Ordinal Utility Functions
An ordinal utility function describing a consumer's preferences over, say, two goods can be written as
where x and y are the quantities of the goods consumed. Both partial derivatives of this function are positive if the consumer prefers more of both goods. But the same preferences could be expressed as another utility function that is a monotonic transformation of u:
where f is any globally increasing function. Utility functions g and u give rise to identical indifference curve mappings. Thus in ordinal utility theory, there is no concept of diminishing marginal utility, which would correspond to the second derivative of utility being negative. For example, even if u has a negative second derivative with respect to x, the equivalent utility function g may have a positive second derivative with respect to x.
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