Advances in MATHEMATICAL ECONOMICS ppsx

Adv. Math. Econ. 7, 47-93 (2005) Advances
in
MATHEMATICAL
ECONOMICS
©Springer-Verlag2005
A method in demand analysis connected
with the Monge—Kantorovich problem
Vladimir L. Levin*
Central Economics and Mathematics Institute of Russian Academy of Sci-
ences,
47 Nakhimovskii Prospect, 117418 Moscow, Russia
(e-mail: )
Received: February 18, 2004
Revised: August 6, 2004
JEL classification: C65, Dil
Mathematics Subject Classification (2000): 91B42
Abstract. A method in demand analysis based on the Monge-Kantorovich
duality is developed. We characterize (insatiate) demand functions that are
rationalized, in different meanings, by concave utility functions with some
additional properties such as upper semi-continuity, continuity, non-decrease,
strict concavity, positive homogeneity and so on. The characterizations are
some kinds of abstract cyclic monotonicity strengthening revealed preference
axioms, and also they may be considered as an extension of the Afriat-Varian
theory to an arbitrary (infinite) set of 'observed data'. Particular attention is
paid to the case of smooth functions.
Key words: demand function, budget set, insatiate demand, utility function,
indirect utility function, rationalizing, strict rationalizing, inducing, strict in-
ducing, Monge-Kantorovich problem (MKP) with a fixed marginal difference,
cost function, constraint set of a dual MKP, concave function, strictly concave
function, positive homogeneous function, superdifferential

Introduction
This article is devoted to concave-utility-rational demand functions. The
problem of demand rationalizing is studied in mathematical economics
Supported in part by Russian Foundation for Humanitarian Sciences
(project 03-02-00027). A part of the material of this paper was presented
at the international conference "Kantorovich memorial. Mathematics and
economics: old problems and new approaches", St Petersburg, January,
8-13,
2004.
48 V.L. Levin
since 1886 (Antonelli); for history and references see [4], [7]. A substan-
tial role here is played by theory of revealed preference; see papers by
Samuelson [31], [32], Houthakker [6], Uzawa [33], Richter [29] and others.
The revealed preference axioms, along with some regularity assumptions,
give conditions for a demand function to be utility-rational. In general,
an utility function rationalizing the demand function need not be con-
cave.
It will be automatically quasiconcave when the revealed preference
relation is convex. If f/ is a quasiconcave utility function that rationalizes
a given demand and if a function /i: R ^' R is (strictly) increasing, then
the composition hoU rationalizes the demand and is quasiconcave as
well. In some cases, such a composition proves to be not merely quasi-
concave but concave [8] (see also [5]). Thus, revealed preference theory
together with the concavifiability criteria enables to obtain conditions
for demand rationalizing by a concave utility provided that the corre-
sponding revealed preference ordering is convex. A different approach to
concave rationalizability is the nonparametric method of rationalizing a
trade statistic; see papers by Afriat [1], [2] and Varian [34], [35].
In [20], [22] we proposed a new method in demand analysis (and in
an abstract rational choice theory). The method is based on duality re-

sults relating to the Monge - Kantorovich mass transportation problem
(MKP), a relaxation of an old 'excavation and embankments' problem
due to Caspar Monge [26]. As is shown in [20], [22], a demand function
is rationalized by a given concave utility function if and only if the cor-
responding indirect utility function belongs to the constraint set of an
infinite linear program, which is dual to the MKP with a fixed marginal
diflFerence and a special cost function. The definition of that constraint
set implies that a function u{p) belongs to it if and only if u solves a
system of inequalities extending the Afriat system to an infinite set of
'observed data'. In [16], [21] we gave a criterion for such a constraint set
to be nonempty. In general, this criterion means some kind of abstract
cyclic monotonicity, and for a special cost function determined by a given
demand, it is closely connected with axioms of revealed preference. In
[27],
[3] the same approach is applied to problem of rationalizing reverse
demand functions by positive homogeneous concave utility functions.
Furthermore, in case of smooth cost function, applying earlier results
by the author [13],[20] yields conditions in diff^erential form (separately,
necessary ones and sufficient ones) for nonemptiness of the corresponding
constraint set hence for concave utility rationalizing the demand.
In the present paper we give further development of that method.
We expose its main features in a more detailed and systematic way than
before and generalize considerably our previous results.
In what follows, P stands for a nonempty subset of
A method in demand analysis 49
intR!^ = {p= {pu' ,Pn) : pi > 0, ,pn > 0},
and P'q:= YliPiQi for any p = {pi, ,Pn), q = (gi,. •., 9n) € R"".
Consider a consumer buying n commodities ^ = (^i, •. •, gn) ^ ^+ at
prices p = (pi,
•

,Pn) ^ ^ and having income I{p) > 0 , and denote by
B{p) her/his budget set,
B{p):={qeRl:p-q<I{p)}.
The function I : P ^ (0, -|-oo) is assumed to be given. Important par-
ticular cases are I{p) = I (constant income; in such a case one can take
/ = 1 by passing to a new unit of money) and I{p) = p-uj (a consumer
with endowment a;
G
M!J:
\ {0}).
Given a set of data M = {{p^q)} C intE!J: x intR!J: is associated
with a multifunction (demand map) D : intWl -^ 2'''^^+,D{p) := {q :
(p^q) € M}. Let P = domD := {p : D{p) ^ 0} and assume M to be
compatible with consumer's budget, so that q e B{p) whenever (p, q) G
M or, equivalently, D{p) C B{p) for all p £ P. Also we suppose that all
D{p),p E P, are closed hence compact.
The economic interpretation of D is as follows. Being faced a price
vector p G P, the consumer prefers each bundle of commodities from
D{p) to any bundle from B{p) \ D{p). When such consumer's choice is
concave-utility-rational? In other words, when there is a concave utility
function U on W^ that has nice properties and rationalizes D? The latter
means that maximum of U on B{p) is attained at bundles q G D{p), i.e.,
D{p) C Arg max
U\B{p)
Vp
G
P.
Of course, any D is rationalized by a constant function, and this is why
we say about a utility function with nice properties. Another way to
exclude this degenerate example is to consider the notion of strict ra-

tionalizing. We say U strictly rationalizes D, ii D{p) = ArgmaxC/|5(p)
whenever p e P.
Rationalizing a trade statistic M = {(p*,^*) : t = 1, ,m} where
p*
^p^ iort ^ s (see [1], [2], [34], [35]) is an important particular case of
the general demand rationalizing problem (this variant of the problem
corresponds to a finite P, which is the p-projection oi M).
Our approach to the problem is as follows. Given a demand map D
(or a single-valued demand function f : P -^ intR!J:) and a Lagrange-
Kuhn-Tucker multiplier A : P ^ R+, we take as a cost function (p one
of five specific functions as follows:
50 V.L. Levin
c\{p,p') : = KP') ( n^in y
•
q - I{p') ) ,
\qeD{p) J
C(p,p'):=p'-(/(p)-/(y)),
cHp,p')
=
{p'-p)-m-
Properties of the corresponding constraint set
Qo{^) := {ueR^: u{p) - u{p') < ^{p,p') \/p,p' € P}
and conditions for this set to be nonempty are the key points in our
study.
The structure of the paper is as follows. Section 1 contains prelim-
inary information on the Monge-Kantorovich problem including condi-
tions for Qo{(p) to be nonempty. Section 2 is devoted to concave-utility-
rational demand maps. We give criteria for rationalizing (strict rational-
izing) a demand map D : P -^ 2'"*"*+ by a concave utility function U
with dorall D D{P) in terms related to nonemptiness of Qo{c\) (Theo-

