Karen's utility of money is u=x^1 / 2 and John's utility of...

Karen's utility of money is $u=x^{1 / 2}$ and John's utility of money is $u=x^{1 / 3}$, where $x$ is the current balance in each person's bank account. Who is most averse against actuarial risks, Karen or John, for large amounts of money?

Key Concepts

Arrow-Pratt Measure of Risk Aversion

This measure provides a quantitative assessment of an individual's risk aversion by using the first and second derivatives of the utility function. For relative risk aversion, it is defined as -x * (u''(x)/u'(x)). A higher value indicates that the individual is more risk averse. For example, in CRRA utility functions, different exponents directly correspond to different levels of risk aversion.

Constant Relative Risk Aversion (CRRA)

CRRA utility functions are a common form used in economics to model preferences under uncertainty. They exhibit a constant coefficient of relative risk aversion regardless of the wealth level. This means that the willingness to take risks relative to one’s wealth remains unchanged even as wealth increases. In power utility functions of the form u(x) = x^?, the relative risk aversion is given by 1 - ?.

Risk Aversion

This concept refers to the degree to which an individual prefers certainty over uncertainty, especially when outcomes involve risk. It is captured by the curvature of the utility function, where more concave functions indicate higher risk aversion, meaning the individual is less willing to accept a gamble for a potentially higher return.

Utility Function

A utility function represents an individual's preferences over different outcomes, typically wealth or monetary gains. In the context of risk, the shape of the utility function—whether it is concave, linear, or convex—helps determine how much utility changes with changes in wealth, thereby revealing the individual’s attitude towards risk.

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