Local and Global Consumer Preferences
Reinhard John∗
Department of Economics
University of Bonn
Adenauerallee 24-42
53113 Bonn, Germany
email: [email protected]
March 2006
Abstract
Several kinds of continuous (generalized) monotone maps are characterized by partial gradient maps of skew-symmetric real-valued bifunctions
displaying corresponding (generalized) concavity-convexity properties. As
an economic application, it is shown that two basic approaches explaining
consumer choice are behaviorally equivalent.
1
Introduction
In order to explain consumer behavior, two basic models coexist in the economic
literature: the local and the global theory [10].
The global theory, described in more detail in Section 2, is the standard
approach which assumes that a consumer is able to rank any two alternative
commodity bundles in a convex consumption set X ⊆ Rn . By contrast with
the usual textbook version, we do not suppose this ranking to be transitive.
According to Shafer[9], it is formalized by a continuous and complete binary
relation R on X that can be numerically represented by a continuous and skewsymmetric real-valued bifunction r on X, i.e. xRy if and only if r(x, y) ≥ 0 where
this is interpreted as ”x is at least as good as y”. We call R a global preference
on X.
∗
I would like to thank Nicolas Hadjisavvas and Juan Enrique Martı́nez-Legaz for helpful
discussions. In particular, I am grateful to N. Hadjisavvas for a valuable suggestion leading to
an improvement over an earlier version.
1
Assume now that the consumer faces a nonempty set of feasible alternatives
Y ⊆ X from which he is allowed to choose. Then, the interpretation of R implies
a choice of an x ∈ Y such that x is as least as good as any other alternative
y ∈ Y . In terms of the numerical representation r of R, the choice set assigned
to Y in the global theory is given by
C r (Y ) = {x ∈ Y | ∀y ∈ Y : r(x, y) ≥ 0},
i.e. C r (Y ) is the solution set of the equilibrium problem EP(r, Y ) defined by r
on Y .
The local theory has been introduced by Allen[1] and further developed by
Georgescu-Roegen[4],[5] and Katzner[6]. By contrast with the global approach,
a local preference only requires that the consumer is able to rank alternatives
in a small neighborhood of a given bundle relative to that bundle. As will be
explained in Section 3, this idea can be represented by a continuous function
g : X → Rn such that y in a neighborhood of x is interpreted to be better than
x if and only if g(x)T (y − x) > 0. For Y ⊆ X the choice set assigned to Y in the
local theory is then characterized by
C g (Y ) = {x ∈ Y | ∀y ∈ Y : g(x)T (y − x) ≤ 0},
i.e. C g (Y ) is the solution set of the Stampacchia variational inequality problem
VI(g, Y ) defined by g on Y .1 Furthermore, as argued by Allen and GeorgescuRoegen, stable choices require g to be at least pseudomonotone.
Conceptually, the local approach seems to be weaker than the global one.
However, as shown in Section 5 (Theorem 2), both are behaviorally equivalent if
the global preferences are represented by a class of bifunctions which are defined
as diagonally pseudo-concave-convex in Section 4. It means that any such global
preference representation r yields a local preference representation gr such that
C gr (Y ) = C r (Y ) for every convex Y ⊆ X and that, conversely, for any given
local preference defined by g there is a diagonally pseudo-concave-convex global
preference representation rg such that grg = g. Put differently, for every convex
Y ⊆ X the solutions to EP(r, Y ) and VI(gr , Y ) coincide as well as those to
VI(g, Y ) and EP(rg , Y ). Since gr is obtained as the partial gradient map given by
gr (x) = ∇1 r(x, x), this result confirms that ”VI can be viewed as the differential
form of EP” (I. Konnov in [7], p. 560).
Taking this view for granted, it is natural to extend the well known relationship between (generalized) convexity properties of real-valued functions and
cyclic (generalized) monotonicity properties of their gradients to the case of nongradient maps. In Section 4 several kinds of diagonally (generalized)-concaveconvex skew-symmetric bifunctions are introduced (Definition 1). It is shown
1
Notice the reverse inequality compared to the usual definition. This convention is employed
throughout the paper.
2
that their partial gradient mappings characterize continuous maps with corresponding (generalized) monotonicity properties (Theorem 1). From a purely
mathematical viewpoint, this section contains the main results. They provide
the essential steps for the intended application to consumer theory in Section 5.