rem 1). Also connections with revealed preference axioms are discussed
and some information on the corresponding Lagrange-Kuhn-Tucker mul-
tiplier is obtained. Section 3 deals with single-valued demand functions.
In Theorem 2 we give a criterion for such a function to be concave-
utility-rational (strict rational), and in Theorem 3 we describe all utiUty
functions (within a broad class of concave functions) that rationalize a
given insatiate demand function. In Theorem 4 we characterize demand
functions that are (strictly) rationalized by non-decreasing strictly con-
cave utility functions. Also we study demand functions that can be ratio-
nalized by continuous (Theorem 5) and by smooth (Theorem 6) utility
functions. In that Section, all the results are based on nonemptiness
conditions for the set QO(CA) and its subset
QI(CA) := {u e QO(CA) : fip) ^ f{p') ^
u(jp)
- u{p') < ^x{p,p')}.
Section 4 is devoted to demand functions that are rationalized by pos-
itive homogeneous utility functions. Here the cost function ^ is used.
In Theorem 7, that generalizes the corresponding variant of the Afriat-
Varian theory, necessary and sufficient conditions are given for an insa-
tiate demand function / to be rationalized by a positive homogeneous
(continuous) concave utility function, which is strictly positive on /(P).
These conditions are equivalent to nonemptiness of Qo(0- ^^ Theorem 8
conditions for Qo(0 ^^ t)e nonempty are established for a smooth /, and
A method in demand analysis 51
in Theorem 9 we describe all positive homogeneous use concave utility
functions (within some natural class) that rationalize a given demand
function. Finally, Section 5 is devoted to a stronger variant of demand
rationalizing. In that variant, the budget constraint is rejected and the
gain to be maximized by a consumer equals utility minus expences. We
say that / is induced (resp. strictly induced) by a utility function U

if f{p) e ArgmaxC/P \/p e P (resp. if f{p) = argmaxC/^ Vp G P),
where the gain U^{q) := U{q)
—
p
-
q. Theorem 10 characterizes func-
tions f : P -^ intM!J: that are induced by upper semi-continuous (use)
concave utility functions U with domU 2 f{P)- Here cost functions
C and C"^ are used. Among other characterizations. Theorem 10 asserts
that / has the stated property iff Qo{0 is nonempty or, equivalently,
iff Qo(C*^) is nonempty. In Theorem 11 conditions for / to be strictly
induced by U are studied. A necessary condition is nonemptiness of the
set Qi(C) := {u G Oo(C) : f{p) ^ f{p') =^ u{p) - u{p') <
C(p,p')},
and
if f{P) is open or convex and closed, that condition is also sufficient. In
Theorems 12, 13, and 14 we deal with the case where P is a convex do-
main and / is smooth (C^). Theorem 12 says that / is induced by a use
concave utility function C/with domU D f{P) (or, equivalently, Qo(C)
is nonempty) if and only if for every p
G
P the matrix {dfi{p)/dpj)ij is
symmetric negative semidefinite. In Theorem 13 we show that if these
matrices are symmetric and negative definite, then / is strictly induced
by an utility function with the stated properties and Qi {Q is nonempty.
Theorem 14 says that in case where f{p) = (/i(pi), , fnipn)), Qi(C)
is nonempty if and only if every fi is non-increasing.
1.
Preliminary information on the
Monge-Kantorovich problem

Let X and Y be closed domains in spaces R"^ and R"^, cri and
G2
positive
Borel measures on them, aiX = a2Y, and c : X x y ^ R a bounded con-
tinuous cost function. The Monge-Kantorovich problem MKP{c\ cri, cr2)
is to minimize a linear functional
{c,ß):= I c{x,y) ß{d{x,y)) (1.1)
JxxY
over the set r(cri,cr2) of positive Borel measures /i on X x y satisfying
TTi/x = <Ji, 7r2/i =
(T2'
Here, 7ri,7r2 are the natural projecting maps of
XxY onto X, y, and TTI//, 7r2// are the corresponding marginal measures:
for any Borel sets Bx C X and By C y,
52 V.L. Levin
{iT2ß)BY
: =
^T^2^{BY)
= fi{X X BY).
The optimal value of MKP{c; 0-1,^2) is denoted as C(c; ai,a2) so that
C(c;ai,a2) := inf{(c,//) :/x
G
r((7i,(72)}. (L2)
This is the Monge-Kantorovich problem with given marginals ai,
(72
and
a cost function c. It is a relaxation of the Monge problem MP{c;
CTI,
(72)
that consists in minimizing the functional

:r(/):= [ c{xj{x))ai{dx) (L3)
Jx
over the set ^(cri,(j2) of measure-preserving Borel maps f : X
—^
Y.
A map / is called measure-preserving if f{(Ti) = (72, that is (72By =
crif~^{BY) for every Borel set By C F. The optimal value of the Monge
problem is thus
V(c;(7i,(72) := inf{jr(/) : / e ^((71,(72)}. (1.4)
Each measure-preserving map / € $((7i,
(72)
is associated with a measure
/jLf
= ßficFi) € r((7i,(72), where /x/ = (idx x /)(cri). That is, for every
Borel set B CX xY,
fXfB = ai(idx X f)-\B) = ai{x: (x, /(x)) G B}, (1.5)
It is clear that
(c,
/x/) =^(/), which implies C(c;
(7i,
(72)
<
V(c;
cri, (72).
In general, this inequality is strict but in some cases it holds with
the equality sign. Measures /x G F((71,(72) are called (feasible) solu-
tions and measures of the form /x/ with / G ^((71,(72) are called
Monge solutions to MKP{c\ai^a2)' If there exists an optimal solution
to Mi(rP(c;c7i,(72), which is a Monge solution /x/, then / is an opti-
mal solution to MP(c;(7i,(72) and C(c;(71,(72) = V(c;(71,(72). This is an

