In economics and consumer theory, a linear utility function is a function of the form:
Contents
- Economy with linear utilities
- Competitive equilibrium
- Existence of competitive equilibrium
- Competitive equilibrium with equal incomes
- Examples
- Uniqueness of utilities in competitive equilibrium
- Calculating competitive equilibrium
- Related concepts
- References
or, in vector form:
where:
A consumer with a linear utility function has the following properties:
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
Suppose that the market prices are represented by a vector
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
In equilibrium, each agent holds only goods for which his utility/price ratio is weakly maximal. I.e, if agent
(otherwise, the agent would want to exchange some quantity of good
Without loss of generality, it is possible to assume that every good is desired by at least one agent (otherwise, this good can be ignored for all practical purposes). Under this assumption, an equilibrium price of a good must be strictly positive (otherwise the demand would be infinite).
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
Proof of "only if" direction: Suppose the economy is in equilibrium with price
Every equilibrium allocation is Pareto efficient. This means that, in the equilibrium allocation
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
The CEEI allocation is important because it is guaranteed to be envy-free: the bundle
One way to achieve a CEEI is to give all agents the same initial endowment, i.e., for every
(if there are
Examples
In all examples below, there are two agents - Alice and George, and two goods - apples (x) and guavas (y).
A. Unique equilibrium: the utility functions are:
The total endowment is
B. No equilibrium: Suppose 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.
C. Many equilibria: Suppose there are two goods and two agents, both agents assign the same value to both goods (e.g. for both of them,
But, in both these equilibria, the total utilities of both agents are the same: Alice has utility 6 in both equilibria, and George has utility 8 in both equilibria. This is not a coincidence, as shown in the following section.
Uniqueness of utilities in competitive equilibrium
Gale proved that:
Proof. The proof is by induction on the number of traders. When there is only a single trader, the claim is obvious. Suppose there are two or more traders and consider two equilibria: equilibrium X with price vector
a. The price vectors are the same up to multiplicative constant:
b. The price vectors are not proportional. This means that the price of some goods changed more than others. Define the highest price-rise as:
and define the highest price-rise goods as those good/s that experienced the maximum price change (this must be a proper subset of all goods since the price-vectors are not proportional):
and define the highest price-rise holders as those trader/s that hold one or more of those maximum-price-change-goods in Equilibrium Y:
In equilibrium, agents hold only goods whose utility/price ratio is weakly maximal. So for all agents in
So in equilibrium X, the
On one hand, in equilibrium X with price
(where
On the other hand, in equilibrium Y with price
Combining these equations leads to the conclusion that, in both equilibria, the
Hence, the agents 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.
Goods with linear utilities are a special case of substitute 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. An extension of Gale's result to this setting is given by Weller's theorem.
Under certain conditions, an ordinal preference relation can be represented by a linear and continuous utility function.