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.
Certainty Equivalent
When the states are wealth, one can ask what fixed amount of wealth {% W' %} yields the same utility as the the random wealth variable {% W %}.
{% \mathbb{E}[u(W)] = u(W') %}
As long as Jensens inequality holds, it can be shown that
{% W' < \mathbb{E}(W) %}
The difference is then termed the risk premium
{% \pi = \mathbb{E}(W) - W' %}