immediate consequence of the identity (c, /x/) =
.?*(/).
Another type of MKP is the Monge-Kantorovich problem with a
fixed marginal difference. It relates to the case Y = X and is formulated
as follows. Given a (signed) Borel measure p(= cri
— G2)
on X such that
pX = 0 and a (not necessarily continuous) universally measurable cost
function (f : X x X ^^R^ the problem is to find the optimal value
A{ip;
p) := inf I I ip{x, z) fi{d(x, z))
:
fi> 0,
TTI/X
- 7r2/x = p\ .
UxxX )
Both types of MKP, with fixed marginals and with a fixed marginal
difference, were posed by L.V. Kantorovich (see [9], [10], [11], [12]) who
A method in demand analysis 53
examined the case where X — Y is a. metric compact space with its
metric as the corresponding cost function c = cp. In such a case, two
problems are equivalent:
C((^;
(71,0-2) = A{(p;ai -(72),
and there exists a measure ^ ET{ai,a2) which is an optimal solution to
both the problems. Moreover, for each of two problems, a duality theorem
holds true, that is optimal values of the corresponding original and dual
infinite linear programs are equal. These equivalence and duality remain
true when X is a (not necessarily metrizable) compact space and
(p

is a,
continuous (or merely a lower semi-continuous) function on X x X that
satisfies the triangle inequality and vanishes on the diagonal
[25].
(Two
variants of MKP cease to be equivalent when the triangle inequality
is not satisfied.) Generalizations of the duality theorems to MKPs on
non-compact (or non-topological) spaces see [14], [17], [18], [19], [23]. A
crucial role in duality theory for general MKPs with a given marginal
difference is played by the reduced cost function
where x^ = x,
x^
= z and (^(x, x) = 0 Vx E X, and by the constraint
set of the dual linear program
Q{(p) := {u e C{X)
:
u{x^) - ^/(x^) < (p{x^,x^), x^,x^
G
X},
where C{X) stands for the space of bounded continuous real-valued func-
tions on X; see Levin - Milyutin [25] and Levin [14]. (Clearly (p^ = (p
when (p satisfies the triangle inequality and vanishes on the diagonal.)
By analogy with Q((/?), we define a broader set
Qo((p) :={'a€R^ :u{x^)-u{x^) <
(p{x^,
x^), x^,
x'^
G
X}, (1.6)
which is the constraint set of an infinite linear program dual to an ab-

stract (non-topological) version of MKP with a fixed marginal difference
[17].
Clearly Q{^) =
Q{(p*)
and Qo(^) = Qo(<^*)- Notice also that if
(p
is
bounded, continuous and vanishes on the diagonal, then Qo{^) = Q{^).
The multifunction Qo arises in a natural way in various topics (mass
transportation, cyclically monotone operators, dynamic optimization,
approximation theory, utility theory, demand theory); see [3], [13], [14],
[15],
[18], [20], [21], [22], [23], [24\\ Many problems in those fields may
^ Most of the corresponding results with references to the original papers by
the author may be found in a book [28], Chapter 5. Also in [28], Chapter
4,
the duality theory from [25], [14] is expounded.
54 V.L. Levin
be reduced to the single question of whether or not the set Qo(^) is
nonempty for some specific cost function ^.
The following theorems are particular cases of more general results
that are contained in [13], [14], [16], [20], [21], [24].
Given a continuous cost function c on X xY and a map f : X —^Y,
we define on X x X the function
(^^(x, z) := c{z, f{x)) - c(x, /(x)). (1.7)
Theorem A (Levin [21], [24]). Suppose that f e ^(cri,cr2) is continuous
and the support ofai,Z:= spt(cri), is compact. The following statements
are equivalent:
(a)iif{ai) is an optimal solution to MKP{c\ai^a2) hence / is an
optimal solution to MP(c;cri,cr2);

(b)the set Qo{(p^\zxz) is nonempty.
Theorem B (Levin [14], [16]). Given an abstract nonempty set X and
a function (^ : X x X
—>
R satisfying
(p{x,x) = 0
\/XGX,
(1.8)
the following statements are equivalent:
(ajQoi^f) is nonempty;
(h)for every positive integer I and every cycle x^,x^,
x^
in X, the inequality holds:
rj.1 ^Z + 1 _
Y,^{x\x^-^^)>Q.
(1.9)
k=i
(c)for every x £ X, (f^{x,x) = 0.
If, in addition, X is a
topological
space and
ip
is continuous
onXxX,
then every u G Qo{y^) is a continuous function on X.
Theorem C (Levin [13], [20]). Let X be a domain in R^. Suppose that
if is C^ on an open set containing the diagonal P = {(x, x) : x
G
X} and
vanishes on D. Then either Qo{^) is empty or there exists a C^ function

u{x),
unique up to a constant term, that satisfies the equation
Vu(x) = -V^(^(x,z)U=^. (1.10)
In the latter case, Qo{^) = {u{')-\-a : a:
G
R} and</?*(x,z) = u{x)—u{z)
for all
X,
2;
G
X.
The next theorem generaUzes an earlier similar result by the author;
see [13], [20], where stronger regularity assumptions were imposed on (p.
A method in demand analysis 55
Theorem D. Let X be a domain in W^. Suppose that: (i)ip vanishes on
the diagonally
J
ie.,(1.8) is satisfied, (ii)(f is C^ on an open neighborhood
ofD, and (Hi) on D the partial derivatives d^(p/dzidxj exist, and
d'^(fi(x,x) d^(p(x,x) . „ . . r^ 1 / -» N
^!ä^ = ^ä^ foram,je{l, ,n}. (1.11)
The following statements hold then true:
(a) IfQoi^) is nonempty, then for every XEX, the matrix
{d'^(p{x,
x)/
dzidxj)ij is symmetric negative semidefinite.
(b) Suppose, in addition to (i), (ii),
(Hi),
that X is convex and on D
there exist the partial derivatives d'^^p/dzidzj satisfying

^^;ä^ = ^^ä^ VMG{l, ,n}. (1.12)
If for every x^z E X the inequality holds:
(1.13)
then Qo(^) is nonempty.
(c) Suppose, in addition to hypotheses of (b), that the derivatives
dtdx ' dtd'z ^^^ continuous functions of x. Then every u G Qoi^p)
isC^!
Proof,
(a) Given u € QoC^)? we consider for every x E X the function
^^ on X as follows:
g'^iz)
:= u{z) + ^{x, z), zeX. (1.14)
Since u G Qo{^), g^ is C^, and we have
p^(a:):=minp^(2:);
therefore Vg^{x) = 0 (this is the first order condition for z = a; to be the
minimum point oi g^) and the matrix {d'^g^{x)/dzidzj)ij is symmetric
positive semidefinite provided these second partial derivatives exist (this
is the second order condition for minimaUty of ^ = x). Taking into
account (1.10), we get
dg'^jz) ^ d(p{x,z) _ dip{z,z)
dzi dzi dzi
56 V.L. Levin
hence Vg^{x) = 0 and the first order minimahty condition is thus sat-
isfied. Moreover, (1.15), along with (iii), imphes that the derivatives
d'^g^{x)/dzidzj exist and
d^g^{x) _ d^ip{x,x)
dzidzj dzidxj
and taking into account (1.11), the result follows.
(b)Let us consider on X the vector field e{x) = (ei(x), ,en(a:)),
where

