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 the consumer. The element $w_{i}$ represents the relative value that the consumer assigns to good $i$. If $w_{i}=0$, this means that the consumer thinks that product $i$ is totally worthless. The higher $w_{i}$ is, the more valuable a unit of this product is for the consumer.

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

Define a linear economy as an exchange economy in which all agents have linear utility functions. Several economists studied the properties of linear economies.

Existence and uniqueness of competitive equilibrium

David Gale proved necessary and sufficient conditions for the existence of a competitive equilibrium in a linear economy. He also proved several other properties of linear economies.

Assume that each agent $A$ has an initial endowment ${\overrightarrow {e_{A}}}$. This is a vector of size $m$ in which the element $e_{A,i}$ represents the amount of good $i$ that is initially owned by agent $A$. Then, the initial utility of this agent is ${\overrightarrow {w}}\cdot {\overrightarrow {e_{A}}}$.

A set $S$ of agents is called self-sufficient if the members of $S$ assign a positive value only for goods that are owned exclusively by members of $S$ (in other words, they assign a value of $w_{i}=0$ to any product $i$ which is place no value on the goods owned by agents that are not in

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.