# Multidimensional commodity structures.

While measurement theory is a fundamental part of analysis in the natural sciences and has had interesting applications in psychology, economists have appealed to measurement theory mainly in the theory of utility. Historically, economic theory has been grounded in the numerical measurement of commodities in an n-dimensional euclidean space with scarcely any intelligible reference to the empirical commodities themselves. It is the purpose of this paper to partially fill this void in the logic of economics. In what follows, one identifies properties of raw commodities that yield a suitable theory of economic commodity measurement, sketches measurement in one-dimensional commodity spaces, and extends informally one-dimensional to n-dimensional spaces [Mirowski, 1991, p. 153].I. One-dimensional Commodity Structures

Section I discusses measurement theory in certain one-dimensional commodity structures. A rigorous presentation of these same structures and of the underlying abstract measurement theory can be found elsewhere [Narens, 1985; Garvin, 1987!. Quantity measurement is applied in this paper to all commodities for which magnitude sets can be obtained. In the Walrasian tradition, the theory of quantity measurement is applied to real and financial fixed and working (variable) capital; the measurement of services is derived from the measurement structure of the capital to which the services correspond [Walras, 1954, p. 237; Debreu, 1959, pp. 30-2].

(A) The Commodity Stock

A commodity, [gamma], is understood to be a set of elements x called elementary economic particles. The particles are regarded as the fundamental entities of which [gamma] is composed. At any moment of time, the totality of commodity y in existence is an economic stock. For any pair of elements, x and y, in [gamma] such that x [conguent] y, where [conguent] means `is qualitatively indistinguishable from.' Any commodity satisfying this relation is called particle-homogeneous (or, simply, homogeneous) by tradition.

A commodity is either finite or infinite and, if infinite, either countable or uncountable, depending, of course, on the number of its particles. The stock of every empirical commodity is evidently bounded in the sense that at any moment of time there is a real number t, such that the quantity measurement of its totality is less than t. Every commodity y is scarce (in the measurement sense), even in the short run. In economic theory, a commodity whose current market supply, s, is so large relative to its current demand, d, that there is no positive price, p, for which d(p) = s(p) is said to be a free good, even though its quantity measurement is bounded. The measurement of a commodity is logically independent of the condition of economic scarcity or superabundance.

At any moment of time, a partition separates commodity stock y into a collection of mutually disjoint magnitudes (portions, market baskets), mutually exclusive subsets of [gamma], whose union is [gamma]. Since there may be more than one partition of [gamma], the partition [phi] is chosen from the set [psi] of all possible partitions of [gamma]. If commodity y is sufficiently fine-grained, a refinement [[phi].sub.*] of [phi] may always be constructed from [phi]. Should the current partition [phi] be incompatible with the measurement requirements for [gamma], the most detailed partition of [phi] is the partition containing only singleton subsets of [gamma]. Let an appropriate refinement [[phi].sub.*] of a partition [phi] be chosen. Let c, called the magnitude set of [gamma], be the set containing the disjoint magnitudes in [[phi].sub.*]., all possible disjoint unions of these magnitudes, all possible disjoint unions of disjoint unions of the same magnitudes, and so on. The magnitude set c is, of course, a subset of P([gamma]), the power set of the commodity stock [gamma]. If the partition [[phi].sub.*] is the set of all singleton subsets of [gama], then c = P([gamma]). The magnitudes in c are empirically measurable. Since it is the measurement of magnitudes of commodity c, not the measurement of the particles of y, which is central to quantity measurement theory, the set c of magnitudes of [gamma] is taken as a primitive of commodity measurement.

(B) The Relation [is less than or equal to] in c

Measurement of commodity [gamma] is a systematic comparing of magnitudes in the magnitude set c in terms of some attribute of those magnitudes. Since quantity theory in economics is a statement about the size of the elements of c, a binary relation [is less than or equal to] is assumed to exist which size-orders some or all of its elements. Thus, c is qualitatively ordered by [is less than or equal to], and [is less than or equal to] is a primitive of commodity quantity measurement.

For any x and y in c and a binary relation [is less than or equal to], if x [is less than or equal to] y and y [is less than or equal to] x ~ y. If x :s y and y < x, then x - y. If x [is less than or equal to] y and [Mathematical Expression Omitted] ( x ~ y), then x < y. If [Mathematical Expression Omitted] (x [is less than or equal to] y), then y < x. The relation <, which is irreflexive and transitive, nay be called a strict preorder. When [is less than or equal] is reflexive and transitive, it is a preorder. If [is less than or equal to! is reflexive, transitive, and antisymmetric, it is an order. If [is less equal to] holds for every pair x and y in c, it is said to be complete. The relation ~ in c is reflexive, transitive, and symmetric; it is, thus, an equivalei relation. If ~ is an equivalence relation in c, the set [x] of all magnitudes y, such that x ~ y, is the equivalence (indifference) class of x. The set c/~ of all equivalence classes [x] of c is called the quotient set derived from c.

Let [is less than or equal to] be a complete preorderig of the magnitude set c and let a be a subset of c. An element x in : is an upper bound of a iff (if, and only if) y [is less than or equal to] x for every y in a. x is a supremum of a iff x is an upper bound of a and x [is less than or equal to] z for any other upper bound z of a. x is maximal in a iff x is a supremum of a and x is in a. x is maximum in a iff x is the unique maximal element in a. An element u in c is a lower bound of a iff u [is less than or equal to] y for every y in a. u is an infimum of a iff u is a lower bound of a and v [is less than or equal to! u for any other lower bound v of a. u is minimal in a iff u is an infimum of a and u is in a. u is minimum in a iff u is the unique minimal element in a. Clearly, c is bounded since the commodity stock y is one of the magnitudes (the maximum magnitude) in c; c is bounded, whether or not y is bounded.

(C) Concatenation

Observing that an n-ary operation r on the magnitude set c may be regarded as an (n + 1)-ary relation on c and allowing for multiple relations on the same magnitude set, the ordered (m + 1)-tuple (c, [r.sub.1], [r.sub.2], ...., [r.sub.m] is called an empirical relational structure. Of central interest to abstract measurement theory is the establishment of the existence and uniqueness properties of a function q, called a representation from an empirical relational structure (c, [r.sub.1.], [r.sub.2], ..., [r.sub.m] to a numeric relational structure ([r.sup.*], [S.sup.1], [S.sup.2], ..., [S.sub.m], where [r..sup.*] is a subset of the nonnegative real numbers [r.sub.0+]. The relation [r.sub.i] in c is preserved as the relation [s.sub.i] in [r.sup.*], i = 1, 2, ..., m. In other words, q is a homomorphism of c into [r.sub.0+]. The set s of all permissible transformations of q is called a scale. If q is a representation of (c, [r.sub.1], [r.sub.2], ,..., [r.sub.m]) in ([r.sup.*], [S.sub.1], [S.sub.2], ..., [S.sub.m]), such that the measurement of magnitudes in c by q is additive or multiplicative, then (c, [r.sub.1], [r.sub.2], ..., [r.sub.m]) is extensively measured.

