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Numerical representability of preferences by economic functions
J. Math. Anal. Appl. 424 (2015) 1223–1236
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Numerical representability of preferences by economic functions Susanne Fuchs-Seliger 1 Institut für Volkswirtschaftslehre (ECON), Karlsruher Institut für Technologie, Kollegium am Schloss Bau IV, 76128 Karlsruhe, Germany
a r t i c l e
i n f o
Article history: Received 19 February 2013 Available online 24 November 2014 Submitted by J.A. Filar Keywords: Preference relation Representability Benefit function Income compensation function Money-metric utility function Distance function
a b s t r a c t This article is concerned with the representation of the individual’s preference relation by numerical functions, which are well-known in economics. The functions to be considered in this paper are the income compensation function, an appropriate distance function, and a Luenberger-type benefit function, which represent the individual’s preferences under slightly different conditions. Since these functions have an appealing economic meaning, the individual’s preferences can be represented in an appropriate economic context. Therefore, the preferences will not only be represented by an abstract unknown utility function, but by a function which can be constructed in the corresponding economic model. By this approach the hedonistic and problematic notion of utility can be avoided. © 2014 Elsevier Inc. All rights reserved.
1. Introduction This article studies the problem of representing preference relations by functions which have an economic meaning. These functions are essential in appropriate economic models like the theory of demand or welfare theory. In his famous book “Theory of Value” [9] G. Debreu has shown that every transitive, complete2 and continuous relation on a connected subset X of Rn can be represented by a continuous utility function u : X → R such that for all x, y ∈ X: xy
⇐⇒
u(x) u(y).
The proof uses profound mathematical knowledge and skills (Debreu [9, pp. 56–59]). Especially, we must be familiar with order-density. If the relation is not continuous, then in general, one has to require that there exists a countable subset of the given set X which is order dense in X (Debreu [9], Fishburn [12]). If the relation is monotone, then the difficulty of the proof reduces considerably (Mas-Colell, Whinston and
1 2
E-mail address: [email protected]. Fax: +49 721 608 43082. For all x, y ∈ X : x y ∨ y x.
http://dx.doi.org/10.1016/j.jmaa.2014.11.052 0022-247X/© 2014 Elsevier Inc. All rights reserved.
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Green [20, pp. 47–48]). In the present article we will reinvestigate the representation of a given preference relation by a function which has an appealing economic meaning, and which is principally constructive. Three different classes of functions will be considered: the income compensation function, the distance function, originally applied by R.W. Shephard to production theory (Shephard [27]), and a modified Luenberger benefit function (Luenberger [18]). Everybody, who is familiar with these functions, will realize that they are intuitive and useful representations of a given relation, when these functions are parts of appropriate economic models. The proofs of representability are quite simple. Only in the case of continuous representability of a given relation by the income compensation function one needs a little more advanced mathematical tools. Moreover, this article revisits the problem of the hedonistic and psychological meaning of utility in a scientific field like economics, which has been discussed for a long time in the economic literature (Samuelson [25], et al.). In his article “A note on the pure theory of consumer’s behaviour” Samuelson developed an economic model “freed from any vestigial traces of the utility concept” (Samuelson [25, p. 71]). The aforementioned functions represent the agent’s preferences without using the utility concept. This article is organized as follows: We will start with the income compensation function as a representation of a given relation. In comparison, properties of will be presented which imply that can be represented by a distance function. Finally, a Luenberger-type benefit function as a representation of will be studied. As we will see, the assumptions imposed on the underlying preferences will differ slightly. 2. The income compensation function Income compensation functions are an important tool in the theory of consumer behavior. They were introduced to demand theory by Lionel McKenzie in 1957 [21]. By means of the income compensation functions, we can develop a model of consumer behavior based on the individual’s preference relation and not on utility functions (Fuchs-Seliger [13]). We will come back to this point at the end of this section. Income compensation functions can be applied in order to describe consumer’s preferences by moneyincome. Therefore, the problematic and hedonistic notion of utility can be avoided in consumer theory, especially for the reason why the utility of goods is not cardinally measurable. In economics ordinal utility functions are usually assumed, meaning that the functions are determined up to a strictly increasing transformation. However, in this respect, utility is just an empty word. In order to avoid the problematic notion of utility, Paul Samuelson in 1938 developed a new description of consumer behavior based on demand functions. He introduces this approach by saying “I propose, therefore, that we start anew in direct attack upon the problem, dropping off the last vestiges of the utility analysis” (Samuelson [25, p. 62]). Paul Samuelson also suggested to measure “utility” by money-income (Samuelson [26, p. 1262]), and introduced the notion of money-metric utility functions (Samuelson [26]). Obviously, measuring the individual’s “utility” by money-income has an intuitive meaning. Therefore, this paper will be especially concerned with income compensation functions. Representability of a given relation by an income compensation function has been investigated by several authors before. Especially, we refer to J. Weymark [29], S. Honkapohja [17], and J. Alcantud and A. Manrique [1]. While an income compensation function is in general defined for a complete, transitive, and continuous relation , the domain of varies, and also the continuity of the income compensation function follows from different conditions. In this paper, we consider a closed set X ⊆ Rn+ , X = ∅, which is interpreted as a set of commodity bundles, and the set Rn++ of strictly positive price vectors p. Firstly, it will only be assumed that is reflexive. Then the income compensation (or minimum income) function can be defined as m0 p0 , x := inf p0 y y x , y∈X
for p0 ∈ Rn++ , x ∈ X.
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m0 (p0 , x) denotes the lowest income at prices p0 needed to purchase a commodity bundle at least as good as x according to the preferences of the individual. If, in addition, is upper semicontinuous, then, by definition, for all x ∈ X the upper contour set {y ∈ X | y x} is closed in X (Alcantud and Mehta [2, p. 593]). In this case m p0 , x = min p0 y y x . y∈X
This result follows from the upper semicontinuity of and X being closed and bounded from below. Moreover, fp (y) = py is a continuous function on X and, therefore, miny∈X {p0 y | y x} exists. One can immediately see Lemma 1. If is a reflexive, transitive and upper semicontinuous relation on a closed set X ⊆ Rn+ , then for every p0 ∈ Rn++ , x 1 x2
=⇒
m p 0 , x1 m p 0 , x2
∀x1 , x2 ∈ X.
Hence m(p0 , · ) weakly represents the relation . Proof. The above assertion easily follows, since x1 x2 together with the transitivity of implies, {p0 y | y ∈ X ∧ y x1 } ⊆ {p0 z | z ∈ X ∧ z x2 }, and by the upper semicontinuity of , min{p0 y | y ∈ X ∧ y x1 } min{p0 z | z ∈ X ∧ z x2 }. 2 Recall that representability of a relation by a real-valued function u : X → R means that xy
⇐⇒
u(x) u(y),
∀x, y ∈ X.
Therefore, we have to show, for any given price system p0 ∈ Rn++ , xy
⇐⇒
m p0 , x m p0 , y ,
∀x, y ∈ X.
We will assume that the relation is continuous, i.e. is upper and lower semicontinuous on X (Hildenbrand and Kirman [16, p. 60]). Upper semicontinuity has already been defined. Lower semicontinuity means that the lower contour set {x ∈ X | y x} is closed in X for all y ∈ X. In addition it will be assumed, that X is a cone with vertex 0,3 and is locally nonsatiated. Recall that a relation on X is locally nonsatiated if for every x ∈ X and every ε > 0 there exists y ∈ Nε (x) ∩ X, where Nε (x) denotes the ε-neighborhood of x, such that y x. We will not only prove that m(p0 , x) represents the relation , but that m(p0 , x) is a continuous representation of . Usually, in economic models of consumer behavior the existence of a continuous utility function describing consumer’s preferences is assumed. Accordingly, it is important to find appropriate assumptions imposed on the individual’s preference relation, implying that the income compensation function m(p0 , x) is a continuous representation of . The present investigation of this problem has been inspired by articles of J. Weymark [29], S. Honkapohja [17], and J. Alcantud and A. Manrique [1]. In accordance with these articles completeness, transitivity and continuity of the relation is assumed. By contrast, Weymark assumes that X ⊂ Rn is convex, closed and bounded from below. In addition, he assumes that any point of local satiation is also a point of global satiation [29, Proposition 2, pp. 228–229]. The latter property is also required in the articles of Honkapohja [17, Proposition 5, p. 548], and Alcantud and Manrique [1, Theorem 2, pp. 371–372]. 3
A set X is a cone with vertex 0, if for all x ∈ X, λx ∈ X, ∀λ 0.
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While Honkapohja investigates the continuity of m(p, x) when X ⊆ Rn is closed, connected and bounded from below [17, p. 546], Alcantud and Manrique assume that X ⊆ Rn is open and bounded from below [1, p. 371]. In our investigation the following correspondence related to m(p, x) is of great importance: Definition. Let be a complete, transitive and upper semicontinuous relation on a closed set X ⊆ Rn+ . Then g : Rn++ × X → 2X ,
y ∈ g(p, x) := arg min{pz | z x}, z∈X
is called the “compensated demand correspondence”. We will only be concerned with g(p0 , x) for some fixed p0 ∈ Rn++ . Under the assumptions of Lemma 1, g(p0 , x) is well defined. As a preliminary result, we will show Lemma 2. Let be a complete, transitive and continuous relation on a closed cone X ⊆ Rn+ with vertex 0. Then for any p0 ∈ Rn++ , and x ∈ X, y ∈ g p0 , x = arg min p0 z z x z∈X
=⇒
y ∼ x,
where ∼ denotes the symmetric part of . Hence, y ∼ x means y is indifferent to x. Proof. Under the assumptions made, m(p0 , x) is well defined. However, there may exist more than one alternative which minimizes {p0 y | y x} (except at x = 0) on X. Therefore, g(p0 , x) = arg miny∈X {p0 y | y x} may be many-valued. By definition, for every y ∈ g(p0 , x), y x. Suppose, by contradiction, there exists y 0 ∈ g(p0 , x) such that y 0 x. By lower semicontinuity of , for every > 0, there exists z ∈ N (y 0 ) ∩ X such that z x. In particular, since X is a cone with vertex 0, for some > 0 there exists z 0 ∈ N (y 0 ) ∩X such that y 0 z 0 , y 0 = z 0 , and z 0 x. However, since z 0 p0 < y 0 p0 , we obtain a contradiction to the definition of g(p0 , x). Hence, y ∼ x for all y ∈ g(p0 , x). 2 To establish the continuous representability of , the upper hemicontinuity of the compensated demand correspondence g(p0 , x) will be exploited. Recall that upper hemicontinuity of a compact-valued correspondence can be characterized by the following property (Hildenbrand and Kirman [16, p. 262]): A compact-valued correspondence ϕ : S → 2T \ {∅} (S, T ⊆ Rn ; S, T = ∅) is upper hemicontinuous at x0 ∈ X, if for every sequence xk in S such that xk → x0 ∈ S and for every sequence y k , y k ∈ ϕ(xk ), there exists a subsequence y kj of y k such that limj→∞ y kj = y 0 ∈ ϕ(x0 ). To prove upper hemicontinuity of g(p0 , x) at x we will first prove that the correspondence R : X → 2X , where R(x) = {y ∈ X | y x}, is lower hemicontinuous. The notion of lower hemicontinuity can be characterized as follows (Hildenbrand and Kirman [16, p. 271]): A correspondence ϕ : S → 2T \ {∅} (S, T ⊆ Rn ; S, T = ∅) is called lower hemicontinuous at x0 , if for every sequence xk in S with xk → x0 ∈ S and for every y 0 ∈ ϕ(x0 ) there exists a sequence y k with y k ∈ ϕ(xk ) and limk→∞ y k = y 0 . ϕ is upper (lower) hemicontinuous on S if it is upper (lower) hemicontinuous at every x ∈ S. ϕ is continuous if it is upper and lower hemicontinuous on S.
