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 x1, x2,..., xn denote
the amounts of the various commodities possessed by
an individual, his total utility is the sum of the single
utilities, U1(x1) + U2(x2) + ... + Un(xn). 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(x1,x2,..., 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,...
be John's utility isolines, a utility isoline being the loci
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.
