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 $i,j$:

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

When there are two goods, the indifference curves are straight lines.

Economy with linear utilities

Define a linear economy as an exchange economy in which all agents have linear utility functions. A linear economy has several properties.

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 ${\overrightarrow {w_{A}}}\cdot {\overrightarrow {e_{A}}}$.

Suppose that the market prices are represented by a vector ${\overrightarrow {p}}$ - a vector of size $m$ in which the element$p_{i}$ is the price of good $i$. Then, the budget of agent $A$ is ${\overrightarrow {p}}\cdot {\overrightarrow {e_{A}}}$. While this price vector is in effect, the agent can afford all and only the bundles ${\overrightarrow {x}}$ that satisfy the budget constraint: ${\overrightarrow {p}}\cdot {\overrightarrow {x}}\leq {\overrightarrow {p}}\cdot {\overrightarrow {e_{A}}}$.

Competitive equilibrium

A competitive equilibrium is a price vector and an allocation in which the demands of all agents are satisfied (the demand of each good equals its supply). In a linear economy, it consists of a price vector ${\overrightarrow {p}}$ and an allocation $X$, giving each agent a bundle ${\overrightarrow {x_{A}}}$ such that:

• $\sum _{A}{\overrightarrow {x_{A}}}=\sum _{A}{\overrightarrow {e_{A}}}$ (the total amount of all goods is the same as in the initial allocation; no goods are produced or destroyed).
• For every agent $A$, its allocation ${\overrightarrow {x_{A}}}$ maximizes the utility of the agent, ${\overrightarrow {w_{A}}}\cdot {\overrightarrow {x}}$, subject to the budget constraint ${\overrightarrow {p}}\cdot {\overrightarrow {x}}\leq {\overrightarrow {p}}\cdot {\overrightarrow {e_{A}}}$.

In equilibrium, each agent holds only goods for which his utility/price ratio is maximal. I.e, if agent $A$ holds good $i$ in equilibrium, then for every other good $j$:

$w_{A,i}/p_{i}\geq w_{A,j}/p_{j}$

(otherwise, the agent would want to exchange some quantity of good $i$ with good $j$, thus breaking the equilibrium).

Existence 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.

A set $S$ of agents is called self-sufficient if all members of $S$ assign a positive value only for goods that are owned exclusively by members of $S$ (in other words, they assign value $w_{i}=0$ to any product $i$ which is owned by members outside $S$). The set $S$ is called super-self-sufficient if someone in $S$ owns a good which is not valued by any member of $S$ (including himself). Gale's existence theorem says that:

A linear economy has a competitive equilibrium if and only if no set of agents is super-self-sufficient.

Proof of "only if" direction: Suppose the economy is in equilibrium with price ${\overrightarrow {p}}$ and allocation $x$. Suppose $S$ is a self-sufficient set of agents. Then, all members of $S$ trade only with each other, because the goods owned by other agents are worthless for them. Hence, the equilibrium allocation satisfies:

$\sum _{A\in S}{\overrightarrow {x_{A}}}=\sum _{A\in S}{\overrightarrow {e_{A}}}$.

Every equilibrium allocation is Pareto efficient. This means that, in the equilibrium allocation $x$, every good is held only by an agent which assigns positive value to that good. By the equality just mentioned, for each good $i$, the total amount of $i$ held by members of $S$ in the equilibrium allocation $x$ equals the total amount of $i$ held by members of $S$ in the initial allocation $e$. Hence, in the initial allocation $e$, every good is held by a member of $S$, only if it is valuable to one or more members of $S$. Hence, $S$ is not super-self-sufficient.

• Example: suppose there are two agents and two goods. Alice holds apples and guavas but wants only apples. George holds only guavas but wants both apples and guavas. The set {Alice} is self-sufficient, because Alice thinks that all goods held by George are worthless. Moreover, the set {Alice} is super-self-sufficient, because Alice holds guavas which are worthless to her. Indeed, a competitive equilibrium does not exist: regardless of the price, Alice would like to give all her guavas for apples, but George has no apples so her demand will remain unfulfilled.

Competitive equilibrium with equal incomes

Competitive equilibrium with equal incomes (CEEI) is a special kind of competitive equilibrium, in which the budget of all agents is the same. I.e, for every two agents $A$ and $B$:

${\overrightarrow {p}}\cdot {\overrightarrow {x_{A}}}={\overrightarrow {p}}\cdot {\overrightarrow {x_{B}}}$

The CEEI allocation is important because it is guaranteed to be envy-free: the bundle $x_{A}$ gives agent $A$ a maximum utility among of all the bundles with the same price, so in particular it gives him at least as much utility as the bundle $x_{B}$.

One way to achieve a CEEI is to give all agents the same initial endowment, i.e, for every $A$ and $B$:

${\overrightarrow {e_{A}}}={\overrightarrow {e_{B}}}$

(if there are $n$ agents then every agent receives exactly $1/n$ of the quantity of every good). In such an allocation, no subsets of agents are self-sufficient. Hence, as a corollary of Gale's theorem:

In a linear economy, a CEEI always exists.

Uniqueness of utilities in competitive equilibrium

A linear economy may have many different equilibria. For example, suppose there are two goods and two agents, both agents assign the same value to both goods (e.g. for both of them, $w_{1}=w_{2}=1$). Then, in equilibrium, the agents may exchange some units of good 1 for an equal number of units of good 2, and the result will still be an equilibrium. For example, if there is an equilibrium in which Alice holds 4 apples and 2 bananas and Bob holds 5 apples and 3 bananas, then the situation in which Alice holds 5 apples and 1 banana and Bob 4 apples and 4 bananas is also an equilibrium. But, in all these equilibria, the total utilities of both agents are the same: Alice has utility 6 and Bob has utility 8. This is not a coincidence:

Gale proved that, if there are two equilibria in a linear economy, all agents are indifferent between the equilibria (every agent has exactly the same utility in each equilibria).

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.

Goods with linear utilities are a special case of Independent goods.

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.