2
Global Preferences
Consider an economy with a finite number n of commodities. A vector x =
(xi )ni=1 ∈ Rn is called a consumption bundle and interpreted as the consumption
of xi units of commodity i for i = 1, ..., n where positive components represent inputs (e.g. consumption goods) and negative components represent outputs
(e.g. labor services). In general, not all consumption bundles are physically possible. We assume that the feasible ones are given by an open convex subset X of
Rn , the consumer’s consumption set. For example, X = Rn++ in case that only
consumption goods are considered.
A consumer’s taste is described by a binary relation R on X where xRy is
interpreted as ”x is at least as good as y” or as ”x is (weakly) preferred to y”. x
is called strictly preferred (resp. indifferent) to y if xRy but not yRx (resp. xRy
and yRx). A basic assumption in the global theory is the completeness of R, i.e.
xRy ∨ yRx for all x, y ∈ X.
Equivalently, any two consumption bundles x and y can be ranked in the sense
that either x is strictly preferred to y or y is strictly preferred to x or x and y
are indifferent.
In contrast to the standard textbook model (see e.g. [8]) we do not assume
that R is transitive, i.e. that R is a total preorder. However, as usual in that
model, we always require R to be continuous, i.e. R is a closed subset of X × X,
and call a complete and continuous relation R a global preference on X.
In case that R is transitive, it is well known (see e.g. [3]), that R can be
represented by a continuous utility function u : X → R in the sense that for all
x, y ∈ X
xRy ⇔ u(x) ≥ u(y).
In the general case, Shafer[9] has shown that R has a continuous numerical representation, i.e. a real-valued function r defined on X ×X such that for all x, y ∈ X
xRy ⇔ r(x, y) ≥ 0
r(x, y) = −r(y, x) .
Observe that a utility function yields such a representation by defining r(x, y) =
u(x) − u(y).
Conversely, it is obvious that any continuous skew-symmetric real-valued bifunction r on X defines a global preference that is represented by r and, therefore,
is called a global preference representation.
3
Let us now describe a consumer’s behavior. If the consumer is allowed to make
a choice between the alternatives in a nonempty subset Y of X, it is natural to
assume that he will choose a bundle x ∈ Y such that xRy for all y ∈ Y . If R
is represented by r this is obviously equivalent to r(x, y) ≥ 0 for all y ∈ Y . Put
differently, the choice set
C r (Y ) = {x ∈ Y | ∀y ∈ Y : r(x, y) ≥ 0}
consists of all solutions to the equilibrium problem EP(r, Y ) defined by the representing bifunction r on Y .
It is clear that, in general, the existence of such a bundle is not guaranteed. In
order that the concept of a preference can be considered as an operational definition of a consumer, it should meet at least the requirement that the choice set is
nonempty for all nonempty, compact, and convex subsets Y of X, in particular,
for all compact budget sets.
In the transitive case, this property is implied by continuity. In general, an
additional assumption is needed. As it is well known, a sufficient condition is
that the representation r is quasiconvex in the second variable or, equivalently,
quasiconcave in the first variable (see e.g. [7], Theorem 13.1). At the same time,
this condition also ensures the convexity of C r (Y ) which is another indispensable
property for developing a meaningful equilibrium theory. For that reason, the
utility function u in the standard model is usually also assumed to be quasiconcave.
Most textbook presentations actually suppose that u is differentiable and
pseudoconcave in order to derive the well known first order characterization
x ∈ C r (Y ) ⇔ ∀y ∈ Y : ∇u(x)T (y − x) ≤ 0.
In the general case with a differentiable and pseudoconcave-pseudoconvex representation r one obtains the analogous characterization (see [7], Theorem 13.10)
x ∈ C r (Y ) ⇔ ∀y ∈ Y : ∇1 r(x, x)T (y − x) ≤ 0
⇔ ∀y ∈ Y : ∇2 r(x, x)T (y − x) ≥ 0.
These results replace the global viewpoint embodied in the definition of C r (Y )
by a local one in the sense that knowledge of R is only required near x. As will be
shown in Section 5, they even hold under weaker conditions on r. Furthermore,
they suggest to base a theory of choice directly on a local preference concept to
which we turn in the following section.
