Risk and Expected Utility

This app illustrates expected utility with binary lotteries

You can:

- visualize the shape of the utility function and the value of certainty equivalent (Panel Utility)

- compute several quantities: certainty equivalent, risk premium, willingess to pay for the lottery... (Panel Quantities)

- visualize the shape of marginal utility and the certainty equivalent (Panel Marginal Utility)

- visualize the shape of the indifference curves in the Hirshleifer-Yaari diagram (Panel Indifference curves)

Modifying sliders allows you:

- to choose the characteristics of the lottery: probability, consequences

- to choose the utility function: CRRA or CARA

- to choose the utility function parameter

Power/CRRA: $$u(x)=\frac{x^{1-\theta}}{1-\theta} \text{ if } \theta \neq 1, u(x)=log(x) \text{ if }\theta=1$$ Expo/CARA: $$u(x)=\frac{1-exp(-\theta x)}{\theta} \text{ if } \theta \neq 0, u(x)=x \text{ if }\theta=0$$

xc denotes the certainty equivalent (see Panel Quantities for evaluation)

The certainty equivalent is such that:

$$u(x_c)=\mathbb{E}[u(\tilde{x})]$$

The risk premium is such that:

$$u(\mathbb{E}[\tilde{x}]-\Pi)=\mathbb{E}[u(\tilde{x})]$$

The prudence premium is such that:

$$u'(\mathbb{E}[\tilde{x}]-\Psi)=\mathbb{E}[u'(\tilde{x})]$$

xp denotes the certainty equivalent on marginal utility

Hirshleifer-Yaari diagram