===== PAGE 1 =====

Zeitschrift far
Nationalokonomie
Journal of Economics

© by Springer-Verlag 1977

Vol. 37 (1977), No. 3-4, pp. 249—280

The Austrian Theory of the Marginal Use
and of Ordinal Marginal Utility
By
J. Huston McCulloch*, Chestnut Hill, Mass., U. S. A.

(Received July 22, 1977)

Introducticn

The Austrian theory of the marginal use and of ordinal marginal
utility has not stood still since its original development in the hands
of Menger, Wieser, and Bohm-Bawerk. Over the past hundred
years, it has moved far beyond their statement of it, even though
this movement sometimes proceeded at a rather leisurely pace.

‘This paper brings the old theory up to dare and extends it. We
insist on a2 new English translation for one of its most important
technical terms and call attention to two crucial assumptions which
were implicit in the old theory, but which were never stated expli-
citly until the past decade. A recent mathematical finding implies
that the Austrian marginal utility concept is not just ordinal, but
in a sense is “intrinsically ordinal™.

The restated theory has many important implications for the
structure of preferences over commodities, implications which do
not follow from the currently orthodox “indifferent” approach. The
theory indicates that preferences over commodities are indeed quasi-
concave {as Hicks and Allen merely assume), that marginal utility
does diminish, even in an ordinalist framework, and that rival and
complementary interactions between goods do lead to the Auspitz
and Lieben-Edgeworth-Pareto criterion. From this criterion we
are then able to deduce chat a negative cross substitution clasticity,

* The author is Scherman Research Fellow at NBER-West and Assistant
Professor at Boston College. He is grateful far helpful suggestions made by
J. R. Meginniss and by various participants in seminars at Boston Col-
lege, at the Universities of Hartford and Chicago, at Stanford University,
and at NBER-West.

Zeitsche. f. Nacionalokonomie, 37, Bd., Hefr 3-4 17


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250 J. H. McCulloch:

while neither a necessary nor a sufficient condition for net com-
plementarity, is not entirely unrelated to the presence of comple-
mentarity. Furthermore, the model provides reason to believe the
von Ncumann-Morgenstern utility index will in fact exhibit
risk-aversion, as modern finance theory merely assumes.

Wants and Utility

What distinguishes the Austrian approach from that of Jevons
or Walras is that the Austrians did not accept the utility or sub-
jective value of commodities as given, but rather derived it from
the importance of the wants that the goods can be used to satisfy?l.
The starting point for inferences about the subjective importance of
goods is a subjective rank-ordering of the set of all wants which
arranges them in the order of their importance to the individual?
Given this scale, if we can determine which want is dependent upon:
the possession of a certain good, we may ascribe the importance of
that want to the good. Thus, the utility of a good will essentially
be a position on this scale of wants.

The Dependent Want and the Implicit Assumptions

The Austrian determination of the dependent want that deter-
mines value, as given in the classic expositions of Menger’s farmer
and Bo6hm-Bawerk’s hunter, runs essentially as follows: Suppose
there are three wants, a, b, and ¢, any one of which can be satisfied
by a unit of a certain good, and that the individual prefers a to b
and b to c. We represent these preferences by at bc, using the
symbol t rather than >, in order to emphasize that this is a pre-
ference ordering rather than a numerical inequality. Obviously if
the individual has only one unit of the good he will use it to satisfy

1 See 0. g. Menger (1950, 116). , Bedsirfnis“ is variously translated as
“want” or “need”.

2 See e. g. Bohm-Bawerk (1959 11, 137).

$ Bohm-Bawecrk points out that “The expression [the ranking of
wants’] may mean the rank and order of categories of wants, or may
mean concrete wants, that is to say, the individual feelings of want”,
(1959 II, 137). He goes on to make it explicit that he has in mind a
ranking on concrete wants. Thus, we are to enter nothing so general as
“the want for food” in the scale of wants, but are to break wants down
into specific uses for each portion and type of food.


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The Austrian Theory of the Marginal Use 251

want a, if he has two units, he will satisfy wants a and b, and if
he has three units he will satisfy all three wants. Therefore the
value of the first unit is the importance of want a, the value of the
second unit is the importance of want b, the value of the third unit
is the importance of want ¢, and any additional units are worthless
unless the individual can come up with more wants the good can
be used to satisfy.

This conclusion may be obvious, but it is not really warranted,
given only the traditional rank-ordering on wants. It is true that if
the individual has only one unit he will, by assumption, use it to
satisfy want a. However, if he has two units, he may satisfy any
two wants, that is, he may choose from a and b, a and ¢, and b and c.
In fact, if he feels like it, he may satisfy only one want, a, b, ¢, or
for that matter, he may satisfy no wants at all if he is so inclined.
If W is the set of all wants, in this case W={a, b,c}, then with
two units he may choose from any subset of W with two or fewer
elements. In order to infer which subset he will choose, we must
be given a preference ordering not just on W, but on W*, the set of
all subsets of W: W*={P|P « W}. In our example, W* ={¢, {a}, {6},
{¢c}, {a,b}, {a,c}, {b,c}, {a, b, c}}, where “¢” is the empty set, the
subset of W which corresponds to the satisfaction of no wants at all.
To eliminate unnecessary clutter, we will omit the braces and com-
mas from the designation of elements of W*, so that W*={¢, 4, b,
¢, ab, ac, be, abe). If W is finite and has n elements, then W* has
27 elements, in this case 23=8. The tirst implicit assumption that
the Austrians made is therefore that the individual's preferences
define a linear ordering on W*, such that an individual with a
quantity of some good or goods will use these goods to satisfy the
highest rated subset which is feasible, given the supply. This implicit
assumption was noted by Georgescu-Roegen (1968, 251), and
was also independently discovered by Young (1969) and the present
author at about the same time.

And the traditional Austrian formulation makes a second implicit
assumption, which somehow implies that if atb}tc and all are
“desirable” so that a} bc + é, then it is ab that will be the highest
rated subset with two or fewer elements. An article by the later
Austrian-school economist Bilimovi& (1934, esp. p. 183) provides
a clue to what they had in mind. He argues in effect that bic
"would imply ab } ac, that at b would imply ac} be, and that ct ¢
would imply ac } a. These inferences, together with the transitivity

4 Neurath (1911, 104—105) performs similar operations on his “con-
stellatious of pleasures” in his interpretation of Menger and Béhm-

17*


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252 J. H. McCulloch:

of the ordering on W*, imply abract bec aud abitactat bic,
so that ab is indeed the highest feasible subset when two units are
available. Bilimovi¢ argues as if these inferences were valid deduc-
tions from a rank-ordering on W, but that is not the case unless we
assume that the wants are unrelated, so that if an additional want
or set of wants is added to both sides of a relationship, the elements
of the additional set not being contained in either of the sets in-
volved in the original relationship, then the relationship remains
undisturbed. We will call this property of the ranking “unrelated-
ness”5. We may loosely refer to the wants as being “unrelated”
provided we keep in mind that it is a property of the subjective
ordering on W*, rather than an objective property of the wants
themselves. In consumer theory this assumption was first made
explicit by Young (1969) and by the present author, working in-
dependently at about the same time.

Ww

——

a —
a
PNA QP
N——

Fig. 1. Unrelatedness in the ranking of suheete of W means that P is preferred
to Q if and only if P—Q is preferred to QP

Unrelatedness is illustrated in the Venn diagram of Fig. 1. The
two circles P and Q represent subsets of W, the set of all relevant
wants. The set difference P—Q is the set of all wants in P but not
in OQ, and Q—P is the set of all wants in QO but not in P. The
intersection P A Q is the set of all wants in both P and Q. The
unrelatedness assumption states that Pn Q, the wants P and Q
have in common, are irrelevant ta the relative ordering of P and Q.
All that matters is the relative importance of P—Q and Q—P, the

Bawerk. However, his inference that bibacy is preferred to arb) is un-
warranted, given only that aitbiter and that ci is equal in value to be.

