Which of the following utility functions have the expected...

Which of the following utility functions have the expected utility property? (a) $u\left(c_{1}, c_{2}, \pi_{1}, \pi_{2}\right)=a\left(\pi_{1} c_{1}+\pi_{2} c_{2}\right)$ (b) $u\left(c_{1}, c_{2}, \pi_{1}, \pi_{2}\right)=\pi_{1} c_{1}+$ $\pi_{2} c_{2}^{2}$ (c) $u\left(c_{1}, c_{2}, \pi_{1}, \pi_{2}\right)=\pi_{1} \ln c_{1}+\pi_{2} \ln c_{2}+17$.

Which of the following utility functions have the expected utility property?
(a) $u\left(c_{1}, c_{2}, \pi_{1}, \pi_{2}\right)=a\left(\pi_{1} c_{1}+\pi_{2} c_{2}\right)$
(b) $u\left(c_{1}, c_{2}, \pi_{1}, \pi_{2}\right)=\pi_{1} c_{1}+$
$\pi_{2} c_{2}^{2}$
(c) $u\left(c_{1}, c_{2}, \pi_{1}, \pi_{2}\right)=\pi_{1} \ln c_{1}+\pi_{2} \ln c_{2}+17$.

Key Concepts

Expected Utility Theory

Expected utility theory is a framework for modeling decision making under risk. It asserts that a decision maker evaluates risky prospects by taking the weighted average of the utilities of outcomes, with the weights being the probabilities of those outcomes. This theory requires that the overall utility is calculated in a way that is linear in the probabilities of various outcomes.

Linearity in Probabilities

A key property of any expected utility representation is linearity in probabilities. This means that if a utility function can be expressed as a sum of terms where each term is the product of a probability and a function of an outcome, then the decision maker’s evaluation of uncertain prospects is simply the expected value of the outcome utilities. This linear combination is crucial for satisfying the independence axiom of expected utility theory.

Additive Separability

Additive separability refers to the ability to decompose a utility function into components that depend solely on the outcomes in each state, with each component being weighted by the respective state’s probability. When a utility function takes the form of a sum of terms like ??v(c?) + ??v(c?), it indicates that the evaluation of a lottery is independent across states, allowing the overall utility to be determined by summing the utilities from each state.

Affine Transformation Invariance

Within expected utility theory, preferences are preserved under positive affine transformations of the utility function. This means that if a utility function u is transformed into a function of the form au + b, where a is a positive constant, the ordering of preferences over risky options remains unchanged. This property is important because it implies that the exact numerical form of the utility function is not unique, as long as it represents the same ordering of choices.

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