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

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Dictionary of the History of Ideas | ||

*IV*

*1.* All founders of the utility theory had some doubts

about the cardinal measurability of utility, but they

took for granted that the utility of each commodity

is independent of other commodities, that the utility

of bacon, for instance, does not depend on how many

eggs one has. This means that, if *x*1, *x*2,..., *xn* denote

the amounts of the various commodities possessed by

an individual, his total utility is the sum of the single

utilities, *U*1(*x*1) + *U*2(*x*2) + ... + *U*n(*x*n). It goes

without saying that the assumption greatly simplifies

the analysis of value. Let John have six bushels of

potatoes which he can trade at the price of four eggs

for a bushel. In Figure 1, let the number of bushels

be measured from O1 to X1 and the number of eggs

from O2 to X2. Let A1C1 represent John's *direct* mar-

ginal utility of potatoes when consumed as such and

let A2C2 represent his *indirect* marginal utility of pota-

toes derived from the eggs obtained by trading. If John

wants to maximize his total utility—a basic assumption

of every utility theory—he should, obviously, trade the

sixth and fifth bushels: their indirect utility is greater

than their direct utility. And he should stop trading

at the point M, where the two curves intersect, because

the direct utility of the fourth bushel is greater for him

than its indirect utility. And if John possessed initially

twenty-four eggs instead of six bushels of potatoes, he

should end up with the same distribution of commodi-

ties, four bushels of potatoes and eight eggs. The same

result obtains in the equivalent case in which John has

twelve dollars and the prices are two dollars for a

bushel and fifty cents for an egg. But if, as it may well

happen, the marginal utilities are such that A1C1 and

A2C2 do not meet, then John must choose to have

either only potatoes or only eggs, according to which

commodity has everywhere a greater marginal utility.

In any case, the optimal distribution of the budget is

unique, which is a direct consequence of the Principle

of Decreasing Marginal Utility.

The fact that a glance at Figure 1 suffices to clarify

many issues of value is the reason why economists still

use this highly unrealistic framework. For example, the

diagram (with A1C1 and A2C2 being drawn as they are)

shows that John's dollar buys more utility when spent

on potatoes than on eggs. This simple point explains

away the paradox of value. The same diagram shows

that as a potato seller John gains the amount of utility

represented by the area MA2C1, and as an egg seller

he gains the greater amount MA1C2 (which may be

infinite if the first potato is indispensable to life). There

can be then no just exchange in Aristotle's sense. And

to know whether there are just exchanges in Turgot's

sense we need the interpersonal comparison of utilities

in which Bentham believed.

*2.* The independence axiom was discarded as Edge-

worth (*Mathematical Psychics,* 1881) proposed to

represent total utility by a general function

*U*(*x*1,*x*2,..., *xn*). The diagram supplied by Edgeworth

for the representation of exchange under these general

conditions has become the most popular in economic

analysis. Let potatoes and eggs be measured on OX1

and OX2, respectively (Figure 2). Let C1, C2, C3,...

of all combinations of potatoes and eggs that have the

same utility. Naturally, utility increases as we move

from an isoline to a “higher” one, from C2 to C3, for

example. The alternatives open to John, whom we may

now assume to have ten bushels and be able to trade

one bushel for six eggs, are represented by the points

of the

*budget*line B1B2. The budget distribution that

maximizes John's total utility is the point M at which

one isoline is tangent to this budget line. Clearly, with

isolines having the shape shown in Figure 2, all other

possible distributions of John's budget lie on

*lower*

isolines. Therefore, John will trade four bushels of

potatoes for twenty-four eggs and retain six bushels

for his own consumption. The same solution is valid

if John has, say, six dollars and the prices are sixty cents

for one bushel and ten cents for an egg.

*3.* Obviously, for the optimal budget distribution to

be unique the utility isolines must be convex toward

O (as they have been drawn in Figure 2). A new

difficulty arises now because the Principle of Decreas-

ing Marginal Utility does not suffice to guarantee this

convexity. The shape of the utility isolines depends,

in addition, on the relation between the commodities.

As Edgeworth noted, commodities may be rival—like

margarine and butter—if an increase in one diminishes

the marginal utility of the other. They may be comple-

mentary—like bread and butter—if an increase in one

increases the marginal utility of the other. However,

there is no way to reduce the convexity property to

a property related to this classification. The convexity

of the isolines had to be added as a new axiom for

which no transparent explanation has yet been offered.

The axiom says that *along any isoline* the marginal

rate of substitution increases in favor of the commodity

that is decreased.

Dictionary of the History of Ideas | ||