Linear utility

In economics and consumer theory, a linear utility function is a function of the form:

$u(x_{1},x_{2},\dots ,x_{m})=w_{1}x_{1}+w_{2}x_{2}+\dots w_{m}x_{m}$

or, in vector form:

$u({\overrightarrow {x}})={\overrightarrow {w}}\cdot {\overrightarrow {x}}$

where:

• $m$ is the number of goods in the economy.
• ${\overrightarrow {x}}$ is a vector of size $m$ that represents a bundle. The element $x_{i}$ represents the amount of good $i$ in the bundle.
• ${\overrightarrow {w}}$ is a vector of size $m$ that represents the subjective preferences of a consumer. The element $w_{i}$ represents the relative value that the consumer assigns to good $i$.

For a consumer with a linear utility function, the marginal rate of substitution of all goods is constant. For every two goods $i,j$:

$MRS_{i,j}=w_{i}/w_{j}$.

Economy with linear utilities

Several economists studied the properties of an economy in which all agents have linear utility functions.

Existence and uniqueness of competitive equilibrium

Gale proved necessary and sufficient conditions for the existence of a competitive equilibrium in such an economy.

Calculating competitive equilibrium

Eaves presented an algorithm for finding a competitive equilibrium in a finite number of steps, when such an equilibrium exists.

Related concepts

Linear utilities functions are a small subset of Quasilinear utility functions.

Suppose the set of goods is not finite but continuous. E.g, the commodity is a heterogeneous resource, such as land. Then, the utility functions are not functions of a finite number of variables, but rather set functions defined on Borel subsets of the land. The natural generalization of a linear utility function to that model is an additive set function. This is the common case in the theory of fair cake-cutting.

Under certain conditions, an ordinal preference relation can be represented by a linear and continuous utility function.