Preferences and Utility Functions
V. Bardis
In this context, utility is "ordinal", i.e., only the ranking established by the numbers is signifcant (not
the numbers themselves and how much they differ).
The function U is "unique up to a positive monotonic transformation", that is, W(x) =g(U(x))
represents the same preferences as U given the function g is strictly increasing since
U(x') > U(x'') <=> g(U(x')) > g(U(x'')) <=> W(x') > W(x'')
U(x') = U(x'') <=> g(U(x')) = g(U(x'')) <=> W(x') = W(x'')
As an example, the following functions represent the same preferences:
since
and
>0
( x' is in the Neighborhood of x, a "ball" of radius epsilon)
Condition (5) implies (6) ( but (6) does not imply (5) ).
Local Non-Satiation (and therefore Monotonicity) implies that the consumer will find optimal to
consume a bundle on her budget line, i.e., every bundle below the budget line is worse than some
bundle on the budget line.
"Convexity of Preference" (related to the "Quasi-concavity of the Utility Function")
Convexity can be interpreted as "averages are preferred to extremes".
With Strict Convexity, the tangency condition identifies the optimum and the Method of Lagrange
can be used (assuming the optimal point is an interior point).
A homothetic Utility function can be defined as ``a monotonic transformation of a H.O.D.(1) function.
The shape of its indifference curves is the same along any line from the origin.
If U(x,y) is differentiable, the MRS depends only on the ratio y/x.
Homotheticity will result to linear income expansion paths, HOD(1) in income òf the market demand
functions, and constant expenditure shares (at given prices).
Examples