dcp{x,x) .
ei{x) = , z =
l, ,n.
OZi
We have
dei{x) _ d^(p{x^x) d^ip{x,x)
dxj dzidxj dzidzj
which, along with (1.11) and (1.12), implies
^=^
«-^^(' ">•
Then there is a C^ function u{x) satisfying (1.10).
It remains to verify that u G
Qoi^)-
To this end, fix x, z G X, denote
x{t) := tx-\- {1
—
t)z, and consider the function
a{t):=g''{x{t)), 0 < t < 1,
where g^ is given by (1.14). We have a(0) = g^{z) = u{z)-^ip{x^ z)^ a{l) =
g^{x)
—
u{x), hence
u{x) - u{z) -
(p{x,
z) = a(l) - a(0). (1.16)
Since a is C^, there is 0,0 < 6 < 1, such that
a(l)-a(0) = ^. (1.17)
Taking into account (1.15) and (1.13), we get
^ = V.^(x(0)-^
^ |, (^9^^(||W) _ Mx{t)Mt))^ (^^ _

^^)
< 0. (1.18)
Now, applying (1.16),(1.17), and (1.18) yields u{x) - u{z) - (p{x,z) <
0,i.e., u G Qo{(f).
(c) Continuity of the functions x i—> d^(p{x, x)/dzidxj and x i->
d^(p{x,x)/dzidzj implies that the right-hand side of (1.10) is C^, and
the result follows. D
A method in demand analysis
57
2.
Concave-utility-rational demand maps
In what follows,
P
stands for
a
nonempty subset of
intR!|: = {p= (pi, ,Pn) : pi
>
0, ,pn
>
0},
and p
•
^ := YliPiQi
^r
any p
=
(pi, ,Pn),^
=
(gi, "-^qn)

e
W^.
The consumer's budget set is defined as
B{p):={q€Rl:p-q<I{p)},
(2.1)
where the income function
/
:
P
—>
(0, +00) is assumed to be given.
Definition 1.
A
nonempty-closed-valued multifunction D :
P -^
2^+
is
called a demand map if
D{p) C B{p)
n
intM!;: for every p
e
P.
(2.2)
A demand map D is called insatiate ifp-Qp = I{p) whenever p E P,qp
£
Dip).
It follows from the above definition
of a
demand map that

all
D{p),p
G
P, are compact subsets in B{p) fl intR!fi.
Definition
2. A
demand map
D is
said to be rationalized by
a
util-
ity function
U : R^ -^ R
U {-00}
if for
each
p e P,
D{p)
C
Arg max C/15(p), i.e.,
D{p)
c\qe B{p) : U{q) = max
[/(g')}
•
(2.3)
A demand map D is said to be strictly rationalized by a utility function
C7
: E!^ -^ R
U
{-00}

if
for each
peP,
D{p)
=
ArgmaxC/|5(p), i.e.,
D{p) =
\qe
B{p) : U{q) = max C/(gO|
•
(2.4)
Definition
3.
Given
a
utility function
U :
W^
-^
R U {—00} and
a
budget map
B
:
P
^^ 2^+, the function u(p)
=
sup^^^^p) U{q) on
P is
called the indirect utility function associated with U.

Let Z) be
a
demand map. For any
A
:
P
—>
R+ we consider on
P
x
P
the cost functions
CA(P,PO
:=
A(PO
(
min p'
•
q
-
I{p'))
(2.5)
and
58 V.L. Levin
c'x{p.p') := cx{p\p) = A(p) ( min p
•
q' - I(p)) (2.6)
(both minima are attained because of compactness of D{p), p e P).
Recall that the superdifferential of a concave function U : W^ -^
R

U
{—00} at a point q e domU is the set
d'Uiq) :={p€R":p-iq- q') < U{q) - U{q') \/q' €
M!^},
or, equivalently, d'U{q) = —d{—U){q), where d stands for the subdiffer-
ential of a convex function; elements p € d'U{q) are called supergradients
of U at q.
Definition 4. Say a function U
:
W^
—^
R
U
{—00} is:
- non-decreasing if U{q + g') > U{q) whenever g,
q^
G W^;
' increasing if U{q -h g') > U{q) whenever q € R!^, g' G int R!J:;
- strictly increasing if U{q + q') > U{q) whenever g, g'
G
R!J:, g' ^ 0.
Theorem 1. Gi^en a demand map D, the following statements hold
true:
(a) Suppose D is rationalized by a concave utility function U : R!fi
—>
RU{-oo} with domUD D{P) where domU := {g
G
R!^ : U{q) > -00}.
Then there exists a function
A

: P
—>
R_|_ such that
X{p)(p
•
q
—
lip)) = 0 for every p e P and every q
G
D{p) (2.7)
and that the indirect utility function u associated with U belongs to
Qo{cx).
In such a case, X{p)p is a supergradient ofU at every q^
G
D{p):
X{p)ped'U{qp) \/qpGD(p).
If in addition, D is strictly rationalized by U, then an implication
holds as follows:
{B{p) \ D{p)) n Dip') ^ 0
=>
u(p) >
u{p').
(2.8)
(b) If
X
: P
—^
R+ satisfies (2.7) and u G Qo{cx), then there exists a
non-decreasing use concave utility function U : R!f:
—>

R U {—00} with
domU D D{P) such that D is rationalized by U and u is the indirect
utility function associated with U. As such a utility function one can take
U{q) := inf {«(p) + A(p)(p
•
q -
/(p))}.
peP
(c) Let
A
and u be as in (b), and suppose, in addition, that u satisfies
(2.8) and that the set D{P) is either open or convex and
closed.
Then
there exists a use concave utility function U such that domU D D{P),
A method in demand analysis 59
u is the indirect utility function associated with f/, and D is strictly
rationalized by U.
(d) Suppose that X: P -^ R+ satisfies (2.7). Then Qo{cx) is nonempty
if and only if, for every positive integer
I
and for every cycle
p^^p^,
-»p^
pi+i _ pi ^^ p^ ^^g inequality holds
I
fc=i
whenever q^ G D{p^)^ fc = 1, , /.
Proof,
(a) Fix p e P and consider the constraint maximization problem:

U{q) -^ max, qeR^, P'q< I{p),
It is clear that each q e D{p) is an optimal solution to this problem,
and that the Slater condition is satisfied: there is an element ^o = 0 in
R!J. such that p-
qo
< lip)- By the Kuhn-Tucker theorem, there exists a
Lagrange-Kuhn-Tucker multiplier A(p) > 0 such that, for every q e D{p),
(2.7) holds and
U{q)>U{q')-^X{p){I{p)-p-q') V^'€ R!^. (2.9)
It follows from (2.9) and (2.7) that U{q) = U{q') for q,q' e D{p);
therefore the equality holds
u{p) = U{qp) for all p e P, (2.10)
where u is the indirect utility function associated with U and
qp
is taken
arbitrarily from D{p). Furthermore, (2.9) implies —u E (5o(c^); therefore
u e QO(CA)- Also, since I{p) > p-qp, (2.9) implies U{qp) > U{q')-\-\{p)p'
{%
—
q') whenever q'
G
R!J:, that is \{p)p
G
d'U{qp).
Suppose now that D is strictly rationalized by U and show that (2.8)
holds true. To this end, take any
qp
G D{p) and any
qp'
G {B{p)\D{p))r\