An additive operation O on the magnitude set c, which combines a finite number of its mutually disjoint magnitudes to form a new magnitude y in c in such a way that the measurement of y is the arithmetic sum of the measurements of the individual magnitudes, is called (additive) concatenation. Placing distinct, short, straight, steel rods in a set of such rods end-to-end to form a new straight, steel rod is an additive concatenation of the short steel rods. Additive concatenation of distinct magnitudes in c is tantamount to the union of mutually disjoint sets but, loosely speaking, not all concatenated magnitudes are distinct. A magnitude x in c may be expressed as a size-multiple n of a magnitude y by concatenating y with itself n times. That is,

x ~ y O y O ... O y = n [multiplied by] y. (1)

The function [b.sub.n]: [i.sup.*] X c [arrow right] c, where n is in a subset i* of nonnegative integers, defined by [b.sub.n](X) = n [multiplied by! x,

(2) (2) is called magnitude autoconcatenation, a metalanguage interpretation of which is "measurement by `meterstick'." Magnitude autoconcatenation [b.sub.n](x) = n [multiplied by] x may be regarded as the multiplication of magnitudes in c by scalars in [i.sub.*]. It is useful to define for any x in c, O [multiplied by] x = [x.sub.0], the null magnitude (empty basket) in c, and 1 [multiplied by! x = x. The coneatenation operation O is a primitive of the system.

For any x and y in c with the complete preorder [is less than or equal to], it is assumed that there exists a positive integer n such that x [is less than or equal to] n [multiplied by] y. Because it is analogous to the Archimedean property of the real numbers, this condition in c is called Archimedean. Since c is bounded, n [multiplied by] y, as such, may not be in c. If n [multiplied by] 9 is otherwise not in c, b.(y) = n y is defined to be a maximal element of c. By such a rule, n [multiplied by] y is in c for every n. Suppose there are two subsets a and b of c such that a U b = c and, for every x in a and y in b, x [is less than or equal to] y. The ordered pair (a, b) is a Dedekind cut of the set c. Suppose, in addition, there is an element u in c such that either u is maximal in a or minimal in b. Then u is a cut element of c. It is easy to see that if (a, b) is a Dedekind cut of c and if u and v are cut elements of c, then u ~ v. If every x in c is a cut element of c, then c is said to be Dedekind-complete (or, simply, Dedekind). Clearly, a set that is Dedekind is also Archimedean. Suppose c is Dedekind and suppose autoconcatenation has been generalized to the nonnegative real numbers t in [r.sub.0+]. If [x.sup.1] the unit magnitude, is specified, then every y in c can be expressed as the autoconeatenation y ~ s[multiplied by][x.sup.1] for some nonnegative real number s.

By construction, the magnitude set c is closed with respect to magnitude concatenation 0. From the equivalence of magnitude concatenation in c with the set-theoretic union of disjoint sets, c is found to be associative and commutative with respect to O. Since the null magnitude x" is in c, it easily follows that [x.sup.0] is the unique zero element (additive identity) of c. On the other hand, c contains no inverses; there is no y in c such that x O y = [X.sup.0] for any magnitude x. Thus, the completely preordered, Dedekind, relational structure (c, [is less than or equal to] O) forms an abelian semi-group (c, O) with identiity [x.sup.0].

(D) Operations on Autoconcatenation in c.

A function e of a commodity relational structure (c, [r.sup.1], [r.sup.2], .... [r.sub.m]), m a nonnegative integer, into itself, preserving [r.sub.1], [r.sub.2], ..., [r.sub.m], is an endomorphism of c. An endomorphism b of (c, [r.sub.1], [r.sub.2], ..., [r.sub.m] onto itself is an automorphism of (c, [r.sub.1], [r.sub.2], ..., [r.sub.m]). Suppose a is a set of endomorphisms of (c, [r.sub.1], [r.sub.2], ..., [r.sub.m]). a is point-unique iff, for any b and c in a and some x in c, b(x) = c(x) implies b = c. a is point-homogeneous iff for every x and y in c there exists a b in a such that b(x) = y. (c, [r.sub.1], [r.sub.2], ... [r.sub.m]) is said to be point-unique and point-homogeneous. Only oblique mention of endomorphism theory is made in this paper. Elegant existence and uniqueness theorems for endomorphisms in c and their applications to measurement theory can be examined in detail elsewhere [Naren, 1985, pp. 44 ff].

Let a be the set of all functions [b.sub.t]: [r.sub.0+] x c [right arrow] c defined by

[b.sub.t](x) = t [multiplied by] x.

Since [b.sub.t](x) [less than or equal to] [b.sub.t] (y) iff t [multiplied by] x [less than or equal to] t [multiplied by] y iff x [less than or equal to! y and since

(4) [b.sub.t](x [circle] y) = t [multiplied by] (x [circle] y) = t [multiplied by x] [circle] t [multiplied by y]

for arbitrary t and for every disjoint x and y in c by definition of concatenation, [b.sub.t], when t is taken as arbitrary, is an order- and addition-preserving function from c onto c. That is, [b.sub.t] with arbitrary t is an automorphism of c.

Let h: a [right arrow] c be defined by

(5) h([b.sub.t]) = [b.sub.t](x)

for arbitrary x in c. Clearly, b, is an autoconcatenation in c. Let [cross in circle] be an additive operation in a. Then, h([b.sub.s] [cross in circle] [b.sub.t]) = ([b.sub.s] [cross in circle] [b.sub.t])(x) for arbitrary x is defined by

(6) ([b.sub.s] [cross in circle] [b.sub.t]) (x) = [b.sub.s](x) [circle] [b.sub.t)(x)

for every s and t in [r.sub.0+]. By what has already been established,

(7) [b.sub.s](x) [circle] [b.sub.t](x) = s [multiplied by] x [circle] t [multiplied by] x = (s + t) [multiplied by x.

That is, the addition [cross in circle] of autoconcatenations in a is preserved by h as the sum of the autoncatenations on an arbitrary x in c.

Suppose for arbitrary [zb.sub.t](z) = t [multiplied by] z ~ y and [b.sub.s](y) = s [multiplied by] y ~ x. Clearly, x is equivalent to the composite function value

(8) [b.sub.s]([b.sub.t](z)) = [b.sub.s] (t [multiplied by] z) = s [multiplied by] (t [multiplied by] z) ~ [b.sub.s](y).

Let [dot in circle] be the multiplicative operation in a defined by

(9) h([b.sub.s] [dot in circle] [b.sub.t]) = ([b.sub.s] [dot in circle] [b.sub.t])(z))

for arbitrary z in c. Thus, the multiplication [dot in circle] of automorphisms in a is preserved by the homomorphism h in c as the scalar product of autoconcatenations on an arbitrary magnitude z. It is worth observing that autoconcatenation is the repeated concatenation of a magnitude z in c with itself. However, this should not obscure the fact that the algebraic and measurement structure; in this paper are built entirely of primitive set and relations.

(E) The Quotient Structure (c/ ~, [less than or equal to]/~, [circle]/~)

Let p be the function that associates each x in c with the equivalence class [x] of x. For all y in c,

(10) p(y) = [x]

iff x ~ y. Thus, for all x and y in c either [x] = [y] or [x] [intersection] [y] = [[x.sup.0]]. p is the projection of the magnitude set c onto its quotient set c/~. It is easily seen that the size relation [less than or equal to]/~ is induced in c/~ from [less than or equal to] in c by p. For all [x] and [y] in c/~, [x] [less than or equal to] y for all x in [x] and y in [y]. Furthermore, if [less than or equal to] is a complete preorder in c, then [less than or equal to]/~ is a complete preorder in c/~. In addition, p induces the concatenation operation [circle]/~ in c/~ from [circle] in c: for all x and y in c,

(11) p(x [circle] y) = [x [circle] y] = [x] [circle]/~[y].

Autoconcatenation also holds in c/~: p(s [multiplied by] x). Then, for arbitrary x in c and all s in [r.sub.0+],

(12) [s [multiplied by] x] = s [multiplied by] [x].

Clearly, if the completely preordered, Dedekind, relational structure (c, [less than or equal to], [circle]) contains an abelian semi-group (c, [circle]) with identity [x.sub.0], then (c/~, [less than or equal to]/~, [circle]/~) is a completely ordered, Dedekind, relational structure and (c/~, [circle]/~) is an abelian semi-group with identity [[x.sup.0]].

