Quasilinear utility


In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function where is strictly increasing and concave. A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for does not depend on wealth and is thus not subject to a wealth effect; The absence of a wealth effect simplifies analysis and makes quasilinear utility functions a common choice for modelling. Furthermore, when utility is quasilinear, compensating variation, equivalent variation, and consumer surplus are algebraically equivalent. In mechanism design, quasilinear utility ensures that agents can compensate each other with side payments.

Definition in terms of preferences

A preference relation is quasilinear with respect to commodity 1 if:
In other words: a preference relation is quasilinear if there is one commodity, called the numeraire, which shifts the indifference curves outward as consumption of it increases, without changing their slope.
In the two dimensional case, the indifference curves are parallel. This is useful because it allows the entire utility function to be determined from a single indifference curve.

Definition in terms of utility functions

A utility function is quasilinear in commodity x if it is in the form
where is an arbitrary function. In the case of two goods this function could be, for example,
The quasilinear form is special in that, if the consumer's income is sufficiently large, then the demand functions for all but one of the consumption goods depend only on the relation between the good and the numeraire good and not on the income.
Example:
then, maximizing utility subject to the budget constraint
yields the first order conditions for internal solution:
.
Therefore, the demand function for good i is
which is independent of the income I. Note, however, that the interior solution exists only if the individual's income is sufficiently large that she can buy enough of each good to bring its marginal surplus down to the point that it equals the marginal surplus of the numeraire. If that is not the case, then demand for all non-numeraire goods will depend on income, and demand for the numeraire will be zero. Returning to the interior solution, note that the good i is a substitute to good x. That is, the demand for good i increases in response to an increase in the price of good x.
The indirect utility function in this case is
which is a special case of the Gorman polar form.

Equivalence of definitions

The cardinal and ordinal definitions are equivalent in the case of a convex consumption set with continuous preferences that are locally non-satiated in the first argument.