A choice may not have a certain outcome.
For example an action could have have 50
We can model any risk preference as:
\(U[L]=\sum_i p_iu(x_i)\)
If the agent is risk neutral we can use:
\(u(x_i)=x_i\)
If the agent is risk averse:
\(u(x_i)=\ln x_i\)
If the agent is risk loving we can use:
\(u(x_i)=x_i^2\)
Given a utility function we can calculate the risk aversion.
\(A(x)=-\dfrac{u''(x)}{u'(x)}\)
Constant Absolute Risk Aversion (CARA) is:
\(A(x)=c\)
\(u(x)=1-e^{\alpha x}\)
Hyperbolic Absolute Risk Aversion (HARA) is:
\(A(x)=\dfrac{1}{ax+b}\)
Increasing and Decreasing Absolute Risk Aversion (IARA and DARA):
Risk aversion increase or decreases in \(x\).
Absolute risk aversion is:
\(A(x)=-\dfrac{u''(x)}{u'(x)}\)
\(R(x)=xA(x)\)
\(R(x)=-\dfrac{xu''(x)}{u'(x)}\)
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
\(E[u(x)]=\)
Subjective expected utility
HARA Hyperbolic Absolute Risk Aversion
CRRA Constant Relative Risk Aversion
Cumulative prospect theory.