MICROECONOMICS LECTURE SUPPLEMENTS
Hajime Miyazaki
File Name: lexico95.usc/lexico99.dok
DEPARTMENT OF ECONOMICS
OHIO STATE UNIVERSITY
Fall 1993/1994/1995
Miyazaki.1@osu.edu
On Lexicographic (Dictionary) Preference
This short note discusses a lexicographic preference over two commodity bundles. The extension to the n-commodity case is
straightforward.
Lexicographic Preference
Let a = (a1 , a2 ) and b = (b1 , b2 ) be two consumption
vectors. We say that a is lexicographically preferred to b, and write
a f b if and only if either (a1 > b1 ) or (a1 = b1 and a2 ≥ b2 ).
~
From this definition, it follows that a f b if and only if either (a1 >
b1 ) or (a1 = b1 and a2 > b2 ). It is straightforward to verify that the
lexicographic preference is complete, transitive, and reflexive. It is
also convex and strictly monotone. Nevertheless, the lexicographic
preference cannot be represented by a utility function.
An important observation is antisymmetry of the lexicographic
preference. That is, (a ∼ b
⇔
a = b) where a ∼ b means that
a f b and b f a . As a result, a is indifferent to b (a ~ b), if
~
~
and only if a1 = b1 and a2 = b2 . In other words, I(a), the
indifference class of a, consists of "a" only.
I(a)
= {x | x ~ a}
= {x | x = a}
= {a}.
The singleton indifference class already suggests a difficulty in
constructing a utility function that represents the lexicographic
preference. The singleton indifference class means that each
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consumption vector must have its own value, different from all others:
that is, no two points in the plane R2 can have the same value
assigned. But, R2 has a way too many more points than R from which
the numerical values have to be assigned. It is just impossible to
assign different numbers to all points in R2 plane from the real line R.
There will not be enough real numbers assignable to all consumption
vectors. This intuition can be formally validated by a reductio ad
absurdum argument. It will become apparent that the existence of
utility function representation and the existence of non-degenerate
indifference curve are completely synonymous.
The other important property of a lexicographic preference is
that it is not a continuous preference. To see this, check whether the
upper contour set
A(a) = {x | x f a } is closed.
~
2
a2
a
A(a)
0
a1
1
Given a = (a1 , a2 ), any vector to the right of the vertical line at a1 is
strictly preferred to a. Similarly, any vector directly above a is also
strictly preferred to a. All other vectors are strictly less preferred to
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a. Thus, A(a) is the shaded half-space with the thick vertical half-line
above the “a” vector included. The upper contour set A(a) is neither
open nor closed.
Since A(a) is not closed, the preference ordering of a
converging sequence can be reversed in the limit. To illustrate,
consider the following sequence.
x ( n) = ( a1 +
1 a2
1
,
+ )
n 2
n
as n = 1, 2, 3, ⋅⋅⋅⋅⋅⋅. For any n, a1 + (1/n) > a1 . Thus, x ( n ) f a
for any n. The limit of x(n) as n goes infinity is
x = ( a1 ,
But, x = ( a1 ,
a2
).
2
a2
) p ( a1 , a 2 ) = a .
2
2
a
a2/
2
0
•a
x x(n x(2
•) • •
a a1+(1/2)
x(1
)•
a1+1
1
The diagram illustrates the following sequence and preference
reversal.
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x (1)
x ( 2)
x (3)
M
x (n )
M
f
f
f
M
f
M
x
p a.
a
a
a
M
a
M
and
Proposition: A lexicographic preference on the two-commodity space
cannot be represented by a utility function.
Here I provide a heuristic proof that relies on reductio ad absurdum.
Suppose that the lexicographic preference can be represented
by a utility function u : R+2 → R . For each x 1 , the lexicographic
preference says that ( x1 , 1) f ( x1 , 0) if and only if u(x 1 , 1) >
u(x 1 , 0). Then, for each x 1 we can assign a nondegenerate interval
R(x 1 ) = [u(x 1 , 0), u(x 1 , 1)].
Next, take x$1 such that x$1 > x1 , so that
u( x$1 , 1) > u( x$1 , 0) .
