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Concept
Linear utility
Summary
In economics and consumer theory, a linear utility function is a function of the form:
or, in vector form:
where:
is the number of different goods in the economy.
is a vector of size that represents a bundle. The element represents the amount of good in the bundle.
is a vector of size that represents the subjective preferences of the consumer. The element represents the relative value that the consumer assigns to good . If , this means that the consumer thinks that product is totally worthless. The higher is, the more valuable a unit of this product is for the consumer.
A consumer with a linear utility function has the following properties:
The preferences are strictly monotone: having a larger quantity of even a single good strictly increases the utility.
The preferences are weakly convex, but not strictly convex: a mix of two equivalent bundles is equivalent to the original bundles, but not better than it.
The marginal rate of substitution of all goods is constant. For every two goods :
The indifference curves are straight lines (when there are two goods) or hyperplanes (when there are more goods).
Each demand curve (demand as a function of price) is a step function: the consumer wants to buy zero units of a good whose utility/price ratio is below the maximum, and wants to buy as many units as possible of a good whose utility/price ratio is maximum.
The consumer regards the goods as perfect substitute goods.
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 has an initial endowment . This is a vector of size in which the element represents the amount of good that is initially owned by agent . Then, the initial utility of this agent is .
Suppose that the market prices are represented by a vector - a vector of size in which the element is the price of good . Then, the budget of agent is . While this price vector is in effect, the agent can afford all and only the bundles that satisfy the budget constraint: .
Official source
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