Economic Behavior Under Uncertainty

Abstract

The chapter explains the expected utility hypothesis as a useful theory of decisions under uncertainty in traditional economics. There have been paradoxes, found in experiments and hypothetical surveys that cannot possibly be explained by the expected utility hypothesis.

Appendices

Appendix 1: Axioms for Expected Utility Theory

This Appendix explains axioms for expressing a preference ordering by an expected utility function. Consider a lottery that gives x dollars (or x units of a good) with probability p and y dollars with probability (1 − p). Let this lottery denoted by $\left( {x,p;y,1 - p} \right)$, p be a real number in [0, 1], and Y be a set of all lotteries. When we mix two lotteries $\left( {x,p;y,1 - p} \right)$ and $\left( {x,r;y,1 - r} \right)$ by a lottery that gives the former lottery with probability q and the latter lottery with probability $\left( {1 - q} \right)$, the mixed lottery is assumed to be the same as $\left( {x,qp + \left( {1 - q} \right)r;y,q\left( {1 - p} \right) + \left( {1 - q} \right)\left( {1 - r} \right)} \right)$. An individual’s preference ordering over the lotteries is denoted by ≿. We write $\left( {x,p;y,1 - p} \right) {\succsim} \left( {x,r;y,1 - r} \right)$ when the individual prefers to $\left( {x,p;y,1 - p} \right)$ to $\left( {x,r;y,1 - r} \right)$ or is indifferent between them. If $\left( {x,p;y,1 - p} \right) {\succsim} \left( {x,r;y,1 - r} \right)$ and $(x,p;y,1 - p){ \precsim }\left( {x,r;y,1 - r} \right)$, then he is indifferent between the two lotteries, and we write $\left( {x,p;y,1 - p} \right) \sim \left( {x,r;y,1 - r} \right)$. If $\left( {x,p;y,1 - p} \right) {\succsim} \left( {x,r;y,1 - r} \right)$, but $\left( {x,p;y,1 - p} \right) { \precsim } \left( {x,r;y,1 - r} \right)$ does not hold, then $\left( {x,p;y,1 - p} \right)$ is strictly preferred, and we write $\left( {x,p;y,1 - p} \right) > \left( {x,r;y,1 - r} \right)$.

• The four axioms of expected utility theory are:

Axiom 1: (Completeness): For any two lotteries $\left( {x,p;y,1 - p} \right),\left( {x,q;y,1 - q} \right)$, either $\left( {x,p;y,1 - p} \right) {\succsim} \left( {x,q;y,1 - q} \right)$ or $\left( {x,p;y,1 - p} \right) { \precsim } \left( {x,q;y,1 - q} \right)$ holds.

Axiom 2 (Transitivity): For any three lotteries $\left( {x,p;y,1 - p} \right),\left( {x,q;y,1 - q} \right)$, if $\left( {x,r; y, 1 - r} \right), if \left( {x,p; y, 1 - p} \right) {\succsim} \left( {x,q;y,1 - q} \right)$ and $\left( {x,q;y,1 - q} \right) {\succsim} \left( {x,r;y,1 - r} \right)$, then $\left( {x,p;y,1 - r} \right) {\succsim} \left( {x,q;y,1 - r} \right)$.

Axiom 3 (Independence): For any two prizes x and y, if $x {\succsim} y$, then for any prize z and for any probability $p\left( {0 \le p \le 1} \right),\left( {x,p;z,\left( {1 - p} \right)} \right) {\succsim} \left( {y,p;z,\left( {1 - p} \right)} \right)$.

Axiom 4 (Continuity): for any three prizes x, y, z, if $x {\succsim} y {\succsim} z$, then there exists a probability p such that $\left( {x,p;z,1 - p} \right) \sim y$.

According to the expected utility theorem under these four axioms, there exists a real valued function U on Y with the following two properties:

• 1. For any two lotteries L, M, a necessary and sufficient condition for $L {\succsim} M$ is $U\left( L \right) \ge U\left( M \right)$.

• 2. Let $U\left( {\left( {x,1} \right)} \right) = u\left( x \right)$. Then $U\left( {\left( {x,p; y, 1 - p} \right)} \right) = pu\left( x \right) + \left( {1 - p} \right)u\left( y \right)$.