Again, R( x$1 ) = [ u( x$1 , 0), u( x$1 , 1)] is nondegenerate. Further,
R( x$1 ) and R(x 1 ) are disjoint intervals, that is R( x$1 ) ∩ R( x1 ) = ∅
because x$1 > x1 implies that u( x$1 , 0 ) > u( x1 ,1) .
Now, each nondegenerate interval contains a rational number,
in fact, infinitely many rational numbers as well as infinitely many
irrational numbers. Let Q(x 1 ) be the set of all rational numbers
contained in R(x 1 ), and Q( x$1 ) the set of all rational numbers
contained in R( x$1 ) .
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Because R(x 1 ) and R( x$1 ) are disjoint, Q(x 1 ) and Q( x$1 ) are
also disjoint. Observe that we can define R(x 1 ) for any nonnegative
real number, and all such R(x 1 )’s are disjoint from each other. There
are as many nondegenerate R(x 1 ) intervals as the number of all
nonnegative real numbers. Since each R(x 1 ) contains Q(x 1 ), it
follows that there are as many such Q(x 1 ) sets as the number of all
nonnegative real numbers. Since each Q(x 1 ) contains rational
numbers, the upshot is that the total number of rational numbers
collected over all Q(x 1 )’s is at least as great as the number of all
nonnegative real numbers.
u( x$1 ,1)
l
l
l
u(x 1, 1) l
l
l
R( x$1 ,0)
u( x$1 ,0 )
l
l
2
l
l
R(x 1, 0)
(0, 1)
l
u(x 1, 0)
1
l
( x$1 ,0 )
l
(x 1, 0)
0
Mathematicians have the notion called "cardinality" to measure
the size of a given by “counting” the number of elements in the set.
Using this terminology, we have just argued that the cardinality of the
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rational-number set is at least as great as the cardinality of the
nonnegative real-number set. But, it is an established fact that the
cardinality of the nonnegative real-number set is the same as the
cardinality of the set of all real numbers. It is also an established fact
that the cardinality of the set of all rational numbers is strictly (far)
less than the cardinality of the set of all real numbers. We have thus
reached a conclusion that is contrary to this established mathematical
fact by supposing that a utility function exists representing the
lexicographic preference.
Q.E.D.
The upshot is that there is no utility function representation of
the lexicographic preference. A rigorous proof of the nonexistence
proposition for the lexicographic case can be found in Theory of Value:
An Axiomatic Analysis of Economic Equilibrium by Gerald Debreu,
(1959) Cowles Foundation Monograph, ISBN 0-300-01558-5. In the
same monograph, Debreu provides a most general existence proof of a
utility function for any continuous preference preordering.
Representation Function
The absence of a utility function does not imply the absence of
a decision function for a lexicographic consumer. Suppose that a
consumer chooses the most preferred bundle from a budget set of the
form
B(p1 , p2 m) = {(x1, x 2 )
p1 x1 + p2 x2 ≤ m} ,
where prices are strictly positive. Consider the decision function
v( B( p1 , p2 m)) = ( max{x1 p1 x1 + p2 x 2 ≤ m for some x2 }, 0 )
= ( m / x1 , 0 ) .
A consumer with this decision function will always choose the same
consumption bundle that a consumer with lexicographic preferences
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would choose on all budget sets of the form B( p1 , p2 m) . In this
sense, this ν function serves as an adequate behavioral rule for the
lexicographic consumer facing these budget sets. To put it more
bluntly, a consumer with the utility function u( x1 , x2 ) = x1 behaves
exactly the same as the lexicographic consumer on all budget sets of
the form B( x1 , x2 m) with p >> 0. But neither this u nor v
above is a utility function for the lexicographic preference, because
neither can be used to rank consumption vectors of the form ( x1 ,⋅ • ).
Integer Consumption
The impossibility of utility representation has crucially
depended on the idea that all real vectors were consumption bundles.
If one commodity can only be consumed in nonnegative integer units,
a utility presentation is possible even for the lexicographic consumer.