D{p').
We have u{p) = U{qp), u{p') = U{qpf) and, as g^/ G B{p) \ D{p),
it follows that U{qpf) < U{qp)^ i.e., u{p^) < u{p).
(b) Since u G Qo{cx), it follows from (2.5) and (2.7) that u{p) = u{p')
whenever D{p) fi D{p') ^ 0. Then the function U{q) := n(p), where
p e P and q = qp is an arbitrary element of D{p), is well-defined and
finite on D{P). The inclusion u G QO(CA) is equivalent to the inclusion
—u G Qo(c^)j which, in turn, can be rewritten as
Uiqp) > U{q) + A(p)(7{p) -p-q) (2.11)
60 V.L. Levin
for all
Qp
e D{p) and all q e D{P). Since, in view of (2.7), c^ vanishes
on the diagonal, we see from (2.11) that, for every q
G
D{P),
U{q) = inf {u{p) + X{p){p
.
q - I{p))} . (2.12)
peP
We extend now U to the whole of W^ by formula (2.12). Obviously
the extended function is non-decreasing use concave, domU 2 D{P),
and D is rationalized by U. Furthermore, in virtue of (2.11) and (2.7),
we have u{p) = U{qp) = max{C/(g) : q
G
B{p)}^ that is u is the indirect
utility function associated with U.
(c) First suppose that D{P) is open and take [/ as in the proof of (b).
We have to verify that D is strictly rationalized by U. Take q^ G D{p)^
q G B{p) \ D{p). U q e D{P), then q G D{p') for some p' G P, and

from (2.8) it follows that U{qp) > U{q). li q ^ D{P), then by (2.11)
U{qp) > U{q), and it remains to show that the inequality is strict. Indeed,
if U{qp) = U{q), then, by concavity of [/,
C/((l - a)qp + aq) > U{qp) whenever 0 < a < 1. (2.13)
Clearly {l
—
a)qp-^aq G B{p) for every a, 0 < a < 1. Furthermore, since
D{P) is open, D{p) is a compact subset in D{P), q ^ D{p), and qp G
D{p),
there exists a, 0 < a < 1, such that {l
—
a)qp-haq
G
D{P)\D{p).
Hence (1 - a)^p + aq e D{p') fl (5(p) \ D{p)) where p' ^ P^p' ^ p,
and applying (2.8) yields
[/(g'p)
> C/((l
—
a)gp
-I-
aq)^ which contradicts
(2.13).
Thus, in the case where D{P) is open the proof is complete.
If now -D(P) is convex and closed, we define
t/(g):=[/i(g)-dist(g,Z?(P)), (2.14)
where
UM
••=
inf{«(p) + A(p)(p

•
q - I{p))}. (2.15)
Then U{q) < Ui{q) whenever q ^ D{P). Clearly [/ is use concave and
domU = domf/i 2 D{P). According with (b), u is the indirect utility
function associated with C/i, and D is rationalized by Ui. Since Ui{q) =
[/(g) for every q G D(P) and f7(g) < Ui[q) for g ^ D{P), it follows that
1/
is the indirect utility function associated with U and D is rationalized
byC/.
It remains to verify that D is strictly rationalized by U. To this end,
take qp G D{p),q G 5(p) \ L)(p). If g G D{P), then g G D{p') for some
/ G P, and from (2.8) it follows that U{qp) = u{p) > u{p') = U{q), If
q ^ D{P) then, taking into account (2.14) and (2.15), we get U{qp) =
Ui{qp) = u{p) > u{p)+\{p){p'q-I{p)) > Ui{q) = U{q)+dist{q,D{P)) >
U{q).
(d) This is an easy consequence of Theorem B. D
A method in demand analysis 61
There is a close connection between nonemptiness of Qo{cx) and
strong axioms of revealed preference.
Recall that Houthakker's strong axiom of revealed preference (SARP)
[6] relates to a single-valued demand. A traditional form of SARP is as
follows: if /> 2 is an integer, p\.,, ,p^ € P, p^+^
•
{D{p') - D{p^-^^)) <
0, i = 1, ,/- 1, and D{p^) ^ D{p^), then p^ • {D{p^) - D{p^)) > 0.
The next formulation is clearly an equivalent restating the axiom: given
a cycle
p^,
,p^p^"*"^
= p^ in P such that, for at least one

z
€
{1, , /},
D{p^) ^ £)(p*'^^), then the inequality holds
max{p2 . (^1 _ ci\ ,,.J- {q'-' - q%p'
•
{q' ~ q')} > 0, (2.16)
where q'' = D{p^), k = 1, ,/. Notice that if D{p') ^ D{p'-^^) and
D{p') ^ P(p^+^), then
p'-^^'D{p')
> /(p^+^) > p^+i •D(p^+i) and (2.16)
proves to be trivial. Therefore SARP may be reformulated equivalently
as follows: if
p^,
,p^p^"^^ = p^ is a cycle in P such that, for at least
one i e
{1, ,/},
D{p') ^ D{p'-^^) and D{p') e
B{p'-^^),
then for
q^ = D{p^), fc = 1, , /, (2.16) holds true.
This formulation of SARP is extended to a multivalued demand as
follows.
Definition 5. Say a demand D satisfies SARP if the following condition
is
valid.
Given a cycle
p^,
,p^p^"^^ = p^ in P such that, for at least
one i G

{1, ,Z},
{B{p'^^)\D{p'-^^))nD{p') ^ 0, then inequality (2.16)
holds whenever q^
G
D{p^)^
A;
= 1, , /.
Definition 6. Following Varian [34], say a demand D satisfies the
generalized axiom of revealed preference (GARP) if, for every cycle
p^,
,p^p'•*•^ = p^ in P, the inequality
max{p2 . (gl - q% J. (g'-i - g'),pi. (g' - q^)} > 0 (2.17)
holds whenever q^ G D{p^),
A:
= 1, , /.
Proposition 1. Given a demand map D : P -^ intR^J:, the following
statements hold true:
(I)If D is insatiate, A is strictly positive, and QQ{C\) is nonempty,
then D satisfies GARP.
(11)If there exist A and u G QO(CA) satisfying (2.7) and (2.8), then
D satisfies SARP.
Proof.
(I) Let us fix a function ?ZG(3O(CA)- Given a cycle p\
,p^,p^"*"^
=
p^
in P, we have
u{p^) - u{p^^^) < cxip^p''^^) < A(p^+i)p^+i
•
{q^ - q^^^)