(F) Ordinal Structures

Let [less than or equal to] be a complete ordering of a magnitude set c. c is order-dense (with respect to the ordering [less than or equal to]) iff for every x and z in c with x < z there is a y in c such that x < y < z. Further, y is an interior point of c iff there exist x and z in c such that x < y < z. A subset a of c is called an order-dense subset of c iff for every x and z in c with x < z there is a y in a such that x < y < z.

A magnitude set c with a complete ordering [less than or equal to] has the Cantorian order-type n iff c is countably infinite, c is order-dense, and every magnitude in c\{[gamma]} is an interior point in c. It is understood that, by a standard topology in c, sets of the form (y: y [epsilon] c [conjunction] ([inverted A]z) (z [epsilon] c [conjunction] [x.sup.0] [less than or equal to] y < z)} are open [Garvin, 1989, p. 35-81. Suppose further that the magnitude set c with the complete ordering [less than or equal to] is Dedekind and that every point in c\{[gamma]} is an interior point of c. Suppose that there exists a subset a of c, such that (a, [less than or equal to!) is an n-order and that for all x and z in c with x < z there is a y in a for which x < y < z. In other words, a is a countably order-dense subset of c. Then c has the Cantorian order-type [zeta] [Narens, 1985, pp. 32-3].

The relational structure (c, [less than or equal to]) is an ordinal structure iff c is non-empty, [less than or equal to] is a binary relation in c, and there exists a representation q of c in [r.sub.0+] such that, for all x and y in c, x [less than or equal to] y iff q(x) [less than or equal to] q(y). The critical existence theorem for an (ordinal) representation q of c in [r.sub.0+] states that (c, [less than or equal to]) is an ordinal structure iff [less than or equal to] is a complete preordering of c and c has a countably order-dense subset [Narens, 1985, p. 38].

The representation q of the ordinal structure (c, [less than or equal to]) in ([r.sub.0+], [less than or equal to]) is not unique. Suppose that q is a representation of (c, [less than or equal to]) in ([r.sub.0+], [less than or equal!) and that f is a strictly increasing function from q(c) into [r.sub.0+]. It can be shown that 1) the composite f * q is a representation of c in ([r.sub.0+], [less than or equal to]) and 2) if m is also a representation of (c, [less than or equal to]) in ([r.sub.0+], [less than or equal to!), there exists a strictly increasing function g from q(c) into [r.sub.0+], such that m(x) = (g * q)(x) for every x in c. The set of all representations of the ordinal structure (c, [less than or equal to]) in (r, [less than or equal to]) is an ordinal scale [Narens, 1985, p. 381.

An ordinal structure is the simplest measurement structure. If the preordering [less than or equal to] is taken as a complete preference relation on the commodity magnitude set c and if c has a countably dense subset, then (c, [less than or equal to]) is an ordinal structure and the representation q of c in [r.sub0+] is a preference (utility) function, reflecting Pareto's claim that ordinal measurement of c is adequate for the economic theory of utility.

(G) Extensive Structures

The theory of extensive measurement is a major achievement of measurement theory. The first successful theorems were written by Helmholtz [1887] and Holder [1901]. An early, but clear, application of the theory of extensive measurement in economics was made by Fisher [1920, pp. 30-4] in the construction of the compensation principle in the theory of money and, more recently, extensive measurement has appeared in the theory of inflation [Hall, 1982, pp. 112-5; Garvin, 1990, pp. 35-6]. The following sketch contains central features of the quantity measurement of all economic commodities.

Consider the abelian semi-group (c, [circle]) with identity [x.sup.0]), where c is a commodity magnitude set. Suppose (c, [circle]) with ordering [less than or equal to] satisfies the following axioms: A(1. 1): [less than or equal to! is a complete preordering of c. A(1.2): c is bounded. A(1.3): c is Dedekind-complete. A(1.4): For every x, y, and z in c, x [less than or equal to] y iff x [circle] z. A(1. 5): For every x and y in c, x [less than equal to! y implies there exists a z in c such that x 0 z ~ y.

The assumption A(1.3) that c is Dedekind-complete allows the current partition-refinement [theta]* of commodity stock [gamma] to be exceedingly fine. A(1.4) is a montonicity law, and A(1.5) s a solvability law. Then, it is clear that a homomorphism q of c into a subset [r.sup.*] of the set [r.sub.0+] of nonnegative real numbers exists such that 1) for all x and y in c, if x [less than or equal to] y, then q(x) [less than or equal to] q(y) and 2) for all disjoint x and y in c and for all s and t in [r.sub.0+],

(13) q(s [multiplied by] x [circle] t [multiplied by] y) = q(s [multiplied by] x) + q(t [multiplied by] y) = sq(x) + tq(y).

Equation (13) defines q as a linear transformation of the subset of disjoint elements in (c, [less than or equal to] [circle]) into the numeric relational structure ([r.sup.*], [less than or equal to], +), the final link between empirical commodities and the real numbers in the quantification of the former. It follows that, for each x in c, the unit magnitude [x.sup.1] and some scalar t,

(14) q(x) = q(t [multiplied by [x.sup.1]]) = t

in [r.sub.0+], revealing the suppression of the measurement dimension of x, according to a convention of measurement theory [deJong, 1967]. The structure ([r.sup.*], [less than or equal to], +), of course, contains an additive representation of (c, [less than or equal to], [circle]). A relational structure (c, [less than or equal to], [circle]) satisfying axioms A(1. 1) - A(1.5) and having an additive representation q in ([r.sup.*], [less than or equal to!, +) is an extensive relational structure. The nonempty set s of representations of the relational structure (c, [less than or equal to], [circle]) in ([r.sup.*], [less than or equal to], +) is a ratio scale iff 1) tq is in s for every q in s and every t in [r.sub.0+], and 2) for each m and q in s, there is a t in [r.sub.0+], such hat m = tq [Narens, 1985, p. 23; Garvin, 1987, pp. 8-10!.

(H) q, p, and Q

Suppose (c, [less than or equal to], [circle]) is the commodity relational structure satisfying axioms A(1.1) - A(1.5) and suppose c/~ is the quotient set factored from c, using the equivalence relation ~. Since the projection p from c to c/~ preserves [less than or equal to] and [circle] in c as [less than or equal to] /~ and [circle] /~ in c /~, p is a homomorphism of c onto c/~. It is easy to show that the c-quotient relational structure (c/~, [less than or equal to]/~, [circle/~) is Dedekind-complete (since the magnitude set c is Dedekind-complete) and satisfies, in addition, axioms A(1.1), A(1.2), A(1.4), and A(1.5) [Garvin, 1987, pp. 20-1]. Since (c/~, [circle]/~) is a completely ordered semi-group, there is an isomorphism Q of (c/~, [circle]/~, [circle]/~) onto the numeric relational structure [r.sup.*,] [less than or equal to], +) in the nonnegative real numbers [r.sub.0+), using a version of Holder's Theorem with the property that

(15) Q[x] = q(x)

for all x in c [Krantz et al., 1971, pp. 53-4].