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First, it will be shown that the correspondence R : X → 2X , R(x) = {y ∈ X | y x}, is lower hemicontinuous. Note that the range of R contains the upper contour set of x for every x ∈ X, which is assumed to be closed in X. If R(x) is closed in X for every x ∈ X, then according to the previous definition, is called upper “semicontinuous” on X. Therefore, we have to discriminate between semicontinuity of a relation and hemicontinuity of a correspondence. Throughout the following analysis, it will be assumed that X ⊆ Rn+ is a closed cone with vertex 0. Lemma 3. Let be a complete, transitive, continuous and locally nonsatiated relation on X. Then, the correspondence R : X → 2X , R(x) = {y ∈ X | y x}, is lower hemicontinuous on X. Proof. Let xk be a sequence in X such that limk→∞ xk = x0 ∈ X, and y 0 ∈ R(x0 ). It has to be shown: There exists a sequence y k in X such that y k ∈ R(xk ) and limk→∞ y k = y 0 . Since 0 y x0 , we will consider two cases: 1. y 0 x0 and 2. y 0 ∼ x0 . 1. Let us first consider the case y 0 x0 . Then, by upper semicontinuity of there exist > 0 and a positive integer N 0 such that for all k N 0 , y 0 xk , and xk ∈ N (x0 ). Defining a sequence by y k = y 0 , for k N 0 , and y k = xk for k < N 0 , we obtain y k ∈ R(xk ) for all k and limk→∞ y k = y 0 . 2. Now, let us consider the case, y 0 ∼ x0 . For n = n1 we will construct a sequence y n such that limn→∞ y n = y 0 and y n ∈ R(xn ). This can be done in the following way: By local nonsatiation, for 1 there exists y˜ ∈ N1 (y 0 ) such that y˜ y 0 and, thus, y˜ x0 . By upper semicontinuity of there exists δ 1 > 0 such that for all x ∈ Nδ1 (x0 ) : y˜ x. In particular, there exist δ 1 > 0 and a positive integer K(δ 1 ) such that for all l K(δ 1 ), y˜ xl . Choose l1 = min{l | y˜ xl , xl ∈ Nδ1 (x0 )}. We will now change the notation and set y l1 := y˜. Then, y l1 ∈ R(xl1 ). For all natural numbers n and for n = n1 there exists y ∈ Nn (y 0 ) such that y y 0 . Since y x0 , there exists δ n such that for all x ∈ Uδn (x0 ) : y x. In particular, there exist δ n > 0 and a positive integer K(δ n ) such that for all l K(δ n ), y xl . Choose ln = min{l | y xl , xl ∈ Nδn (x0 )} and set y ln := y. Thus, y ln ∈ R(xln ). Now we have constructed a sequence y ln such that y ln ∈ R(xln ) and limn→∞ xln = x0 . For all k such that ln k < ln+1 we can take y k = y ln and obtain y k xk . For k < l1 we can set y k = xk and obtain y k ∈ R(xk ). We have thus constructed a sequence y k with the properties y k y 0 , y k ∈ R(xk ) for all k, and limk→∞ y k = y 0 . This completes our proof. 2 Now we will establish that the compensated demand correspondence g(p0 , x) is upper hemicontinuous at x for every x ∈ X. Lemma 4. Let be a complete, transitive, continuous and locally nonsatiated relation on X. Then for any fixed p0 ∈ Rn++ , g(p0 , x) is upper hemicontinuous at every x ∈ X. Proof. First, it will be shown that g(p0 , x) is compact-valued for every x ∈ X. Therefore, for any x ∈ X let y k be a sequence such that y k ∈ g(p0 , x) and limk→∞ y k = y 0 ∈ X. As y k ∈ g(p0 , x), it follows that y k x. By upper semicontinuity of , we can conclude that y 0 x. Moreover, since p0 y k+1 = p0 y k for all k, and limk→∞ (p0 y k ) = p0 y 0 , it follows that p0 y 0 = miny∈X {p0 y | y x}. Thus, y 0 ∈ arg miny∈X {p0 y | y x}. It follows that g(p0 , x) is closed for all x ∈ X. Now suppose that g(p0 , x) is not bounded. Then a sequence y k , y k ∈ g(p0 , x), exists such that limk→∞ y k = ∞ and limk→∞ p0 y k = ∞. However, y k ∈ g(pk , x) means p0 y k p0 x, contradicting this result. Now it will be shown that g(p0 , x) is upper hemicontinuous at x. Therefore, let us consider a sequence k x in X with limk→∞ xk = x0 ∈ X, and let y k be a sequence such that y k ∈ g(p0 , xk ). Since by definition, p0 y k p0 xk , and xk is bounded, y k is also bounded. Consequently, there exists a convergent subsequence y kj of y k such that limj→∞ y kj = y 0 .
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By way of contradiction, suppose y 0 ∈ / g(p0 , x0 ). By the definition of g(p0 , x0 ), there exists y˜ ∈ g(p0 , x0 ). kj 0 kj kj From y ∈ g(p , x ) it follows that y xkj , and, thus, the continuity of implies y 0 x0 . This together with y 0 ∈ / g(p0 , x0 ) yields p0 y 0 > p0 y˜. Hence, there exists ε1 , ε2 > 0 such that for all y ∈ Nε1 (y 0 ) ∩ X and for all z ∈ Nε2 (˜ y ) ∩ X : p0 y > p0 z. By Lemma 3, R(x) is lower hemicontinuous at x. Therefore, in view of 0 y˜ ∈ R(x ), there exists a sequence z kj such that z kj ∈ R(xkj ) and limj→∞ z kj = y˜. Since limj→∞ y kj = y 0 , we can find kr such that y kr ∈ Nε1 (y 0 ) and z kr ∈ Uε2 (˜ y ) and p0 y kr > p0 z kr . From y kr ∈ g(p0 , xkr ) and kr kr 0 z ∈ R(x ) a contradiction to the definition of g(p , · ) arises. This completes our proof. 2 Based on the previous lemmata we obtain that the income compensation function represents the given preference relation of the individual continuously. This is the main result of this article. Theorem 1. Assume the following hypotheses: (M1) X ⊆ Rn+ is a closed cone with vertex 0. (M2) is a transitive, complete and continuous relation on X. (M3) is locally nonsatiated. Then (a) for all x1 , x2 ∈ X and for any p0 ∈ Rn++ , x1 x2
⇐⇒
m p 0 , x1 m p 0 , x2 .
(b) m(p0 , x) is a continuous representation of for any p0 ∈ Rn++ . Proof of Theorem 1(a). In view of Lemma 1, it suffices to show, m p 0 , x1 m p 0 , x2
=⇒
x1 x2 .
First, assume m(p0 , x1 ) > m(p0 , x2 ), and suppose, by way of contradiction, x2 x1 . An application of Lemma 1 yields m(p0 , x2 ) m(p0 , x1 ), contradicting the above assumption. Hence, x1 x2 . In the case of m(p0 , x1 ) = m(p0 , x2 ) > 0, suppose, by way of contradiction, x2 x1 . By Lemma 2 there exist y 1 ∈ g(p0 , x1 ) such that y 1 ∼ x1 , and y 2 ∈ g(p0 , x2 ) such that y 2 ∼ x2 . Hence, y 2 y 1 . Since is lower semicontinuous, for every > 0 there exists an ε-neighborhood of y 2 , Nε (y 2 ), such that for all y ∈ Nε (y 2 ) ∩ X, y y 1 . In particular, there exists y 0 ∈ Nε (y 2 ) ∩ X such that y 0 y 2 , y 0 = y 2 , and y 0 y 1 . Such a y 0 exists, because X is a closed cone with vertex 0. Since y 0 y 2 and y 0 = y 2 , we also have p0 y 0 < p0 y 2 = minx∈X {p0 x | x x2 } = m(p0 , x1 ). This together with y 0 x1 is a contradiction to the definition of m(p0 , x1 ). If m(p0 , x1 ) = m(p0 , x2 ) = 0, then x1 ∼ 0 ∼ x2 by Lemma 2. 2 Proof of Theorem 1(b). In order to show that m(p0 , x) is continuous on X for every fixed p0 ∈ Rn++ , let us consider x0 ∈ X and let xk be a sequence in X such that limk→∞ xk = x0 ∈ X. It has to be shown that limk→∞ m(p0 , xk ) = m(p0 , x0 ). By definition, m(p0 , xk ) = min{p0 x | x xk } = p0 y k for some y k ∈ g(p0 , xk ). From y k xk , it follows 0 m(p0 , xk ) = p0 y k p0 xk . Convergence of xk implies that it is also bounded and thus y k is also bounded. Hence, there exists a convergent subsequence y kj of y k such that limj→∞ y kj = y 0 . As limj→∞ xkj = x0 , y kj xkj , y kj ∈ g(p0 , xkj ) and limj→∞ y kj = y 0 , upper hemicontinuity of g(p0 , ·) implies y 0 ∈ g(p0 , x0 ). From this follows limj→∞ m(p0 , xkj ) = limj→∞ (p0 y kj ) = p0 y 0 = m(p0 , x0 ). Consider any accumulation point m0 of m(p0 , xk ). There exists a subsequence
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m(p0 , xks ) of m(p0 , xk ) converging to m0 , and lims→∞ (p0 , xks ) = (p0 , x0 ). Since lims→∞ m(p0 , xks ) = lims→∞ (p0 y ks ) = m0 , for y ks ∈ g(p0 , xks ), the sequence y ks is bounded, and, thus, possesses a convergent subsequence z kr . As (p0 z kr ) is a subsequence of (p0 y ks ), it also follows limr→∞ (p0 z kr ) = m0 . Without loss of generality let y ks and z kr be identical, so that we have lims→∞ y ks = y˜. By the same arguments as used previously, upper hemicontinuity of g(p0 , ·) implies y˜ ∈ g(p0 , x0 ). From this also follows m0 = lims→∞ m(p0 , xks ) = lims→∞ (p0 y ks ) = p0 y˜ = m(p0 , x0 ). Therefore, the sequence m(p0 , xk ) only has m(p0 , x0 ) as an accumulation point. This completes our proof. 2 Thus, we have seen that the income compensation function represents a given relation continuously under appropriate conditions. Such a representation is particularly appealing, because it evaluates the preferences of the individual by money income. It is therefore a money-metric utility function. This has been pointed out firstly by P. Samuelson [26]. It should be noted that based on hypotheses on the individual’s preference relation, one can develop a model of consumer behavior by means of the income compensation function. I have shown this result in a former article (Fuchs-Seliger [13]). In particular, I will recall Theorem 2, pp. 300–302, which states: Assume the conditions (S1) X ⊆ Rn+ , X = ∅, is a closed set. (S2) is a reflexive and upper semicontinuous relation on X. Moreover, if g(p, x) is single-valued and continuous with respect to p, then (i1 ) ∂m(p,x) = gj (p, x), where gj (p, x) denotes the jth component of g(p, x). ∂pj (i2 ) If m(p, x) is twice continuously differentiable at p, then ∂gi (p, x) ∂gj (p, x) = . ∂pi ∂pj One can even demonstrate (Fuchs-Seliger [14, Lemma 1, pp. 26–27]) that the conditions (S1) and (S2) imply that g(p, x) is continuous with respect to p, if g(p, x) is single-valued. Combining these results we obtain the following conclusion: Assume (S1) and (S2) and let g(p, x) be single-valued. Then Shephard’s Lemma (i1 ) follows, and, if m(p, x) is twice continuously differentiable at p, then in view of (i2 ) the symmetry of the Slutsky-matrix follows. These are important laws in the theory of demand. g(p, x) is single-valued, if the relation is strictly convex. This means: if x y, and x = y, then λx + (1 − λ)y y, ∀λ ∈ ]0, 1[, where is the asymmetric part of . Taking into consideration that by definition m(p0 , x0 ) = p0 · g(p0 , x0 ), if g is single-valued, and if we set M 0 = m(p0 , x0 ), then we can see that z 0 = g(p0 , x0 ) is just the Marshallian demand at prices p0 and income M 0 . The function h(p, M ) is called the Marshallian demand function depending on prices and income (Varian [28, pp. 105, 125–126]). Since by m(p0 , x) we have won a continuous utility function, we could now proceed, as usual in the literature, to describe consumer’s behavior. Thus, the preceding analysis has shown that the income compensation function represents a given preference relation continuously under appropriate conditions. Moreover, the related compensated demand correspondence g(p0 , x) satisfies important laws of the theory of demand like Shephard’s Lemma and the symmetry of the Slutsky-matrix, if g(p0 , x) is single-valued.