5 Our “unrelatedness” is the same as “additivity” in the nomenclature
of Kraft et al. (1959, 408).


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The Austrian Theory of the Marginal Use 253

wants P and OQ do not have in common. Formally defined, a set
of subsets W* is unrelatedly ordered if for any two subsets P and
O of W, we have Py Q if and only if P—-Q + QP.

The Law of the Marginal Use

Given that wants are unrelated and that at bt ct, it follows
that an individual in possession of one unit of our good will use. it
to satisfy want a, that the use of two units would be the satisfaction
of wants a and b, and that the use of three or more units will be
the satisfaction of all three wants. Hence, if the individual has only
one unit, the use which depends on possession of the Jast unit will
be a, and therefore the value or utility of one unit will be that of a,
that is, the place of the set containing only 4 in the rank-ordering
of W*. If he has two units, the use dependent on either of these
units will be the satisfaction of b, the less important of the two
uses covered by two units, and hence the utility of one unit will
be the position of b on the scale. And if he has three units, the
dependent use is the satisfaction of ¢, the least important of the
three uses covered by three units, and hence the utility of the third
unit will be the position of ¢ on the scale.

Menger had no name for this use which determines utility, but
Wieser proposed one which was subsequently adopted by B6hm-
Bawerk:

1 will henceforth refer to that use of a good which is decisive
for the value of a single unit of that good as the economically
marginal use, or simply as the marginal use, since it stands at
the margin of the economically permissible employments... It
will be shown that in every instance in which we are concerned
with the value of a single unit which is part of a supply of a
good, the marginal use determines the magnitude of the value.
Economic value is marginal value®,

If P, is the set of wants that will be satisfied by » units and Pn
the set that will be satisfied by n—1 units, then the set difference
Pn—Pa-1 will be the dependent set of wants or the marginal use of
the nth unit. If the individual has » units of the good, “the marginal
use of one unit” is somewhat ambiguous, since it can refer either
6 Wieser (1884, 128), We insist on “marginal use” as the proper trans-
lation of “Grenznutzen”, at least as used by Wieser here and by Bohm-
Bawerk. It corresponds to the “Grenzverwendung” of Rosenstein-
Rodan (1927, 1199, 1202; 1960, 85, 90). For reasons of space, we defer
discussion of this point to another paper,


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254 J. H. McCulloch:

to the marginal use of the last (nth) unit or the marginal use of one
additional (the + 1st) unit. When necessary, the former may be
referred to as the inner marginal use and the latter as the outer
marginal use. The Austrians’ Law of the Marginal Use, then, which
does not appear in the English literature, is that the value or utility
of a goods-increment is determined by the position of its marginal
use on the scale of sets of wants. A generalized proof of this theorem
is given by McCulloch and Smith (1975).

Wieser's Grenzwerth or marginal value is the closest term the
Austrians had to “marginal utility”?. It corresponds cxactly to
Bernardelli’s “conditional utility” (1938). Thus, it makes sense in
their framework to speak of the (marginal) value of two units of a
good, which is determined by the marginal use of two units, in turn
the satisfaction of the two least important wants covered by the
total supply. Because the Austrians thought in terms of realistic
discretely divisible goods instead of hypothetical continuously divisible
goods, their “value” corresponds to a non-infinitesimal increment.
When only a single unit is at stake, their “value” can be thought of
as “marginal utility”, provided a distinction is kept in mind between
outer marginal utility, corresponding to the outer marginal use, and
inner marginal utility, corresponding to the inner marginal use.

Wicser turns the law of the marginal use about to get what
might be called a “law of marginal utility”, which determines
which uses are permissible and which are not:

Fach desire whose importance lies above or is equal to the
[marginal] value will be permitted, each whose importance lies
below it will be rejected. All economically permissible employ-
ments will be included by the marginal value [Grenzwerth], and
all impermissible ones excluded. (1884, 136, my trans.)

Marginal utility or value thus serves the individual as a mental
short-cut to facilitate everyday decisions.

The Law of Diminishing Marginal Utility

An immediate consequence of the Law of the Marginal Use is
the fact that if we have a greater quantity of a good, the dependent
want will have a lower rank on the scale, and therefore the (mar-
ginal) utility of one unit will be lower8, If the marginal uses decline

7 Alt's concept of Grenzivert (1936, 163) has no relation to Wieser's.
After World War I, “Grenznutzen” was confused with marginal utility.

B See ¢. g. Menger (1950, 151).


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The Austrian Theory of the Marginal Use 255

in importance as the available quantity increases, and it is the im-
portance of the marginal use which determines marginal utility,
then marginal utility must decrease as the available quantity in-
creases, Note that the Austrian principle of diminishing marginal
utility ts a theorem, rather than an assumption as with Gossen,
Jevons, and Walras®.

To illustrate this law, suppose thar there is one good, “X", and
that there are four unrelated wants, a, b, ¢ and d that can be
satisfied by a unit of X. (An imaginative individual might be able
to think up an infinite number of wants he would like to satisfy
with a certain commodity. For the sake of brevity, however, we
will restrict our examples to finite cases.) A ranking of W* which
could describe an individual’s subjective preferences and which
satisfies “unrelatedness” is given in Table 1. The sixteen positions

Table 1. Hypothetical Prefer- Table 2. Total Use and Total Utility
ence Ordering of W*, With of Various Quantities of X
Assigned Ordinal Utility
Levels
. Unita Use Ordinal
Set of Ordinal of X Ueiliey
Wants Utdlicy
0 é Oth
abed 15¢ch 1 a 5th
abe l4th 2 ab 11th
abd 13th 3 abc lath
acd 12th 4 abcd 15th
ab 11s Ss abed 15th
bed 10th
ac 9th
ad 8th
be 7th Table3. Marginal Use and Marginal
bd orn Utility of 1 Unit of X
a Sth
cd 4th
b 3rd Unit Marginal Ordinal Marginal
c 2nd , of X Use of 1X | Utility of 1X
d lst
|) Oth lst a 3th
2nd b 3rd
3rd c 2nd
4th d lat
Sth p Oth

on this scale have been numbered from “Oth” to “15th”, starting
with the lowest position and proceeding up to the highest. These
numbers comprise an ordinal utility index, where each number
designates a certain utility level, These utility indices are »ot meant
to mean that the twelfth utility level is in any sense “twice” as high

® Cp. Mises (1966, 243) and Pirou (1945, 64).


===== PAGE 8 =====

256 J. H. McCulloch:

as the sixth level, or that the utility of the tenth level equals that
of the third level “plus” that of the seventh level. The indices simply
give us a convenient method of referring to higher or lower posi-
tions vn the scale. It scems appropriate to give the empty set ¢
(the set with no elements) the zeroeth position, though it could just
as logically be assigned the ninety-seventh, or any other position.

If an individual has » units of X, unrelatedness implies his nse of
them will be to satisfy the n most important wants, as indicated in
Table 2. The utility level of this total use naturally increases with
X, as long as we have additional “desirable” wants (that is, ones
that are preferred to the empty set).

The marginal use of one unit of X for different quantities is
shown in Table 3, along with the utility of this use, which in turn
is the marginal utility of a unit of the good. The marginal utility
of one additional unit is found to decline from fifth to third to
second to first to zeroeth as X increases from 0 to 4.

Notice that the marginal utility is not the arithmetic difference
in the utility level, Rather, the Austrian concept of marginal utility
is the utility level of the set difference of the respective uses. When
von Mises insists, “There are in the sphere of values and valuations
no arithmetical operations; there is no such thing as a calculation
of values”, (1966, 122) he has therefore only gotten at half the
truth, for there are, we argue, set operations implicit in the Austrian
utility analysis. Since the algebra of set manipulation is only a
formalization of elementary categories of logic and since it has only
recently come into fashion to use set notation, even in mathematics,
it is understandable that the Austrians did not make these opera-
tions explicit, and in fact, were probably not even consciously aware
that they were using them.

