# Linear utility

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. A linear economy has several properties.

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).

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

In all examples below, there are two agents - Alice and George, and two goods - apples (x) and guavas (y).

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.

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In a linear economy, all agents are indifferent between all the equilibria

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:

Linear utilities functions are a small subset of Quasilinear utility functions.