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3. Representability by a distance function In comparison with the income compensation function, we will now consider another constructive “utility” function, which is the distance function originally applied by R. Shephard to producer theory (Shephard [27]). Various applications of distance functions to producer and consumer theory have been studied in the literature (Deaton [8], Diewert [10], Blackorby, Primont and Russell [4], Färe [11], Cornes [7], FuchsSeliger [15], et al.). The distance function is also an important tool in index number theory, and in welfare theory (see for instance, Deaton [8] and Malmquist [19]). Given a relation on a set X ⊆ Rn+ the distance function d(x, x ) can be defined by
d x, x
:= max t ∈ R++
x x t
∀x, x ∈ X,
under the supposition that the maximum exists. According to its definition, d(x, x ) is defined as the maximum value of t into which x can be divided x so that d(x,x ) attains a level of well-being not worse than x . In order to establish conditions under which d(x, x ) is well defined, we will, preliminarily, consider the utility function u(x) = x1 x2 for x ∈ R2+ , as an example. The relation on R2+ is defined as xy
:⇐⇒
x1 x2 y1 y2 .
It immediately follows x x 1 2 x1 x2 . du x, x = max t ∈ R++ t If x = (0, 1) and x = (1, 2), then du (x, x ) is not defined. If x = (0, 1) and x = (1, 0), then du (x, x ) will be defined only for t ∈ R++ ∪ {∞}. If x, x ∈ R2++ , then du (x, x ) is well defined. Before turning to the representability of by the distance function we will first establish an auxiliary lemma which is concerned with the definition of d(x, x ). It is based on the following conditions: (D1) Let X ⊆ Rn+ be a closed cone with vertex 0. (D2) is a reflexive and continuous relation on X. (D3) is strictly monotone, i.e. x > y ⇒ x y, ∀x, y ∈ X (x > y means xi > yi , ∀i n). As we can see, the strict monotonicity condition (D3) replaces local nonsatiation, assumed in the previous section as a basic property of the preference relation. Lemma 5. Assume the conditions (D1) to (D3). Then for all x, x ∈ X ∩ Rn++ , (a)
d x, x ∈ R++ ,
(b)
x ∼ x , d(x, x )
where Rn++ consists of the strictly positive n-tuples of Rn . Proof of Lemma 5(a). In view of the strict monotonicity of the set {t | xt x } is bounded from above and, by upper semicontinuity of , {t | xt x } is closed. Hence, max{t | xt x } exists, and t ∈ R++ . 2 x Proof of Lemma 5(b). Suppose, by contradiction, that d(x,x ) x . Then, in view of the lower semicontinuity x of there exists > 0 such that for all z ∈ N ( d(x,x ) ) it follows that z x , in particular for some
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x s > d(x, x ) for which xs ∈ N ( d(x,x ) ) and the definition of the distance function. 2
x s
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x , by strict monotonicity of . However, this contradicts
Remark. In order to prove Lemma 5(b) we need the assumption that is also lower semicontinuous. Therefore, we assumed the continuity of in contrast to the former article (Fuchs-Seliger [15]). Note that if we do not assume to be strictly monotone, then d(x, x ) may be ∞ even for x, x ∈ X ∩ Rn++ . Reflexivity implies that d(x, x ) is also defined for x = x . We will establish two representability theorems by means of the distance function. For our first representability theorem we additionally assume that is homothetic, this means xy
⇐⇒
λx λy,
∀λ > 0.
It will be shown now that for any fixed x , d(· , x ) is a continuous representation of on X ∩ Rn++ , where x is the reference commodity bundle. Theorem 2. In addition to (D1) to (D3) let be complete, transitive, and homothetic. Then (a) for every x ∈ X ∩ Rn++ , x1 x2
⇐⇒
d x 1 , x d x 2 , x ,
∀x1 , x2 ∈ X ∩ Rn++ .
(b) d(· , x ) is a continuous representation of on X ∩ Rn++ . Proof of Theorem 2(a). Suppose x1 x2 . Since d(x1 , x ) and d(x2 , x ) are well defined, we can set k∗ = 1 2 d(x1 , x ) and l∗ = d(x2 , x ). By definition, kx∗ x and xl∗ x . Since x1 x2 , homotheticity of implies x1 x2 x1 1 ∗ ∗ l∗ l∗ . Thus, by transitivity of , l∗ x . From the definition of d(x , x ), k l follows. 1 2 ∗ 1 2 ∗ To show the converse, suppose k = d(x , x ) d(x , x ) = l . By definition, we have kx∗ x and xl∗ x . 2 2 First, suppose k∗ > l∗ . Hence, ¬( kx∗ x ). Completeness of implies x kx∗ . Thus, by transitivity of , x1 x2 1 2 k∗ k∗ . Homotheticity of finally yields x x . 1 2 x1 x2 ∗ ∗ Finally, consider k = l . Since k∗ ∼ x and l∗ ∼ x , transitivity of implies kx∗ ∼ xl∗ . Application of the homotheticity of implies x1 ∼ x2 . 2 Proof of Theorem 2(b). By Lemma 5 d(· , x ) is well defined for all x ∈ X ∩Rn++ . First, we will show that the function d(· , x ) is semicontinuous from above. By way of contradiction, suppose there exists x0 ∈ X ∩ Rn++ , at which d(· , x ) is not semicontinuous from above. Then, there exist > 0 and a sequence xk , xk ∈ X∩Rn++ such that limk→∞ xk = x0 and d(xk , x ) d(x0 , x ) + for almost all xk . As x0 , x ∈ Rn++ and d(xk , x ) = xk max{l(k) ∈ R++ | l(k) x } for all k, strict monotonicity of implies that the sequence d(xk , x ) is bounded from above. Hence, there exists a convergent subsequence of d(xk , x ). Without loss of generality, let this be d(xk , x ) itself. Consequently, there exists k∗ such that limk→∞ d(xk , x ) = k∗ . By definition, 0 xk x and, by upper semicontinuity of , kx∗ x . From this k∗ d(x0 , x ) follows, in view of the d(xk ,x ) definition of d(· , x ). Since k∗ = limk→∞ d(xk , x ), a contradiction to the inequality d(xk , x ) d(x0 , x ) + for nearly all k follows. Suppose now, by way of contradiction, that the function d(x, x ) is not semicontinuous from below at 0 x ∈ X ∩ Rn++ . Then, there exist δ > 0 and a sequence y l , y l ∈ X ∩ Rn++ such that liml→∞ y l = x0 and l
l
d(y l , x ) d(x0 , x )−δ for nearly all y l . Due to d(yyl ,x ) ∼ x we also have d(yyl ,x ) x . Proceeding analogously to the previous case, there exists a subsequence d(y lk , x ) of d(y l , x ) such that limk→∞ d(y lk , x ) = l∗ , and 0 x xl∗ by lower semicontinuity of . Consequently, l∗ d(x0 , x ) follows. Owing to the above inequality,
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we have limk→∞ d(y lk , x ) = l∗ d(x0 , x ) − δ for nearly all y lk , and, therefore, a contradiction arises. Thus, d(x, x ) is semicontinuous from above and from below at every x0 ∈ X ∩Rn++ , and, therefore, it is continuous on X ∩ Rn++ . 2 Strengthening the assumptions, it will be now required that is convex.4 Then, one can show that the distance function is a concave function on X ∩ Rn++ . Theorem 3. In addition to the conditions of Theorem 2, let X and be convex. Then d · , x is concave on X ∩ Rn++ . Proof. Since it can easily be seen that d(x, x ) is linear5 in x, it is sufficient to demonstrate that d(· , x ) is quasiconcave6 on X ∩ Rn++ (see Newman [24]). This has been done in (Fuchs-Seliger [15, p. 86]). 2 The assumption of homotheticity is rather strong. One can delete this assumption when we introduce the notion of “inverse” representability. By this we mean a real-valued function v : X → R represents the relation “inversely” if for all x, y ∈ X : x y ⇔ v(x) v(y). One can prove the following theorem (Fuchs-Seliger [15, p. 77]). Theorem 4. In addition to (D1) to (D3), let be transitive and complete. Then, d(x0 , · ) represents inversely, i.e., for every reference commodity bundle xa ∈ X ∩ Rn++ , x 1 x2
⇐⇒
d x a , x1 x a , x2 ,
∀x1 , x2 ∈ X ∩ Rn++ .