3
Local Preferences
Consider a consumer with a consumption set X as in the previous section who
faces alternative consumption bundles in a neighborhood of some given bundle
4
x ∈ X. Assume that the consumer is able to distinguish three kinds of possible
directions in which he can move away from x according to his taste: Preference,
nonpreference, and indifference. Allen[1] has formalized this idea by postulating
the existence of a vector g(x) ∈ Rn such that a direction v ∈ Rn is a preference
(resp. nonpreference, resp. indifference) direction if and only if
g(x)T v > 0 (resp. < 0, resp. = 0).
Observe that multiplying g(x) by some positive number does not change the
local preference, i.e. provided that g(x) 6= 0 for all x ∈ X we could normalize
g by requiring ||g(x)|| = 1 for all x ∈ X. However, in order to be as general
as possible, we do not exclude the case g(x) = 0 which is interpreted as a local
satiation point.
According to Allen[1] and Georgescu-Roegen[4],[5], a consumer’s choice is
described as follows. Assume that x is contained in some given set Y ⊆ X that
represents the consumer’s feasible bundles. They have called x an ”equilibrium
position” relative to Y if no direction away from x to any other alternative y in
Y is one of preference, i.e. if
g(x)T (y − x) ≤ 0 for all y ∈ Y.
Thus, x is an equilibrium consumption bundle iff x is a solution to the Stampacchia variational inequality problem VI(g, Y ) defined by g on Y .
Allen also required the equilibrium to be stable. He added an assumption on
g that was made more precise by Georgescu-Roegen who called it the ”principle
of persisting nonpreference”:
If the consumer moves away from an arbitrary bundle x to a bundle x + ∆x
such that ∆x is not a preference direction, then ∆x is a nonpreference direction
at x + ∆x. Formally stated, this principle says
g(x)T ∆x ≤ 0 implies g(x + ∆x)T ∆x < 0,
or, by denoting x + ∆x = y,
g(x)T (y − x) ≤ 0 implies g(y)T (y − x) < 0.
Clearly, this property is now called strict pseudomonotonicity of g, where, in
contrast to most of the literature, we use this notion in the sense of generalizing
a decreasing real valued function of one real variable.
Later, Georgescu-Roegen[5] generalized his principle of persisting nonpreference by only requiring g to be pseudomonotone, i.e.
g(x)T (y − x) ≤ 0 implies g(y)T (y − x) ≤ 0.
Not surprisingly, he also assumed continuity of g. Thus, we call a continuous and
pseudomonotone mapping g : X → Rn a local preference representation on X.
5
It is important to notice that the local approach yields a satisfactory choice
behavior. Indeed, continuity and pseudomonotonicity of g imply that for any
nonempty, convex, and compact subset Y of X the choice set in the local theory
C g (Y ) = {x ∈ Y | ∀y ∈ Y : g(x)T (y − x) ≤ 0}
is nonempty, convex, and compact (see e.g. [7], Theorem 13.6).
4
Main results
In the sequel, X denotes a nonempty, open, and convex subset of Rn .
Let r : X × X → R be a bifunction on X and define for arbitrary x ∈ X and
h ∈ Rn the single variable real valued functions r1 (x, h) and r2 (x, h) on the set
Ix,h = {t ∈ R| x + th ∈ X} by
r1 (x, h)(t) = r(x + th, x) and r2 (x, h)(t) = r(x, x + th).
We call r diagonally differentiable if for all x, h and t the derivatives r1 (x, h)0 (t)
and r2 (x, h)0 (t) exist and are continuous in x, h, t and if the partial gradients
∇1 r(x, x) and ∇2 r(x, x) exist for all x ∈ X, i.e. for all x, h
r1 (x, h)0 (0) = ∇1 r(x, x)T h and r2 (x, h)0 (0) = ∇2 r(x, x)T h.
A basic derivation of such a bifunction from a continuous mapping g : X → Rn
is given in
Proposition 1
Let g : X → Rn be continuous. Then the bifunction rg : X × X → R defined by
Z 1
g
g (x + s(y − x))T (y − x) ds
r (y, x) =
0
is skew-symmetric, continuous, and diagonally differentiable.2 More precisely, for
every x ∈ X and every h
r1g (x, h)0 (t) = g(x + th)T h = −r2g (x, h)0 (t)
and, in particular,
∇1 rg (x, x) = g(x) = −∇2 rg (x, x).
2
This function was suggested by Nicolas Hadjisavvas.