An empirical relational structure (c, [r.sub.1], [r.sub.2], ..., [r.sub.m]) is a scalar structure iff c has order-type [zeta] and (c, [r.sub.1], [r.sub.2], ... [r.sub.m]) is point-unique and point-homogeneous. It is not difficult to prove that a ratio scale [s.sub.~] for the quotient relational structure ([c/~-, [less than or equal to]/~, [circle]/~) exists such that some Q in [s.sub.~] is onto [r.sup.*] iff 1) (c/~, [less than or equal to]/~, [circle]/~) is a scalar structure and 2) each endomorphism of (c/~, [less than or equal to]/~, [circle]/~ is an automorphism of (c/~, [less than or equal to]/~, [circle]/~) [Narens, 1985, pp. 49-50; Garvin, 1987, p. 15]. Thus, the measurement properties that originate in the magnitude set c are preserved both in [r.sup.*] and in the quotient set c / ~ in such a way that the numeric sturucture in [r.sup.*] and the empirical structure in the quotient set c/~ are measurement copies of each other. A consequence of the nexus between q, p, and Q is that order, addition, and ordinary multiplication in [r/.sup.*] that represent order, magnitude concatenation, and magnitude autoconcantenation in the magnitude set c can be carried out without explicit reference to c. It is a startling fact that discourse about commodity quantities in economic theory which depends on the numeric structure ([r.sup.*], [less than or equal to], +) and ignores the empirical structure (c, [less than or equal to], [circle]) may be at least partially successful without comprehending the logic which clarifies this success.

A general competitive economy containing a monetary system operating on a gold standard in which the unit of account (perhaps one dollar) is the measure of the standard unit [x.sup.1] (or [[x.sup.1]]) of monetized gold c (perhaps 1/42.22 troy ounces of fine gold) by a representation q (or Q) in [r.sup.*] is a historically important application of scalar measurement in economics. Generalizing the scalar measurement of monetized gold to the real collateral underlying bank portfolio assets in a way that exploits the scalar properties of its measurement, it is clear that inflation provoked (with other things equal) by the denigration of this collateral may be characterized as a measurement phenomenom [Garvin, 1990].

II. Multidimensional Commodity Structures

(A) Conjoint Measurement Structures

Every set of objects examined for measurement cannot be concatenated and, thus, cannot be extensively measured. For example, the failure of concatenation in the measurement of utility in economic theory is well-known, casting a penetrating light on Pareto's assertion that cardinal measurement of commodities with respect to utility is (in the state of knowledge in Pareto's time) impossible. On the other hand, an empirical set that reveals multiple attributes, each of which is subject to distinct measurement, may sometimes be measured conjointly with the measurement of its component structures. If [c.sub.1], [c.sub.2], ..., [c.sub.n] are empirical sets, their product [II.sub.i][c.ub.i], i = 1, 2, ..., n, when it exhibits an appropriate structure of its own, is a conjoint measurement structure.

Let the cartesian product [II.sub.i][c.sub.i], i = 1, 2, ..., n, be an n-dimensional empirical set and let [less than or equal to] be a complete preorder in [II.sub.i][c.sub.i]. It is a consequence of the complete preordering of the product set [II.sub.i][c.sub.i] that the elements of the individual components [c.sub.1] can be selected independently of each other. More formally, [II.sub.i][c.sub.i] is component-independent (with respect to [less than or equal to]) iff for any k [subset] l, where l = {1, 2, ..., n}, [[less than or equal to].sub.k], induced in [II.sub.j][c.sub.j], J in k, by[less than or equal to] in c, with [[less than or equal to].sub.i], in [c.sub.i], i in l\k, held fixed of the choice of k [Krantz et al., 1972, p. 30!.

Let ([II.sub.i][c.sub.i], [less than or equal to]) be an ordinal structure. Thus, [x.sup.1], [x.sup.2], ..., [x.sup.n]) [less than or equal to] ([y.sub.1], [y.sub.2] ..., [y.sub.n]) iff q([x.sub.1], [x.sub.2], ..., [x.sub.n]) [less than or equal to] q([y.sub.1], [y.sub.2], ..., [y.sub.n]) for all ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and ([y.sub.1], [y.sub.2], ..., [y.sub.n]) in [II.sub.i][c.sub.i], and some representation q. A component-independent ordinal structure is called a decomposable structure. An n-commodity space [II.sub.i][c.sub.i] with the complete preference preordering [less than or equal to] is a well-known decomposable structure. As utility theory asserts, if [II.sub.i][c.sub.i] is an n-commodity space and [less than or equal to] is the complete preference preordering of [II.sub.i][c.sub.i], then [[less than or equal to].sub.i] completely preorders the magnitude set [c.sub.i] for all i. Moreover, the set of all magnitudes [y.sub.i] in the magnitude set [c.sub.i] such that, for some [x.sub.i] in [c.sub.i], [x.sub.i] ~ [y.sub.i], is the indifference equivalence) class [[x.sub.i]] of [x.sub.i]. It is easy to see that the quotient set [c.sub.i]/[~.sub.i] of indifference classes [[x.sub.i]] is completely ordered by the preference relation [[less than or equal to].sub.i]/[~.sub.i] and that the product set [II.sub.i][c.sub.i]/[~.sub.i] is a decomposable structure [Krantz et al., pp. 245-6].

A decomposable structure [II.sub.i][c.sub.i] with the property that, for all elements ([x.sub.1], [x.sub.2], ..., [x.sub.n]) in [II.sub.i][c.sub.i], there are representations [q.sub.i] of [c.sub.i] in [r.sub.0+] such that ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [less than or equal to] ([y.sub.1], [y.sub.2], ..., [y.sub.n]) iff [[sigma].sub.i][q.sub.i]([y.sub.i]) is an (additive) conjoint structure. It can be shown that if the decomposable structure [II.sub.i][c.sub.i] is Dedekind-complete, then each component magnitude set [c.sub.i] is Dedekind-complete (and conversely). [II.sub.i][c.sub.i] is unrestrictedly solvable iff, given all but one of [x.sub.1], [y.sub.1] in [c.sub.1], [x.sub.2], [y.sub.2] in [c.sub.2], ..., [x.sub.n], [y.sub.n] in [c.sub.n], the missing element exists so that (x.sub.1], [x.sub.2], ..., [x.sub.n]) ~ ([y.sub.1], [y.sub.2], ..., [y.sub.n]). Taking [II.sub.i][c.sub.i] to be unrestrictedly solvable is a very strong assumption that requires [II.sub.i][c.sub.i] to be at least countably infinite. [II.sub.i][c.sub.i] is restrictedly solvable iff, given ([x.sub.1], ..., [text with no conversion], ..., [x.sub.n) [less than or equal to ([y.sub.1], ..., [y.sub.i], ..., [y.sub.n]) [less than or equal to] ([x.sub.i], ..., [bar][x.sub.i], ..., [x.sub.n]), there exists [x.sub.i] in [c.sub.i], with [text with no conversion] [less than or equal to] [x.sub.i] [less than or equal to] [bar][x.sub.i], such that ([x.sub.1], ..., [x.sub.i], ..., [x.sub.n]) ~ ([y.sub [y.sub.n] [Krantz et al., 1971, p. 30].

Suppose the empirical n-commodity set [II.sub.i][c.sub.i], i = 1, 2, ..., n, n [greater than or equal to] 3, with nonempty components [c.sub.i] and the binary relation [less than or equal to] satisfies the following axioms: A(2.1): [less than or equal to] is a complete preordering of [II.sub.i][c.sub.i]. A(2.2): [II.sub.i][c.sub.i] is bounded. A(2.3): [II.sub.i][c.sub.i] is component-independent. A(2.4): [II.sub.i][c.sub.i] is Dedekind-complete. A(2.5): [II.sub.i][c.sub.i] is unrestrictedly solvable.

A magnitude set [II.sub.i][c.sub.i] satisfying axioms A(2.1) - A(2.5) is an additive conjoint measurement structure. Axioms (2.1) - (2.5) are consistent with conditions for extensive measurement of bounded commodity magnitude sets [c.sub.i] that are Dedekind-complete and unrestrictedly solvable, as discussed in Section I. In particular, since [II.sub.i][c.sub.i] is bounded, each [c.sub.i] is bounded (and conversely).