62 V.L. Levin
whenever q^ G D{p^)^ k = 1, ,l^q^'^^ := q^. Summing up these in-
equalities yields
I
Yl A(/+i)p'^+i .
{q^
-
q^+^)
> 0,
k=i
and as p^+i =p\g'^^ = q\ and all X{p^) > 0, (2.17) follows.
(II) Taking into account (2.7) and (2.8) we have
«(p'=)-u(/+i) < CA(p^p'=+^) < A(p'=+i)p'=+i-(g'=-g'^+i), k ^ i (2.18)
and
u(p'+^) - u{p') > 0. (2.19)
Summing up inequalities (2.18) and taking into account (2.19), we get
Yl A(p*+')p'=+i
•
(g* - q''+^) > w(p'+i) - u{p') > 0;
kik^i
hence
max A(/+i)/+^
•
(g'^ -
q^-^^)
> 0,
and, as all A(p^) > 0, (2.16) holds true. D
Remark 1. Suppose P and D{P) are finite, and D is insatiate. In such
a case, the relation u G Qo{cx) ioi X : P —> intM^. means positive
solvability of an appropriate Afriat's system of inequalities, and GARP

proves to be equivalent to the existence of a strictly positive
A
such that
Qo{cx) is nonempty; see the proof of (2)=>(3) in [34], p.969.
Dejßnition 7. Given a utility function U : W^
—>
Mu{—oo} rationalizing
a demand map D, we say a function X : P -^ M4. is compatible with U
if it is a Lagrange-Kuhn-Tucker multiplier with regard to U, i.e., if {2.7)
and (2.9) hold true whenever p
G
P and q G D{p).
The next result is an immediate consequence of statements (a), (b)
of Theorem 1.
Corollary 1. Suppose D is rationalized by a concave utility function
U :Wl^ MU{-oo} with domC/ D D{P). LetX:P-^R^ be compatible
with Uj and let u be the indirect utility function associated with U. Then
there exists a non-decreasing use concave utility function U' : W^ -^
R U {-00} such that U'{q) > U{q) for every q G M!;:, U'{q) = U{q)
for q G D{P) (hence
dovaU'
D doiaU), D is rationalized by U', X is
compatible with U', and u is the indirect utility function associated with
U'. The function U' is as follows:
A method in demand analysis 63
U'{q) = inijuip) + X{p){p
•
q - I{p))}.
p€P
Example 1. Let n = 1, P = {p

:
p >
0},
I(j>)
= 1, and
[[1,2],
ifp<i,
D{p)=l[l,l], if|<P<l,
l{^},
ifP>l-
Then D{P) =
(0,2].
It is easily seen that D is (strictly) rationaUzed
by a concave utility function C/, as follows:
U{q)
q, ifO<g<l,
1,
if
1
< ^ < 2,
S-q,
if g > 2.
The indirect utility function associated with U is given by
u{p) =
while the function
1,
ifO<p<l,
^^, ifp>l,
X{p) =
[0,

ifO<p<l,
.p'
ifp>l,
is compatible with U. An easy calculation shows that a non-decreasing
function t/' from Corollary 1 is as follows:
U\q)
\q,
ifO<g<l,
11,
if9>l.
Remark 2. It is clear that if D is rationalized by an increasing utility
function, then it is insatiate.
Proposition 2. Suppose D is insatiate. Let U be an increasing concave
utility function such that: (i)domU D D{P), (ii) for every p e P, U is
continuous on domU f) B{p)j and (Hi) the interior of domU 0 B{p) is
nonempty. The following statements are then equivalent:
(a) D is rationalized by U;
(b)for every p
G
P, the inequality holds
sup
U{q')-u{p)
<
inf
u{p) - U{q')
92€R!f :p-92>/(p)
p-q'^-
lip) q^mi:p-g^<I(p) /(p) - p
•
q^ '

(2.20)
64 V.L. Levin
where
u is the
indirect utility function associated with
U.
If these equivalent statements hold true, then
any
number lying be-
tween
the
left-hand and the right-hand sides
of
(2.20) can be taken
as a
multiplier X{p) compatible with
U.
Proof (a)=^(b).
By the
Kuhn-Tucker theorem, there
is a
Lagrange-
Kuhn-Tucker multipHer A
: P
—>
R+ such that
Uiqp)>U{q) + Xip){I{p)-p-q)
Hence
U{q^)
-

U{qj,)
< A(p)
<
1 Vgp€
U{qp)
-
HP)-
<m
•:D{p),qeKl.
p-q^
,p-q^
>
m,
and
as
p-q^
- I{p)
whenever qp
G
D{p)^q^^q^
^
1^+5
P
•
q^
U{qp)
=
u{p) (see (2.10)), (2.20) follows.
(b)^(a).
Given

p e P
and
q
G
B{p)
we have
to
show that U{q)
<
U{qp),
where qp
is
some (any) element
of
D{p). We assume
q
G domC/,
otherwise
the
inequality
is
obvious.
Fix
qp
G D{p) and
take
g^ > g^;
then P'{q^
— qp)
>

0, and
as
C/ is increasing, U{q'^)
>
U{qp). Taking into
account that u{p)
=
U{qp) (see (2.10) and
p-
qp
=
I{p), we get
„p £(4^>HÖ_^>0.
(2.21)
q^eRl:pq^>I{p)
P
'
Q'^
' I{p) P
'
{Q^
-
Qp)
li
P' q < I{p)
then,
for q^ = q,
from (2.20)
and
(2.21)

it
follows
that U{q)
<
U{qp).
li p
-
q = I{p),
then we find
a
convergent sequence
q^ G int (dom U H B{p)),
q^ -^ q.
Since
q^
G int B{p), we have
p
-
q^
<
I{p);
therefore U{q^)
<
U{qp), and
as
C/
is
continuous on domC/fi 5(p),
Uiq)
= lim

Uiq")
<
U{qp).
D
k—^oo
The next result generalizes Proposition
2 to
the case where
U is
not
supposed
to be
increasing.
Proposition 3. Suppose
D
is insatiate. Suppose also that U
is a
concave
utility function with domC/
D D{P)
and,
for
every
p e P, U is
contin-
uous
on
domU
f)
B{p)

and the
interior
of
domC/
fl
B{p)
is
nonempty.
The following statements are then equivalent:
(a) D
is
rationalized by
U;
(b)for every
p
G
P the
inequality holds
max
<
0
sup
^^'i")-<P)
q-'eRl:
P'q^-I{p)
P'q^>I{p)
K<
inf
y-^(^;), (2.22)
' q^eR^: I{p)-P'q^

p-q^<I(p)
A method
in
demand analysis
65
where
u is the
indirect utility function associated with
U.
If these equivalent statements hold true, then
any
number lying
be-
tween
the
lefl-hand
and the
right-hand sides
of
(2.22)
can be
taken
as a
multiplier X{p) compatible with
U.
Proof,
(a)=^(b). This follows from
the
Kuhn-Tucker theorem.
(b)=>(a).

