Question

An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take two trips with 6 eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly?

Short answer

Answer: Strategy 2 is preferable if the primary goal is to minimize the risk of breaking all the eggs. This is because it has a more diverse range of possible unbroken eggs (0, 6, or 12) and distributes the eggs across two trips, reducing the probability of all eggs breaking in a single trip compared to Strategy 1, which only has two possible outcomes (0 or 12 unbroken eggs).

Step-by-step solution

Calculate the probability of each outcome for Strategy 1

For the first strategy, we need to find the probabilities of having 0, 12, or any number in between unbroken eggs. There are two possible outcomes: either all eggs will remain unbroken or all will break during the trip. The probability of all eggs remaining unbroken is 50% (1/2), and the probability of all eggs breaking is also 50% (1/2).

Calculate the probability of each outcome for Strategy 2

For the second strategy, the person makes two trips with 6 eggs each. In each trip, either all eggs will remain unbroken (50%) or all will break (50%). We have the following possible combinations of outcomes for each trip: - Both trips are successful, and all eggs are unbroken (probability: 0.5 * 0.5 = 0.25) - The first trip is successful, but the second is not (probability: 0.5 * 0.5 = 0.25) - The second trip is successful, but the first is not (probability: 0.5 * 0.5 = 0.25) - Both trips are unsuccessful, and all eggs are broken (probability: 0.5 * 0.5 = 0.25)

Calculate the expected number of unbroken eggs for each strategy

For the first strategy, the expected number of unbroken eggs is: E(Strategy 1) = 12 eggs * 0.5 + 0 eggs * 0.5 = 6 eggs For the second strategy, we calculate the expected number of unbroken eggs as: E(Strategy 2) = 12 eggs * 0.25 + 6 eggs * 0.25 + 6 eggs * 0.25 + 0 eggs * 0.25 = 6 eggs Thus, on average, 6 eggs will remain unbroken under either strategy.

Develop a graph to show utility under each strategy

To develop a graph, we need to represent the different outcomes and their probabilities on the x and y-axes. For example, for Strategy 1, we can plot a bar for the probability of having 0 or 12 unbroken eggs. Similarly, for Strategy 2, we can plot bars for the probabilities of having 0, 6, or 12 unbroken eggs. By comparing the heights of the bars, we can determine which strategy will be preferable.

Identify the preferable strategy

Examining the graph, we can see that both strategies result in an expected outcome of 6 unbroken eggs. However, Strategy 2 may be preferable as it has a more diverse range of possible unbroken eggs (0, 6, or 12), while Strategy 1 may not be suitable for those who cannot afford risking all eggs breaking in one trip.

Discuss the effect of additional trips on utility

Increasing the number of trips may further improve utility by distributing eggs across trips and decreasing the probability of all eggs breaking in a single trip. However, if additional trips were costly, the individual might have to trade off between the benefit of increased utility and the cost of more trips. The optimal number of trips would then depend on the individual's preferences, risks, and costs involved in the decision-making process, as well as the margin of increase in utility for every additional trip.

Key concepts

Expected Utility Theory

Expected Utility Theory is a cornerstone of microeconomics, particularly when examining how individuals make decisions under uncertainty. Its central tenet is to marry the notions of probability with the utility derived from different outcomes. Instead of focusing merely on the most likely outcome, the theory suggests that individuals consider all possible outcomes weighted by their respective probabilities and the utility that each outcome provides. For example, an individual may be faced with a gamble that could either lead to a substantial payoff or a significant loss. According to expected utility theory, the individual should calculate the 'expected utility' of each scenario by multiplying the utility of the outcome with its probability and then adding these figures together.

In the context of our egg-carrying problem, to determine which strategy is preferable – carrying all 12 eggs in one trip or dividing them into two trips – one should calculate the expected utility rather than the expected number of unbroken eggs alone. This involves considering the level of satisfaction (utility) from having a certain number of eggs unbroken after the trip. If the fear of losing all eggs at once is overwhelmingly dissatisfying, carrying eggs over two trips might yield a higher expected utility, despite the expected number of unbroken eggs being the same for both strategies.

Risk and Decision-Making

Risk and decision-making are intimately connected in the economic analysis of human behavior. When faced with uncertainty, individuals are often called to make decisions that involve risk. The question then becomes: how much risk are they willing to accept? In microeconomics, this risk preference is analyzed through the lens of an individual's utility function—in other words, how much happiness or satisfaction they derive from different outcomes.

Some individuals may be risk-averse, preferring more certain yet potentially less rewarding outcomes to avoid the possibility of a loss, while others may be risk-seeking, willing to gamble for the chance of a larger payoff despite the risk of greater loss. Even those who are neutral to risk will have their decisions influenced by the probabilities and the utility outcomes associated with their options.

Understanding Risk Preference

For the individual deciding how to carry the eggs home, risk preference plays a crucial role. If each broken egg brings great unhappiness and diminishes utility significantly, the individual may opt for multiple trips to minimize the chance of breaking all eggs. On the other hand, if their utility is not greatly affected by the loss, they may risk carrying all eggs in one trip.

Probability Outcomes

Probability outcomes are the foundation of assessing risks and benefits in various scenarios. They quantify the likelihood of different events occurring and allow individuals to plan accordingly. In a microeconomic setting, understanding probability outcomes is crucial for predicting the behavior of economic agents. Each potential event is assigned a probability value between 0 and 1, with 0 indicating impossibility and 1 signifying certainty.

In our exercise, we analyze probability outcomes in the context of carrying eggs. By looking at each potential outcome and its respective probability, we can deduce the average number of unbroken eggs after each strategy. This approach does not only provide numerical expectations for outcomes but also shapes the decision-making process when combined with the individual's utility preference.

Simplifying Complexity

By breaking down complex situations into a series of simple probability outcomes, individuals can make more informed and rational decisions. For instance, considering the cost of additional trips home and their diminishing returns might influence the decision about how many trips to take to transport the eggs. Balancing these probabilities with potential utility gains or losses helps ensure that the decision is both economically and personally rational.