Question

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}$$(\mathrm{c}) u\left(c_{1}, c_{2}, \pi_{1}, \pi_{2}\right)=\pi_{1} \ln c_{1}+\pi_{2} \ln c_{2}+17\)

Short answer

Options (a) and (c) have the expected utility property.

Step-by-step solution

Understanding Expected Utility Property

A utility function has the expected utility property if it can be represented as a weighted sum of utilities of outcomes, where weights are probabilities. That means a utility function should be of the form \(u(X) = \sum_{i} \pi_i u(c_i)\), where \(\pi_i\) are probabilities and \(c_i\) are outcomes.

Analyze option (a)

The utility function is given by \(u(c_{1}, c_{2}, \pi_{1}, \pi_{2}) = a(\pi_{1} c_{1} + \pi_{2} c_{2})\). This can be rewritten as \(u(c_{1}, c_{2}) = a\sum_{i=1}^{2}(\pi_i c_i)\). The function represents a weighted sum of outcomes \(\pi_1 c_1\) and \(\pi_2 c_2\), thus it has the expected utility property.

Analyze option (b)

The utility function is \(u(c_{1}, c_{2}, \pi_{1}, \pi_{2}) = \pi_{1} c_{1} + \pi_{2} c_{2}^{2}\). Here, the term \(\pi_{2} c_{2}^{2}\) cannot be written in the standard form where outcomes are simply multiplied by probabilities. Since \(c_2\) is squared, it does not hold the expected utility property.

Analyze option (c)

The utility function is \(u(c_{1}, c_{2}, \pi_{1}, \pi_{2}) = \pi_{1} \ln c_{1} + \pi_{2} \ln c_{2} + 17\). This can be rewritten as \(\sum_{i=1}^{2}(\pi_i \ln c_i) + 17\), aligning with the standard form of expected utility because it's a weighted sum of the \(\ln c_i\) terms, thus it has the expected utility property.

Key concepts

Utility Function

A utility function is a mathematical representation of a decision-maker's preferences. It assigns a numerical value to each possible outcome of a decision. These numbers express the utility or satisfaction the decision-maker derives from different choices.
For example, in the context of the given exercise, the utility functions under examination are trying to express how beneficial different combinations of outcomes and their probabilities are. The focus is on expressing these benefits in a way that reflects the decision maker's preferences. This numerical representation helps economists and decision-makers predict choices and analyze behavior under uncertainty.
A critical aspect of utility functions is they serve as the foundation of expected utility theory, which we'll delve into. Understanding utility functions helps in assessing which decision will yield the most satisfaction or utility, given various uncertain outcomes.

Probabilities

Probabilities play an essential role in expected utility theory by serving as weights for different outcomes. They quantify the likelihood of each possible outcome occurring. In simpler terms, probabilities reflect how likely it is for an event to happen compared to other events.
In our exercise, each term within the utility functions is related to a probability \(\pi_i\). This probability is a crucial component of determining expected utility as it helps ensure that outcomes are evaluated correctly. A probability must be a number between 0 and 1 where 0 indicates no chance of the outcome, and 1 means the outcome is certain. The sum of all probabilities in a complete scenario should equal 1.
By using probabilities to weigh different outcomes, we can factor in both the level of satisfaction each outcome provides and the chance of that outcome occurring. This combination helps to establish a clearer expectation or average satisfaction.

Weighted Sum of Utilities

The concept of a weighted sum of utilities forms the cornerstone of expected utility theory. It involves calculating the total utility of a decision by considering both the utility of each possible outcome and its probability. Essentially, you multiply the utility value of each outcome by its probability and then sum all these products.
In mathematical terms, this can be expressed as: \(u(X) = \sum_{i} \pi_i u(c_i)\). Here, \(u(c_i)\) could represent the utility of getting a certain sum of money or experiencing a certain level of happiness if an event occurs, whereas \(\pi_i\) is the probability of that event.
This approach of calculating expected utility ensures that more likely outcomes weigh more heavily in the decision-making process. It helps individuals and businesses choose options that offer the highest expected level of utility, based on their preferences and the probabilities of different events. This concept is crucial in various fields, notably in economics and finance, where decision-making under uncertainty is a common challenge.