For
every
p E P we
take
X{p)
from
the
condition
max
<
0,
sup ^ÄZ^
<A(p)<
inf
"(^)-^(^^)
g-GR?: P'Q^-HP)
I
q'eR^:
I{p) -
P
'
0^
pq^>I{p)
J
P-Q^<HP)
(2.23)
It follows from (2.23) that
A(p) > 0 and
U{qp)
= u{p) > U{q) +

\{p){I{p)
-p-q)
(2.24)
whenever
qp G
D{p),q
G
Wl;.,p
•
q ^ I{p)' If we
show that (2.24) holds
true
for all q € M!f:,
then
the
implication will follow from
the
Kuhn-
Tucker theorem.
For q ^ dom U
(2.24)
is
trivial.
If now p
-
q = I{p)
and
q e domf/,
then
q G

domU
D B{p) and we can
find
a
sequence
q^
G
int
{dom.Ur[B{p)) such that
q^
converges
to q.
Since
q^
G intB{p)^
we have
p
-
q^ < I(p)\
consequently, every
q^
satisfies (2.24).
Now, by
using continuity
of U on dom [/
D
5(p), we get
U{qj,)
> lim
(C/(g'=)

+
A(p)(/(p)
- p
•
q")) = U{q) +
Xip){I(p)
-p-q),
fc—>00
and
the
proof
is
completed.
D
Remark
3.
Clearly (2.20) along with (2.10) implies (2.22); therefore
Proposition
2
proves
to be a
consequence
of
Proposition
3.
3.
Concave-utility-rational demand functions
In this Section, demand
is
considered

as a
single-valued function
of
prices.
Definition
8.
Given
a set P C intW^ and a
budget
map B : P -^
2^+,
B \py-^ B{p)j ^^
^^2/ that
f
:
P ^^ W^ is a
demand function
if
f{p)
e Bip) n
intM!J:
for all p e P. (3.1)
A demand function
f is
called insatiate
if p
•
f[p) = I{p) for all p E P.
Remark
4-

Here
/ is not
assumed
to be
continuous. Moreover,
any
f
: P -^ intR!f. can be
considered
as a
demand function with regard
to
66 V.L. Levin
a budget set B{p)
=
Bf{p) := {q
e Wl
:
p
-
q <
p
-
fip)}
determined by
the income function I{p)
=
If{p) :=
p
•

f{p). In such
a
case, condition
(3.1) is satisfied automatically.
Definition
9.
We say
a
demand function
f is
rationalized by
a
utility
function U : W^
—^
R
U
{—oo}
if for
each
p
G
P,
f{p)eAvgmaxU\B{p),
(3.2)
i.e.j U{f{p)) > U{q)
for
all q
e
B(p). We say

f is
strictly rationalized
by
a
utility function U
:
W^
—>
E
U
{—oo}
if for
each p
£ P,
f{p) = 8iTgmaxU\B{p),
(3.3)
^.e.
U{f{p)) > U{q) for all q
e
B{p)
\
{/(p)}.
Let
/
be
a
demand function. For any A
: P
—>
R_|_ we consider on

P X
P
the cost function
<xip,p') ••=
WW
•
m -
lip'))-
(3.4)
Theorem 2. Given a demand function
f,
the following statements hold
true:
(a)
If f is
rationalized by
a
concave utility function
U :
W\.
-^
M
U
{—oo} with domC/
D f{P),
then there exists a function X
: P -^
R4.
such that
mip-m-Hp))=o

^pep (3.5)
and that
the
indirect utility function
u
associated with
U
belongs
to
Qo(Cx)'
If^ ^^c/i
a
case J X{p)p
is
a supergradient
ofU
at f{p):
Xip)ped'U{f{p)) VpeP.
(3.6)
If
in
addition^
f is
strictly rationalized by U, then an implication
holds as follows:
fip') 7^
m, fip') e
B{p)
^
u(p) >

u(p').
(3.7)
(b)
If
X
:
P
—^
R+ satisfies (3.5) and u G (5O(CA); then there exists
a
non-decreasing use concave utility function
U :
W^
—>
R U {—00} with
domU D
f{P)
such that
f is
rationalized by U, X
is
compatible with
U,
and
u is
the indirect utility function associated with U. As such a utility
function one can take
U{q)
=
inUuip) + X{p){p

•
q
-
/(p))}.
(3.8)
peP
A method in demand analysis 67
(c) Let
A
and u he as in (b), and suppose, in addition, that u satisfies
(3.7) and that the set f{P) is either open or convex and
closed.
Then
there exists a use concave utility function U such that domU D f{P), u
is the indirect utility function associated with U, and f is strictly ratio-
nalized by U.
(d) Suppose that
X
:
P -^ R_f- satisfies (3.5). Then QO(CA) is nonempty
if and only if for every positive integer
I
and for every cycle
p^^p^,
"",p\
pZ+i _ pi
j^rjri
p^ the inequality holds
J2 xip'^')?'^'
•

ifip") - fip"^'))
>
0.
fc=l
Proof This is a direct consequence of Theorem 1 taking into account
that, for a single-valued multifunction D = f, the function
CA,
as given
by (2.5), turns into
CA-
D
Remark 5. If / is rationalized by a concave function C/, which is differ-
entiable at /(p), p £ P, then (3.6) impUes
w . V^(/(P))-/(P) ^^p
P'
f{P)
In the next theorem we consider a class of concave functions U :
R^ -^ E with nonempty sets d'U{q)nmtR%, q G intR!^, and completely
describe those functions inside the class that rationalize a given insatiate
demand function.
Theorem 3. Suppose U : R!^ —^Risa use concave utility function
such that, for every q e
intR!f.,
d'U{q) flintR!J: is nonempty. Given an
insatiate demand function / : P
—>
int R!f:, the following statements are
equivalent:
(a) U rationalizes f, and there is a strictly positive multiplier X{p),p
G

P,
compatible with U;
(b) U is represented in form
U{q) = iniJu'ip') + \'{p')p'
•
{q -
f'{p'))},
q € R!J:, (3.9)
where P C P' C intM!J:, f : P' -^ intE!J:, f'\P = f, X' : P' -y intR+,
u' e <5O(CA')>
CA'(P,P') :=
X'{p')p'
•
if'ip) -
f'ip'))
^p,p' e P'.
(Without loss of generality, one can take A'(p') = 1 for allp' ^ P' \ P.)
V.L. Levin
Proof.
(a)=^(b) For every q G int R!J:\/(P) we chose
p^
€ d'U{q)nmtWl,
define
P':=PU{p,:qemtRl\f{P)},
consider the multifunction D : P' -^
c^intR^
^
where D{p) = f{p) for
p € P, D{p) = {q:p = pq} for p e P'\P, and take
f'\P'-^