Here, by Property (2), U is an expected utility. By Property (1), this expected utility represents the individual’s preference ordering.

Appendix 2: Properties of the Measure of Absolute Risk Aversion

This appendix explains two propositions regarding properties of the measure of absolute risk aversion. As in the text, we consider a lottery (E(X) + h, 0.5; E(X) − h, 0.5) with $E\left( X \right) = 0$. Because risk premium depends on the values of e and h, we write it as a function $\rho \left( {e,h} \right)$.

Proposition 3.1

Given e, if h is sufficiently small, then

$$R\left( e \right) \simeq \frac{2}{{h^{2} }}\rho \left( {e,h} \right)$$

This is a special case of Pratt’s (1964) Eq. (5).¹⁰ According to Proposition 3.1, the measure of absolute risk aversion is approximately positively proportional to risk premium for small h. Therefore, an individual with a higher measure of absolute risk aversion requires a higher risk premium for a lottery. Since this property holds for small h, it is a local property of the measure of absolute risk aversion.

Global properties that hold for any h as long as $z - h > 0$ are also known. Let $i = 1,2$ denote two individuals. For an individual with utility function $u_{i} \left( z \right)$. Let ${\text{R}}_{\text{i}} \left( {\text{z}} \right)$ be her measure of absolute risk aversion and $\uprho_{\text{i}} \left( {{\text{z}},{\text{h}}} \right)$ be her risk premium.

Proposition 3.2

The following two conditions are equivalent.

• i.
  $$R_{a1} \left( z \right) > R_{a2} \left( z \right)$$

• ii. For all he, $\rho_{1} \left( {z,h} \right) > \rho_{2} \left( {z,h} \right)$

This proposition is a special case of part of Pratt’s (1964) Theorem 1.¹¹ According to this proposition, an individual with a higher measure of absolute risk aversion requires a higher risk premium for a lottery. Conversely, an individual who requires a higher risk premium has a higher measure of absolute risk aversion.

Appendix 3: Properties of the Measure of Relative Risk Aversion

This appendix explains two propositions regarding properties of the measure of relative risk aversion. As in the text, we consider a lottery $\left( {E\left( X \right) + eh^{*} ,0.5;E\left( X \right) - eh^{*} ,0.5} \right)$ with $E\left( X \right) = 0$ when endowment e is given. Because relative risk premium depends on the values of e and $h^{*}$, we write it as a function $\rho^{*} \left( {e,h^{*} } \right)$.

Proposition 3.3

Given e, if $h^{*}$ is sufficiently small, then

$$R^{ *} \left( e \right) \simeq \frac{2}{{h^{*2} }}\rho^{*} \left( {e,h^{*} } \right)$$

This is a special case of Pratt’s (1964) Eq. (42).¹² According to Proposition 3.3, the measure of relative risk aversion is approximately positively proportional to relative risk premium for small $h^{*}$. Therefore, an individual with a higher measure of relative risk aversion requires a higher risk premium for a lottery. Because this property holds for small $h^{*}$, it is a local property of the measure of relative risk aversion.

Global properties that hold for any $h^{*}$ as long as $z - eh^{*} > 0$ are also known. Let i = 1, 2 denote two individuals. For an individual with utility function $u_{i} \left( z \right)$, let $R_{i}^{*} \left( z \right)$ be her measure of relative risk aversion and $\rho_{1}^{*} \left( {z,h^{*} } \right)$ be her risk premium.

Proposition 3.4

The following two conditions are equivalent.

• i.
  $$R_{1}^{*} \left( z \right) > R_{2}^{*} \left( z \right)$$

• ii. For all $h^{*} ,\rho_{1}^{*} \left( {z,h^{*} } \right) > \rho_{2}^{*} \left( {z,h^{*} } \right)$

This proposition is a special case of part of Pratt’s (1964) Theorem 6.¹³ According to this proposition, an individual with a higher measure of relative risk aversion requires a higher relative risk premium for a lottery. Conversely, an individual who requires a higher relative risk premium has a higher measure of relative risk aversion.