# Lotteries and risk aversion

## Introduction

### Lotteries

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:

\(u(x_i)=x_i\)

If the agent is risk averse:

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

If the agent is risk loving we can use:

\(u(x_i)=x_i^2\)

## Absolute risk aversion

### Absolute risk aversion

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 \(x\).

## Relative risk aversion

### Relative risk aversion

Absolute risk aversion is:

\(A(x)=-\dfrac{u''(x)}{u'(x)}\)

\(R(x)=xA(x)\)

\(R(x)=-\dfrac{xu''(x)}{u'(x)}\)

## Risk

### 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

\(E[u(x)]=\)

Subjective expected utility

### Risk aversion

HARA Hyperbolic Absolute Risk Aversion

CRRA Constant Relative Risk Aversion

### Prospect theory

Cumulative prospect theory.

### Knightian uncertainty

### Bounded rationality