6
Proof: It is easy to see that rg is skew-symmetric and continuous. Moreover, for
x, h and t such that x + th ∈ X we obtain
r1g (x, h)(t) = rg (x + th, x) =
Z 1
Z t
T
g(x + sth) th ds =
g(x + sh)T h ds.
0
0
Since the derivative of the definite integral with respect to the upper limit of
integration is equal to the value of the integrand at that limit, g(x + th)T h =
r1g (x, h)0 (t) = −r2g (x, h)0 (t) which is continuous in x, h, t. By definition of the
partial gradients, ∇1 rg (x, x) = g(x) = −∇2 rg (x, x).
The next proposition shows that (generalized) monotonicity properties of g
imply (generalized) concavity-convexity properties of the bifunction rg which are
introduced in the following definition. We emphasize that all notions of (generalized) monotonicity are defined in the sense of generalizing a nonincreasing real
valued function of one real variable.
Definition 1
A skew-symmetric and diagonally differentiable bifunction r on X is called
(i) diagonally quasi-concave-convex, if for every x ∈ X and h 6= 0 the function
r1 (x, h) is quasiconcave,
(ii) diagonally (strictly) pseudo-concave-convex, if for every x ∈ X and h 6= 0
the function r1 (x, h) is (strictly) pseudoconcave,
(iii) diagonally (strictly) concave-convex, if for every x ∈ X and h 6= 0 the
function r1 (x, h) is (strictly) concave.
Proposition 2
Let g : X → Rn be a continuous function.
(i) If g is quasimonotone, then rg is diagonally quasi-concave-convex.
(ii) If g is (strictly) pseudomonotone, then rg is diagonally (strictly) pseudoconcave-convex.
(iii) If g is (strictly) monotone, then rg is diagonally (strictly) concave-convex.
Proof: It is well known (see e.g. [2]) that all mentioned (generalized) monotonicity properties of g are characterized by the corresponding property of the
single variable functions gx,h : Ix,h → R, defined by gx,h (t) = g(x + th)T h. By
7
Proposition 1, gx,h is the derivative of r1g (x, h) which implies the corresponding
(generalized) concavity property of r1g (x, h) (see e.g. Proposition 2.5 in [2]). Conversely, we shall show the (generalized) monotonicity of the partial gradient map x 7→ ∇1 r(x, x) for a skew-symmetric and diagonally (generalized)
concave-convex bifunction r. It turns out that this is already implied by weaker
(generalized) concavity-convexity conditions on r. These are, in a sense, local
versions of the well known characterizations for differentiable functions and introduced in
Definition 2
Let r be a skew-symmetric bifunction on X. If the partial function r(·, x) is
differentiable at x ∈ X then r is called
(i) quasi-concave-convex at x, if for all y ∈ X
∇1 r(x, x)T (y − x) < 0 ⇒ r(y, x) < 0,
(1)
(ii) pseudo-concave-convex at x, if, in addition to (1), for all y ∈ X
∇1 r(x, x)T (y − x) ≤ 0 ⇒ r(y, x) ≤ 0,
(2)
(iii) strictly pseudo-concave-convex at x, if for all y ∈ X, y 6= x
∇1 r(x, x)T (y − x) ≤ 0 ⇒ r(y, x) < 0,
(3)
(iv) (strictly) concave-convex at x, if for all y ∈ X, y 6= x
r(y, x) ≤ (<) ∇1 r(x, x)T (y − x).
(4)
Observe that (ii) in Definition 2 also requires (1) to be satisfied. Actually, it
is not difficult to show that in general (1) does not follow from (2). On the other
hand, (2) trivially implies
∇1 r(x, x)T (y − x) < 0 ⇒ r(y, x) ≤ 0
(5)
which means that in general (5) is weaker than (1).
However, in the special case where r(y, x) = f (y)−f (x) such that (2) describes
pseudoconcavity of f , (1) and (5) are equivalent and characterize quasiconcavity
of f (see e.g. Proposition 2.1 in [2]). For our purpose, the stronger condition (1) is
the appropriate one as shown by the next two results. On the one hand, (1) turns
out to be necessary for a diagonally quasi-concave-convex r, on the other hand,
it is needed in the proof of quasimonotonicity of the partial gradient mapping in
Proposition 4.
