In economics and consumer theory, a linear utility function is a function of the form: or, in vector form: where: A consumer with a linear utility function has the following properties: Define a linear economy as an exchange economy in which all agents have linear utility functions.
Suppose that the market prices are represented by a vector
While this price vector is in effect, the agent can afford all and only the bundles
In a linear economy, it consists of a price vector
such that: In equilibrium, each agent holds only goods for which his utility/price ratio is weakly maximal.
Under this assumption, an equilibrium price of a good must be strictly positive (otherwise the demand would be infinite).
David Gale[1] proved necessary and sufficient conditions for the existence of a competitive equilibrium in a linear economy.
assign a positive value only for goods that are owned exclusively by members of
Gale's existence theorem says that: Proof of "only if" direction: Suppose the economy is in equilibrium with price
trade only with each other, because the goods owned by other agents are worthless for them.
: The CEEI allocation is important because it is guaranteed to be envy-free:[2] the bundle
One way to achieve a CEEI is to give all agents the same initial endowment, i.e., for every
Hence, as a corollary of Gale's theorem: In all examples below, there are two agents - Alice and George, and two goods - apples (x) and guavas (y).
Unique equilibrium: the utility functions are: The total endowment is
Without loss of generality, we can normalize the price vector such that
The price vector depends on the initial allocation.
E.g., if the initial allocation is equal, [(3,3);(3,3)], then both agents have the same budget in CE, so
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.
Suppose there are two or more traders and consider two equilibria: equilibrium X with price vector
This means that in both equilibria, all agents have exactly the same budget set (they can afford exactly the same bundles).
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.
experienced the highest price-rise, when the price vector is
This allows us to do some budget calculations: On one hand, in equilibrium X with price
-goods, so: Combining these equations leads to the conclusion that, in both equilibria, the
This means that equilibrium X is composed of two equilibria: one that involves only
is a proper subset of the agents, the induction assumption can be invoked and the theorem is proved.
Suppose the set of goods is not finite but continuous.
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.