Lotteries and risk aversion



A choice may not have a certain outcome.

For example an action could have have 50

Von Neumann-Morgenstern utility theorem

We can model any risk preference as:

\(U[L]=\sum_i p_iu(x_i)\)

If the agent is risk neutral we can use:


If the agent is risk averse:

\(u(x_i)=\ln x_i\)

If the agent is risk loving we can use:


Absolute risk aversion

Absolute risk aversion

Given a utility function we can calculate the risk aversion.


Constant Absolute Risk Aversion (CARA) is:


\(u(x)=1-e^{\alpha x}\)

Hyperbolic Absolute Risk Aversion (HARA) is:


Increasing and Decreasing Absolute Risk Aversion (IARA and DARA):

Risk aversion increase or decreases in \(x\).

Relative risk aversion

Relative risk aversion

Absolute risk aversion is:





Expected utility

If an agent faces an uncertain world they can make decisions under uncertainty. For example, how would an agent value £10 relative to a 50 There are many different attitudes an agent could have – we need a form which can capture these. A standard approach is expected utility.

We start by taking


Subjective expected utility

Risk aversion

HARA Hyperbolic Absolute Risk Aversion

CRRA Constant Relative Risk Aversion

Prospect theory

Cumulative prospect theory.

Knightian uncertainty

Bounded rationality