8
Proposition 3
Let r be a skew-symmetric and diagonally differentiable bifunction on X. Then
the partial gradients ∇1 r(x, x) are continuous in x and
(i) If r is diagonally quasi-concave-convex, then r is quasi-concave-convex at
every x ∈ X.
(ii) If r is diagonally (strictly) pseudo-concave-convex, then r is (strictly) pseudoconcave-convex at every x ∈ X.
(iii) If r is diagonally (strictly) concave-convex, then r is (strictly) concaveconvex at every x ∈ X.
Proof: The partial gradients are continuous by the definition of diagonal differentiability. We only prove (i) and (iii). The statement (ii) is shown analogously.
(i): By Definition 1, r is diagonally quasi-concave-convex if for every x ∈ X
and every h 6= 0 the function r1 (x, h) is quasiconcave, i.e. for all t1 , t2 ∈ Ix,h the
inequality r1 (x, h)0 (t1 )(t2 − t1 ) < 0 implies that r1 (x, h)(t2 ) < r1 (x, h)(t1 ). By
setting h = y − x for y ∈ X, y 6= x and t1 = 0, t2 = 1 we obtain r1 (x, h)0 (t1 )(t2 −
t1 ) = ∇1 r(x, x)T (y − x), r1 (x, h)(t2 ) = r(y, x) and r1 (x, h)(t1 ) = r(x, x) = 0
which yields (1).
(iii): r is diagonally (strictly) concave-convex if for every x ∈ X and every
h 6= 0 the function r1 (x, h) is (strictly) concave, i.e. for all t1 , t2 ∈ Ix,h such that
t1 6= t2 the inequality r1 (x, h)(t2 ) − r1 (x, h)(t1 ) ≤ (<) r1 (x, h)0 (t1 )(t2 − t1 ) holds.
Setting h, t1 , t2 as before yields (4).
Proposition 4
If the bifunction r on X is quasi-concave-convex (resp. (strictly) pseudo-concaveconvex, resp. (strictly) concave-convex) at every x ∈ X then the gradient mapping
gr defined by
gr (x) = ∇1 r(x, x)
is quasimonotone (resp. (strictly) pseudomonotone, resp. (strictly) monotone).
Proof: If r is quasi-concave-convex at every x, ∇1 r(x, x)T (y − x) < 0 implies
r(y, x) ≤ 0 by (1). From skew-symmetry of r it follows that r(x, y) ≥ 0 and,
again by (1), that ∇1 r(y, y)T (x − y) ≥ 0 or, equivalently, ∇1 r(y, y)T (y − x) ≤ 0.
Hence, gr is quasimonotone.
Assume now that r is pseudo-concave-convex at every x. If ∇1 r(x, x)T (y −
x) ≤ 0 then, by (2), r(y, x) ≤ 0 or, equivalently, r(x, y) ≥ 0. Hence, (1) implies
∇1 r(y, y)T (x − y) ≥ 0, i.e. ∇1 r(y, y)T (y − x) ≤ 0. Thus, we have proved that gr
is pseudomonotone.
9
In the strict case, by (3), ∇1 r(x, x)T (y − x) ≤ 0 and x 6= y imply r(y, x) < 0
or, equivalently, r(x, y) > 0. It follows again from (3) that ∇1 r(y, y)T (x − y) > 0
or ∇1 r(y, y)T (y − x) < 0. Hence, gr is strictly pseudomonotone.
Finally, assume that r is (strictly) concave-convex at every x. (4) implies the
inequalities
r(y, x) ≤ (<) ∇1 r(x, x)T (y − x)
r(x, y) ≤ (<) ∇1 r(y, y)T (x − y).
Adding these inequalities and using skew-symmetry yields
0 ≤ (<) [∇1 r(x, x) − ∇1 r(y, y)]T (y − x)
or, equivalently,
[∇1 r(x, x) − ∇1 r(y, y)]T (x − y) ≤ (<) 0.
Thus, gr is (strictly) monotone.
The previous results can be summarized by the following
Theorem 1
Let g : X → Rn be defined on an open and convex subset X of Rn . Then the
following statements are equivalent:
(i) g is continuous and quasimonotone (resp. (strictly) pseudomonotone, resp.
(strictly) monotone).
(ii) There is a skew-symmetric, continuous, and diagonally quasi-concave-convex
(resp. diagonally (strictly) pseudo-concave-convex, resp. diagonally (strictly)
concave-convex) bifunction r : X × X → R such that for every x ∈ X
∇1 r(x, x) = g(x).
