Question

In some cases individuals may care about the date at which the uncertainty they face is resolved. Suppose, for example, that an individual knows that his or her consumption will be 10 units today \((Q)\) but that tomorrow's consumption \(\left(\mathrm{C}_{2}\right)\) will be either 10 or \(2.5,\) depending on whether a coin comes up heads or tails. Suppose also that the individual's utility function has the simple Cobb-Douglas form \\[U\left(Q, C_{2}\right)=V^{\wedge} Q.\\] a. If an individual cares only about the expected value of utility, will it matter whether the coin is flipped just before day 1 or just before day \(2 ?\) Explain. More generally, suppose that the individual's expected utility depends on the timing of the coin flip. Specifically, assume that \\[\text { expected utility }=\mathbf{f}^{\wedge}\left[\left(\mathrm{fi}\left\\{17\left(\mathrm{C}, \mathrm{C}_{2}\right)\right\\}\right) \text { ) } \text { ? } |\right.\\] where \(E_{x}\) represents expectations taken at the start of day \(1, E_{2}\) represents expectations at the start of day \(2,\) and \(a\) represents a parameter that indicates timing preferences. Show that if \(a=1,\) the individual is indifferent about when the coin is flipped. Show that if \(a=2,\) the individual will prefer early resolution of the uncertainty - that is, flipping the coin at the start of day 1 Show that if \(a=.5,\) the individual will prefer later resolution of the uncertainty (flipping at the start of day 2 ). Explain your results intuitively and indicate their relevance for information theory. (Note: This problem is an illustration of "resolution seeking" and "resolution-averse" behavior. See D. M. Kreps and E. L. Porteus, "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica [January \(1978]: 185-200\).

Short answer

If yes, how do preferences for timing of resolving uncertainty affect an individual's desire for when the coin is flipped? Answer: Yes, the expected utilities for both scenarios are the same. Preferences for timing of resolving uncertainty affect an individual's desire for when the coin is flipped, as some individuals may be risk-averse and prefer early resolution (flipping at the start of day 1), while others may be more risk-seeking and prefer later resolution of uncertainty (flipping at the start of day 2).

Step-by-step solution

State the utility function and expected utility representation

The utility function is given by \\[U(Q, C_2) = Q^{\wedge} C_2.\\] The expected utility representation is given by \\[\text { expected utility } = \textbf{f}^{\wedge}\left[\left(\mathrm{fi}\left\\{17\left(\mathrm{C},\mathrm{C}_{2}\right)\right\\}\right) \text { ) } \text { ? } |\right.\\] where \(E_{x}\) represents expectations taken at the start of day 1, \(E_{2}\) represents expectations at the start of day 2, and \(a\) represents a parameter that indicates timing preferences.

Compute the expected utility for both coin flip scenarios

For each scenario, we need to compute the expected utility when consumption is 10 or 2.5. When the coin is flipped just before day 1: \\[E_{x}[U(Q, C_2)] = \frac{1}{2} U(10, 10) + \frac{1}{2} U(10, 2.5)\\] \\[E_{x}[U(Q, C_2)] = \frac{1}{2}(10^{\wedge} 10) + \frac{1}{2}(10^{\wedge} 2.5)\\] When the coin is flipped just before day 2: \\[E_{2}[U(Q, C_2)] = \frac{1}{2} U(10, 10) + \frac{1}{2} U(10, 2.5)\\] \\[E_{2}[U(Q, C_2)] = \frac{1}{2}(10^{\wedge} 10) + \frac{1}{2}(10^{\wedge} 2.5)\\] The expected utility for both scenarios is the same.

Determine the individual's preference for coin flip timing, based on the value of 'a'

For \(a = 1\), the individual is indifferent about when the coin is flipped because the expected utilities are identical. For \(a = 2\), the individual will prefer early resolution of the uncertainty (flipping the coin at the start of day 1). For \(a = .5\), the individual will prefer later resolution of the uncertainty (flipping at the start of day 2).

Explain the results intuitively and indicate their relevance for information theory

Intuitively, if an individual's utility only depends on the expected value of consumption, then it doesn't matter when the coin is flipped. However, as preferences for the timing of resolving uncertainty vary, so do preferences for when the coin is flipped. In information theory, this concept is relevant as it highlights the importance of the timing of information and how different individuals will place different value on when uncertainty is resolved. Depending on preferences, some individuals may be risk-averse and prefer early resolution, while others may be more risk-seeking and prefer later resolution of uncertainty.

Key concepts

Expected Utility

The concept of expected utility is central to understanding decision-making under uncertainty. It captures the idea that when faced with various outcomes, each with a certain probability, individuals will opt for the option that offers the highest expected satisfaction, or utility. To calculate expected utility, you multiply the utility of each possible outcome by its probability, and then sum these products.

Referring to our exercise, the individual is evaluating the uncertainty of tomorrow’s consumption and is looking to maximize their expected utility. Whether to flip the coin before day 1 or day 2 becomes a question of when they prefer to deal with uncertainty. The calculated expected utility for both scenarios was identical, meaning if the individual were only concerned with expected utility, they would be indifferent to the timing of the coin flip. However, preference over timing indicates that factors beyond the expected utility, such as risk preferences and the psychological value of information, are in play.

Cobb-Douglas Utility Function

The Cobb-Douglas utility function is a widely used form to represent preferences in economics. It is characterized by a simply multiplicative form, often used to model the utility that a consumer gets from consuming different goods. In our particular problem, the Cobb-Douglas utility function is represented as a function of two consumption goods, today's and tomorrow's consumption (Q and C₂).

The utility from consumption for each day is expressed as the product of the consumption amounts raised to a specific power, typically reflecting the proportion of expenditure on each good. This form captures the trade-offs consumers are willing to make between goods and has an implication of constant elasticity of substitution, which in simpler terms means individuals are able to substitute one good for another to some extent without a change in the level of happiness or utility.

Risk Preferences

The term risk preferences refers to an individual's tolerance for risk and uncertainty in their choices. In the context of our exercise, the risk preferences of the individual are revealed through the timing of the coin flipping that determines consumption. If the individual prefers to flip the coin earlier (a = 2), they are demonstrating a preference for early resolution of risk, which aligns with risk-averse behavior. Conversely, a preference for flipping the coin later (a = 0.5) indicates a tolerance for uncertainty, often associated with risk-seeking behavior.

It's important to realize that each person's risk preferences are unique and can greatly affect their decision-making process, especially in economic contexts where uncertainty is prevalent. The utility function and the concept of expected utility help to model these preferences in a way that we can analyze and predict behavior.

Information Theory

Lastly, information theory is a field that examines the processing, transmission, and communication of information. In economic decisions, the timing and resolution of uncertainty are relevant topics within information theory. The individuals’ preferences for when they would like to resolve uncertainty relates closely to the value of information and how it is processed.

Our exercise demonstrates that the value of information is not just about the content of what is known, but also about when it is known. Depending on a person's risk preferences, the timing of resolving uncertainty can alter their perceived utility. In information theory, this is important because it asserts that the timing of information can be just as valuable as the information itself. For instance, knowing the outcome of a risky event ahead of time can reduce anxiety and lead to better decision-making, which is directly correlated with the individual's utility and preferences.