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.
