Expected Utility Theory

Suppose the space of alternatives has a little more structure: $$ X = X_1 \times X_2 \times \cdots \times X_n. $$

We define Lottery Space on this $X$: Let $\mathcal{L} = \Delta(X)$ = set of lotteries on $X$ (probability distributions).

Expected Utility Representation

Definition. A preference $\succeq$ on lottery space $\mathcal{L}$ admits an expected utility form if $\exists u:X\to\mathbb{R}$ such that $\forall L,L’\in\mathcal{L}$:

  1. $U(L) = \sum_{x\in X} p^L_x u(x)$,
  2. $L\succeq L’ \iff U(L)\geq U(L’)$.

The von Neumann Morgenstein Theorem

Theorem Consider the space of lotteries $\mathcal{L} = \Delta(X)$ over a finite set of outcomes $X$. For any preference relation $\succeq$ on $\mathcal{L}$, iff. it satisfies:

  1. Rationality (completeness + transitivity).

  2. Continuity (note: one can define continuity on different abstract levels. For the problem we are working with right now, the following suffices)

    Continuity Axiom. (VNM version) If $L \succeq L’ \succeq L’’$, then there exists some probability $\alpha \in [0,1]$ such that $$ L’ \sim \alpha L + (1-\alpha)L’’. $$

  3. Independence

    Independence Axiom. $\succeq$ on $\mathcal{L}$ satisfies independence if
    $$ L \succeq L’ \iff \beta L+(1-\beta)L’’ \succeq \beta L’ + (1-\beta)L" $$ for all $L,L’,L’’\in\mathcal{L}$ and $\beta\in(0,1)$.

Then, there exist a utility function $u:X\to\mathbb{R}$ such that preferences are represented by the expected utility: $$ U(x) = \sum_i U_i(x_i) $$

Remark: “expected utility” is also called von Neumann–Morgenstern (VNM) utility because John von Neumann and Oskar Morgenstern were the first to give a rigorous axiomatization of when preferences over lotteries can be represented by the expected value of some numerical utility function.

Linearity

Definition. $U:\mathcal{L}\to\mathbb{R}$ is linear if
$$ U(\beta L+(1-\beta)L’) = \beta U(L)+(1-\beta)U(L’) \quad \forall L,L’, \beta\in[0,1]. $$

Proposition. $U$ has an expected utility form $\iff$ $U$ is linear.

Uniqueness of Bernoulli Utility

Suppose
$$ U(L) = \sum_{x\in X} p^L_x u(x),\qquad \hat U(L) = \sum_{x\in X} p^L_x \hat u(x) $$ both represent the same $\succeq$.

Then $\exists \alpha\in\mathbb{R}, \beta>0$ such that
$$ U = \alpha + \beta \hat U, \qquad u = \alpha + \beta \hat u. $$

Risk Aversion

Let $X$ be a compact subset of $\mathbb{R}$.

Definition. An individual is

  • risk-neutral if $ \forall P: \mathbb{E}_P[u(x)] = u(\mathbb{E}_P[x])$,

  • risk-averse if $ \forall$ nondegenerate $P: \mathbb{E}_P[u(x)] < u(\mathbb{E}_P[x])$,

  • risk-seeking if $ \forall P: \mathbb{E}_P[u(x)] > u(\mathbb{E}_P[x])$.

Mathematically, these risk-attitudes relates to utility function’s convexity/concavity.

(Jensen’s Inequality) Let $f:\mathbb{R}\to\mathbb{R}$. Then

  • $f$ is linear $\iff \forall P: \mathbb{E}_P[f(x)] = f(\mathbb{E}_P[x])$,
  • $f$ is strictly concave $\iff \forall$ nondegenerate $P: \mathbb{E}_P[f(x)] < f(\mathbb{E}_P[x])$,
  • $f$ is strictly convex $\iff \forall P: \mathbb{E}_P[f(x)] > f(\mathbb{E}_P[x])$.

Subtlety: Risk Attitudes and Wealth Utility

Note that when we’re defining risk attitude, the utility function $u(\cdot)$ is de facto, defined relative to a 1-dimensional utility of wealth:

$$ u(w) := \max_{x: p\cdot x \leq w} u(x,p), $$

which is like optimal outcome of the lower-level utility function, implicitly fixing a price vector $p$.

However, if the primitive preferences are defined on alternatives $X=\mathbb{R}^2$, the induced wealth utility (and therefore risk-attitude) can depend on the price system.

$$ \textbf{Example:}\quad u(a,b) = a^{1/2} + b^2 - 10^{10} \cdot \mathbb 1_{\lbrace a>0,b>0\rbrace}. $$

Here, depending on whether the consumer faces $p^1$ or $p^2$, the indirect utility $u(w)$ constructed via

$$ u(w) = \max_{x: p\cdot x \leq w} u(x) $$

may exhibit different risk attitudes.

Takeaway.
Risk aversion is a property of the induced 1-dimensional wealth utility. But since wealth utility itself comes from maximizing over a richer choice set, the observed risk attitude can vary with the environment (prices, feasible allocations), even if the underlying $u(x)$ is fixed.

Measures of Risk Aversion

Let $u:\mathbb{R}\to\mathbb{R}$. The Arrow–Pratt Relative Risk Aversion index:

$$ \begin{align*} \textit{(Absolute)}\quad \Gamma^A(x) & = -\frac{u’’(x)}{u’(x)}\cr \textit{(Relative)}\quad \Gamma^R(x) & = -x\frac{u’’(x)}{u’(x)}\cr \end{align*} $$

Not too much difference between these two notions. For two utilities $u_A,u_B$ $A$ is more risk-averse than $B$ if $\Gamma_A(x) \geq \Gamma_B(x) \quad \forall x$.

Common Utility Forms that is Useful to Know

CARA (Constant Absolute Risk Aversion):

$$ u(x) = -e^{-ax}, \quad r^A(x) = a. $$

CRRA (Constant Relative Risk Aversion):

$$ u(x) = \frac{x^{1-\delta}-1}{1-\delta}, \qquad r^R(x) = \delta. $$

Special case: $\delta\to 1 \Rightarrow u(x)=\ln(x)$.

Comparative Concavity

Definition. Given $f,g:\mathbb{R}\to\mathbb{R}$, say $f$ is more concave than $g$ if $\exists$ concave $h:\mathbb{R}\to\mathbb{R}$ s.t.
$$ f(x) = h(g(x)). $$

Theorem. If $r_A^R(x) \geq r_B^R(x)$ $ \forall x$, then $u_A$ is more concave than $u_B$.