebudget assumption (H) and the strong axiom(SF) .

V

Derivation of the
Indirect andDirect Utility Functions

The duality relation between the direct andindirect utility functions has especially beenstudied in connection with special functional
forms of separability. Although such a separability approach is of some mathematicalinterest, its deeper and wider significance ineconomic theory remains to be still debatable7).
In this section, we would like to investigate
more fundamental questions than the separability question aforementioned. First, we will
discuss the problem whether the indirect revealed favorability relation on the normalized
space can be represented by a real-valued function, i.e., the indirect utility function. Second,
we will explore the related problem whether
there is a way of making a bridge between theindirect and direct utility functions from theperspective of the revealed favorability approach taken in the present research.
In this connection, we will state and prove
the following result of very rich substance.
THEOREM 5.5 (the derivation of the indirect utility)
Suppose that the demand function h satisfies the budget assumption (H) and the strong
axiom of revealed favorability (SF). Then,
there exists a real-valued function v on Q,
namely the indirect utility function, such thatthe following series of properties hold.

(i) (minimality) For any q ∈ Q,
h(q) = { x ∈ X : qx ≦ 1, andv(r) ≧ v(q) for any r ∈ Q such that rx ≦ 1}.
(ii)(closeness) For any q ∈ Q, the set { r ∈Q : v(q) ≧ v(r) } is closed in Q.
(iii)(strict concavity) If v (q1) ≧ v(q0), q1,
q0 ∈ Q, q1 ≠ q0, and
qt = (1- t) q1 + t q0, then v (q1) ＞ v(qt).
(iv)(monotonicity) If q1 ≤ q0 and q1, q0 ∈Q, then v (q1) ＞ v(q0).
(v)(heritability) For any q1, q0 ∈ Q,
q1F q 0 implies v (q1) ＞ v(q0).
Since the proof of this theorem is long andcomplicated, it is omitted here. For details,
please see Sakai (2023).
Theorem 5 is a very important theorem,
deriving the indirect utility function and its
many properties. Property (i) indicates thatfor a given normalized-price vector q, thechoice set h(q) constitutes bundles x that minimize the indirect utility of normalized-pricevectors r subject to the budget constraint rx ≦1. Property (ii) means that for a given q, the"inferior" set {r : v(q) ≧ v(r)} is closed in Q ;namely, the indirect utility function is lower
semi-continuous. Property (iii) shows that v isstrictly quasi-concave, whereas Property (iv)says that v is monotonous. Finally, it follows
from Property (v) that the revealed favorability
relation F can be well-represented by the indirect utility v. In other words, there is a sort ofheritability relationship between revealed favorability and indirect utility,
Now, we are in a position to make a bridge
between the indirect and direct utility functions. To carry out such a nice task, we find it

7) For the duality results, see Lau (1969).

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necessary to introduce an additional strong assumption on the demand function h :namely the direct utility function, such that thefollowing series of properties hold.

(D) For any q ∈ Q, h(q) is a singleton, or aset which contains exactly one element.

(i) (maximality) For any q ∈ Q,
h(q) = { x : x ∈ b(q) ∩ X, and u(r) ≧u(q) for any y such that y ∈ b(q) ∩ X }.
(ii) (closeness) NOT DERIVABLE(iii)(strict concavity) If u(x1) ≧ u(x0), x1,
x0 ∈ X, x1≠x0, and xt = (1- t) x1 + t x0,
then u(x1) ＞ u(xt). ,(iv)(monotonicity) If x1 ≥x0 and x1, x0 ∈X, then u(q1) ＞ u(q0). ,(v)(heritability) For any x1, x0 ∈ X, x1H
x0 implies u(q1) ＞ u 0 (q0 ).

Let us recall that the strong axiom (SF) of
reveled favorability implies the unique invertibility of h. Therefore, if h satisfies both (SF)and (D), then h is a one-to-one correspondence
between Q and X.
Let us define that a function g on X as follows:
For any x ∈ X, g(x) = h -1 (x). (17)Then, it is clear that g(h(q)) = q for all q ∈Q, and that h(g(x)) = x for all x ∈ X. Since g is
the inverse of h on X, it immediately follows
from THEOREM 5 that for any x ∈ X, thefollowing equation holds:
g(x) = { q ∈ Q: qx ≦ 1, and v(r) ≧ v(q)for any r ∈ Q such that rx ≦ 1 }. 
(18)Now, we are in a position to define a realvalued function (a direct utility function) u on
X, i.e. that h uniquely maximizes u over B.
More specifically, we would like to establish thefollowing important theorem.
THEOREM 6 (the derivation of the direct
utility)
Suppose that the demand function h satisfies the budget assumption (H) and the strong
axiom of revealed favorability (SF) together
with the strong demand assumption (D). Then,
there exists a real-valued function u on Q,