In case n = 2, then ([II.sub.i][c.sub.i], [less than or equal to]) is an additive conjoint structure, if axioms A(2.1) - A(2.5) are satisfied along with a double cancellation axiom called the Thomsen condition. The structure ([II.sub.i][c.sub.i], [less than or equal to]), i = 1, 2, satisfies the Thomsen condition iff for every [x.sup.1], [x.sup.2], [x.sup.3] in [c.sub.1] there exist magnitudes [y.sup.1], [y.sup.2], [y.sup.3] in [c.sub.2] such that if (x.sup.1], [y.sup.2]) ~ ([x.sup.2], [y.sup.1]) and ([x.sup.2], [y.sup.3]) ~ [x.sup.3], [y.sup.2]), then ([x.sup1], [y.sup.3]) ~ (x.sup.3], [y.sup.1]). In other words, if ([x.sup.1], [y.sup.2]) and ([x.sup.2], [y.sup.1]) are in an indifference class and ([x.sup.2], [y.sup.3]) and ([x.sup.3], [y.sup.2]) are in a second, distinct indifference class, then [x.sup.2] and [y.sup.2] may be cancelled (the double cancellation), leaving ([x.sup.1], [y.sup.3]) and ([x.sup.3], [y.sup.1]) in a third distinct indifference class. If n [less than or equal to! 3, however, then the Thomsen condition can be confirmed under axioms A(2.1) - A(2.5) for any [c.sub.i]x [c.sub.j], i = 3, ..., n, j = 3, ..., n, (text with no conversion] [Krantz et al., 1971, pp. 251-3].

If ([II.sub.i][c.sub.i], [less than or equal to]), i = 1, 2, ..., n, n [greater than or equal to] 3, is an additive conjoint structure, then there are representations [q.sub.1] from [c.sub.1] to [r.sub.0+] such that, for all ([x.sub.1, [x.sub.2], ..., [x.sub.n]) and ([y.sub.1], [y.sub.2], ..., [y.sub.n]) in [II.sub.1][c.sub. [x.sub.2, ..., [x.sub.n]) [less than or equal to] ([y.sub.1], [y.sub.2], ..., [y.sub.n]) iff [[sigma].sub.i][q.sub.i]([y.sub.i]). Each [q.sub.i] is unique up to linear transformation; that is, if {[m.sub.i]: i = 1, 2, ..., n} is another set of component representations such that ((m,: i = 1, 2, ..., n} = {[q.sub.i]: i = 1, 2, ..., n}), then for each i there are constants [s.sub.i] in r and t in [r.sub.+] such that [m.sub.i] = [s.sub.i] + [tq.sub.i] [Krantz et al., 1971, p. 302!.

A theorem for additive conjoint measurement was stated and proved by Debreu for the first time in 1960, using topological methods. Walras wrote late in the last century that the utility measurement of each market basket x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) of consumer goods was equal to the sum of the utility measurements of the individual commodities [x.sub.1] in the market basket x [Walras, 1954, pp. 164-72]. Walras' use of conjoint measurement for the utility measurement of an n-commodity structure, however remarkable for the intuitive quality of his work as a whole, has long been rejected in economics, since additive utility of market baskets containing complementary commodities is surely meaningless. Fisher [1927] made use of conjoint measurement by recognizing decomposability, solvability, and other measurement laws in an intuitive, fragmentary way his application of scalar measurement to gold in monetary theory and policy has already been noted.

While the size-ordering of n-commodity market baskets, when n = 1, yields the quantity measurement of Section I, an attempt to apply conjoint measurement to an n-commodity cartesian product space when n > 1 evidently is futile. Other things equal, a market basket containing 2 apples and 3 oranges is unambiguously larger than one containing 2 apples and 2 oranges. but defies size-comparison with a basket containing 3 apples and 2 oranges. One, therefore, concludes that the size-ordering of market baskets in an n-commodity cartesian product space, n > 1, is impossible; there is no size-ordering [less than or equal to] and, thus, no conjoint measurement, not even ordinal measurement, in such spaces.

(B) n-commodity Quantiiy Measurement

Although conjoint quantity measurement fails in an n-commodity cartesian product space, all is not lost. Recalling that [c.sub.i] of Section I is a bounded magnitude set derived from commodity stock [[gamma].sub.i] = 1, 2, ..., n, and assuming that ([c.sub.i], [[circle].sub.i]) is a Dedekind-complete, abelian semi-group with identity [Mathematical Expression Omitted] and that ([c.sub.i], [less than or equal to], [[circle].sub.i]) is a scalar structure, the product space [II.sub.i][c.sub.i] has measurement properties universally utilized in economic theory [Debreu, 1959, pp. 28-36]. Clearly, [II.sub.i][c.sub.i] is bounded, since each [c.sub.i] is bounded; since each [c.sub.i] is Dedekind, so is [II.sub.i][c.sub.i]. Regarding the distinct market baskets ([x.sub.1], [x.sub.2], .... [x.sub.n]) and ([y.sub.1], [y.sub.2], ..., [y.sub.n]) as vectors in [II.sub.i][c.sub.i[, define

(16) ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] ([y.sub.1], [y.sub.2], ..., [y.sub.n]) = ([x.sub.1] [[circle].sub.1] [y.sub.1], [x.sub.2] [[circle].sub.2] [y.sub.2], ..., [x.sub.n] [[circle].sub.n] [y.sub.n]).

The operation [circle] in [II.sub.i][c.sub.i], called vector concatenation, is well-defined, since [x.sub.i] [[circle].sub.i] [y.sub.i] is pairwise concatenation of disjoint magnitudes in [c.sub.i]. By construction, [II.sub.i][c.sub.i] is closed under vector concatenation, since each [c.sub.i] is closed under magnitude concatenation. The concatenation of ([x.sub.1], [x.sub.2], ..., [x.sub.n]) with itself m times, m in [i.sub.0+], given by

(17) ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] ... [circle] ([x.sub.1], [x.sub.2], ..., [x.sub.n])

= ([x.sub.1] [[circle].sub.1] ... [[circle].sub.1] [x.sub.1], [x.sub.2] [[circle].sub.2] ... [[circle].sub.2] [x.sub.2], ... [x.sub.n] [[circle].sub.n] ... [[circle].sub.n] [x.sub.n])

= (m [[multiplied by].sub.1] [x.sub.1], m [multiplied by].sub.2] [x.sub.2], ..., m [multiplied by].sub.n] [x.sub.n])

= (m [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n]),

is called vector autoconcatenation. It is obvious that 0 [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n]) = [Mathematical Expression Omitted] and 1 [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n]) = ([x.sub.1], [x.sub.2], ..., [x.sub.n]), since 0 [multiplied by] [x.sub.i] = [Mathematical Expression Omitted] and 1 [ each i. Since, for each i and each m, m [multiplied by] [x.sub.i] is in [c.sub.i] by the convention that m [multiplied by] [x.sub.i] is maximal in [c.sub.i] when [[gamma].sub.i] [less than or equal to] m [multiplied by] [x.sub.i], every m [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n]) is in [II.sub.i][c.sub.i]. (The commodity stock [[gamma].sub.i], is, the maximum magnitude in [c..sub.i].) Since vector autoconcatenation in [II.sub.i][c.sub.i] can be generalized to multiplication of vectors in [II.sub.i][c.sub.i] by elements in [r.sub.0+], autoconcatenation in [II.sub.i][c.sub.i] may be regarded as the operation of scalar multiplication in [II.sub.i][c.sub.i] with scalars in [r.sub.0+]. Since [[multiplied by].sub.i] is well-defined as magnitude autoconcatenation in [c.sub.i], [multiplied by] is well-defined as vector autoconcatenation in [II.sub.i][c.sub.i]. While [c.sub.i] and [[circle].sub.i], are primitives of ([c.sub.i], [less than or equal to], [[circle].sub.i]), [II.sub.i][c.sub.i] is the product of the [c.sub.i] and [circle] is induced in [II.sub.i][c.sub.i] by [[circle].sub.i] in [c.sub.i], i = 1, 2, ..., n.

