Risk Aversion and Utility
Overview
The agent is assumed to maximized the
expected
utility.
{% max \; \mathbb{E}[u(\omega)] %}
When the outcomes are states of wealth, the utility then becomes a function of a real number, {% u(x) %}.
Risk Aversion and Wealth Outcomes
When the outcome of a choice is a numeric value, such as wealth, the notion of covexity or concavity can
be applied.
When {% u %} is concave,
Jensens Equality
gives
{% u(\mathbb{E}[x]) \geq \mathbb{E}[u(x)] %}
The concave utility function demonstrates decreasing marginal utility. That is, as wealth level goes up 1 dollar, the utility increases by a smaller
and smaller amount up the curve. This effectively means that an individual prefers to avoid a $10 loss more than gaining $10.
