On Lexicographic (Dictionary) Preference

Transcription

1 MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 On Lexicographic (Dictionary) Preference This short note discusses a lexicographic preference over twocommodity bundles. The extension to the n-commodity case is straightforward. Let a = (a, a ) and b = (b, b ) be two consumption vectors. We say that a is lexicographically preferred to b, and write a f ~ b if and only if either (a > b ) or (a = b and a b ). From this definition, it follows that a f b if and only if either (a > b ) or (a = b and a > b ). 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 a = b and a = b. 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

2 Fall 93/94/95/ Hajime Miyazaki consumption vector must have its own value, different from all others: that is, no two points in the plane R can have the same value assigned. But, R 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 R 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. ~ a a A(a) 0 a Given a = (a, a ), any vector to the right of the vertical line at a 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

3 Fall 93/94/95/ Hajime Miyazaki 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. a x( n) = ( a +, + ) n n as n =,, 3,. For any n, a + (/n) > a. Thus, x( n) f a for any n. The limit of x(n) as n goes infinity is x = ( a, a ). But, x a a = (, ) p ( a, a) = a. a a a / 0 x x(n x( x( ) ) a a +(/) a + The diagram illustrates the following sequence and preference reversal.

4 Fall 93/94/95/ Hajime Miyazaki and x( ) f a x( ) f a x( 3) f a M M M x( n) f a M M M x p a. 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+ R. For each x, the lexicographic preference says that ( x, ) f ( x, 0) if and only if u(x, ) > u(x, 0). Then, for each x we can assign a nondegenerate interval R(x ) = [u(x, 0), u(x, )]. Next, take $x such that $x > x, so that u( x$, ) > u( x$, 0). Again, R( x$ ) = [ u( x$, 0), u( x$, )] is nondegenerate. Further, R( x$ ) and R(x ) are disjoint intervals, that is R( x$ ) R( x) = because $x > x implies that u( x$, 0) > u( x, ). Now, each nondegenerate interval contains a rational number, in fact, infinitely many rational numbers as well as infinitely many irrational numbers. Let Q(x ) be the set of all rational numbers contained in R(x ), and Q( x$ ) the set of all rational numbers contained in R( x$ ).

5 Fall 93/94/95/ Hajime Miyazaki Because R(x ) and R( x$ ) are disjoint, Q(x ) and Q( x$ ) are also disjoint. Observe that we can define R(x ) for any nonnegative real number, and all such R(x ) s are disjoint from each other. There are as many nondegenerate R(x ) intervals as the number of all nonnegative real numbers. Since each R(x ) contains Q(x ), it follows that there are as many such Q(x ) sets as the number of all nonnegative real numbers. Since each Q(x ) contains rational numbers, the upshot is that the total number of rational numbers collected over all Q(x ) s is at least as great as the number of all nonnegative real numbers. u( x$, ) u(x, ) R( x$, 0 ) (0, ) R(x, 0) u(x, 0) u( x$, 0 ) ( x$, 0 ) 0 (x, 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

6 Fall 93/94/95/ Hajime Miyazaki 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, (959) Cowles Foundation Monograph, ISBN 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(p, p m) = {(x x ) p x + p x m},, where prices are strictly positive. Consider the decision function v( B( p, p m)) = ( max { x p x + p x m for some x }, ) 0 = ( m / x, 0 ). A consumer with this decision function will always choose the same consumption bundle that a consumer with lexicographic preferences

7 Fall 93/94/95/ Hajime Miyazaki would choose on all budget sets of the form B( p, p 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( x, x ) = x behaves exactly the same as the lexicographic consumer on all budget sets of the form B( x, x 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 ( x, ). 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. a 0 a a+ a+ a+3

8 Fall 93/94/95/ Hajime Miyazaki Suppose that the consumer has a lexicographic preference over two-commodity bundles, but that the first commodity x can only be consumed in nonnegative integer amounts. x = 0,,, 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 3 0 Consider the function u( x, x ) = ( x + ) x + where x is a nonnegative integer. Note that for each integer x, x u( x, ) < x +

9 Fall 93/94/95/ Hajime Miyazaki and u(x, ) increases asymptotically in x to the value x + (as x ). Since x + u( x +, ) < x +, we have u( x, ) < u( x +, ). As a result, 0 = u(0, 0) < u(0, ) < u(, ) < u(, ) <. Once again, we have disjoint intervals R(x ). But, the previous conundrum does not arise. The cardinality of Ζ + is decisively far less than the cardinality of R +. Note that u(x, x ) = (x +) (/e x ) 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 +) (/e x ) is a monotone transform of (x +) /(x +). The above exercise can be extended to a case in which x can be consumed only in increments of /k. That is, x = 0, /k, /k,, n/k,. This k can be taken as large as one wishes, so that /k can be made as small as one wishes. Consider ( k) u( x, x) = ( x + ) k x +. Then, x u( x, ) < x +, k and u(x, x ) increases in x approaching the asymptotic value of x + as x. Once again, k

10 Fall 93/94/95/ Hajime Miyazaki 3 0 = u( 0, 0) < u( 0, x) < u(, ) < u(, ) < u(, ) k k k + < L < u( n, ) < u( n, ) < L. k k Relevance of Despite the logically compelling example of a lexicographic preference, economists have generally dismissed it as curiosum. For example, Malinvaud (97, p. 0) 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 is immeasurably more important than the good. 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 be a red apple and the good a yellow apple, we can have a modified lexicographic preference given by (x, x ) f (x ~, x ) if either (x + x > x + x ) or ( x + x = x + x and x > x ) 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

11 Fall 93/94/95/ Hajime Miyazaki the consumer with the continuous preference given by the utility function as u(x, x ) = x + x on every linear budget except when the relative prices of yellow and red apples are unitary ( p / p. = ). When p = p, the lexicographic consumer chooses all red apples (x = m/ p. and x = 0 ), but the consumer with u(x, x ) = x + x 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 function. 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 > x ) 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 > x ) holds. For example, by this method, a Cobb-Douglas utility function can be modified into a Cobb-Douglas type lexicographic preference. Expenditure Minimization 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.

12 Fall 93/94/95/ Hajime Miyazaki 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 defined unless x o is on the horizontal axis, i.e., x o = (x o, 0). In contrast, the consumer with u( x, x ) = x has the expenditure function e(p, x o ) = p x o = 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 p > p. Again, for the consumer with u(x, x ) = x + x the expenditure function is well defined for all p and x, namely, e(p, p, x, x ) = (x + x ) Min {p, p }. 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 well- If we redefine the expenditure function as the infimum, rather than the minimum, expenditure, then we have it well-defined. e(p, x o ) = inf{p x x f ~ x o } = p x o for all nonnegative x o. 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.

13 Fall 93/94/95/ Hajime Miyazaki 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 p p. Even so, the minimizer is of the form x = (x, 0 ) = (m/ p, 0 ), and whenever x o is not on the boundary, the minimizer is strictly preferred to x o : x = (x, 0 ) f x o = (x o, x o ). 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.