Linear utility

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: .