2
a
1
0
a
a+1
a+2 a+3
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Suppose that the consumer has a lexicographic preference over
two -commodity bundles, but that the first commodity x 1 can only be
consumed in nonnegative integer amounts. x 1 = 0, 1, 2, 3, ⋅⋅⋅The
relevant consumption set is Ζ+ × R+ where Ζ+ is the set of
nonnegative integers and R+ the set of nonnegative real numbers. The
upper contour set for a is A(a) = {x in Ζ+ × R+ | x f a }. For
~
each nonnegative integer "a", A(a) is a closed set. It is then possible to
define a numerical representation of the lexicographic preference
defined on Ζ+ × R+.
u
x
x
2
1
3
1
2
1
0
Consider the function
u( x1 , x 2 ) = ( x1 + 1) −
1
x2 + 1
where x 1 is a nonnegative integer. Note that for each integer x 1 ,
x1 ≤ u( x1 , ⋅ ) < x1 + 1
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and u(x 1 , ⋅) increases asymptotically in x 2 to the value x 1 + 1 (as x 2
→ ∞). Since x1 + 1 ≤ u( x1 + 1, ⋅) < x1 + 2 , we have
u( x1 , ⋅) < u ( x1 + 1, ⋅) . As a result,
0 = u(0, 0) < u(0, ⋅) < u(1, ⋅) < u(2, ⋅) < ⋅⋅⋅.
Once again, we have disjoint intervals R(x 1 ). But, the previous
conundrum does not arise. The cardinality of Ζ+ is decisively far less
than the cardinality of R+. Note that u(x 1 , x 2 ) = (x 1 +1) − (1/ex 2 )
will be another utility representation of the lexicographic preference
when the first commodity is integer-indivisible. It is left to the reader
to confirm that (x 1+1) – (1/ex 2 ) is a monotone transform of (x 1+1) –
1/(x 2+1).
The above exercise can be extended to a case in which x 1 can
be consumed only in increments of 1/k. That is,
x 1 = 0, 1/k, 2/k, ⋅⋅⋅, n/k, ⋅⋅⋅.
This k can be taken as large as one wishes, so that 1/k can be made
as small as one wishes. Consider
u( x1 , x 2 ) = ( x1 +
1
(1 k )
) −
.
k
x2 + 1
Then,
x1 ≤ u( x1 , ⋅) < x1 +
1
,
k
and u(x 1 , x 2 ) increases in x 2 approaching the asymptotic value of
x1 +
1
as x 2 → ∞. Once again,
k
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1
2
3
0 = u( 0, 0) < u( 0, x 2 ) < u( , ⋅) < u( , ⋅) < u( , ⋅)
k
k
k
n
n +1
< L < u( , ⋅) < u(
, ⋅) < L .
k
k
Relevance of Lexicographic Preference
Despite the logically compelling example of a lexicographic
preference, economists have generally dismissed it as curiosum. For
example, Malinvaud (1972, p. 20) says, “Such a preference relation
has sometimes been considered; it hardly seems likely to arise in
economics, since it assumes that, for the consumer, the good 1 is
immeasurably more important than the good 2. We loose little in the
way of realism if we eliminate this and similar cases which do not
satisfy [the continuity axiom].” (italics and brackets mine). But,
modified forms of the basic lexicographic preference can arise and
rather plausible.
Consider for example, the case of yellow and red apples. I will
not be surprised that if my neighbor compares two bushels of apples
first by the total quantity of apples and, only if two baskets contain the
same number of apples, she prefers the one with more red apples than
yellow apples. Letting the good 1 be a red apple and the good 2 a
yellow apple, we can have a modified lexicographic preference given
by (x 1, x 2) f (x′1, x′2) if either (x 1 + x 2 > x′1 + x′2 ) or (x 1 + x 2 =
~
x′1 + x′2 and x 1 > x′1) holds. Clearly, this consumer’s upper
contour set is neither open nor closed as in the standard lexicographic
preference. Note that this consumer’s demand behavior is identical to
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the consumer with the continuous preference given by the utility
function as u(x 1, x 2) = x 1 + x 2 on every linear budget except when
the relative prices of yellow and red apples are unitary ( p1 / p2. = 1).