Proof. Assume x1 x2 . Since by Lemma 5(b), x1 ∼
xa d(xa , x1 )
and x2 ∼
xa , d(xa , x2 )
transitivity and completeness of yields xa xa . d(xa , x1 ) d(xa , x2 ) Suppose d(xa , x1 ) > d(xa , x2 ). By strict monotonicity, xa xa ≺ , d(xa , x1 ) d(xa , x2 ) contradicting our previous result. Therefore, d(xa , x1 ) d(xa , x2 ) follows. Conversely, assume d(xa , x1 ) d(xa , x2 ). Then, Lemma 5(b) and strict monotonicity imply x1 ∼
xa xa ∼ x2 . a 1 d(x , x ) d(xa , x2 )
Therefore, x1 x2 , by transitivity and completeness of . 2 4 5 6
A relation on a convex set X is convex if x1 x2 ⇒ λx1 + (1 − λ)x2 x2 , ∀λ ∈ [0, 1], ∀x1 , x2 ∈ X. A function f (x, y) is linear in x, if f (λx, y) = λf (x, y) for all λ > 0. A function f on a convex set X is called quasiconcave if f (x1 ) f (x2 ) ⇒ f (λx1 + (1 − λ)x2 ) f (x2 ), ∀λ ∈ [0, 1], ∀x1 , x2 ∈ X.
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Summarizing the above results, Theorem 2 shows that the function d(·, x ) represents the relation , while Theorem 4 shows that the function d(x , ·) represents “inversely”. The next theorem establishes an important relationship between the income compensation function and the distance function. Theorem 5. In addition to (D1) to (D3), let be a complete and transitive relation on X, where X is a closed cone with vertex 0. Then, for every fixed p0 ∈ Rn++ , m p 0 , x =
min
x∈X∩Rn ++
p0 · x d x, x = 1 ,
∀x ∈ X ∩ Rn++ .
Proof. The above result immediately follows since by the above assumptions m(p0 , x ) and d(x, x ) are well defined, and by Lemma 5. 2 Theorem 5 is important, since it establishes a direct relationship between the distance function and demand behavior. It also stresses the duality between modeling consumer behavior by income compensation functions or by distance functions. In view of the above equality we can start by income compensation functions or by distance functions in order to develop a model of consumer behavior. In economics, another distance function representing consumer’s preferences is well known. This is the Euclidian distance function which has been pioneered by Wold [30] and Arrow and Hahn [3] as a utility function. Since then, many different articles have been concerned with that function (Bridges and Mehta [5], Mehta [22], Alcantud and Mehta [2], et al.), and especially by G.B. Mehta [23]. Using the Euclidian distance-method, Arrow and Hahn proved the existence of a continuous utility function on a convex subset X of Rn representing any complete, transitive, continuous and locally nonsatiated relation (Arrow and Hahn [3, Theorem 1]). First, Arrow and Hahn choose an arbitrary point x0 , and then define a utility function U (x) by the Euclidian distance between x0 and the upper contour set R(x) by D x, x0 = min y − x0 . y∈R(x)
In a second step, they extend this function to the whole consumption possibility set X. If we compare the distance function chosen by Shephard with that one chosen by Arrow and Hahn, we observe that by the Euclidian distance-method an important information is lost: the distances between the components of y and x0 are aggregated to one number, which has little information, while d(y, x0 ) defines a real number by which every component of y has to be scaled (up or down) so that the “utility” level of x0 is attained. Therefore, d(y, x0 ) has a concrete meaning and is more relevant to the purpose of the present article, where appropriate measures of prosperity or well-being in respective economic models are explored. 4. Representation by a Luenberger-type benefit function Based on a given continuous utility function u : X → R, D.G. Luenberger introduced a benefit function, defined on a closed convex subset X ⊆ Rn , bounded from below. According to Luenberger, this benefit function is defined as follows (Luenberger [18, p. 463]): For any q ∈ Rn+ \ {0}, x ∈ X, s ∈ u(X): b(q; x, s) =
max{β | x − βq ∈ X, u(x − βq) s},
if u(x − βq) s for some β
−∞
otherwise.
Relationships between distance and benefit functions and their role in production theory have been studied by Chambers, Chung and Färe [6].
S. Fuchs-Seliger / J. Math. Anal. Appl. 424 (2015) 1223–1236
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Fig. 1. Illustration of bq (x, x ).
In contrast to Luenberger, we will start with a complete, transitive and continuous relation on Rn+ . Then, a benefit function, similar to the one above, will be defined for q ∈ Rn+ \ {0} as:
bq x, x
:=
sup{β | x x + βq},
if x x + βq for some β ∈ R
−∞
otherwise.
The n-tuple q is an arbitrary reference bundle of commodities. The benefit function bq (x, x ) picks the highest value of β such that the sum of x and bq (x, x )q attains a level of well-being not higher than x. In practice, the benefit function can be applied in welfare theory. Fig. 1 explains how bq (x, x ) = −∞ should be interpreted. As we can see, (x + βq) can never fall to the same level of well-being as x. Therefore, there does not exist a real number β such that x x + βq. However, if, for instance, the reference commodity bundle q is chosen, then for x0 a real number β exists. According to the above definition, bq (x, x ) can be also +∞. Given a relation such that x ∼ y for all x, y ∈ X, then bq (x, x ) = ∞. In view of the continuity of , bq (x, x ) = max{β | x x + βq}, if bq (x, x ) is finite. If x = (1, 1), x = (1000, 1000) and q = (0, 1), then bq (x, x ) = sup{β | (1, 1) (1000, 1000) + β(0, 1)} may be −∞, depending on . By definition, bq (x, x ) = max{β | x x + βq} equals that share of the reference commodity bundle q which, if added to x (or subtracted from x ), will move the individual’s level of well-being from x to x. In order to show that the above benefit function represents the individual’s preferences the following hypotheses will be assumed. (L1) (L2) (L3) (L4)
X = Rn+ . is a complete and transitive relation on Rn+ . is continuous. q ∈ Rn+ \ {0} is a reference bundle of commodities.
As a preliminary result, the following lemma will be established. Lemma 6. Assume (L1) to (L4) and let +∞ = bq (x, x ) = −∞ for x, x ∈ Rn+ . Then x ∼ x + bq (x, x )q. Proof. By definition x x + bq (x, x )q. Suppose, by way of contradiction, that x x + bq (x, x )q. By upper semicontinuity of there exists an > 0 such that for all z ∈ N (x + bq (x, x )q) ∩ X : x z. Consequently, there exists λ > bq (x, x ) such that x + λq ∈ Rn+ and x x + λq, contradicting the definition of bq (x, x ). 2 As our main result of this section, we will now turn to a representation theorem by the benefit function.
S. Fuchs-Seliger / J. Math. Anal. Appl. 424 (2015) 1223–1236
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Theorem 6. Assume (L1) to (L4). Then for x1 , x2 , x ∈ Rn+ : x1 x2
⇐⇒
bq x1 , x bq x2 , x
if bq x1 , x , bq x2 , x both are finite.
Proof. Firstly, assume x1 x2 . Since bq (x2 , x ) is finite and is continuous, bq (x2 , x ) = max{β | x2 x + βq}. From x1 x2 and x2 x + bq (x, x )q it follows that x1 x + bq (x2 , x )q. Therefore, by the definition of b, bq (x1 , x ) bq (x2 , x ). In order to prove the converse, assume bq (x1 , x ) bq (x2 , x ). By way of contradiction, suppose x2 x1 . Applying Lemma 6 gives x + bq x2 , x q x + bq x1 , x q.
(1)
In view of the first part of the proof, x2 x1 gives bq (x2 , x ) bq (x1 , x ), and thus, in view of the above assumption, bq (x2 , x ) = bq (x1 , x ). From this a contradiction to (1) follows. 2 Considering the above figure, we can realize that bq (x, x ) = bq (x, x ) = −∞, although x x . It will be shown now that bq (x, x ) is a continuous representation of , if is weakly monotone.7 Theorem 7. Assume (L1) to (L4), and additionally let be weakly monotone. Then bq (x, x ) is continuous at every x ∈ X where bq (x, x ) is finite. Proof. The function bq (· , x ) is semicontinuous from above if for each α ∈ R the set {x ∈ Rn+ | bq (x, x ) α} is closed. Therefore, let xk be a sequence with the properties that limk→∞ xk = x0 ∈ Rn+ , and bq (xk , x ) α, for α ∈ R. It has to be shown that bq (x0 , x ) α. In view of xk x + bq (xk , x )q x + αq, and since is weakly monotone, it follows, xk x + αq, and by continuity of , x0 x + αq. By the definition of bq it follows that bq (x0 , x ) α. In order to demonstrate that the function bq (· , x ) is semicontinuous from below we can proceed analogously to the above proof regarding that xk ∼ x + bq (xk , x )q. Then, if limk→∞ xk = x0 and bq (xk , x ) α for any α ∈ R we obtain xk x + bq (xk , x )q x + αq. By weak monotonicity, xk x + αq follows. Regarding continuity of , x0 x + αq follows. Since we have shown that bq (·, x ) is semicontinuous from above and from below, it is continuous. 2 We have the following Corollary. Under the above hypotheses, bq (· , x ) is a continuous representation of . By the previous analysis, we have seen that the benefit function defined above indicates an increase or decrease of the individual’s welfare when he or she moves from situation x to a new situation y, compared with a reference situation x . Therefore, the benefit function is an appropriate measure in welfare theory. 5. Summary In the preceding sections we have considered three economic functions as representations of consumer’s preferences in economic models. These functions represent the individual’s preferences under slightly different conditions. Income compensation function, distance function, and benefit function are useful representations of individual’s preferences in economic models. They show the common feature that the problematic 7
A relation is weakly monotone, if x y ⇒ x y, ∀x, y ∈ X.
