Quasilinear utility

In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. This utility function has the representation $u(x_{1},x_{2},\ldots ,x_{n})=x_{1}+\theta (x_{2},\ldots ,x_{n})$. If $\theta$ is concave, this has the interpretation that the marginal rate of substitution is diminishing, which is typical of a utility function.

Informally, an agent has quasilinear utility if it can express all its preferences in terms of money and the amount of money it has will not create a wealth effect. As a practical matter in mechanism design, quasilinear utility ensures that agents can compensate each other with side payments. In regard to surplus, quasilinear preferences entail that Marshallian surplus will equal Hicksian surplus since there would be no wealth effect for a change in price.

Definition in terms of preferences

A preference relation $\succsim$ is quasilinear with respect to commodity 1 (called, in this case, the numeraire commodity) if:

• All the indifference sets are parallel displacements of each other along the axis of commodity 1. That is, if a bundle "x" is indifferent to a bundle "y" (x~y), then $\left(x+\alpha e_{1}\right)\sim \left(y+\alpha e_{1}\right),\forall \alpha \in \mathbb {R} ,e_{1}=\left(1,0,...,0\right)$
• Good 1 is desirable; that is, $\left(x+\alpha e_{1}\right)\succ \left(x\right),\forall \alpha >0$

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 two dimensional case, the indifference curves are parallel; which is useful because the entire utility function can be determined from a single indifference curve.

Definition in terms of utility functions

A utility function is quasilinear in commodity 1 if it is in the form:

$u\left(x\right)=x_{1}+\theta \left(x_{2},...,x_{L}\right)$

where $\theta$ is a function. In the case of two goods, this function could be, for example, $u\left(x\right)=x_{1}+{\sqrt {x_{2}}}$.

The quasilinear form is special in that the demand function for the consumption goods depends only on the prices and not on the income. E.g, with two commodities, if:

$u(x,y)=x+\theta (y)$

then the demand for y is derived from the equation:

$\theta '(y)=p_{y}$

so:

$y(p,I)=(\theta ')^{-1}(p_{y})$

which is independent of the income I.

The indirect utility function in this case is:

$v(p,I)=v(p)+I$

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.

See also

• Quasiconvex function
• Linear utility function - a special type of a quasilinear utility function.