int
W\.
to be
an arbitrary selection of D. Also we set \'{p) = A(p) for p G P,\\p) = 1
for p e P'\P, and define tz'(pO = U{f{p')) for all p' G P'. Since,
by Theorem 2, A(p)p G d'U{f{p)) for every p e P and, by definition,
p'
G d'U{f'{p')) for every p' G P' \ P, we have
U{q)-U{f{p'))<\\p')p'-{q-f\p'))
whenever q
G
M!}:,p' G P'. This implies
C/(g) < if^,{u'{p') + A'(p'y
•
{q - /'{p'))} Vg € Rl- (3.10)
It follows from [30], Theorem 10.2, that U and the right-hand side of
(3.10) are continuous functions of q. Therefore, it suffices to prove (3.9)
for all q
G
intR!f:.
For q = f{p) G /(P) we have
U{q)
=
u'ip)
=
min
Wip') + ^(^0^
•
{q
- f{p%

and (3.9) is thus satisfied. If q e intRlJ. \ f{P), then pg € d'U{q) D
d'Uif'ipg)). We get C/(/'(p,)) - U{q) < p, • (fipg) - q) and U{q) -
Uif'iPg)) <P,-iq-
f'ip,))
hence C/(g) - C/(/'(p,)) = p,
•
(g - /'(p,))
or, equivalently,
U{q) = u'ipg) +pg-{q-
f'ipg))
= imniu'ip') +
X'{p')p'
•
(q -
f'{p'))},
p GJr"
and (3.9) is satisfied in this case, as well. The relation u'
G
QO(CAO is an
immediate consequence of (3.9).
(b)=^(a) For q
G
B{p),p
G
P, (3.9) implies
U{q) < u'{p) + \'{p)p . {q -
f{p)).
(3.11)
Furthermore, it follows from (3.9) that U{f{p)) < u'{p), and taking into
account that u' G QoiCy)^ we get

Uif'ip)) < u'ip) < u'ip') + \'(p')p'
•
if'ip) -
f'ip'))
whenever p' € P'. Since, again by (3.9), infp'6p/{u'(p')+A'(p')p'-(/'(p)-
f'ip'))}
= Uif'ip)), we get
u'{p) = Uif'ip)). (3.12)
A method in demand analysis 69
Now (3.11) together with (3.12) imply
t/(9)<C/(/(p))+A'(p)p.(^-/(p))<f/(/(p))-fV(p)(p.g-/(p))<f/(/(p)),
hence / is rationalized by U and A = A'|P is strictly positive and com-
patible with U. D
Remark 6. Clearly /' can be considered as an insatiate demand function
with regard to the price set P' and the income I'{p') := 'p''f'(p')^p' G P'-
Statement (b) impUes that /' is rationalized by 17, A' is compatible with
C/, and u' is the indirect utility function associated with U.
Definition 10. Given a concave function U : W^
—^ M U
{—oo} and a
(not necessarily convex) open set M C dom U, we say that U is strictly
concave on M if, for every q £ M and for every supergradient p €
d'U{q)j the inequality holds:
Uiq)-Uiq')>p-{q-q') V9'€ R^ \ {g}.
If M is convex, this definition turns into the standard one:
U{{l-t)q-\-tq')>{l-t)U{q)-htU{q') whenever 0<t<l,
q,q'eM,
q ^ q\
Given a function X : P -^ M-|_ satisfying (3.5), we consider the set
QI(CA) := {u e QO(CA) : u{p)-u{p') < Cx{p.p') whenever f{p) ^

f{p')},
(3.13)
Theorem 4. Suppose f{P) is open, the following statements hold then
true:
(a) If f is rationalized by a concave utility function U : R!f.
—^
R U
{—oo} such that domU D f{P) and U is strictly concave on f{P) (in
such a case, f is strictly rationalized by U), and if X : P -^ M^. is
compatible with U, then the indirect utility function associated with U
belongs to QiiCx)-
(bJIfXiP—^ R-i- satisfies (3.5) andQi{(^x) is nonempty, then for ev-
ery u
G
QI(CA) there exists a non-decreasing use concave utility function
U : R!f: -^ RU {—oo} such that: (i)domU 2 f{P) and U is strictly con-
cave on f{P), (ii)X : P -^ R+ is compatible with U,
(Hi)
f is (strictly)
rationalized by U, and (iv) u is the indirect utility function associated
with U.
Proof (a) It follows from Theorem 2 that X{p)p e &U{f{p)) \/p e P
(see (3.6)), and as f/ is strictly concave on /(P), we get
U{f{p)) > U{q) - X(p)p
•
{q - f(p)) (3.14)
70 V.L. Levin
whenever p e P,q eW^.q ^ f{p). Taking into account (3.5), inequality
(3.14) is rewritten for q = /(p') in form
uifip')) - uifip)) < \{p)p

• {f{p')
- f(p)) = UP\P)
^P,p'eP,f{p)^fip'),
that is U{f{-)) € (3I(CA)- It remains to notice that U{f{')) is the indirect
utiUty function associated with U.
(b) Let us consider some u E Qi{(^x). According with Theorem 2 (b),
the function f/, as given by (3.8), has all the stated properties with the
exception of strict concavity. Let us verify that U is strictly concave on
/(P).
To this end, take p e P and q
G
R!f:, q ^ f{p), and consider
q':=f{p) + t{q-f{p)),teR^.
Since f{P) is open, q* = f{p') e f{P) when ^ > 0 is small enough. For
such a t we have f{p^) ^ f{p) and /(p') - f{p) = t{q -
f{p)).
Taking
into account that u € Qi{Cx)^ we get
Uifip)) =
uip)
>
uip')
-
Cxip',?)
= Uifip')) -
HP)P
•
ifip') - fip))
> (1 - t)Uifip)) + tUiq) - tXip)p
•

iq -
fip)),
hence tU{f{p)) > tU{q)
—
tX{p)p
•
{q
—
/(p)), which, after dividing by t,
shows that U is strictly concave on f{P). D
Theorem 5. Suppose P is compact^ / : P
—^
int
1R!J:
and X: P -^R^ are
continuous^ and (3.5) is satisfied. IfQoiCx) ^^ nonempty, then it consists
of continuous functions. In such a case, for every u
G
QO(CA) there exists
a non-decreasing continuous concave utility function U : R!f: -^ R such
that: (i) A is compatible with U, (ii) f is rationalized by U, and (Hi) u
is the indirect utility function associated with U.
If, in addition, A is strictly positive on P (in such a case, (3.5) means
that f is insatiate), then for every u
G
QO(CA) there exists a function U
that has all the stated properties and satisfies the strict inequality:
Uifip) + q)> Uifip)) Vp 6 P, 9 e
E;,
9 7^ 0.

Proof.
It follows from the continuity of / and A, along with (3.5), that
CA
is a continuous function on P x P. Then every u
G
QO(CA) is a continuous
function on P, and w{p,q) := u{p)
-\-
X{p)p
•
{q
—
f{p)) is a continuous
function on P x R!J Now compactness of P together with (3.5) implies
that the function
U{q) := mm w{p, q) = mm{u{p) + X{p){p
•
q - I{p))} (3.15)