(iii) There is a skew-symmetric and continuous bifunction r : X × X → R which
is quasi-concave-convex (resp. (strictly) pseudo-concave-convex, resp. (strictly) concave-convex) at every x ∈ X such that ∇1 r(x, x) = g(x) is continuous in x.
Proof: By Propositions 1 and 2, (i) implies (ii) by choosing r = rg . (iii) follows
from (ii) by Proposition 3. Finally, by Proposition 4, (iii) implies (i).
Remark: In an earlier version of this paper the implication (i) ⇒ (iii) was
obtained more easily by assigning to g the bifunction rg defined by
1
rg (y, x) = (g(y) + g(x))T (y − x).
2
10
The use of rg enables the proof of the stronger implication (i) ⇒ (ii). The question
remains whether (ii) can be further sharpened.
5
Behavioral equivalence of the local and the
global approach
In order to prove the behavioral equivalence of the local and the global theory
we need a further step that is provided by the following
Proposition 5
Let r be a skew-symmetric and continuous bifunction on X that is pseudoconcave-convex at every x ∈ X and let Y be a convex subset of X with x̄ ∈ Y .
Then the following statements are equivalent:
(i)
r(x̄, y) ≥ 0 for all y ∈ Y .
(ii)
∇1 r(x̄, x̄)T (y − x̄) ≤ 0 for all y ∈ Y .
(iii)
∇2 r(x̄, x̄)T (y − x̄) ≥ 0 for all y ∈ Y .
Proof: By (2) and skew-symmetry of r, (i) is immediately implied by (ii).
Assume now that (i) holds. Convexity of Y implies that x̄ + t(y − x̄) ∈ Y for
t ∈ [0, 1]. Thus, it follows from (i) that r(x̄, x̄ + t(y − x̄)) ≥ 0 for all t ∈ [0, 1].
Hence, for all t > 0
1
[r(x̄, x̄ + t(y − x̄)) − r(x̄, x̄)] ≥ 0
t
and, consequently,
1
∇2 r(x̄, x̄)T (y − x̄) = lim+ [r(x̄, x̄ + t(y − x̄)) − r(x̄, x̄)] ≥ 0,
t→0 t
i.e., we have shown that (i) implies (iii).
Finally, ∇2 r(x̄, x̄) = −∇1 r(x̄, x̄) by skew-symmetry of r, i.e. (iii) and (ii) are
equivalent.
Assume that the behavior of a consumer with the consumption set X is described by a choice correspondence C on X which assigns to each nonempty and
convex subset Y of X a (possibly empty) choice set C(Y ) ⊆ Y . Then the local
and the global approach are behaviorally equivalent in the sense of the following
11
Theorem 2
If C is a choice correspondence on X, the following conditions are equivalent:
(i) C = C g for some local preference representation g on X.
(ii) C = C r for some diagonally pseudo-concave-convex global preference representation r on X.
Moreover, the equivalence also holds for strictly pseudomonotone local representations and diagonally strictly pseudo-concave-convex global representations.
Proof: If (i) holds then, by Theorem 1,(i)⇒(ii), there is some diagonally pseudoconcave-convex global preference representation rg such that ∇1 rg (x, x) = g(x)
for every x ∈ X. This implies for every Y that x̄ ∈ C g (Y ) iff g(x̄)T (y − x̄) ≤ 0
for all y ∈ Y , i.e. ∇1 rg (x̄, x̄)T (y − x̄) ≤ 0 for all y ∈ Y . By Proposition 5, the
g
latter statement is equivalent to rg (x̄, y) ≥ 0 for all y ∈ Y , i.e. to x̄ ∈ C r (Y ).
Conversely, if (ii) is satisfied then we obtain for every Y that x̄ ∈ C r (Y ) iff
r(x̄, y) ≥ 0 for all y ∈ Y . By Proposition 5, this is equivalent to the inequality
∇1 r(x̄, x̄)T (y − x̄) ≤ 0 for all y ∈ Y , i.e. to x̄ ∈ C gr (Y ) for gr (x) = ∇1 r(x, x). By
Theorem 1,(ii) ⇒ (i), gr is continuous and pseudomonotone, i.e. a local preference
representation.
The strict case is proved analogously.
References
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