The Austrian theorem of diminishing ordinal marginal utility
points up the substantial difference between the Austrian tradition
and the orthodox theory of utility. Hicks tells us that if we reject
cardinal utility and purge our analysis of

all concepts which arc tainted by quantitative utility,. .. the first
victim must be marginal utility itself. If total utility is arbitrary
so is marginal utility... The second victim (a more serious one
this time) must be the principle of Diminishing Marginal Utility.
If marginal utility has no exact sense, diminishing marginal
utility can have no exact sense either. (1946, 19—20).

Yet the Austrians had an ordinal concept of utility in which marginal
utility does have a meaning, and furthermore, their marginal utility
does diminish. For example, in Table T we could square each of the


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The Austrian Theory of the Marginal Use 257

ordinal utility index values so that from the top down they read
152 =225th, 142 =196th, etc. The marginal utilities in Table 3 would
still decline, from 25th to 9th to 4th to 1st to Oth. The Austrian law
of diminishing marginal utility is thus invulnerable to monotonic
transformations of the utility index.

The Utility of 'I'wo Independent Goods

Let us suppose that there are two kinds of goods, X and Y, and
that one unit of X will satisfy want a, ¢ or ¢, and that one unit of

Y will satisfy b or d. We then have W=1{a, b,c, d,e}

Table 4. Hypothetical Prefer- Table 5. Derived Utility of Combi-
ence Ordering of W* nations of X and Y
Set of |Ordinal Units of Ordinal
Wants |Utdilicy Use Ueildicey
X Y
abcde - Jlst
sbed 30th 0 oj Oth
abce 29th 1 Y a 9th
abc 28th z [4] Lac 19th
abde 27th 3 0 ace 21st
acde 26th 0 1 b 5th
abd 25th 1 1 ab 20th
acd 23rd 3 1 dbce 29th
bede 22nd 0 2 bd 10th
ab 20th 2 2 abcd 30th
bed 19th 3 2 abcde 31st
ac 18th
ade 17ch
ad 16th
bee 15th Table 6. Derived Preference Order-
be 1heh ing of Combinations of X and Y
bde 13th
ae 12th
cde 11th Units of Ordinal
bd 10th Use Ueiliey
a Jth x ky
be Bth
ed 7th 3 2 abcde 3lat
ce bh 2 2 | abcd 30th
b Sth 3 1 abce 29th
c 4th 2 1 abc 28tn
de 3rd 1 2 abd 25th
d 2nd 3 0 ace 21st
a 1st 1 1 ab 20th
¢ Oth 2 Q ac 18th
0 2 bd juth
1 0 a 9th
Q 1 b 5th
0 0 ) oth

A conceivable preference ordering of W* is shown in Table 4,
along with an ordinal utility index identifying the positions on the


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258 J. H. McCulloch:

scale from zeroeth to thirty-first. If an individual with the preferences
of Table 4 has m units of X and # units of Y, unrelatedness of the
wants implies that he will use them to satisfy the 7 most important
elements of the set {g, c, e}, and the » most important elements of
the set {b, d}. Table 5 shows the optimal use which would be made
of various combinations of X and Y and the respective utility levels.
These utility levels imply a derived preference ordering on the com-
modity bundles. In Table 6 the commodity bundles are arranged in
decreasing order of utility. Table 7 lays out the total use and total
utility of these bundles in two-dimensional tabular form. In Fig. 2

Table7. Total Use and Total Utility
of Combinations of X and Y

: Y
3 bd abd abcd abcde [abcde
10th | 25th | 30eh | 31st | 3lat 3 py
Pe 2 bd abd abcd abcde | abcde
“ 10th 25th 30th lat 31st 2
i Ll ab abe abce |abce
pa 5th 20th | 28th [29th [29th
= 1
0 B a ac ace ace
Oth 9th 18th 21st 21st 0
0 1 2 3 4 0 1 2 3 4 x
Units of X . .
nite © Fig. 2. “Indifference Curves” sepa-
rating more preferred combinations
Table 8. Marginal Use and from less preferred combinations
Marginal Utihity of 1 Unit
of X . }
Table9. Marginal Use and Marginal
Utility of 1 Unit of Y
2 a c e [}
9th 4th lat Oth
NERERERIEL
* 212 ¢ & ¢ td | on | och | Oth fOth | Oth
yo 9th 4th ist Oth Ig
5. d d d d d
3 1 8 c € 0 . me 2nd 2nd 2nd | 2nd 2nd
[2 9th 4th | 1st | Oth
p=1 [+]
= b b b b b
0 a c e 9 let Sth 5th 5th | Sth Sch
Qth arh Tat Oth RE i
1st 2nd . 3rd 4th 0 1 2 3 4
Unit of X Units of X

the horizontal axis represents units of X and the vertical axis units
of Y. Lines have been drawn on this graph corresponding to dif-
ferent utility levels. These lines have dic property that any point
below and to the left of the line has a utility level lower than that
corresponding to that of the line, while all points on the line have


===== PAGE 11 =====

‘The Austrian Theory of the Marginal Use 259

exactly this utility, and all points above and to the right have at least
this utility. Our lines roughly correspond to the “indifference curves”
of conventional utility theory. The only difference is that our com-
modities are “lumpy”, rather than infinitely divisible, and therefore
the lines usually go through only one point. The reader may, if he
objects to indifference curves, think of these lines as “preference
curves”.

Table 8 shows the marginal use and marginal utility of one unit
of X as the total quantities of X and Y vary. Table 9 shows the
marginal use and marginal utility of onc unit of Y. The marginal
utility of X is found to diminish from 9th to 4th to 1st to Oth as X

increases from 1 to 4, regardless of the quantity of Y available.
Similarly, the marginal utility of Y diminishes from Sth to 2nd to Oth,

regardless of the quantity of X available. It could not be otherwise
in this case, for the quantity of one good has no bearing on the
use that will be made of the other, and therefore no effect on the
marginal use. We may therefore state as a general rule that when X
and Y are independent in consumption, i. e., when W may be parti-
tioned into two categories of wants such that a unit of X and only
a unit of X will satisfy the wants in one category, and a unit of Y
and only a unit of Y will satisfy the wants in the other category,
the marginal utility of one good will be independent of the quantity
available of the other?

It should be noted that when there is only one good, the concept
of marginal utility has no operational significance. So what if a
unit of a good has a certain desirability, if there is nothing to com-
pare it to? But when there is more than one good, we have the
seemingly trivial but actually important rule, that if an individual
is offered a choice between a unit of one good or a unit of another,
he will always choose the one with the higher marginal utility, as
determined by the marginal nse.

The Austrian Resolution of the Paradox of Value

Before the Austrians came on the scene, economists were troubled
by the so-called paradox of value. As Adam Smith expressed it,

The things which have the greatest value in usc have frequently

little or no value in exchange; and on the contrary, those which

10 Sirotz’s concept of a “utility tree” (1957) is undoubtedly related to
independence of the goods in question, as is the concept of “additively
separable” preferences. The exact connection deserves to be examined in
greater derail.


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260 J. H. McCulloch:

have the greatest value in exchange have frequently little or no
value in use. Nothing is more useful than water: but it will
purchase scarce any thing; scarce any thing can be had in ex-
change for it. A diamond, on the contrary, has scarce any value
in use; but a very great quantity of other goods may frequently
be had for it. (1776/1937, 28).

The Austrians argued thar, if righty qualified, the value of a
good is in fact determined by the importance of its usefulness. In
Wieser's words, or rather in our translation of Wieser’s words,

For most goods a distinction must be made between the mag-
nitude of their value [ihres Werthes] and the magnitude of their
use [ihres Nutzens]. Only for those goods that are actually em-
ployed to bring about the marginal use-performance will the
good’s own use be the source of its value and will there be
agreement between the two judgments. For any other good a
different use, which must nevertheless be a use characteristic of
that sort of good, will be the basis for the estimate of its value,
which accordingly will differ from the estimate of the use-effect
it actually brings about; for such a good, the actual use is higher
than the dependent use and therefore higher than its value!l.