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and hedonistic notion of utility can be avoided. However, we should quote Samuelson again who argues “This does not preclude the introduction of utility by any who may care to do so, nor will it contradict the results attained by use of related constructs” (Samuelson [25, p. 62]). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
J.C.R. Alcantud, A. Manrique, Continuous representation by a money-metric function, Math. Social Sci. 41 (2001) 365–373. J.C.R. Alcantud, G.B. Mehta, Constructive utility functions on Banach spaces, J. Math. Anal. Appl. 350 (2009) 590–600. K.J. Arrow, F.H. Hahn, General Competitive Analysis, North-Holland, Amsterdam, 1991. C. Blackorby, D. Primont, R.R. Russell, Duality, Separability and Functional Structure: Theory and Economic Applications, American Elsevier, New York, 1978. D.S. Bridges, G.B. Mehta, Representation of Preference Orderings, Springer-Verlag, Heidelberg, 1995. R.G. Chambers, Y. Chung, R. Färe, Benefit and distance functions, J. Econom. Theory 70 (1996) 407–419. R. Cornes, Duality and Modern Economics, Cambridge University Press, Cambridge, 1992. A. Deaton, The distance function in consumer behavior with applications to index numbers and optimal taxation, Rev. Econom. Stud. 46 (1979) 391–405. G. Debreu, Theory of Value, John Wiley and Sons, New York, 1959. W.E. Diewert, Application of duality theory, in: M. Intriligator, D. Kendrik (Eds.), Frontiers of Quantitative Economics, vol. 2, North-Holland, Amsterdam, 1974. R. Färe, Fundamentals of Production Theory, Springer, Heidelberg, 1988. P.C. Fishburn, Utility Theory for Decision Making, John Wiley and Sons, New York, 1970. S. Fuchs-Seliger, Compensated and direct demand without transitive and complete preferences, Ann. Oper. Res. 23 (1990) 299–310. S. Fuchs-Seliger, On Shephard’s Lemma and the continuity of compensated demand functions, Econom. Lett. 48 (1995) 25–28. S. Fuchs-Seliger, An analysis of distance functions in consumer theory, in: Conference ‘Utility Theory and Applications’, Trieste, 26–29 June 2002, Dipartimento di Matematica Applicata ‘Bruno de Finetti’, pubblicazione n. 7, pp. 69–91. W. Hildenbrand, P. Kirman, Equilibrium Analysis, North-Holland, Amsterdam, 1988. S. Honkapohja, On continuity of compensated demand, Internat. Econom. Rev. 28 (1987) 219–232. D.G. Luenberger, Benefit function and duality, J. Math. Econom. 21 (1992) 461–481. S. Malmquist, Index numbers and indifference surfaces, Trab. Estad. 4 (1953) 209–242. A. Mas-Colell, M.D. Whinston, J. Green, Microeconomic Theory, Oxford University Press, New York, 1995. L. McKenzie, Demand theory without a utility index, Rev. Econom. Stud. 24 (1957) 185–214. G.B. Mehta, Metric utility functions, J. Econ. Behav. Organ. 26 (1995) 289–298. G.B. Mehta, Preferences and utility, in: S. Barbera, P. Hammond, C. Seidel (Eds.), Handbook of Utility Theory, Kluwer Academic Publishers, Dortrecht, 1998, pp. 1–47. P. Newman, Gauge functions, in: The New Palgrave – A Dictionary of Economics, vol. II, 1987, pp. 484–488. P.A. Samuelson, A note on the pure theory of consumer behaviour, Economica 5 (1938) 61–71. P.A. Samuelson, Complementarity – an essay on the 49th anniversary of the Hicks–Allen revolution in demand theory, J. Econ. Lit. 12 (1974) 1255–1289. R.W. Shephard, Cost and Production Functions, Princeton University Press, Princeton, 1953. H.R. Varian, Microeconomic Analysis, third edition, W.W. Norton & Company Ltd, Castle House, 1992. J.A. Weymark, Money-metric utility functions, Internat. Econom. Rev. 26 (1985) 219–232. H. Wold, A synthesis of pure demand analysis I, II and III, Scand. Actuar. J. 26 (1943–1944) 85–118, 220–263, 69–120.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Numerical representability of preferences by economic functions Susanne Fuchs-Seliger 1 Institut für Volkswirtschaftslehre (ECON), Karlsruher Institut für Technologie, Kollegium am Schloss Bau IV, 76128 Karlsruhe, Germany
a r t i c l e
i n f o
Article history: Received 19 February 2013 Available online 24 November 2014 Submitted by J.A. Filar Keywords: Preference relation Representability Benefit function Income compensation function Money-metric utility function Distance function
a b s t r a c t This article is concerned with the representation of the individual’s preference relation by numerical functions, which are well-known in economics. The functions to be considered in this paper are the income compensation function, an appropriate distance function, and a Luenberger-type benefit function, which represent the individual’s preferences under slightly different conditions. Since these functions have an appealing economic meaning, the individual’s preferences can be represented in an appropriate economic context. Therefore, the preferences will not only be represented by an abstract unknown utility function, but by a function which can be constructed in the corresponding economic model. By this approach the hedonistic and problematic notion of utility can be avoided. © 2014 Elsevier Inc. All rights reserved.
1. Introduction This article studies the problem of representing preference relations by functions which have an economic meaning. These functions are essential in appropriate economic models like the theory of demand or welfare theory. In his famous book “Theory of Value” [9] G. Debreu has shown that every transitive, complete2 and continuous relation on a connected subset X of Rn can be represented by a continuous utility function u : X → R such that for all x, y ∈ X: xy
⇐⇒
u(x) u(y).
The proof uses profound mathematical knowledge and skills (Debreu [9, pp. 56–59]). Especially, we must be familiar with order-density. If the relation is not continuous, then in general, one has to require that there exists a countable subset of the given set X which is order dense in X (Debreu [9], Fishburn [12]). If the relation is monotone, then the difficulty of the proof reduces considerably (Mas-Colell, Whinston and
1 2
E-mail address: [email protected]. Fax: +49 721 608 43082. For all x, y ∈ X : x y ∨ y x.
http://dx.doi.org/10.1016/j.jmaa.2014.11.052 0022-247X/© 2014 Elsevier Inc. All rights reserved.
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Green [20, pp. 47–48]). In the present article we will reinvestigate the representation of a given preference relation by a function which has an appealing economic meaning, and which is principally constructive. Three different classes of functions will be considered: the income compensation function, the distance function, originally applied by R.W. Shephard to production theory (Shephard [27]), and a modified Luenberger benefit function (Luenberger [18]). Everybody, who is familiar with these functions, will realize that they are intuitive and useful representations of a given relation, when these functions are parts of appropriate economic models. The proofs of representability are quite simple. Only in the case of continuous representability of a given relation by the income compensation function one needs a little more advanced mathematical tools. Moreover, this article revisits the problem of the hedonistic and psychological meaning of utility in a scientific field like economics, which has been discussed for a long time in the economic literature (Samuelson [25], et al.). In his article “A note on the pure theory of consumer’s behaviour” Samuelson developed an economic model “freed from any vestigial traces of the utility concept” (Samuelson [25, p. 71]). The aforementioned functions represent the agent’s preferences without using the utility concept. This article is organized as follows: We will start with the income compensation function as a representation of a given relation. In comparison, properties of will be presented which imply that can be represented by a distance function. Finally, a Luenberger-type benefit function as a representation of will be studied. As we will see, the assumptions imposed on the underlying preferences will differ slightly. 2. The income compensation function Income compensation functions are an important tool in the theory of consumer behavior. They were introduced to demand theory by Lionel McKenzie in 1957 [21]. By means of the income compensation functions, we can develop a model of consumer behavior based on the individual’s preference relation and not on utility functions (Fuchs-Seliger [13]). We will come back to this point at the end of this section. Income compensation functions can be applied in order to describe consumer’s preferences by moneyincome. Therefore, the problematic and hedonistic notion of utility can be avoided in consumer theory, especially for the reason why the utility of goods is not cardinally measurable. In economics ordinal utility functions are usually assumed, meaning that the functions are determined up to a strictly increasing transformation. However, in this respect, utility is just an empty word. In order to avoid the problematic notion of utility, Paul Samuelson in 1938 developed a new description of consumer behavior based on demand functions. He introduces this approach by saying “I propose, therefore, that we start anew in direct attack upon the problem, dropping off the last vestiges of the utility analysis” (Samuelson [25, p. 62]). Paul Samuelson also suggested to measure “utility” by money-income (Samuelson [26, p. 1262]), and introduced the notion of money-metric utility functions (Samuelson [26]). Obviously, measuring the individual’s “utility” by money-income has an intuitive meaning. Therefore, this paper will be especially concerned with income compensation functions. Representability of a given relation by an income compensation function has been investigated by several authors before. Especially, we refer to J. Weymark [29], S. Honkapohja [17], and J. Alcantud and A. Manrique [1]. While an income compensation function is in general defined for a complete, transitive, and continuous relation , the domain of varies, and also the continuity of the income compensation function follows from different conditions. In this paper, we consider a closed set X ⊆ Rn+ , X = ∅, which is interpreted as a set of commodity bundles, and the set Rn++ of strictly positive price vectors p. Firstly, it will only be assumed that is reflexive. Then the income compensation (or minimum income) function can be defined as m0 p0 , x := inf p0 y y x , y∈X
for p0 ∈ Rn++ , x ∈ X.
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m0 (p0 , x) denotes the lowest income at prices p0 needed to purchase a commodity bundle at least as good as x according to the preferences of the individual. If, in addition, is upper semicontinuous, then, by definition, for all x ∈ X the upper contour set {y ∈ X | y x} is closed in X (Alcantud and Mehta [2, p. 593]). In this case m p0 , x = min p0 y y x . y∈X
This result follows from the upper semicontinuity of and X being closed and bounded from below. Moreover, fp (y) = py is a continuous function on X and, therefore, miny∈X {p0 y | y x} exists. One can immediately see Lemma 1. If is a reflexive, transitive and upper semicontinuous relation on a closed set X ⊆ Rn+ , then for every p0 ∈ Rn++ , x 1 x2
=⇒
m p 0 , x1 m p 0 , x2
∀x1 , x2 ∈ X.
Hence m(p0 , · ) weakly represents the relation . Proof. The above assertion easily follows, since x1 x2 together with the transitivity of implies, {p0 y | y ∈ X ∧ y x1 } ⊆ {p0 z | z ∈ X ∧ z x2 }, and by the upper semicontinuity of , min{p0 y | y ∈ X ∧ y x1 } min{p0 z | z ∈ X ∧ z x2 }. 2 Recall that representability of a relation by a real-valued function u : X → R means that xy
⇐⇒
u(x) u(y),
∀x, y ∈ X.
Therefore, we have to show, for any given price system p0 ∈ Rn++ , xy
⇐⇒
m p0 , x m p0 , y ,
∀x, y ∈ X.
We will assume that the relation is continuous, i.e. is upper and lower semicontinuous on X (Hildenbrand and Kirman [16, p. 60]). Upper semicontinuity has already been defined. Lower semicontinuity means that the lower contour set {x ∈ X | y x} is closed in X for all y ∈ X. In addition it will be assumed, that X is a cone with vertex 0,3 and is locally nonsatiated. Recall that a relation on X is locally nonsatiated if for every x ∈ X and every ε > 0 there exists y ∈ Nε (x) ∩ X, where Nε (x) denotes the ε-neighborhood of x, such that y x. We will not only prove that m(p0 , x) represents the relation , but that m(p0 , x) is a continuous representation of . Usually, in economic models of consumer behavior the existence of a continuous utility function describing consumer’s preferences is assumed. Accordingly, it is important to find appropriate assumptions imposed on the individual’s preference relation, implying that the income compensation function m(p0 , x) is a continuous representation of . The present investigation of this problem has been inspired by articles of J. Weymark [29], S. Honkapohja [17], and J. Alcantud and A. Manrique [1]. In accordance with these articles completeness, transitivity and continuity of the relation is assumed. By contrast, Weymark assumes that X ⊂ Rn is convex, closed and bounded from below. In addition, he assumes that any point of local satiation is also a point of global satiation [29, Proposition 2, pp. 228–229]. The latter property is also required in the articles of Honkapohja [17, Proposition 5, p. 548], and Alcantud and Manrique [1, Theorem 2, pp. 371–372]. 3
A set X is a cone with vertex 0, if for all x ∈ X, λx ∈ X, ∀λ 0.
