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Linear utility
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Linear utility : ウィキペディア英語版
Linear utility
In economics and consumer theory, a linear utility function is a function of the form:
::u(x_1,x_2,\dots,x_m) = w_1 x_1 + w_2 x_2 + \dots w_m x_m
or, in vector form:
::u(\overrightarrow) = \overrightarrow \cdot \overrightarrow
where:
* m is the number of different goods in the economy.
* \overrightarrow is a vector of size m that represents a bundle. The element x_i represents the amount of good i in the bundle.
* \overrightarrow is a vector of size m that represents the subjective preferences of the consumer. The element w_i represents the relative value that the consumer assigns to good i. If w_i=0, this means that the consumer thinks that product i is totally worthless. The higher w_i 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 i,j:
::MRS_ = w_i/w_j .
* 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.
== 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 A has an initial endowment \overrightarrow. This is a vector of size m in which the element e_ represents the amount of good i that is initially owned by agent A. Then, the initial utility of this agent is \overrightarrow\cdot \overrightarrow.
Suppose that the market prices are represented by a vector \overrightarrow - a vector of size m in which the elementp_i is the price of good i. Then, the ''budget'' of agent A is \overrightarrow\cdot \overrightarrow. While this price vector is in effect, the agent can afford all and only the bundles \overrightarrow that satisfy the budget constraint: \overrightarrow\cdot \overrightarrow \leq \overrightarrow\cdot \overrightarrow.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Linear utility」の詳細全文を読む



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