Both associativity and commutativity (with respect to [circle]) are induced in [II.sub.i][c.sub.i] by those same properties in [c.sub.i] (with respect to [[circle].sub.i]). Since [Mathematical Expression Omitted], the null magnitude in [c.sub.i], is the additive identity in ([c.sub.i], [[circle].sub.i]) and since ([x.sub.i], [x.sub.2], ..., [x.sub.n]) [circle] [Mathematical Expression Omitted] = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) for every ([x.sub.1], [x.sub.2], ..., [x in [III.sub.i][c.sub.i], [Mathematical Expression Omitted] is the additive identity (empty basket) in [II.sub.i][c.sub.i]. [Mathematical Expression Omitted] is unique in [II.sub.i][c.sub.i], because [Mathematical Expression Omitted] is unique in [c.sub.i]. Since for each ([x.sub.1], [x.sub.2], ..., [x.sub.n]) in [II.sub.i][c.sub.i] there is no ([y.sub.1], [y.sub.n]) in [II.sub.i][c.sub.i] such that ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] ([y.sub.1], [y.sub.2], ..., [y.sub.n]) = ([Mathematical Expression Omitted]), [II.sub.i][c.sub.i] contains no inverses. (Each [c.sub.i] contains no inverses. One concludes that ([II.sub.i][c.sub.i], [circle]) is an abelian semi-group with identity ([Mathematical Expression Omitted]).

The following distributive laws hold for the set [II.sub.i][c.sub.i] of vectors ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and for scalars s in [r.sub.0+]. First, multiplication by scalars is distributive over vector concatenation:

(18) s [multiplied by] ((x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] ([y.sub.1], [y.sub.2], ..., [y.sub.n]))

= s [multiplied by] ([x.sub.1] [[circle].sub.1] [y.sub.1], [x.sub.2] [[circle].sub.2] [y.sub.2], ..., [x.sub.n] [[circle].sub.n] [y.sub.n])

= (s [multiplied by] ([x.sub.1] [[circle].sub.1] [y.sub.1], [x.sub.2] [[circle].sub.2] [y.sub.2], ..., s [multiplied by] ([x.sub.n] [[circle].sub.n] [y.sub.n]))

= (s [multiplied by] [x.sub.1] [[circle].sub.1] s [multiplied by] [y.sub.1], s [multiplied by] [x.sub.2] [[circle].sub.2] s [multiplied by] [y.sub.2], ..., s [multiplied by] [x.sub.n] [[circle].sub.n] s [multiplied by] [y.sub.n])

= (s [multiplied by] [x.sub.1], s [multiplied by] [x.sub.2], ..., s [multiplied by [x.sub.n]) [circle] (s [multiplied by] [y.sub.1], s [multiplied by] [y.sub.2], ..., s [multiplied by] [y.sub.n])

= s [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] s [multiplied by] ([y.sub.1], [y.sub.2], ..., [y.sub.n]),

using (16), (17), and other formulae. Second, multiplication by vectors is distributive over scalar addition:

(19) (s + t) [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n])

= (s + t [multiplied by] [x.sub.1], (s + t [multiplied by] [x.sub.2], ..., (s + t [multiplied by] [x.sub.n])

= (s [multiplied by] [x.sub.1] [[circle].sub.1] t [multiplied by] [x.sub.1], s [multiplied by] [x.sub.2] [[circle].sub.2] t [multipied by] [x.sub.2], ..., s [multiplied by] [x.sub.n] [[circle].sub.n] t [multiplied by] [x.sub.n])

= (s multiplied by] [x.sub.1], s [multiplied by] [x.sub.2], ..., s [multiplied by] [x.sub.n]) [circle] (t multiplied by] [x.sub.1], t [multiplied by] [x.sub.2], ..., t [multiplied by] [x.sub.n])

= s [multiplied by ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] t [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n])

by (16), (17) and other formulae. Scalar multiplication (vector autoconcatenation) hold; for distribution in [II.sub.i][c.sub.i], since scalar multiplication (magnitude concatenation) holds in [c.sub.i]. Since [II.sub.i][c.sub.i], [circle]) is a semi-group, [II.sub.i][c.sub.i] is a semi-vector space over [r.sub.0+].

Of special interest is the scalar multiplication of any vector z in [II.sub.i][c.sub.i] by a product of scalars in [r.sub.0+]:

(20) st [multiplied by] ([z.sub.1], [z.sub.2,] ..., [z.sub.n])

= (st [multiplied by] [z.sub.1], st [multiplied by] [z.sub.2], ..., st [multiplied by] [z.sub.n])

= (s [multiplied by] (t [multiplied by] [z.sub.1]), (s [multiplied by] (t [multiplied by] [z.sub.2]), ..., (s [multiplied by] (t [multiplied by] [z.sub.n]))

= (s [multiplied by] [b.sub.t]([z.sub.1]), s [multiplied by] [b.sub.t]([z.sub.2]), ..., s [multiplied by] [b.sub.t]([z.sub.n]))

= ([b.sub.s]([b.sub.t]([z.sub.1])), [b.sub.s]([b.sub.t]([z.sub.2])), ..., [b.sub.s]([b.sub.t]([z.sub.n])))

~ ([b.sub.s] ([y.sub.1]), [b.sub.s] ([y.sub.2]), ..., [b.sub.s] ([y.sub.n]))

~ ([x.sub.1], [x.sub.2], ..., [x.sub.n),

using (8) and (9) and other formulae. Thus, multiplication of any vector in [II.sub.i][c.sub.i] by a product of scalars in [r.sub.0+] is equivalent to the application of successive autoconcatenations, one for each of the scalars in their product, beginning on the right, to each component of the vector.

(C) q, p, and Q in [II.sub.i][c.sub.i]

Since ([c.sub.i], [less than or equal to], [[circle].sub.i]) is a bounded scalar structure for each i, there is a representation [q.sub.i] of [c.sub.i] in [Mathematical Expression Omitted] such that, for every [x.sub.i] in [c.sub.i], [q.sub.i]([x.sub.i]) is in [Mathematical Expression Omitted]. Let q be the vector function defi [x.sub.n]) = (q.sub.1]([x.sub.1), [q.sub.2[([x.sub.2]), ..., [q.sub.n]([x.sub.n])). The vector function q exists and is well-defined because each linear transformation [q.sub.1] exists and is well-defined. Thus, q is a mapping of [II.sub.i][c.sub.i] onto [Mathematical Expression Omitted]. Furthermore, since, for distinct vectors (x.sub.1], [x.sub.2], ..., [x.sub.n]) and (y.sub.1], [y.sub.2], ..., [y.sub.n]) in [II.sub.i][c.sub.i] and for scalars s and t in [r.sub.0+],

(21) q(s [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] t [multiplied by] ([y.sub.1], [y.sub.2], ..., [y.sub.n]))

= q(s [multiplied by] [x.sub.1] [[circle].sub.1] t [multiplied by] [y.sub.1], s [multiplied by] [x.sub.2] [[circle].sub.2] t [multiplied by] [y.sub.2], ..., s [multiplied by] [x.sub.n] [[circle].sub.n] t [multiplied by] [y.sub.n])

= ([q.sub.1](s [multiplied by] [x.sub.1] [[circle].sub.1] t [multiplied by] [y.sub.1]), [q.sub.2] (s [multiplied by] [x.sub.2] [[circle].sub.2] t [multiplied by] [y.sub.2], ..., [q.sub.n] (s [multiplied by] [x.sub.n] [[circle].sub.n] t [multiplied by] [

= (sq([x.sub.1]) + tq([y.sub.1]), sq([x.sub.2]) + tq([y.sub.2]), ..., sq([x.sub.n]) + sq([y.sub.n])),

vector concatenation and scalar multiplication in [II.sub.i][c.sub.i] are preserved by q as addition of component representations in each factor of [Mathematical Expression Omitted]. Thus, q is a vector homomorphism of [II.sub.i][c.sub.i] onto [Mathematical Expression Omitted]. In the light of the existence of the representation q, ([II.sub.i][c.sub.i], [circle]) is understood to be a quasi-measurement structure.