To illustrate the paradox and its resolution, let us look again
at the individual of Tables 4 through 9. Suppose he has 3X and 2Y.
The total use of X (ace, twenty-first position) is more important
that the total use of Y (bd, tenth position). Furthermore, the highest
use of X (a, ninth position) is more important than the highest use
of Y (b, fifth position). Yet the subjective value, the utility, of a unit
of X, even of the very unit that will satisfy want a, is lower (first
position) than the utility of a unit of Y (second position). ‘The
Austrians’ answer to this paradox is that the value of a unit of a
good is determined, not by the total use of goods of that sort, and
not necessarily even by its own use, but rather by its marginal use.
Goods do not obtain value from the labor they “contain”. Rather,
labor derives its value from use-value of the goods it is used to
produce.

Rival Goods

In the example given above, it was assumed that the two goods
were used independently of one another. However, the law of the

—————

1L Wieser (1884, 128). Sec also Bohm-Bawerk (1959 1, 135-—136 and
1909 A, 234). Note however that Lindgren (1976) puts a completely dif-
ferent interpretation on Smith's meaning.


===== PAGE 13 =====

The Austrian Theory of the Marginal Use 261

marginal use is also applicable if the goods must be used together
to satisfy some wants or if they can be utilized in place of one another.
Suppose there are two goods, X and Y, and that a unit of either
may be used to satisfy some want or wants, say c. Let there be other
wants, a and e, which a unit of X can satisfy, and still others, b
and d, which may be satisfied by a unit of Y. X and Y are then
rivals, at least with respect to want c¢'2,

Table 10. Hypothetical Table 11. Use of X, Use of Y, Total Use
' Preference Ordering and Total Utility
Set of | Ordinal Units of Use of Total Totel
Wants | Ucility Use Ueilicy
X Y X Y
abcde 31st
abcd 30th 0 0 [1 J B Oth
abce 29¢th 1 0 a ¢ a 7th
abde 28th 2 0 ac 8 ac 15th
ahe 27th k| 0 ace | @ ace 19th
abd 26th 0 1 ? b b etn
acde 25th 1 1 a b ab 20th
bede 24th 2 1 ac b abc 27th
acd 23rd 3 1 ace |b abce 29th
abe 22nd 9 z 9 be be 13th
a C abc
bed 2st 2 | 2 | ac |bd abcd 30th
ace 19th 3 2 ace | bd abcde Jlst
ade 18th 0 | 9 bed bed 2ist
bce 17th 1 3 a bed abed 30th
bde 16th 2 3 ae bed abede 31th
ac 15th
ad lath
be 13th Table 12. Implied Preference Ordering on
bd 12th Combinations of X and Y
cde 11th
ae 10th
cd 9¢h X|Y Utility
oe Te z 3 Jlsrc
. Jen 32 3lat
ce Sth 1 3 30th
de 4th 2 2 30th
c 3ed 301 29th
d 2nd 1 2 27th
e lst : ; ae
st
a oth 1]1 20th
3 4 19¢ch
2 1] 15th
0 2 13th
1 0 7th
0 1 6th
0 0 Oth

Let an individual preference-rank the subsets of W={a, b, c, d, e}
as shown in Table 10. For various combinations of X and Y, Table 11
12 The dictionary definition of “rival” is “one of two or more striving
to rcach or obtain that which only one can possess” (Webster 1963, 743).


===== PAGE 14 =====

262 J. H. McCulloch:

shows the wants X will be used to satisfy, the wants Y will be used
to satisfy, the collective use of Y and X, and the corresponding total
utility. Want ¢ is sometimes satisfied by X and sometimes by Y.
Table 12 shows the implied preference ordering on the commodity
space. (Note that this ordering is now only semi-linear; it sometimes
happens that two different commodity bundles have the same util-
ity.) Tables 13—15 and Fig. 3 are constructed in the same manner

as Tables 7—9 and Fig. 2.

l'able 13. Total Use and Total Utility
of Combinations of X and Y

4 bcd abcd abcde | sbede | abcde
21st 30th let Jlst jlst

bed abcd abcde { abede | abcde

oo 2 | 21st | j0en [ater [31er | 31a
8 2 bc avec abcd ” abude | abode
« 13th | 27th | 30th | dlet | 3lat
Ea
[=1
= 1 b ab abc abce abce
6th 20th 27th 29th 29th
0 B a ac ace ace
Oth 7th 15ch 19th 19th
0 1 2 1 4 . .
Units of X Fig. 3. Indifference curves
Table 14. Marginal Use and Table 15. Marginal Use and Mar-
Marginal Utility of X ginal Utility of Y
a e f 9 4th | 8 ¢ # [J 2
4 | yen | 1sc { oon | orn och | och | oth | Och | Och
a e ) ] Ps rd | 94 d e 9 #
2 7en] tac | own | och “ 20d | 2nd | lst | Oth | Oth
[=]
a a a d Q @ i! 20d | © c d d d
w 7th ud lat 0th I rd 3rd znd 2nd 2nd
“
wd
£ 1 | a c e # let | b b b b b
7th rd let 0th 6th 6th 6th 6th 6th
0 a c e # 0 1 2 3 4
7th rd let Oth Units of X

1st 2nd 3rd 4th
Unit of X

As in the case of independent goods, the marginal utility of cach
good decreases with its own quantity. However, three phenomena
in Tables 14 and 15 are different from the case of independent goods
and are worthy of note. First, the marginal utility of one good is


===== PAGE 15 =====

The Austrian Theory of the Marginal Use 263

not independent of the quantity of the other good available. For
example, the marginal utility of the second unit of X falls from the
third to the second to the first position as Y increases from 1 to 3.
Similarly, the marginal utility of the third unit of Y falls from
second to first to zeroeth as X increases from 1 to 3. Thus, when
goods are rivals in consumption, the marginal utility of one good
tends to fall off as the quantity of the other increases.

Second, it sometimes happens that the marginal use of one good
is the satisfaction of a want which that good cannot itself satisfy.
For instance, when the individual has (1X, 2Y), the marginal use
of one additional X is d, a want that can only be satisfied by Y.
When he has (2X, 2Y), the marginal use of one more Y is e, a want
that can only be satisfied by X. Thus Wieser’s assertion above
(p. 260), that the marginal use must be a use characteristic of the
good in question, is not always true.

And third, when goods are rivals, it often happens that their
marginal uses coincide. Thus, when (1X, 1Y) is available, the outer
marginal use of both X and Y is the satisfaction of want ¢. When
(2X, 2Y) is available, the outer marginal use of both X and Y 1s
the satisfaction of want e.

Complementary Goods

The dictionary definition of “complementary” is “serving to fill
out or complete: mutually supplying each other's lack” 13, Let there
be two goods, X and Y, and suppose that one unit of each is re-
quired to satisfy some want, b, so that they are complementary
with respect to this want. Let both goods have alternative uses in
which they are not complements: a and d for X, and ¢ and e for Y.
Table 16 shows an “unrelated” ranking of W* which might reflect
an individual’s preferences. For various combinations of X and Y,
Table 17 shows the best use that can be made of the combination
if the satisfaction of b is excluded, the best use if & is included, and
the utilities or subjective values of both these uses. The best overall
use is the better of these two and is shown with its utility, the
derived utility of the combination, in the last two columns.

When we try to derive the marginal uses of X and Y from
Table 18, we encounter a new difficulty. For instance when our
individual has (2X, 1Y) we find that there is not a simple want

13 Webster (1973, 169). “Complémentaire” has a similar meaning in
French. Bohm-Bawerk (1909 A, 276) attributes the Germanization “kom-
plementdr” 10 Menger.



