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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, while risk seeking individuals have convex utility functions and risk averse individuals have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function.
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.
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.
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]
Take their supreme sense of smell, for example, which can not only help them sniff out the best dog food but also allow them to detect medical conditions, sometimes even saving lives, ...
A Missouri couple has been charged with child abuse after police claim they performed a circumcision on a child at their home despite not having the medical training to do so.
The first woman was elected to lead a country 64 years ago. Here’s a look at where, and when, women have secured national leadership positions since then.