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Most utility functions used for modeling or theory are well-behaved. They are usually monotonic and quasi-concave. However, it is possible for rational preferences not to be representable by a utility function. An example is lexicographic preferences which are not continuous and cannot be represented by a continuous utility function. [8]
Leontief utility functions represent complementary goods. For example: For example: Suppose x 1 {\displaystyle x_{1}} is the number of left shoes and x 2 {\displaystyle x_{2}} the number of right shoes.
E.g., the commodity is a heterogeneous resource, such as land. Then, the utility functions are not functions of a finite number of variables, but rather set functions defined on Borel subsets of the land. The natural generalization of a linear utility function to that model is an additive set function.
A possible solution is to calculate n one-dimensional cardinal utility functions - one for each attribute. For example, suppose there are two attributes: apples and bananas (), both range between 0 and 99. Using VNM, we can calculate the following 1-dimensional utility functions:
Standard utility functions represent ordinal preferences. The expected utility hypothesis imposes limitations on the utility function and makes utility cardinal (though still not comparable across individuals). Although the expected utility hypothesis is standard in economic modeling, it is violated in psychological experiments.
A utility function is considered to be measurable, if the strength of preference or intensity of liking of a good or service is determined with precision by the use of some objective criteria. For example, suppose that eating an apple gives to a person exactly half the pleasure of that of eating an orange.
In decision theory, the von Neumann–Morgenstern (VNM) utility theorem demonstrates that rational choice under uncertainty involves making decisions that take the form of maximizing the expected value of some cardinal utility function. This function is known as the von Neumann–Morgenstern utility function.
A consumer's indirect utility (,) can be computed from their utility function (), defined over vectors of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector (,) by solving the utility maximization problem, and second, computing the utility ((,)) the consumer derives from that ...