===== PAGE 16 =====

264 J. H. McCulloch:

dependent on the possession of anuiher unit of X. Rather, an addi-
tional unit of X enables him to replace want ¢ with the higher
rated want b. We represent this sort of marginal use by the ordered
pair (b, ¢), where the first entry (b) represents the additional want
satisfied and the second entry (c) represents the want (if any) whose
satisfaction is omitted. Clearly in this case, the unit of X will have
higher utility to the individual, the more important b is, and the

Table 16. Hypo-
thetical Preter-
ence Ordering

Set offOrdinal
Wanta (Utility

Table 17. Best Use of X and Y With and Without

abcde 3ith

abed | 30th Want b, and Best Overall Use

abce 29th

abde 28th

abc 27th Units of Beat Ham if b-- Bent Utiliey
abd 26th Excluded Included Use of Best
acde 25th XY Use | Utilicy Use
acd 24th

abe 23rd oo |(@ Oth = —-- 9 Oth
bede 22nd 110 |= Sth -- -— n grh
ab 21st 2 0 ad 15th -—— -— ad 15th
ace 20th 3| 0 [ad 15th | --~ —— ad 15th
ade 19¢ch 0 1 c 3rd ——— -—— c 3rd
hed 18ch 1 1 ac 17th |b 6th ac 17th
ac 17th 2 1 acd 24th ab 21st acd 24th
bee 16th 31 1° |acd 24th | abd 26th abd 26th
ad 15th 0 2 ce Sth ——— — ce 5th
bde 14th 1 2 ace 20th | be 12th ace 20th
ne 11th 2 2 |acde 25th | abc 27th abc 27th
be 12th K) 2 acde 25th abcd 30th abed 10¢h
bd 11th [4] 3 ce Sth -— -— ce Sth
cde 10th 1 3 ace 20th § bce 16th ace 20th
a 9th 2 3 acde 25th abce 29th abca 29th
ed Bth 3 3 acde 25th abcde| 31st abcde 3lat
be 7th

b 6th

ce 5th

de 4th

[4 Jrd

d 2nd

e let

[] Oth

less important ¢ is. Menger carelessly describes such a utility as
the difference between the utility of b and the utility of ¢, without
telling us what we are to make of this concept (1950, 165).

However, by extending “unrelatedness”, we are able to
place such “differences” accurately enough for our needs without
resorting to cardinality. From Table 16, we have b¢ cd. If we “de-
lete” ¢ from both sides of this relation, we obtain (b, ¢) { d (second


===== PAGE 17 =====

The Austrian Theory of the Marginal Use 265

utility level). Similarly, since bt ce, we must have (b,c) te (first
utility level). Therefore (b, ¢} is intermediate between the first and
second positions, In Table 19 we have indicated this by arbitrarily

Table18. Total Use and Total Utility
of Combinations of X and Y

Y
4 ce ace abce | abcde abede 49
Sth 20th 29th Jet 3lat
3 ce ace abce abcde| abcde a é
5¢h | 20th | 29th | 31st | lst
Lad
“ 2 ce ace abe abcd abcd p
: Sth | 20th | 27th | 30th | 30th 2
-
7 1 c ac acd abd abd
= 3rd | 17th | 24th | 26¢h | 26th 16
0 Pp a ad ad ad
Nth 9rh 15th | 15th | 15th 03 , 2 : a
X
0 2 3 4 , .
1 Fig. 4. Indifferences curves
Units of X
Table 19. Marginal Use and Mar- Table 20. Marginal Use and Mar-
ginal Utility of 1 Unit of X ginal Utility of 1 Unit of Y
a |b a | 8 wel ® Te Te Jo J¢
4 oth | 6th | 20d | Oth Och | Oth | Oth | Oth | Oth
0 b d [] 4 [4 # e e e
3 sn | ews | zua |] own - 3rd] nen | orn | lat | 1st | lst
Lad [+]
we e e {b,d) c Pi
o a (bie) | d p 8 2nd ’
o 2 | gn | 2st] 20a | 0en | E let | lst |1.7th | 3rd | 3rd
~
wt h b
5 a d (b,c) | # 1st ¢ « ¢
1 9ch 2nd 1.5th | Oth rd 3rd 3rd 6th 6th
0 a a # [4 0 1 2 3 4
9th 2nd Oth Jth Units of X

lst 2nd 3rd 4th
Unit of X

giving it the “1.5th” position. As with a Dewey decimal classifi-
cation, this is not intended to mean that it is half way between the
first and second positions, but merely that it is somewhere in be-
tween them.

The marginal use of the second unit of X, given 2Y, is (b, e).
This use is not so easy to place on the scale of wants. By com-
parison to ¢ it can be shown to be higher than the third position.
To find an upper bound is more difficult. Because ae } bc, we have
(a, ¢)  (b, ¢). Furthermore, cde’ a implies det (a, c). Therefore,

Zeitschr. f. Nationalékonomie, 37, Bd., Heft 34 18


===== PAGE 18 =====

266 J. H. McCulloch:

(b, €) t de (fourth position). Since (U, €) lies between the third and
fourth position, we assign it the “3.5th” position of Table 16.

The marginal utility of the second unit of Y, given 2X, is the
importance of replacing the satisfaction of d with that of b, or of
(b,d) in our notation. This use presumably has the same relation
to (b, ¢) that bc has to bd. Therefore (b, d) t (b, ¢) (1.5th position).
The reader may confirm that (6, d) { (c, €) { d (second position). We
therefore assign (b, d) the “1.7th” utility level4,

It is still true that the marginal utility of either good always falls
as its own quantity increases. When we have 2 units of VY, the
marginal utility of a unit of X falls from 9th to 3.5th to 2nd, and
finally to Oth. When we have 2 X, the marginal utility of Y falls
from 3rd to 1.7th to 1st, and then to Oth.

Furthermore, we note that with complements, we get exactly
the opposite of what happens with rival goods: as the quantity of
one good increases, the marginal utility of the other tends to increase,
instead of decrease as was the case with rivals. For instance, when
there are 2X available, the marginal utility of another unit of X
rises from Oth to 1.5th to 2nd as Y increases from 0 to 2.

Net Rivals and Complements: The ALEP Criterion

Rivalness and complementarity are not mutually exclusive con-
cepts, Two goods may be rivals with respect to one want and com-
plements with respect to another. Or using them as complements
in one proportion may be rival with using them in another pro-
portion, as in the case of production under variable proportions.

Since when rivalness is the only interaction, the marginal utility
of one good falls as the quantity of the other increases, and since

14 The problems that arise when we introduce complementarity indicate
that W* does not contain all of the “uses” of interest. We must consider a
more complicated set, say W**, the set of all ordered pairs of disjoint
subsets of W. Given (P, QO) in W** P is to be interpreted as the additional
wants that are to be satisfied, and Q the wants whose satisfaction is to
be omitted. W** then contains all marginal uses, in the broad sense that
we need for complementary (and jointness). It appears that the lincar
ordering on W*, together with our extended application of unrelatedness,
defines a partial ordering on W** which is sufficient to say which of two
goods will be valued more highly in any conceivable situation, to prove
diminishing marginal utility, and to establish the ALEP criterion, to be
discussed below. See McCulloch and Smith (1975) for a proof of the
law of the marginal use involving this extended concept. Cp. Neurath
(1911, 96) with respect to “differences in pleasure”.