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While Honkapohja investigates the continuity of m(p, x) when X ⊆ Rn is closed, connected and bounded from below [17, p. 546], Alcantud and Manrique assume that X ⊆ Rn is open and bounded from below [1, p. 371]. In our investigation the following correspondence related to m(p, x) is of great importance: Definition. Let be a complete, transitive and upper semicontinuous relation on a closed set X ⊆ Rn+ . Then g : Rn++ × X → 2X ,
y ∈ g(p, x) := arg min{pz | z x}, z∈X
is called the “compensated demand correspondence”. We will only be concerned with g(p0 , x) for some fixed p0 ∈ Rn++ . Under the assumptions of Lemma 1, g(p0 , x) is well defined. As a preliminary result, we will show Lemma 2. Let be a complete, transitive and continuous relation on a closed cone X ⊆ Rn+ with vertex 0. Then for any p0 ∈ Rn++ , and x ∈ X, y ∈ g p0 , x = arg min p0 z z x z∈X
=⇒
y ∼ x,
where ∼ denotes the symmetric part of . Hence, y ∼ x means y is indifferent to x. Proof. Under the assumptions made, m(p0 , x) is well defined. However, there may exist more than one alternative which minimizes {p0 y | y x} (except at x = 0) on X. Therefore, g(p0 , x) = arg miny∈X {p0 y | y x} may be many-valued. By definition, for every y ∈ g(p0 , x), y x. Suppose, by contradiction, there exists y 0 ∈ g(p0 , x) such that y 0 x. By lower semicontinuity of , for every > 0, there exists z ∈ N (y 0 ) ∩ X such that z x. In particular, since X is a cone with vertex 0, for some > 0 there exists z 0 ∈ N (y 0 ) ∩X such that y 0 z 0 , y 0 = z 0 , and z 0 x. However, since z 0 p0 < y 0 p0 , we obtain a contradiction to the definition of g(p0 , x). Hence, y ∼ x for all y ∈ g(p0 , x). 2 To establish the continuous representability of , the upper hemicontinuity of the compensated demand correspondence g(p0 , x) will be exploited. Recall that upper hemicontinuity of a compact-valued correspondence can be characterized by the following property (Hildenbrand and Kirman [16, p. 262]): A compact-valued correspondence ϕ : S → 2T \ {∅} (S, T ⊆ Rn ; S, T = ∅) is upper hemicontinuous at x0 ∈ X, if for every sequence xk in S such that xk → x0 ∈ S and for every sequence y k , y k ∈ ϕ(xk ), there exists a subsequence y kj of y k such that limj→∞ y kj = y 0 ∈ ϕ(x0 ). To prove upper hemicontinuity of g(p0 , x) at x we will first prove that the correspondence R : X → 2X , where R(x) = {y ∈ X | y x}, is lower hemicontinuous. The notion of lower hemicontinuity can be characterized as follows (Hildenbrand and Kirman [16, p. 271]): A correspondence ϕ : S → 2T \ {∅} (S, T ⊆ Rn ; S, T = ∅) is called lower hemicontinuous at x0 , if for every sequence xk in S with xk → x0 ∈ S and for every y 0 ∈ ϕ(x0 ) there exists a sequence y k with y k ∈ ϕ(xk ) and limk→∞ y k = y 0 . ϕ is upper (lower) hemicontinuous on S if it is upper (lower) hemicontinuous at every x ∈ S. ϕ is continuous if it is upper and lower hemicontinuous on S.
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First, it will be shown that the correspondence R : X → 2X , R(x) = {y ∈ X | y x}, is lower hemicontinuous. Note that the range of R contains the upper contour set of x for every x ∈ X, which is assumed to be closed in X. If R(x) is closed in X for every x ∈ X, then according to the previous definition, is called upper “semicontinuous” on X. Therefore, we have to discriminate between semicontinuity of a relation and hemicontinuity of a correspondence. Throughout the following analysis, it will be assumed that X ⊆ Rn+ is a closed cone with vertex 0. Lemma 3. Let be a complete, transitive, continuous and locally nonsatiated relation on X. Then, the correspondence R : X → 2X , R(x) = {y ∈ X | y x}, is lower hemicontinuous on X. Proof. Let xk be a sequence in X such that limk→∞ xk = x0 ∈ X, and y 0 ∈ R(x0 ). It has to be shown: There exists a sequence y k in X such that y k ∈ R(xk ) and limk→∞ y k = y 0 . Since 0 y x0 , we will consider two cases: 1. y 0 x0 and 2. y 0 ∼ x0 . 1. Let us first consider the case y 0 x0 . Then, by upper semicontinuity of there exist > 0 and a positive integer N 0 such that for all k N 0 , y 0 xk , and xk ∈ N (x0 ). Defining a sequence by y k = y 0 , for k N 0 , and y k = xk for k < N 0 , we obtain y k ∈ R(xk ) for all k and limk→∞ y k = y 0 . 2. Now, let us consider the case, y 0 ∼ x0 . For n = n1 we will construct a sequence y n such that limn→∞ y n = y 0 and y n ∈ R(xn ). This can be done in the following way: By local nonsatiation, for 1 there exists y˜ ∈ N1 (y 0 ) such that y˜ y 0 and, thus, y˜ x0 . By upper semicontinuity of there exists δ 1 > 0 such that for all x ∈ Nδ1 (x0 ) : y˜ x. In particular, there exist δ 1 > 0 and a positive integer K(δ 1 ) such that for all l K(δ 1 ), y˜ xl . Choose l1 = min{l | y˜ xl , xl ∈ Nδ1 (x0 )}. We will now change the notation and set y l1 := y˜. Then, y l1 ∈ R(xl1 ). For all natural numbers n and for n = n1 there exists y ∈ Nn (y 0 ) such that y y 0 . Since y x0 , there exists δ n such that for all x ∈ Uδn (x0 ) : y x. In particular, there exist δ n > 0 and a positive integer K(δ n ) such that for all l K(δ n ), y xl . Choose ln = min{l | y xl , xl ∈ Nδn (x0 )} and set y ln := y. Thus, y ln ∈ R(xln ). Now we have constructed a sequence y ln such that y ln ∈ R(xln ) and limn→∞ xln = x0 . For all k such that ln k < ln+1 we can take y k = y ln and obtain y k xk . For k < l1 we can set y k = xk and obtain y k ∈ R(xk ). We have thus constructed a sequence y k with the properties y k y 0 , y k ∈ R(xk ) for all k, and limk→∞ y k = y 0 . This completes our proof. 2 Now we will establish that the compensated demand correspondence g(p0 , x) is upper hemicontinuous at x for every x ∈ X. Lemma 4. Let be a complete, transitive, continuous and locally nonsatiated relation on X. Then for any fixed p0 ∈ Rn++ , g(p0 , x) is upper hemicontinuous at every x ∈ X. Proof. First, it will be shown that g(p0 , x) is compact-valued for every x ∈ X. Therefore, for any x ∈ X let y k be a sequence such that y k ∈ g(p0 , x) and limk→∞ y k = y 0 ∈ X. As y k ∈ g(p0 , x), it follows that y k x. By upper semicontinuity of , we can conclude that y 0 x. Moreover, since p0 y k+1 = p0 y k for all k, and limk→∞ (p0 y k ) = p0 y 0 , it follows that p0 y 0 = miny∈X {p0 y | y x}. Thus, y 0 ∈ arg miny∈X {p0 y | y x}. It follows that g(p0 , x) is closed for all x ∈ X. Now suppose that g(p0 , x) is not bounded. Then a sequence y k , y k ∈ g(p0 , x), exists such that limk→∞ y k = ∞ and limk→∞ p0 y k = ∞. However, y k ∈ g(pk , x) means p0 y k p0 x, contradicting this result. Now it will be shown that g(p0 , x) is upper hemicontinuous at x. Therefore, let us consider a sequence k x in X with limk→∞ xk = x0 ∈ X, and let y k be a sequence such that y k ∈ g(p0 , xk ). Since by definition, p0 y k p0 xk , and xk is bounded, y k is also bounded. Consequently, there exists a convergent subsequence y kj of y k such that limj→∞ y kj = y 0 .
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By way of contradiction, suppose y 0 ∈ / g(p0 , x0 ). By the definition of g(p0 , x0 ), there exists y˜ ∈ g(p0 , x0 ). kj 0 kj kj From y ∈ g(p , x ) it follows that y xkj , and, thus, the continuity of implies y 0 x0 . This together with y 0 ∈ / g(p0 , x0 ) yields p0 y 0 > p0 y˜. Hence, there exists ε1 , ε2 > 0 such that for all y ∈ Nε1 (y 0 ) ∩ X and for all z ∈ Nε2 (˜ y ) ∩ X : p0 y > p0 z. By Lemma 3, R(x) is lower hemicontinuous at x. Therefore, in view of 0 y˜ ∈ R(x ), there exists a sequence z kj such that z kj ∈ R(xkj ) and limj→∞ z kj = y˜. Since limj→∞ y kj = y 0 , we can find kr such that y kr ∈ Nε1 (y 0 ) and z kr ∈ Uε2 (˜ y ) and p0 y kr > p0 z kr . From y kr ∈ g(p0 , xkr ) and kr kr 0 z ∈ R(x ) a contradiction to the definition of g(p , · ) arises. This completes our proof. 2 Based on the previous lemmata we obtain that the income compensation function represents the given preference relation of the individual continuously. This is the main result of this article. Theorem 1. Assume the following hypotheses: (M1) X ⊆ Rn+ is a closed cone with vertex 0. (M2) is a transitive, complete and continuous relation on X. (M3) is locally nonsatiated. Then (a) for all x1 , x2 ∈ X and for any p0 ∈ Rn++ , x1 x2
⇐⇒
m p 0 , x1 m p 0 , x2 .
(b) m(p0 , x) is a continuous representation of for any p0 ∈ Rn++ . Proof of Theorem 1(a). In view of Lemma 1, it suffices to show, m p 0 , x1 m p 0 , x2
=⇒
x1 x2 .