When p1 = p2, the lexicographic consumer chooses all red apples (x 1
=
m/ p1. and x 2 = 0 ), but the consumer with u(x 1, x 2) = x 1 + x 2
can chooses any vector on the budget line. The plausibility of such a
lexicographic preference is not eclipsed when the consumer’s choice
includes three types of apples: red, green and yellow.
In fact, from any consumer that has a continuous preference, I
can create a modified lexicographic preference that behaves in the
identical fashion on every linear budget set except when the original
consumer’s choice is a demand correspondence, rather than a demand
function1. Let f be an original preference that is continuous in the
~
sense that its upper contour set is closed. Then, induce a modified
lexicographic preference L defined as follows: xLx’ if and only if
either ( x f x’ ) or (x ∼ x’ and x 1 > x′1) holds. Letting U be a
utility function that represents the continuous preference, we define
xLx’ if and only if either U(x) > U(x’) or ( U(x) = U(x’) and x 1 >
x′1) holds. For example, by this method, a Cobb-Douglas utility
function can be modified into a Cobb-Douglas type lexicographic
preference.
Expenditure Minimization
1
If a consumer’s choice is a demand correspondence, it means that the consumer is indifferent
among multiple vectors, but the lexicographic choice on the same demand set will select the one that
has the largest first component. A nonconvex preference can induce a demand correspondence. The
previous example produced the case of a demand set when the relative price of yellow and red
apples was unity.
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The duality theory of consumption is built on the use of
expenditure function. The expenditure function calculates the
minimum expenditure necessary for a consumer to attain a given level
of preference. That is, e(p, x o) = min{p· x | x f x o}. The
~
expenditure function does not generally exist unless the upper contour
set A(x o) = {x | x f x o} is closed. The need for this technical
~
requirement is fairly transparent if we express the expenditure
function as e(p, x o) = min{p· x | x in A(x o)}. In fact, for the case of
the original lexicographic preference, there is no expenditure
minimizer and the expenditure function is not defined2 unless x o is on
the horizontal axis, i.e., x o = (x o1, 0 ). In contrast, the consumer with
u( x1 , x2 ) = x1 has the expenditure function e(p, x o) = p1 x o1 = min
{p· x | u(x) ≥ u(x o)} for all x o.
Similarly, for the lexicographic preference for yellow and red
apples, there is no expenditure minimizer, and the expenditure
function is not defined, whenever p1 > p2. Again, for the consumer
with u(x 1, x 2) = x 1 + x 2 the expenditure function is well defined
for all p and x, namely, e(p1, p2, x 1, x 2) = (x 1 + x 2) Min {p1, p2 }.
Since the expenditure function is itself a utility function, called
a money-metric utility function, we can heuristically confirm
Debreu’s Theorem on the existence of utility function. The existence
of the expenditure function is tantamount to the existence of a utility
function. Because a closed upper contour set guarantees a well2
If we redefine the expenditure function as the infimum, rather than the minimum, expenditure,
then we have it well-defined. e(p, xo) = inf{p· x | x f xo} = p 1 xo1 for all nonnegative xo. But,
~
the preference still fails to have any utility function representation. In this sense, I consider the
infimum version an isolated technical remedy, and prefer using the minimum definition to
underscore the economic-theoretic underpinning of the relations among preference, demand and
utility function.
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defined expenditure function, we know that a continuous preference,
as it necessarily has closed upper contour sets, can be represented by a
utility function.
Remark: We can minimize expenditure on the lexicographic
preference over yellow and red apples, provided that p1 ≤ p2 . Even
so, the minimizer is of the form x = (x 1, 0 ) = (m/ p1 , 0 ), and
whenever x o is not on the boundary, the minimizer is strictly
preferred to x o: x = (x 1, 0 ) f x o = (x o1, x o2) . In that sense, the
constraint x f x o is not binding at the minimization solution. We
~
often say casually that the indifference curve of a given consumption
vector x f x o is the boundary of the upper contour set {x | x f x o}.
~
~
If the preference is lexicographic, the boundary definition of an
indifference curve does not hold. The boundary of the upper contour
set, even where the boundary is included in the upper contour set, is
not necessarily an indifference curve.