# Quasilinear utility

In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. This utility function has the representation $U(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n)$. 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. If $\theta$ is concave, this has the interpretation that the marginal rate of substitution is diminishing, which is typical of a utility function.

## Definition in terms of preferences

Quasilinearity can also be defined as a property of preferences directly; $\preceq$ is quasilinear if (1) if $x \sim y$ then $(x + \alpha e_1) \sim (y + \alpha e_1)$, where $e_1 = (1, 0, 0, \ldots, 0)$ and $\alpha$ is a real number. The two definitions are equivalent in the case of a convex consumption set with continuous preferences that are locally non-satiated in the first argument.

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.[citation needed] As a practical matter in mechanism design, quasilinear utility ensures that agents can compensate each other with side payments. In regards to surplus, quasilinear preferences entail that Marshallian surplus will equal Hicksian surplus since there would be no wealth effect for a change in price.

## Quasilinearity in Microeconomics

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. It is possible to extend this definition to utility functions: a continuous preference relation is quasilinear in commodity 1 if there is a utility function that represents it in the form $u \left ( x \right ) = x_1 + \theta \left (x_2, ..., x_L \right )$, where $\theta$ is a function.[1] In the case of two goods, this function could be, for example, $u \left ( x \right ) = x_1 + \sqrt{x_2}$

More formally, the preference relation $\succsim$ on a set $X= \left ( - \infty, \infty\right )\times \mathbb{R}^{L-1}_+$ 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 $\left ( x+ \alpha e_1 \right ) \sim \left ( y+ \alpha e_1 \right ), \forall \alpha \in \mathbb{R}, e_1= \left ( 1,0,...,0 \right )$[2]
• good 1 is desirable; that is, $\left ( x+ \alpha e_1 \right ) \succ \left ( x \right ), \alpha>0$

## 2 common examples

Suppose $U(x, v) = \ln(x) + v$

This function is also quasi-linear; the function is linear in $v$ and non-linear in $x$

Suppose $F(L,K) = K^2 + L$

This is a quasi-linear production function; the function is linear in $L$ and non-linear in $K$