First, assume m(p0 , x1 ) > m(p0 , x2 ), and suppose, by way of contradiction, x2 x1 . An application of Lemma 1 yields m(p0 , x2 ) m(p0 , x1 ), contradicting the above assumption. Hence, x1 x2 . In the case of m(p0 , x1 ) = m(p0 , x2 ) > 0, suppose, by way of contradiction, x2 x1 . By Lemma 2 there exist y 1 ∈ g(p0 , x1 ) such that y 1 ∼ x1 , and y 2 ∈ g(p0 , x2 ) such that y 2 ∼ x2 . Hence, y 2 y 1 . Since is lower semicontinuous, for every > 0 there exists an ε-neighborhood of y 2 , Nε (y 2 ), such that for all y ∈ Nε (y 2 ) ∩ X, y y 1 . In particular, there exists y 0 ∈ Nε (y 2 ) ∩ X such that y 0 y 2 , y 0 = y 2 , and y 0 y 1 . Such a y 0 exists, because X is a closed cone with vertex 0. Since y 0 y 2 and y 0 = y 2 , we also have p0 y 0 < p0 y 2 = minx∈X {p0 x | x x2 } = m(p0 , x1 ). This together with y 0 x1 is a contradiction to the definition of m(p0 , x1 ). If m(p0 , x1 ) = m(p0 , x2 ) = 0, then x1 ∼ 0 ∼ x2 by Lemma 2. 2 Proof of Theorem 1(b). In order to show that m(p0 , x) is continuous on X for every fixed p0 ∈ Rn++ , let us consider x0 ∈ X and let xk be a sequence in X such that limk→∞ xk = x0 ∈ X. It has to be shown that limk→∞ m(p0 , xk ) = m(p0 , x0 ). By definition, m(p0 , xk ) = min{p0 x | x xk } = p0 y k for some y k ∈ g(p0 , xk ). From y k xk , it follows 0 m(p0 , xk ) = p0 y k p0 xk . Convergence of xk implies that it is also bounded and thus y k is also bounded. Hence, there exists a convergent subsequence y kj of y k such that limj→∞ y kj = y 0 . As limj→∞ xkj = x0 , y kj xkj , y kj ∈ g(p0 , xkj ) and limj→∞ y kj = y 0 , upper hemicontinuity of g(p0 , ·) implies y 0 ∈ g(p0 , x0 ). From this follows limj→∞ m(p0 , xkj ) = limj→∞ (p0 y kj ) = p0 y 0 = m(p0 , x0 ). Consider any accumulation point m0 of m(p0 , xk ). There exists a subsequence
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m(p0 , xks ) of m(p0 , xk ) converging to m0 , and lims→∞ (p0 , xks ) = (p0 , x0 ). Since lims→∞ m(p0 , xks ) = lims→∞ (p0 y ks ) = m0 , for y ks ∈ g(p0 , xks ), the sequence y ks is bounded, and, thus, possesses a convergent subsequence z kr . As (p0 z kr ) is a subsequence of (p0 y ks ), it also follows limr→∞ (p0 z kr ) = m0 . Without loss of generality let y ks and z kr be identical, so that we have lims→∞ y ks = y˜. By the same arguments as used previously, upper hemicontinuity of g(p0 , ·) implies y˜ ∈ g(p0 , x0 ). From this also follows m0 = lims→∞ m(p0 , xks ) = lims→∞ (p0 y ks ) = p0 y˜ = m(p0 , x0 ). Therefore, the sequence m(p0 , xk ) only has m(p0 , x0 ) as an accumulation point. This completes our proof. 2 Thus, we have seen that the income compensation function represents a given relation continuously under appropriate conditions. Such a representation is particularly appealing, because it evaluates the preferences of the individual by money income. It is therefore a money-metric utility function. This has been pointed out firstly by P. Samuelson [26]. It should be noted that based on hypotheses on the individual’s preference relation, one can develop a model of consumer behavior by means of the income compensation function. I have shown this result in a former article (Fuchs-Seliger [13]). In particular, I will recall Theorem 2, pp. 300–302, which states: Assume the conditions (S1) X ⊆ Rn+ , X = ∅, is a closed set. (S2) is a reflexive and upper semicontinuous relation on X. Moreover, if g(p, x) is single-valued and continuous with respect to p, then (i1 ) ∂m(p,x) = gj (p, x), where gj (p, x) denotes the jth component of g(p, x). ∂pj (i2 ) If m(p, x) is twice continuously differentiable at p, then ∂gi (p, x) ∂gj (p, x) = . ∂pi ∂pj One can even demonstrate (Fuchs-Seliger [14, Lemma 1, pp. 26–27]) that the conditions (S1) and (S2) imply that g(p, x) is continuous with respect to p, if g(p, x) is single-valued. Combining these results we obtain the following conclusion: Assume (S1) and (S2) and let g(p, x) be single-valued. Then Shephard’s Lemma (i1 ) follows, and, if m(p, x) is twice continuously differentiable at p, then in view of (i2 ) the symmetry of the Slutsky-matrix follows. These are important laws in the theory of demand. g(p, x) is single-valued, if the relation is strictly convex. This means: if x y, and x = y, then λx + (1 − λ)y y, ∀λ ∈ ]0, 1[, where is the asymmetric part of . Taking into consideration that by definition m(p0 , x0 ) = p0 · g(p0 , x0 ), if g is single-valued, and if we set M 0 = m(p0 , x0 ), then we can see that z 0 = g(p0 , x0 ) is just the Marshallian demand at prices p0 and income M 0 . The function h(p, M ) is called the Marshallian demand function depending on prices and income (Varian [28, pp. 105, 125–126]). Since by m(p0 , x) we have won a continuous utility function, we could now proceed, as usual in the literature, to describe consumer’s behavior. Thus, the preceding analysis has shown that the income compensation function represents a given preference relation continuously under appropriate conditions. Moreover, the related compensated demand correspondence g(p0 , x) satisfies important laws of the theory of demand like Shephard’s Lemma and the symmetry of the Slutsky-matrix, if g(p0 , x) is single-valued.
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3. Representability by a distance function In comparison with the income compensation function, we will now consider another constructive “utility” function, which is the distance function originally applied by R. Shephard to producer theory (Shephard [27]). Various applications of distance functions to producer and consumer theory have been studied in the literature (Deaton [8], Diewert [10], Blackorby, Primont and Russell [4], Färe [11], Cornes [7], FuchsSeliger [15], et al.). The distance function is also an important tool in index number theory, and in welfare theory (see for instance, Deaton [8] and Malmquist [19]). Given a relation on a set X ⊆ Rn+ the distance function d(x, x ) can be defined by
d x, x
:= max t ∈ R++
x x t
∀x, x ∈ X,
under the supposition that the maximum exists. According to its definition, d(x, x ) is defined as the maximum value of t into which x can be divided x so that d(x,x ) attains a level of well-being not worse than x . In order to establish conditions under which d(x, x ) is well defined, we will, preliminarily, consider the utility function u(x) = x1 x2 for x ∈ R2+ , as an example. The relation on R2+ is defined as xy
:⇐⇒
x1 x2 y1 y2 .
It immediately follows x x 1 2 x1 x2 . du x, x = max t ∈ R++ t If x = (0, 1) and x = (1, 2), then du (x, x ) is not defined. If x = (0, 1) and x = (1, 0), then du (x, x ) will be defined only for t ∈ R++ ∪ {∞}. If x, x ∈ R2++ , then du (x, x ) is well defined. Before turning to the representability of by the distance function we will first establish an auxiliary lemma which is concerned with the definition of d(x, x ). It is based on the following conditions: (D1) Let X ⊆ Rn+ be a closed cone with vertex 0. (D2) is a reflexive and continuous relation on X. (D3) is strictly monotone, i.e. x > y ⇒ x y, ∀x, y ∈ X (x > y means xi > yi , ∀i n). As we can see, the strict monotonicity condition (D3) replaces local nonsatiation, assumed in the previous section as a basic property of the preference relation. Lemma 5. Assume the conditions (D1) to (D3). Then for all x, x ∈ X ∩ Rn++ , (a)
d x, x ∈ R++ ,
(b)
x ∼ x , d(x, x )
where Rn++ consists of the strictly positive n-tuples of Rn . Proof of Lemma 5(a). In view of the strict monotonicity of the set {t | xt x } is bounded from above and, by upper semicontinuity of , {t | xt x } is closed. Hence, max{t | xt x } exists, and t ∈ R++ . 2 x Proof of Lemma 5(b). Suppose, by contradiction, that d(x,x ) x . Then, in view of the lower semicontinuity x of there exists > 0 such that for all z ∈ N ( d(x,x ) ) it follows that z x , in particular for some
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x s > d(x, x ) for which xs ∈ N ( d(x,x ) ) and the definition of the distance function. 2
x s
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x , by strict monotonicity of . However, this contradicts
Remark. In order to prove Lemma 5(b) we need the assumption that is also lower semicontinuous. Therefore, we assumed the continuity of in contrast to the former article (Fuchs-Seliger [15]). Note that if we do not assume to be strictly monotone, then d(x, x ) may be ∞ even for x, x ∈ X ∩ Rn++ . Reflexivity implies that d(x, x ) is also defined for x = x . We will establish two representability theorems by means of the distance function. For our first representability theorem we additionally assume that is homothetic, this means xy
⇐⇒
λx λy,
∀λ > 0.
It will be shown now that for any fixed x , d(· , x ) is a continuous representation of on X ∩ Rn++ , where x is the reference commodity bundle. Theorem 2. In addition to (D1) to (D3) let be complete, transitive, and homothetic. Then (a) for every x ∈ X ∩ Rn++ , x1 x2
⇐⇒
d x 1 , x d x 2 , x ,
∀x1 , x2 ∈ X ∩ Rn++ .
(b) d(· , x ) is a continuous representation of on X ∩ Rn++ . Proof of Theorem 2(a). Suppose x1 x2 . Since d(x1 , x ) and d(x2 , x ) are well defined, we can set k∗ = 1 2 d(x1 , x ) and l∗ = d(x2 , x ). By definition, kx∗ x and xl∗ x . Since x1 x2 , homotheticity of implies x1 x2 x1 1 ∗ ∗ l∗ l∗ . Thus, by transitivity of , l∗ x . From the definition of d(x , x ), k l follows. 1 2 ∗ 1 2 ∗ To show the converse, suppose k = d(x , x ) d(x , x ) = l . By definition, we have kx∗ x and xl∗ x . 2 2 First, suppose k∗ > l∗ . Hence, ¬( kx∗ x ). Completeness of implies x kx∗ . Thus, by transitivity of , x1 x2 1 2 k∗ k∗ . Homotheticity of finally yields x x . 1 2 x1 x2 ∗ ∗ Finally, consider k = l . Since k∗ ∼ x and l∗ ∼ x , transitivity of implies kx∗ ∼ xl∗ . Application of the homotheticity of implies x1 ∼ x2 . 2 Proof of Theorem 2(b). By Lemma 5 d(· , x ) is well defined for all x ∈ X ∩Rn++ . First, we will show that the function d(· , x ) is semicontinuous from above. By way of contradiction, suppose there exists x0 ∈ X ∩ Rn++ , at which d(· , x ) is not semicontinuous from above. Then, there exist > 0 and a sequence xk , xk ∈ X∩Rn++ such that limk→∞ xk = x0 and d(xk , x ) d(x0 , x ) + for almost all xk . As x0 , x ∈ Rn++ and d(xk , x ) = xk max{l(k) ∈ R++ | l(k) x } for all k, strict monotonicity of implies that the sequence d(xk , x ) is bounded from above. Hence, there exists a convergent subsequence of d(xk , x ). Without loss of generality, let this be d(xk , x ) itself. Consequently, there exists k∗ such that limk→∞ d(xk , x ) = k∗ . By definition, 0 xk x and, by upper semicontinuity of , kx∗ x . From this k∗ d(x0 , x ) follows, in view of the d(xk ,x ) definition of d(· , x ). Since k∗ = limk→∞ d(xk , x ), a contradiction to the inequality d(xk , x ) d(x0 , x ) + for nearly all k follows. Suppose now, by way of contradiction, that the function d(x, x ) is not semicontinuous from below at 0 x ∈ X ∩ Rn++ . Then, there exist δ > 0 and a sequence y l , y l ∈ X ∩ Rn++ such that liml→∞ y l = x0 and l
l
d(y l , x ) d(x0 , x )−δ for nearly all y l . Due to d(yyl ,x ) ∼ x we also have d(yyl ,x ) x . Proceeding analogously to the previous case, there exists a subsequence d(y lk , x ) of d(y l , x ) such that limk→∞ d(y lk , x ) = l∗ , and 0 x xl∗ by lower semicontinuity of . Consequently, l∗ d(x0 , x ) follows. Owing to the above inequality,
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we have limk→∞ d(y lk , x ) = l∗ d(x0 , x ) − δ for nearly all y lk , and, therefore, a contradiction arises. Thus, d(x, x ) is semicontinuous from above and from below at every x0 ∈ X ∩Rn++ , and, therefore, it is continuous on X ∩ Rn++ . 2 Strengthening the assumptions, it will be now required that is convex.4 Then, one can show that the distance function is a concave function on X ∩ Rn++ . Theorem 3. In addition to the conditions of Theorem 2, let X and be convex. Then d · , x is concave on X ∩ Rn++ . Proof. Since it can easily be seen that d(x, x ) is linear5 in x, it is sufficient to demonstrate that d(· , x ) is quasiconcave6 on X ∩ Rn++ (see Newman [24]). This has been done in (Fuchs-Seliger [15, p. 86]). 2 The assumption of homotheticity is rather strong. One can delete this assumption when we introduce the notion of “inverse” representability. By this we mean a real-valued function v : X → R represents the relation “inversely” if for all x, y ∈ X : x y ⇔ v(x) v(y). One can prove the following theorem (Fuchs-Seliger [15, p. 77]). Theorem 4. In addition to (D1) to (D3), let be transitive and complete. Then, d(x0 , · ) represents inversely, i.e., for every reference commodity bundle xa ∈ X ∩ Rn++ , x 1 x2
⇐⇒
d x a , x1 x a , x2 ,
∀x1 , x2 ∈ X ∩ Rn++ .
