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 $i,j$:

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

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

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

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

proved conditions under which a preference relation can be represented by a linear and continuous utility function.