Let p be the relation defined by p([x.sub.1], [x.sub.2], ..., [x.sub.n]) = ([p.sub.1]([x.sub.1]), [p.sub.2]([x.sub.2]), ..., [p.sub.n]([x.sub.n])) = ([[x.sub.1]], [[x.sub.2]], ..., [[x.sub.n]]), where p, is the projection of [c.sub.1] onto its quotient set [c.sub.i] /[~.sub.i]. p exists and is well-defined because each pi exists and is well-defined. Furthermore, since for any two distinct vectors ([x.sub.1], [x.sub.2], .., [x.sub.n]) and ([y.sub.1], [y.sub.2], ..., [y.sub.n) in [II.sub.i][c.sub.i] and for scalars s and t in [r.sub.0+],

(22) p(s [multiplied by] ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [circle] t [multiplied by] ([y.sub.1], [y.sub.2], ..., [y.sub.n]))

= p((s [multiplied by] [x.sub.1] [[circle].sub.1] t [multiplied by] [y.sub.1]), (s [multiplied by] [x.sub.2] [[circle].sub.2] [t [multiplied by] [y.sub.2]), ..., (s [multiplied by] [x.sub.n] [[circle].sub.n] t [multiplied by] [y.sub.n]))

= ([p.sub.1] (s [multiplied by] [x.sub.1] [[circle].sub.1] t [multiplied by] [y.sub.1]), [p.sub.2](s [multiplied by] [x.sub.2] [[circle].sub.2] t [multiplied by] [y.sub.2]), ..., [p.sub.n](s [multiplied by] [x.sub.n] [[circle].sub.n] t [multiplied by] [

= ([s [multiplied by] [x.sub.1] [[circle].sib.1] t [multiplied by] [y.sub.1]], [s [multiplied by] [z.sub.2] [[circle].sub.2] t [multiplied by] [y.sub.2], ..., [s [multiplied by] [z.sub.n] [[circle].sub.n] t [multiplied by] [y.sub.n]))

= (s [multiplied by] [[x.sub.1]] [[circle].sub.1] / [~.sub.1] t [multiplied by] [y.sub.1], (s [multiplied by] [[x.sub.2]] [[circle].sub.2] / [~.sub.2] t [multiplied by] [y.sub.2], ..., (s [multiplied by] [[x.sub.n]] [[circle].sub.n] / [~.sub.n] t [multi

= ((s [multiplied by] [[x.sub.1]], s [multiplied by] [[x.sub.2]], ..., s [multiplied by] [[x.sub.n]]) [circle] / ~ (t [multiplied by] [[x.sub.1]], t [multiplied by] [[x.sub.2]], ..., t [multiplied by] [[x.sub.2]]))

= s [multiplied by] ([[x.sub.1], [[x.sub.n]], ..., [[x.sub.n]) [circle] / ~ t [multiplied by] ([[y.sub.1], [[y.sub.2]], ..., [[x.sub.n]),

vector concatenation and scalar multiplication in [II.sub.i][c.sub.i] are preserved as vector concatenation [circle] /~ and scalar multiplication in [II.sub.i][c.sub.i]/[~.sub.i]. Thus, p is a vector homomorphism of [II.sub.i][c.sub.i] onto [II.sub.i][c.sub.i].

Finally, let Q be the relation defined by Q([[x.sub.1]], [[x.sub.2]], ..., [[x.sub.n]]) = ([Q.sub.1]([[x.sub.1]], [Q.sub.2]([[x.sub.2]]), ..., [Q.sub.n] ([[x.sub.n]])) = ([q.sub.1]([x.sub.1]), [q.sub.2]([x.sub.2]), ..., [q.sub.n]([x.sub.n])). Q exists and is well-defined because each [Q.sub.i] and each [q.sub.i] exist and are well-defined. The last equality follows from the construction, [Q.sub.i]([[x.sub.i]]) = q([x.sub.1]) = q([x.sub.i]) for every [x.sub.i] in [c.sub.i]. Thus, Q is a mapping of [II.sub.i][c.sub.i]/[~.sub.i] onto a subset [Mathematical Expression Omitted]. Furthermore, for any two distinct vectors ([[x.sub.1]], [[x.sub.2]], ..., [[x.sub.n]]) and ([[y.sub.1]], [[y.sub.2]], ..., [[y.sub.n]]) in [II./sub.i][c.sub.i]/~, and for all scalars s and t in [r.sub.0+] it follows that

(23) Q(s [multiplied by] ([[x.sub.1]], [[x.sub.2]], ..., [[x.sub.n]]) [circle] / ~ t [multiplied by] ([[y.sub.1]], [[y.sub.2]], ..., [[y.sub.n]]))

= Q((s [multiplied by] ([[x.sub.1]], s [multiplied by] [[x.sub.2]], ..., s [multiplied by] [[x.sub.n]]) [circle] / ~ (t [multiplied by] ([[y.sub.1]], t [multiplied by] [[y.sub.2]], t [multiplied by] [[y.sub.n]]))

= Q(s [multiplied by] [[x.sub.1]], [[circle].sub.1] / [~.sub.1] (t [multiplied by] [[y.sub.1]], s [multiplied by] [[x.sub.2]], [[circle].sub.2] / [~.sub.2] t [multiplied by] [[y.sub.2]], ..., s [multiplied by] [[x.sub.n]], [[circle].sub.n] / [~.sub.n] t

= Q([s [multiplied by] [x.sub.1]] [[circle].sub.1] / [~.sub.1] [t [multiplied by] [y.sub.1]], [s [multiplied by] [x.sub.2]] [[circle].sub.2] ' [~.sub.2] [t [multiplied by] [y.sub.2]], ..., [s [multiplied by] [x.sub.n]] [[circle].sub.n] / [~.sub.n] [t [m

= Q([s [multiplied by] [x.sub.1] [[circle].sub.1] t [multiplied by] [y.sub.1], [s [multiplied by] [x.sub.2] [[circle].sub.2] t [multiplied by] [y.sub.2], ..., [s [multiplied by] [x.sub.n] [[circle].sub.n] t [multiplied by] [y.sub.n] )

= ([Q.sub.1] ([s [multiplied by] [x.sub.1] [[circle].sub.1] t [multiplied by] [y.sub.1]), [Q.sub.2] ([s [multiplied by] [x.sub.2] [[circle].sub.2] t [multiplied by] [y.sub.21]), ..., [Q.sub.n] ([s [multiplied by] [x.sub.n] [[circle].sub.n] t [multiplied

= ([Q.sub.1] ([s [multiplied by] [x.sub.1]) + [Q.sub.1] ([t [multiplied by] [y.sub.1]), [Q.sub.2] ([s [multiplied by] [x.sub.2]) + [Q.sub.2] ([t [multiplied by] [y.sub.2]), ..., [Q.sub.n] ([s [multiplied by] [x.sub.n])

+ [Q.sub.n] ([t [multiplied by] [x.sub.n]))

= (s[Q.sub.1] ([[x.sub.1]]) + t[Q.sub.1]([[y.sub.1]]), s[Q.sub.2] ([[x.sub.2]]) + t[Q.sub.2]([[y.sub.2]]), ..., s[Q.sub.n] ([[x.sub.n]]) + t[Q.sub.n]([[y.sub.n]]))

= (s[q.sub.1]([x.sub.1]) + t[q.sub.1]([y.sub.1]), s[q.sub.2]([x.sub.2]) + t[q.sub.2]([y.sub.2]), ..., s[q.sub.n]([x.sub.n]) + t[q.sub.n]([y.sub.n])).