Proof. Assume x1 x2 . Since by Lemma 5(b), x1 ∼
xa d(xa , x1 )
and x2 ∼
xa , d(xa , x2 )
transitivity and completeness of yields xa xa . d(xa , x1 ) d(xa , x2 ) Suppose d(xa , x1 ) > d(xa , x2 ). By strict monotonicity, xa xa ≺ , d(xa , x1 ) d(xa , x2 ) contradicting our previous result. Therefore, d(xa , x1 ) d(xa , x2 ) follows. Conversely, assume d(xa , x1 ) d(xa , x2 ). Then, Lemma 5(b) and strict monotonicity imply x1 ∼
xa xa ∼ x2 . a 1 d(x , x ) d(xa , x2 )
Therefore, x1 x2 , by transitivity and completeness of . 2 4 5 6
A relation on a convex set X is convex if x1 x2 ⇒ λx1 + (1 − λ)x2 x2 , ∀λ ∈ [0, 1], ∀x1 , x2 ∈ X. A function f (x, y) is linear in x, if f (λx, y) = λf (x, y) for all λ > 0. A function f on a convex set X is called quasiconcave if f (x1 ) f (x2 ) ⇒ f (λx1 + (1 − λ)x2 ) f (x2 ), ∀λ ∈ [0, 1], ∀x1 , x2 ∈ X.
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Summarizing the above results, Theorem 2 shows that the function d(·, x ) represents the relation , while Theorem 4 shows that the function d(x , ·) represents “inversely”. The next theorem establishes an important relationship between the income compensation function and the distance function. Theorem 5. In addition to (D1) to (D3), let be a complete and transitive relation on X, where X is a closed cone with vertex 0. Then, for every fixed p0 ∈ Rn++ , m p 0 , x =
min
x∈X∩Rn ++
p0 · x d x, x = 1 ,
∀x ∈ X ∩ Rn++ .
Proof. The above result immediately follows since by the above assumptions m(p0 , x ) and d(x, x ) are well defined, and by Lemma 5. 2 Theorem 5 is important, since it establishes a direct relationship between the distance function and demand behavior. It also stresses the duality between modeling consumer behavior by income compensation functions or by distance functions. In view of the above equality we can start by income compensation functions or by distance functions in order to develop a model of consumer behavior. In economics, another distance function representing consumer’s preferences is well known. This is the Euclidian distance function which has been pioneered by Wold [30] and Arrow and Hahn [3] as a utility function. Since then, many different articles have been concerned with that function (Bridges and Mehta [5], Mehta [22], Alcantud and Mehta [2], et al.), and especially by G.B. Mehta [23]. Using the Euclidian distance-method, Arrow and Hahn proved the existence of a continuous utility function on a convex subset X of Rn representing any complete, transitive, continuous and locally nonsatiated relation (Arrow and Hahn [3, Theorem 1]). First, Arrow and Hahn choose an arbitrary point x0 , and then define a utility function U (x) by the Euclidian distance between x0 and the upper contour set R(x) by D x, x0 = min y − x0 . y∈R(x)
In a second step, they extend this function to the whole consumption possibility set X. If we compare the distance function chosen by Shephard with that one chosen by Arrow and Hahn, we observe that by the Euclidian distance-method an important information is lost: the distances between the components of y and x0 are aggregated to one number, which has little information, while d(y, x0 ) defines a real number by which every component of y has to be scaled (up or down) so that the “utility” level of x0 is attained. Therefore, d(y, x0 ) has a concrete meaning and is more relevant to the purpose of the present article, where appropriate measures of prosperity or well-being in respective economic models are explored. 4. Representation by a Luenberger-type benefit function Based on a given continuous utility function u : X → R, D.G. Luenberger introduced a benefit function, defined on a closed convex subset X ⊆ Rn , bounded from below. According to Luenberger, this benefit function is defined as follows (Luenberger [18, p. 463]): For any q ∈ Rn+ \ {0}, x ∈ X, s ∈ u(X): b(q; x, s) =
max{β | x − βq ∈ X, u(x − βq) s},
if u(x − βq) s for some β
−∞
otherwise.
Relationships between distance and benefit functions and their role in production theory have been studied by Chambers, Chung and Färe [6].
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Fig. 1. Illustration of bq (x, x ).
In contrast to Luenberger, we will start with a complete, transitive and continuous relation on Rn+ . Then, a benefit function, similar to the one above, will be defined for q ∈ Rn+ \ {0} as:
bq x, x
:=
sup{β | x x + βq},
if x x + βq for some β ∈ R
−∞
otherwise.
The n-tuple q is an arbitrary reference bundle of commodities. The benefit function bq (x, x ) picks the highest value of β such that the sum of x and bq (x, x )q attains a level of well-being not higher than x. In practice, the benefit function can be applied in welfare theory. Fig. 1 explains how bq (x, x ) = −∞ should be interpreted. As we can see, (x + βq) can never fall to the same level of well-being as x. Therefore, there does not exist a real number β such that x x + βq. However, if, for instance, the reference commodity bundle q is chosen, then for x0 a real number β exists. According to the above definition, bq (x, x ) can be also +∞. Given a relation such that x ∼ y for all x, y ∈ X, then bq (x, x ) = ∞. In view of the continuity of , bq (x, x ) = max{β | x x + βq}, if bq (x, x ) is finite. If x = (1, 1), x = (1000, 1000) and q = (0, 1), then bq (x, x ) = sup{β | (1, 1) (1000, 1000) + β(0, 1)} may be −∞, depending on . By definition, bq (x, x ) = max{β | x x + βq} equals that share of the reference commodity bundle q which, if added to x (or subtracted from x ), will move the individual’s level of well-being from x to x. In order to show that the above benefit function represents the individual’s preferences the following hypotheses will be assumed. (L1) (L2) (L3) (L4)
X = Rn+ . is a complete and transitive relation on Rn+ . is continuous. q ∈ Rn+ \ {0} is a reference bundle of commodities.
As a preliminary result, the following lemma will be established. Lemma 6. Assume (L1) to (L4) and let +∞ = bq (x, x ) = −∞ for x, x ∈ Rn+ . Then x ∼ x + bq (x, x )q. Proof. By definition x x + bq (x, x )q. Suppose, by way of contradiction, that x x + bq (x, x )q. By upper semicontinuity of there exists an > 0 such that for all z ∈ N (x + bq (x, x )q) ∩ X : x z. Consequently, there exists λ > bq (x, x ) such that x + λq ∈ Rn+ and x x + λq, contradicting the definition of bq (x, x ). 2 As our main result of this section, we will now turn to a representation theorem by the benefit function.
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Theorem 6. Assume (L1) to (L4). Then for x1 , x2 , x ∈ Rn+ : x1 x2
⇐⇒
bq x1 , x bq x2 , x
if bq x1 , x , bq x2 , x both are finite.
Proof. Firstly, assume x1 x2 . Since bq (x2 , x ) is finite and is continuous, bq (x2 , x ) = max{β | x2 x + βq}. From x1 x2 and x2 x + bq (x, x )q it follows that x1 x + bq (x2 , x )q. Therefore, by the definition of b, bq (x1 , x ) bq (x2 , x ). In order to prove the converse, assume bq (x1 , x ) bq (x2 , x ). By way of contradiction, suppose x2 x1 . Applying Lemma 6 gives x + bq x2 , x q x + bq x1 , x q.
(1)
In view of the first part of the proof, x2 x1 gives bq (x2 , x ) bq (x1 , x ), and thus, in view of the above assumption, bq (x2 , x ) = bq (x1 , x ). From this a contradiction to (1) follows. 2 Considering the above figure, we can realize that bq (x, x ) = bq (x, x ) = −∞, although x x . It will be shown now that bq (x, x ) is a continuous representation of , if is weakly monotone.7 Theorem 7. Assume (L1) to (L4), and additionally let be weakly monotone. Then bq (x, x ) is continuous at every x ∈ X where bq (x, x ) is finite. Proof. The function bq (· , x ) is semicontinuous from above if for each α ∈ R the set {x ∈ Rn+ | bq (x, x ) α} is closed. Therefore, let xk be a sequence with the properties that limk→∞ xk = x0 ∈ Rn+ , and bq (xk , x ) α, for α ∈ R. It has to be shown that bq (x0 , x ) α. In view of xk x + bq (xk , x )q x + αq, and since is weakly monotone, it follows, xk x + αq, and by continuity of , x0 x + αq. By the definition of bq it follows that bq (x0 , x ) α. In order to demonstrate that the function bq (· , x ) is semicontinuous from below we can proceed analogously to the above proof regarding that xk ∼ x + bq (xk , x )q. Then, if limk→∞ xk = x0 and bq (xk , x ) α for any α ∈ R we obtain xk x + bq (xk , x )q x + αq. By weak monotonicity, xk x + αq follows. Regarding continuity of , x0 x + αq follows. Since we have shown that bq (·, x ) is semicontinuous from above and from below, it is continuous. 2 We have the following Corollary. Under the above hypotheses, bq (· , x ) is a continuous representation of . By the previous analysis, we have seen that the benefit function defined above indicates an increase or decrease of the individual’s welfare when he or she moves from situation x to a new situation y, compared with a reference situation x . Therefore, the benefit function is an appropriate measure in welfare theory. 5. Summary In the preceding sections we have considered three economic functions as representations of consumer’s preferences in economic models. These functions represent the individual’s preferences under slightly different conditions. Income compensation function, distance function, and benefit function are useful representations of individual’s preferences in economic models. They show the common feature that the problematic 7
A relation is weakly monotone, if x y ⇒ x y, ∀x, y ∈ X.
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