===== PAGE 19 =====

The Austrian Theory of the Marginal Use 267

the opposite is true when complementarity is the only interaction,
we propose that X and Y be designated net rivals in a certain
region of the X—Y plane if in that region the marginal utility of the
one decreases as rhe quantity of the other increases holding the
quantity of any other goods constant, net complements if the oppo-
site is true, and on net independent if the marginal utility of one is
independent of the quantity available of the other, (It can be shown
that these concepts are well defined, that is, that X will have quali-
tatively the same effect on the marginal utility of Y as Y has on
the marginal utility of X, cvcn in the Austrian framework of ordinal
marginal utility.) This is actually the definition of rival and com-
plementary goods proposed, though in terms of the cross partial
derivatives of a smooth cardinal utility function, by Auspitz and
Lieben, Edgeworth, and Pareto!s. We therefore designate it the
“ALEP criterion”. Note, however, that while these authors used
the ALEP criterion as the definition of complements and rivals, the
approach of the marginal use theory is to adopt the common English
definitions of these concepts in terms of how the goods are used,
and then to demonstrate a relationship to the ALEP criterion.
Hicks claims that the “Edgeworth-Pareto definition sins
against Pareto’s own principle of the immeasurability of utility. If
utility is not a quantity, but only an index of the consumer's scale
of preferences, his definition of complementary and competitive goods
will differ according to the arbitrary measure of utility which is
adopted”, (1946, 43). However, we have shown that in the Austrian
concept of ordinal marginal utility, the criterion does indeed have
a precise meaning that is invariant with respect to monotonic trans-
formations of the utility index, so Hicks’ objection is invalid.
Hicks and Allen instead defined the complementarity of X and
Y in terms of Allen's “partial” elasticity of substitution ozy, which
is relared to the curvature of the indifference surfaces (Allen 1962/
38, 504—505). If it is positive they call the two goods “substitutes”
and if it is negative they call them “complements”. However, it has
never heen demonstrated that the sign of the substitution elasticity

15 Auspitzand Lieben (1889, 482), Edgeworth (1897/1925, 117 n. 1),
Pareto (1906/1927, 268—269). It is not actually clear that the functions
Auspitz and Lieben and Edgeworth differentiate are really what we
would call utility functions. For instance, Edgeworth equates his first
derivative to a price. Nevertheless the basic idea is definitely there. While
Auspitz and Lieben were Austrians by nationality, they are not con-
sidered part of the Austrian school. Their approach was closer to that of
Edgeworth. The ALEP criterion has recently been rediscovered by
Samuelson (1974, 1264—1264).

1e8*


===== PAGE 20 =====

268 J. H. McCulloch:

has anything to do with whether X and Y are used in combination
with one another or in place of one another. It is about time this
question be investigated.

We have demonstrated above that the ALEP criterion is related
to whether the goods are rivals or complements in the English sense,
if not in the Hicks-Allen sense. One implication of the ALEP cri-
terion for the structure of commodity preferences, an implication
that was not recognized by Hicks, is that if there is a third good,
Z, which is completely independent of the first two goods, then the

marginal rate of substitution between X and Z will change in one
direction as Y increases holding X and Z constant if X and Y are
net complements, and will change in the opposite direction as Y in-
creases if X and Y are net rivals. If goods and wants are finely
divisible so that the Allen elasticities exist and are well defined,

this implies that
E (P./P.) F (P,/P.)
EY xz 2" TEX vz

will both be positive, negative or zero, depending on whether X
and Y are net complements, rivals, or independents, where E re-
presents the logarithmic differentiation operator:

EX =dlog X =dX/X, ctc., (1)

and Pz, Py, and Pg represent the prices facing a competitive buyer.
It can be shown that

EPafPy) | —kyns ow {I= Rye) Oy. 2)
FY XxX, 7 = ky (022 Ouy— Osu?)
and
EPy/Py) _ Tham oyy— (1 = k:nx) Tay 3)
EX v.2 ky (022 0yy— 0x?)

where the k’s and 7's are respectively the budget shares and income
elasticities of demand for the three goods!®. Setting (2) and (3) equal
to zero as in the case of independent goods and cmploying the
familiar conditions

3

2 ks gi = 0 (4)

jet
implies that :
Hx Oyz = M2 Oxy = Ny Tze.

18 Expressions (2) and (3) do not necessarily have the same sign unless
Z is net independent of X and Y.


===== PAGE 21 =====

The Austrian Theory of the Marginal Use 269

Since independent goods will always have positive income elasticities,
Eq. (5) implies that all three cross substitution elasticities will have
the same sign. Since at most one can be negative, it follows that
they must all be positive. Therefore if X and Y are independent

(and the third good Z is also independent of both X and Y),
ozy will be positive.

It follows that there will be some small amount of ALEP net
complementarity between X and Y for which ozy remains positive.
Therefore 05, being negative is not a necessary condition for com-
plementarity. Note, however, that as ky gocs to zero, (2) takes on
the sign of —oazy, and that as kz goes to zero, (3) does likewise.
Therefore if the budget shares of the two goods in question are
negligible (and if third goods arc independent), a negative ay is a
necessary and sufficient condition for X and Y to be net com-
plements!?,

Even without the shares going to zero, it may still be a sufficient
condition. If oy is negative, the numerators of both (2) and (3) will
be positive except in unusual cases when some of the income
elasticities are negative. Furthermore, it can be shown that the quasi-
concavity of preferences implies that the denominators are necessarily
positive (McCulloch 1977, 7). Therefore if none of the three goods
is inferior (and if the third good is nct independent), a negative
cross substitution elasticity is a sufficient, if not necessary, condition
for net complementarity.

Even though there is some connection between uzy and the net
complementarity or rivalness of X and Y, we cannot say anything
for certain unless we know Z's ALEP relation to X and Y. The
attempt of Hicks and Allen to infer the complementarity of X
and Y from demand parameters alone was therefore futile. But,
what is more useful, we can make inferences about demand rela-
tionships from what we know about how X, Y, and Z are uscd.

In any event, the Hicks-Allen “definition” of complementarity,
which Samuelson (1974, 1528) calls the SHAS definition (after
Slutsky, Hicks, Allen and Schultz), should now be rejected
once and for all. In its place we propose restoring what might be
called the WOLM definition (after Webster, Oxford, Larousse,
and Menger), the Websterian version of which we have quoted
above. If ozy is positive, X and Y may be called positive substitutes,
and if it is negative, they may be called negative substitutes, pro-

17 Cp. Hicks (1946, 44). When the shares vanish, income effects can
be ignored.


===== PAGE 22 =====

270 “J. H. McCulloch:

vided the word “substitute” in this sense is not confused with “rival”.
We have demonstrated above that noninferior negative substitutes
are extremely complementary, relative to third goods, but this is a
deduction, not a definition!8,

The treatment of complementarity illustrates the substantial meth-
odological difference between the Austrian approach to consumer
theory and the current Hicks-Allen orthodoxy. This orthodoxy
might appropriately be called the “indifferent” approach, because
of its preoccupation with indifference curves and its refusal to look
beneath them to examine the relation of goods to underlying wants.
The indifferent approach suffers from the positivistic prejudice that
science can only take note of “observable” phenomena, and must
never attribute human-like motives to its objects of study. The
Austrian school, on the other hand, realizes that there is nothing
unscientific about attributing human-like motives to human beings.
Animism may be impermissible in the natural sciences, but it is
indispensable to the social sciences, In any case, the fact that people
use water to irrigate their lawns and gasoline to fuel their auto-
mobiles, instead of the other way around, is far more observable
than the cross substitution elasticities between water, lawns, gasoline
and automobiles. |

The sterility of the indifferent approach to consumer choice has
led many economists working independently of the Austrian school
to move in a similar direction. Lancaster (1966, 1971) investigates
how goods are used to provide “characteristics”, similar to Austrian
wants, that are the ultimate objects of consumer preference. Becker
and his school (e. g. Michael and Becker 1973) have developed a
model in which market goods are combined in a “household pro-
duction function” to create observable or hypothetical “commodities”
which are the ultimate preference objects. However, neither of these
approaches insists, as the Austrians do, that preferences on market
goods can be broken down in terms of ultimate unrelated preference-
objects, nor do they develop the ALEP criterion and its implications,
or recognize the ordinal character of marginal utility. Nevertheless,
these approaches do belong with the Austrians in the camp of
animistic economics.