Thus, vector concatenation and scalar multiplication in [II.sub.i][c.sub.i]/[~.sub.i] are preserved as the addition of component representations in each factor of [Mathematical Expression Omitted]. Since [Q.sub.1] is a bijection of [c.sub.i]/[~.sub.i] onto [Mathematical Expression Omitted!, Q is a vector isomorphism of the semi-vector space [II.sub.i][c.sub.i]/[~.sub.i] onto [Mathematical Expression Omitted].

Obvious reasoning establishes the quotient space [II.sub.i][c.sub.i]/[~.sub.i] as a quasi-measurement structure. Thus, quasi-measurement properties that originate in the empirical n-commodity space [II.sub.i][c.sub.i] are preserved both in a numeric (euclidean) structure in [Mathematical Expression Omitted] and in the semi-vector space [II.sub.i][c.sub.i]/[~.sub.i] of equivalence class vectors ([[x.sub.1]], [[x.sub.2]], ..., [[x.sub.n]]) in such a way that the numeric structure in [Mathematical Expression Omitted] and the. quotient structure [II.sub.i][c.sub.i]/[~.sub.i] are quasi-measurement copies of each other. A consequence of the nexus between q, p, and Q is that addition and ordinary multiplication in each factor [Mathematical Expression Omitted] of [Mathematical Expression Omitted], representing concatenation and autoconcatenation in the corresponding [c.sub.i], can be executed without explicit reference to the empirical n-commodity space [II.sub.i][c.sub.i] or to the corresponding quotient space [II.sub.i][c.sub.i]/[~.sub.i]. Thus, discourse in economic theory which depends on the vector space properties of [Mathematical Expression Omitted] and ignores the quasi-measurement structure ([II.sub.i][c.sub.i], [circle]) or its corresponding quotient structure ([II.sub.i][c.sub.i]/[~.sub.i], [circle]/~) may be at least partially successful without comprehending the logic which clarifies this success.

III. Recapitulation

Commodity stock [gamma] has the dominant characteristics of being bounded and very finely grained (Dedekind). Surely many commodities satisfy these properties, even though many do not, so that quantity measurement theory is not empirically empty. Other less restricted versions of economic quantity theory may treat the magnitude set c as merely Archimedean or as incomplete with respect to [less than or equal to]; or concatenation may be treated as a partial operation on c. Measurement of finite commodities yields only a counting measure; the commodities considered in this paper are all infinite. In fact, properties that make the theory interesting for its appeal to dogmatic empiricism may contribute less to an understanding of economic behavior than the naive measurement structures considered in this paper [Musgrove, 1991, pp. 245-61.

Conjoint measurement fails in general quantity measurement of commodities since there s no rational basis for the size-ordering of the elements of the n-dimensional commodity space [II.sub.i][c.sub.i]. It is possible to take [II.sub.i][c.sub.i] as a quasi-measurement space, however, in which each component [c.sub.i] is subject to scalar measurement and to represent this space in [mathe [Debreu, 1984, pp, 267-8].

The operation [circle] of concatenation is the basis of the elaborate scalar measurement structure in c already examined, permitting empirical commodities to be at least partially quantified, evidently for the first time. On these grounds, the claim that the numeric (eulidean) space [Mathematical Expression Omitted], not the commodity space [II.sub.i][c.sub.i], underlies virtually the whole of existing mathematical discourse in economics appears to be justified. Given the limitations of idealized set and measurement commodity structures, the purpose of this paper of informally joining economic theory in euclidean n-space to interesting measurement foundations in an empirical n-commodity semi-vector space is now complete.

REFERENCES

Debreu, Gerard. Theory of Value, New Haven: Yale University Press, 1959. --. "Topological Methods in Cardinal Utility Theory," in S. Karlin and P. Suppes, eds., Mathematical Methods in the Social Sciences, Stanford, CA: Stanford University Press, 1960, pp. 16-26. --. "Economic Theory in the Mathematical Mode," American Economic Review, 74, 1984, pp. 267-78. deJong, F. J. Dimension Analysis for Economists, Amsterdam: New Holland, 1967. Fisher, Irving. Stabilizing the Dollar, New York: Macmillan, 1920. --. "A Statistical Method for Measuring 'Marginal Utility' and Testing the Justice of a Progressive Income Tax," in J. H. Hollander, ed., Economic Essays Contributed in Honor of John Bates Clark, New York: Macmillan, 1927, pp. 157-93. Garvin, Alexander. "Quantity Measure in Economics," Presented to the Atlantic Economic Society, London, England, April 1987. --. "Topological Quantity Spaces in Economic Theory," Presented to the Western Economic Association, Lake Tahoe, NV, 1989. --. "The Numeraire Commodity in a Mature Walrasian Economy," Presented to the Eastern Economic Association, Pittsburgh, PA, 1990. Hall, Robert E. "Explorations in the Gold Standard and Related Policies for Stabilizing the Dollar," Inflation: Causes and Effects, Chicago: University of Chicago Press for National Bureau of Economic Research, 1982. Helmholtz, H. V. "Zahlen und Messen erkenntnis-theoretisch betrachet," Philosophische Aufsatze Eduard Zeller gewidmet, Leipzig, 1887, reprinted in Gesammelte Abhandt., 1895, pp. 356-91, translation by C. L. Bryan, Counting and Measuring, Princeton, New Jersey: van Nostrand, 1930. Holder, O. "Die Axiome der Quantitat and die Lehre vom Mass," Ber. Verh. Kgl. Sachsis. Ges. Wiss. Lepzig, Math. -Phys. Classe, 53, 1901, pp. 1-64. Krantz, David H.; Luce, Duncan R.; Suppes, Patrick; Tversky, Amos. Foundations of Measurement, New York: Academic Press, 1971. Mirowski, Philip. "The When, the How, and the Why of Mathematical Expression in the History of Economic Analysis," The Journal of Economic Perspectives, 5, 4, 1991, pp. 145-57. Musgrove, Philip. "Correspondence: Physical Models and Economics," The Journal of Economic Perspectives, 5, 4, 1991, pp. 245-6. Narens, Louis. Abstract Measurement Theory, Cambridge, MA: MIT Press, 1985. Walras, Leon. Elements d'economie politique pure, Edition Difinitive, 1926, William Jaffe translation, Homewood, IL: Richard D. Irwin, Inc., 1954.

(*) Indiana University of Pennslyvania. This paper was prepared for a special session in Applied Economic Theory at the Atlantic Economic Society sessions at the 1992 meeting of the American Economic Association in New Orleans, LA. The author is grateful to Douglas H. Frank, Mathematics Department, Indiana University of Pennslyvania, for the suggestions that have improved the mathematical content of the paper. The errors that remain are, of course, the

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Author: | Garvin, Alexander |
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Publication: | Atlantic Economic Journal |

Date: | Jun 1, 1992 |

Words: | 10533 |

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