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The utility function u(c) is defined only up to positive affine transformation – in other words, a constant could be added to the value of u(c) for all c, and/or u(c) could be multiplied by a positive constant factor, without affecting the conclusions. An agent is risk-averse if and only if the utility function is concave.
The risk attitude is directly related to the curvature of the utility function: risk-neutral individuals have linear utility functions, risk-seeking individuals have convex utility functions, and risk-averse individuals have concave utility functions. The curvature of the utility function can measure the degree of risk aversion.
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
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]
A single-attribute utility function maps the amount of money a person has (or gains), to a number representing the subjective satisfaction he derives from it. The motivation to define a utility function comes from the St. Petersburg paradox: the observation that people are not willing to pay much for a lottery, even if its expected monetary gain is infinite.
Cumulative prospect theory avoids the St. Petersburg paradox only when the power coefficient of the utility function is lower than the power coefficient of the probability weighting function. [13] Intuitively, the utility function must not simply be concave, but it must be concave relative to the probability weighting function to avoid the St ...
Yakar Kannai treats the question in depth in the context of utility functions, giving sufficient conditions under which continuous convex preferences can be represented by concave utility functions. [4] His results were later generalized by Connell and Rasmussen, [3] who give necessary and sufficient conditions for concavifiability.
In the case of two goods this function could be, for example, (,) = +. The quasilinear form is special in that the demand functions for all but one of the consumption goods depend only on the relation between the good and the numeraire good (x) and not on the income. Example: