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Risk aversion (red) contrasted to risk neutrality (yellow) and risk loving (orange) in different settings. Left graph: A risk averse utility function is concave (from below), while a risk loving utility function is convex.
Prospects are coded as gains and losses from a zero point (e.g. using current wealth, rather than total wealth as a reference point), leading people to be risk averse for gains and risk seeking for losses. [5] B. Concave in the domain of gains (risk aversion) and convex in the domain of losses (risk seeking). [1]
The interplay of overweighting of small probabilities and concavity-convexity of the value function leads to the so-called fourfold pattern of risk attitudes: [7] risk-averse behavior when gains have moderate probabilities or losses have small probabilities; risk-seeking behavior when losses have moderate probabilities or gains have small ...
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.
However, in practice it would be difficult to use since quantifying the risk aversion for an individual is difficult to do. The entropic risk measure is the prime example of a convex risk measure which is not coherent. [1] Given the connection to utility functions, it can be used in utility maximization problems.
The utility function whose expected value is maximized is concave for a risk averse agent, convex for a risk lover, and linear for a risk neutral agent. Thus in the risk neutral case, expected utility of wealth is simply equal to the expectation of a linear function of wealth, and maximizing it is equivalent to maximizing expected wealth itself.
In expected utility theory for choice under uncertainty, cardinal utility functions of risk averse decision makers are concave. In microeconomic theory, production functions are usually assumed to be concave over some or all of their domains, resulting in diminishing returns to input factors. [7]
An optimal basket of goods occurs where the consumer's convex preference set is supported by the budget constraint, as shown in the diagram. If the preference set is convex, then the consumer's set of optimal decisions is a convex set, for example, a unique optimal basket (or even a line segment of optimal baskets).