Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*V*

*1.* Before the emergence of the modern school of

utility, thoughts on value, demand, and exchange ordi-

narily reflected the economic conditions prevailing

during each period. A turning point in this respect took

place following the process of mathematization fos-

tered by that school. Utility theory has ever since been

its own source of new ideas, suggested primarily (and,

at times, exclusively) by its mathematical framework.

An excellent illustration is the observation made by

Irving Fisher in his doctoral dissertation (1892). The

pure geometry of Edgeworth's diagram led Fisher to

note that in order to determine the optimal budget

distribution we do not need to know how many utils

each isoline represents: the knowledge of the isolines

as such suffices. This simple geometrical truth caused

the first serious dent in the idea that a cardinally

measurable utility is indispensable for explaining value.

It was, however, Vilfredo Pareto (*Manuale di econo-
mia politica,* 1905) who first constructed a consumer

theory which does not require the notion of utility at

all. His point of departure is that an individual con-

fronted with two baskets of commodities will always

either prefer one basket or be indifferent as to which

one he gets. Given this faculty of

*binary*choice, Pareto

reasoned that, by asking the individual to choose be-

tween M and every other possible basket, we can

determine an

*indifference*curve, i.e., a curve that rep-

resents the loci of all baskets “indifferent” in relation

to M. The procedure does not refer in any way to

utility. And once the indifference curves are deter-

mined, they help determine the optimal distribution

of any budget in exactly the same manner as the utility

isolines. Furthermore, we can construct a function

*V*(

*x*1,

*x*2,...,

*xn*) such that its value is constant on each

indifference curve, just as the utility function

*U*(

*x*1,

*x*2,...,

*xn*) is constant on each isoline. The only

difference is that

*V*is not uniquely determined—any

increasing function of

*V,*say

*V*2, would do.

It is for the function *V* that Pareto coined the term

“ophelimity.” But, as was argued in subsequent devel-

opments, we may still speak of utility and of *V* as its

*ordinal,* instead of cardinal, measure. This means that

the value of *V* simply *orders* all baskets according to

the individual's preferences. Today the notion of an

ordinal utility dominates consumer theory, the central

problem of which is how to derive an ophelimity

function from directly observable budget data.

*2.* In fact, this problem is relatively old. It was first

formulated in a neglected memoir of an Italian engi-

neer, G. Antonelli (1886). And, as happens quite often,

the glory went to the more famous rediscoverer of the

idea, in this case to Pareto (1905). In simple terms,

Pareto's idea was this: if the optimal distribution of

every possible budget has been determined by obser-

vation, every indifference curve can be determined by

the tangential artifice shown in Figure 2 for C3. But

he ignored the fact that this artifice (which in mathe-

matics is called “integration”) is not always available

for more than two commodities. An obvious para-

dox—known as the integrability problem—thus arose

to intrigue many a mathematical economist. Some light

was cast on Pareto's theory of choice and the integra-

bility problem when it was shown (Georgescu-Roegen,

1936) that Pareto's argument failed to include two

axioms (1) that any commodity may be substituted for

another so that the first and the second basket be

*completely* indifferent, and (2) that the binary choice

is transitive. (Choice is transitive if A being chosen over

B and B over C, A is chosen over C.)

*3.* In a signal contribution, Paul A. Samuelson (1938)

parison between two baskets, but on observable budget

data. His point of departure is that John, by choosing

the budget distribution M, reveals that he prefers M

to any other distribution (such as M′) compatible with

his budget. To this transparent definition, Samuelson

added only an equally transparent axiom:

*If a budget*

reveals that the basket A is preferred to B, no budget

can reveal that B is preferred to A.Samuelson claimed

reveals that the basket A is preferred to B, no budget

can reveal that B is preferred to A.

that this axiom alone suffices for deriving by integration

the indifference varieties and hence for constructing

an ophelimity function. In fact, the axiom expresses

only a condition equivalent to the Principle of De-

creasing Marginal Rate of Substitution. And as shown

first by Jean Ville (1946) and later, but independently,

by H. S. Houthakker (1950), Samuelson's idea calls for

a stronger axiom (analogous to the transitivity of binary

choice). But soon thereafter it was proved that even

this stronger axiom does not entail the existence of an

ophelimity function. It still leaves large domains for

which there is no comparability among the commodity

baskets (Georgescu-Roegen, 1954b). The problem of

what set of economically meaningful postulates would

make the Antonelli-Pareto idea work still awaits its

solution. If it is ever solved, it will very probably cause

a greater stir in mathematics than in economics.

Dictionary of the History of Ideas | ||