18 Sato and Koizumi (1973) have shown, in the context of a constant-
returns-to-scale production function, that negative substitutes imply a positive
“elasticity of complementarity”, which in turn has the sign of the ALEDP-like
cross partial derivative of the production function. We approve of this
“elasticity of complementarity”, but it is unclear what its analog is in
terms of utility theory,


===== PAGE 23 =====

The Austrian Theory of the Marginal Use 271

Joint Satisfaction

Yet another type of technological interrelationship between goods
and wants is that of jointness, which arises when one unit of a good
can satisfy more than one want simultancously. This relationship is
important when one of the wants can also be satisfied by a second
good!?. The relation between jointness and inferiority (in the sense
of having a negative income elasticity) deserves careful analysis. It
would appear that inferiority (in some qualified sense applicable to
discretely divisible goods) cannot arise in the absence of joint want
satisfaction, although we have not been able to demonstrate it20,

The assumption that wants are unrelated is perfectly natural
until it is made explicit. Then it becomes apparent what a restrictive
assumption it is. Are we really justified in assuming that preferences
have a certain structure? Lancaster allows his underlying “char-
acteristics”, which correspond roughly to Austrian wants, to be
highly interrelated. However, if we reflect on the types of inter-
relationships among goods that are likely to occur, we ordinarily
find that they can be reduced to purely technological interrelation-
ships affecting the satisfaction of nunrelated ultimate wants. It would
appear that the categories of rivalness, complementarity, and joint-
ness are sufficient to explain any such technological interrelationship.

Convexity of the Indifference Curves

Notice that in Figs. 2, 3, and 4 we were always able to draw
indifference curves that were convex to the origin and were never
forced to draw a backward-bending portion?!. We conjecture that
it can be proven that convex indifference curves may always be
found in the Austrian system of utility, subject only to the reserva-
tions given in the next section. This has already been demonstrated
in the case of two independent goods by Jeffrey Smith (McCul-
loch and Smith, 1975). This issue is of great interest for economic
theory, Hicks is not satisfied that he has given adequate justifi-

1% In this case, the first and second goods would correspond to Menger’s
goods of “superior” (hoher) and “inferior” (minderer or niederer) quality
(1950, 144—145 and 1934, 118—119).

® Grossman (1974, 13—18) demonstrates that jointness can give

market goods a different income elasticity than the corresponding “house-
hold commodities”.

21 Mathematical economists call this property “quasi-concavity”.


===== PAGE 24 =====

272 J. H. McCulloch:

cation for his bald assumption of convexity, or what is the same
thing, of diminishing marginal ratc of substitution:

Since we know from experience that some points of possible
equilibrium do exist on the indifference maps of nearly every
one..., it follows that the principle of diminishing marginal
rate of substitution must sometimes be true.

However, for us to make progress in economics, it is not enough
for us to know that the principle should be true sometimes;
we require a more general validity than that. (1946, 22).

Fortunately, the Austrian utility theory leads to a more satisfying
development of this important proposition than does the orthodox
“indifferent” approach.

Convex indifference curves were first developed by Edgeworth
as an implication of diminishing marginal utility, provided the goods
were on net independent or were net complements. Note, however,
that they work out to have the usual curvature even in our exa mple
of rival goods.

Instances of Increasing Marginal Utility

Suppose that one unit of X will satisfy want a, but that it takes
no less than two units to satisfy want &; one unit cannot “half-way”
satisfy b. Suppose that b is “much greater” than a. If an individual
has one unit of X he will use it to satisfy a. If he has two, he will
satisfy b. The marginal utility of the first unit is then the importance
of a, while the marginal utility of the second is (b, a), the importance
of replacing a with b. If b is sufficiently important and a sufficiently
unimportant, the marginal utility of the second unit may actually
be higher than that of the first unit?2. Mises (1966, 125) has recog-
nized that circumstances like these may arise when several units of
a good must bc uscd together to provide a given effect, and that
they provide valid exceptions to the general principle of diminishing
marginal utility.

If more than one unit of a good must be combined to produce
a given effect, either by itself or in a complementary package with
another good, we would similarly expect to find instances where
we are forced to draw concave segments of our indifference curves.
Therefore any proof of convexity arising from the Austrian theory

22 Ye may say for certain that the second unit has higher marginal
utility than the first if there is a third want ¢, and bac? ct a} ¢. By
deleting a from both sides of b tac, we get (b,a)}c, whence {(b, a) } a.


===== PAGE 25 =====

The Austrian Theory of the Marginal Use 273

of the marginal use must be qualified to hold only if for each good
there is a single quantity in which it enters into the consumption
technology.

Nevertheless, we would still expect diminishing marginal utility
and convexity to hold for a given individual as a general rule, if not
in every instance. Furthermore, when we look at masses of individ-
uals, we might find that any “lumpiness” in the consumption be-
havior of any individual becomes insignificant in examining the
behavior of the group as a whole. Consequently, when describing
the reaction of large numbers of individuals to price changes, in-
come transfers, etc., we might expect them to behave, as a general
rule, as if for each one decreasing marginal utility and convexity
held, even though this may not be exactly true in each individual case.

Is 1t Really Ordinal?

It may have occurred to the reader that the easiest way to
generate an “unrelated” ordering on a set of subsets is to assign a
real number, say m1 (a¢), to each element ay of W, i=1,2,...,n
For each subset P of W define m (P)= X m (ai). Then for each pair

a ep

P and Q of subsets of W, let P+ Q whenever m (P)>m (Q). We
will call an ordering generated in this manner “essentially cardinal”.
Clearly such an ordering obeys unrelatedness, for if PQ, then
m(P)y=m(P-Q)+mPNQ)>m(Q)=m(Q-P)+m (PQ), whence
m (PQ) >m (Q—P), so that P— Qt Q—P. Similarly, '=Q+ QP
implies P} Q, so that any essentially cardinal ordering also obeys
unrelatedness.

For example, it is easy to show how the orderings of Tables 1
and 10 can arise from such a cardinal measure. The reader may
confirm that measures of 11, 8, 6, and 4 for a, b, ¢, and d respect-
ively will generate the ordering of Table 1. Similarly, the five wants
a, b, ¢, d, and e can be given the measures 9.5, 8.5, 5.7, 5.2, and 2.2
to generate the ordering of Table 10. Such numbers will in general
not be unique if W is finite.

It seems plausible that all unrelated orderings, at least on finite
sets and sufficiently reasonable infinite sets, must be essentially
cardinal. In fact, in 1949 the Italian statistician B. de Finetti con-
jectured that this is truc, If so, it would seem to be mere quibbling
to retain an ordinal approach once it is assumed that wants are
unrelated, for we could then derive all properties of the ordering
from a few numbers which we can manipulate in familiar ways.
In any case, it could then be argued that the Austrian utility theory


===== PAGE 26 =====

274 J. H. McCulloch:

is only superficially ordinal, that their assumptions amount to the
same thing as cardinality.

For a decade de Finetti’s conjecture remained unsolved. In
1959, Kraft, Pratt and Scidenberg finally proved it false by
publishing a counter-example. Take for instance the ordering of
Table 4. It contains the four relations be} cd, bc tae, cet b, and
ad t bee. If the ordering arose from a measure m ( ), we would have
m (Dy+m (e)>m (c)+m (d), m (b)+m (c)>m (a) +m {c}, 111 (c) 4
m (e)>m (b), and m (a) +m (d) >m (b) +m (c) +m (e}. Adding these
four numerical inequalities together we get that mi (a) +2m (b) +
2m (c} +m (d) +2m (e) must be strictly greater than itself, a contra-
diction. Therefore the unrelated ordering of Table 4 cannot be es-
sentially cardinal. Similarly, the ordering of Table 16 contains the
four relations b t ce, cd t be, ae} be, and bee t ad, which would also
imply a contradiction if the ordering were essentially cardinal®3,

Since unrelatedness does not imply measurability, it follows that
the Austrian theory of the marginal usc is intrinsically ordinal. It
admits of situations where no cardinal utility function is possible.