Revealed Favorability and Indirect Utility

The proof of this theorem seems to beanalogous to that of the proof of THEOREM
5.5, it is omitted here. For details, see Sakai(2023).
Comparison of Theorems 5 and can indicate the fundamental duality that exists between
the indirect and direct utility functions.
(i) First of all, minimizing the indirect utility v of normalized prices q is equivalent to thedirect utility u of commodities x, with the identical budget constraint q x ≦ 1 being imposedin both instances.
(ii) Second, v is strictly quasi-convex on Q
whereas u is stricly quasi-concave on X. v is decreasing on Q whereas u is increasing on X.
(iii) Third, v represents the indirect revealed favorability relation F on Q whereas u
represents the direct revealed preference relation H on X.
It should be noticed, however, that such
nice symmetry between the indirect and direct
utilities is not perfect, and may possibly break
down under the present assumptions including(H), (SF) and (D) .

Yasuhiro Sakai

057

Let us pay attention to Property (ii) of thepresent THEOREM 6, which is unfortunately
not derivable under the present assumptions.
Indeed, although the indirect utility function v
is lower semi-continuous (see the last THEOREM 5(ii), the direct utility function u may not
be upper semi-continuous under the present
assumptions only. For details of this point, see
Sakai (2023).

VI Choice and Rationality:

Effectiveness and Limitations

In the above, we have been assumed thatthe consumer's choice behavior is always rational and consistent. In fact, the weak and strong
axioms of revealed favorability on the normalize d-price space and those of re vea le dpreference on the commodity space may wellrepresent such rationality and consistency.
Besides, as was shown above, the convexity of
the range of the demand function is also a good
indicator of the wise and sensible human judgment. It is recalled that Richter (1971) discussed
many possible kinds of the consumer's rationality including "transitive rational" and "reflexive
rational." In line with his way of argument, we
may also add that the consumer is "convex rational" if his demand range is wide enough tosatisfy the convex condition.
In the light of the long history of economic
theory, there lies the academic struggle between "Econs" and "Humans." According toRichard H. Thaler (2015), our good friend andrespected Rochester graduate, many standard
models tend to use a fictional creature called
homo economics , or simply Econs. Econs are
generally supposed in the majority of economic058

books and papers including Samuelson (1955,
7th edition 1967) and the present book per se.
However, we have to be very careful of the possible danger of going too far or too much. If
we may put it in the strongest terms, we are allowed to regard Econs as a sort of "rational
fools" a la Amartia Sen (1987) in the sense thatpeople rationally choose goods and services byfollowing the weak and strong axioms of revealed favorability or those of revealed
preference. Or equivalently, people are fictionally assumed to optimize their indirect ordirect utilities.
In contrast, Humans are just human beings, or homo sapience. Compared withfictional Econs, Humans are supposed to havea lot of non-rational feelings such as envy, hatre d , optim ism, p essim ism, s ymp athy,
compassion, and the like. In the world inwhich many Humans live, the traditional economic theory might be far from satisfactory.
We need to establish a more comprehensivemodel of human behavior including a varietyof complicated psychologies. As is stressed by
George A. Akerlof and Robert J. Shiller(2009), a nice bridge between economic theory and human psychology must urgently bebuilt.
It is true that getting economic theory
back on its feet again will not an easy task, presumably requiring a very long arduous way
ahead. We believe, however, that where there
is a strong will, there is a nice way out.
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Revealed Favorability and Indirect Utility
Extending the Samuelson Approach the Price-Income Space

Yasuhiro Sakai

This chapter studies the axiomatic foundations of the indirect utility function, based on a
revealed preference approach a la Paul A. Samuelson. We look at a chain of comparisons ofbudgets as if it gives a relation on the normalized-price space (namely, a revealed favorability
relation) rather than a relation on the commodity space (namely, a revealed preference relation).
In analogy to the weak and strong axioms of
revealed preference, the weak and strong axioms of re vea le d favorabilit y are newly
introduced, and a fundamental theorem concerning the relationship between the latter two
axioms is established. Then, the indirect anddirect utility functions are effectively derived
on the basis of the strong axiom of revealed favorability. It is noted that neither thecontinuity of the demand function nor theconvexity of its range is required for the approach taken here.
Keywords: revealed favorability, weak andstrong axioms, indirect and direct
utility functions, revealed preference, Paul A. Samuelson.

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