The Austrian literature is full of contradictory statements as to
whether utility is expressible cardinally, On the ordinal side we
may cite Bohm-Bawerk (1959 II, 423, n. 17 to p. 141) and Wieser
(1884, 180—181). On the cardinal side, we have Bohm-Bawerk
(195911, 197-198; 124—136) and Wieser (1884, 196). A much-cited
passage in Menger (1950, 183 n.) is often used as evidence that he
was an ordinalist, but his meaning is clearly cardinalist if we read it in
context. See also Menger (1950, 179, 293, un. 1). Only thc later
Austrian school economists, such as Mises (1966, 122), Bilimovi¢
(1934), and Rothbard (1956), can be said to take an adamantly
ordinal position.

The persistent inconsistency of the older Austrians on the car-
dinality question is understandable in light of the close relation
between measurability and their implicit assumption that wants are
unrelated. They can hardly be taken to task for being unclear in the
nineteenth century about a distinction which mathematicians did
not even state until 1949 and did not resolve until 1959. Tt is natural
to draw on cardinal illustrations to force unrelatedness, cven if the

23 Kraft et al. (1959) attribute this conjecture to B. de Finetti (1951,
1—10). The ordering of Table 16 is due to Kraft et al. That of Table 4
has, to the best of our knowledge, never been published. Sce Krantz ef al.
(1971, chapt_ 5) for theorems relating to unrelatedness. In McCulloch and
Smith (1975) it is demonstrated that if W has § elements, there are at
least 1920 different intrinsically ordinal unrelated orderings on W*.


===== PAGE 27 =====

The Austrian Theory of the Marginal Use 275

cardinality has no necessary place in the theory. Perhaps Bohm-
Bawecrk had this in thc back of his mind when he added the
proviso “or something very much like it” to his statement that
utilities may be expressed in multiples of one another.

One situation that docs lead to essentially cardinal preferences
is the hypothetical one in which goods and wants are perfectly
divisible. It can be shown that if W* is unrelatedly ordered in such
a way that W can bc partitioned into arbitrarily insignificant subsets,
its ordering must be essentially cardinal (Krantz et, al.,, 1971,
206—207)24. Thus, if a good such as an automobile could be divided
into arbitrarily small pieces satisfying arbitrarily trifling wants which
when put together would comprise the important wants satisfied
by the whole automobile, utility would be essentially cardinal. Such
an assumption is not very realistic, to be sure. We would not want
to make it a fundamental postulate of all utility theory. Nevertheless
in some applications this convenient simplification might be harm-
less, provided we recognize it as the simplification it is. When we do
indulge in it, the unrelatedness of wants, together with the Austrian
logic of choice, will imply as a theorem that the derived cardinal
marginal utility diminishes.

Probabilistic Cardinalization of Utility

It is a fairly straightforward exercise to adapt the well-known
von Neumann-Morgenstern probabilistic axioms?® to the Aus-

24 Similarly, Alc (1936) demonstrates that any Bernardelli utility
index (1938) rhat is expressible as a continuous function on commodity
space can be monotonically transformed in such a way that Bernardelli’s
conditional utility is the arithmetic difference of his total utility. If the
Bernardelli utility index is not continuous, however, it cannot necessarily
be so transformed. As a counterexample, consider the derived commodity
preferences that would arise when there are several different indivisible
goods, each one of which is capable of satisfying a different basic want,
when the ordering on W* happens to be intrinsically ordinal. (There must
be five or more goods for this to happen.) In a published comment on
Bernardelli’s paper, Samuelson (1939) called attention to crucial flaws
in a functional example Bernardelli attempted to work out in his mathe-
matical appendix. Samuelson’s comments, however, do not reflect on the
text of Bernardelli's paper.

25 See von Neumann and Margenstern (1953, Appendix) or any
advanced text on microeconomics, and Morgenstern (1976, 809). It is a
curious inconsistency in the state of economic doctrine that the leaders of


===== PAGE 28 =====

276 J. H. McCulloch:

trian framework and come up with a cardinal utility index for
the wants and therefore for commodities. In fact, unrelatedness
can be integrated into the traditional von Neumann-Morgenstern
axiom system in a way that virtually eliminates one of the tradi-
tional axioms. When this is done, the Austrian wants-structure will
imply that the resulting cardinal utility index on commodity space
will be mathematically concave, and therefore exhibit diminishing
marginal utility and indifference curves that are convex toward the
origin. What's more, it will imply that consumers really are risk-
averse (as is conventionally merely assumed), in terms of their von
Neumann-Morgenstern utility index?®.

However, doing this rules out intrinsically ordinal rankings on
W*, Therefore economists cannot have both the von Neumann-
Morgenstern axioms and the possibility of intrinsically ordinal
preferences. One or the other has to go. Several economists have
questioned thevonNeumann-Morgensternsystem. Georgescu-
Roegen (1954) argues that perhaps preferences are lexicographic
and linear, ruling out the possibility of indifference that is crucial
to the von Neumann-Morgenstern approach. Taking a different
tack, Quandt (1960) and Meginniss (1976) have questioned whether
expected utility maximization is necessary for rationality. These
authors argue that there is nothing irrational about consumers who
instead maximize expected utility plus a term that depends on the
standard error of the utility of the gamble (Quandt), or on the
entropy of the gamble (Meginniss). Intrinsically ordinal preferences
might not be ruled out for consumers like these.

In summary, the issue of probabilistic cardinalization of utility
is still up in the air. We personally find intrinsically ordinal pre-
ferences and the von Neumann-Morgenstern axiom system
about equally plausible. Until this inconsistency is resolved, however,
it should be remembered that the purely Austrian approach does
admit intrinsically ordinal marginal utility.

the profession acknowledged soon after 1944 that the von Neumann-
Margenstern cardinalization of utility was plausible, yet refused for
decades to grant that it meant that the 1934 Hicks-Allen objections to
the ALEP criterion were no longer valid. Only thirty years later was
Samuelson willing to draw this obvious conclusion (1974, 1264—1265).
Even so, in the same paper he took pains to deny that he was “back-
sliding” from the indifferent tradition (1285, n. 23}.

28 Furthermore, the circumstances described above, under which in-
stances of increasing marginal utility can arise, provide a rationale for the
Friedman-Savage hypothesis (1948).


===== PAGE 29 =====

The Austrian Theory of the Marginal Use 277

Conclusion

The Austrian theory of the marginal use raises almost as many
problems as it has solved. We list here a few of these unsolved
problems,

Complementarity and rivalness do lead to the ALEP criterion
in the examples we worked out above, but we have made no at-
tempt to formalize this rule into a general theorem. Intuitively, the
ALEP condition must appear when the complementary or rival
relationships are somehow active in the inner or outer marginal
uses, but it is not clear exactly what the circumstances are under
which this holds.

Although the theory leads to quasi-concavity of commodity pre-
ferences over goods in the particular cases we worked out, even
when rival or complementary interactions are present, it has only
been proven that this must be generally true when there are two
goods, and then only in the case when the two goods are independent.
Perhaps preferences do not really have to be quasi-concave after all.

And finally, it must be resolved whether the possibility of in-
trinsically ordinal preferences nullifies the von Neumann-Morgen-
stern axiom system, or if instead the validity of those axioms rules
out intrinsically ordinal preferences.

After over a century, the Austrian theory is still in its youth.
Perhaps the day has come for Felix Kaufmann’s young Grenz-
nutzler to return from the netherworld of economic doctrine:

There I will quietly lie in wait,

Amid my neglected writings,

Until I hear the trumpet call

of Complementary Goods.

Then through the sky will gallop Bshm-Bawerk,
Polemics will thunder and flash!

Then armed with a quill I'll rise up from the grave,
To fight for the Grenznutzen school!??

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¢7 Kaufmann and Machlup (1935), “Die Grenznutzenschule” (my
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===== PAGE 30 =====

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Address of author: Prof. Dr. J. Huston McCulloch, Economics
Department, Boston College, Chestnut Hill, MA 02167, U. S. A.

Printed in Austria