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 of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: Strategy 1: Take all 12 eggs in one trip. Strategy 2: Take two trips with 6 in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that on the 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 possi bility be affected if additional trips were costly?

In this strategy, there are two possible outcomes: 1. All eggs are unbroken, which has a probability of 50% (\(0.5\)). 2. All eggs are broken, which has a probability of 50% (\(0.5\)).

In this strategy, there are four possible outcomes: 1. All eggs are unbroken on both trips, which has a probability of 25% (\(0.5 \times 0.5 = 0.25\)). 2. All eggs are unbroken on the first trip and all broken on the second trip, which has a probability of 25% (\(0.5 \times 0.5 = 0.25\)). 3. All eggs are broken on the first trip and all unbroken on the second trip, which has a probability of 25% (\(0.5 \times 0.5 = 0.25\)). 4. All eggs are broken on both trips, which has a probability of 25% (\(0.5 \times 0.5 = 0.25\)).

For Strategy 1: - \(0.5 \times 12 + 0.5 \times 0 = 6\) unbroken eggs on average. For Strategy 2: - \(0.25 \times 12 + 0.25 \times 6 + 0.25 \times 6 + 0.25 \times 0 = 6\) unbroken eggs on average. Under either strategy, the average number of unbroken eggs is 6. #b. Graph to show the utility under each strategy# Since this is a theoretical exercise, I will describe how to develop a graph instead: 1. On the horizontal axis, represent the number of unbroken eggs ranging from 0 to 12. 2. On the vertical axis, represent the utility. 3. Plot the possible outcomes and their probabilities for each strategy, such as \;(0, 0.5)\; and \;(12, 0.5)\; for Strategy 1 and \;(0, 0.25)\;, \;(6, 0.5)\;, \;(12, 0.25)\; for Strategy 2. The strategy that will be preferable depends on the individual's personal preferences and their utility function. If the individual has a risk-averse attitude, they may prefer Strategy 2 since it offers a 50% chance of having 6 unbroken eggs. If the individual is more risk-seeking, they may prefer Strategy 1 since it has a 50% chance of having all 12 unbroken eggs. The preference can be determined by comparing the different combinations of unbroken eggs and their respective probabilities. #c. Utility improvement with more than two trips# To determine if taking more than two trips could improve utility, we would analyze additional strategies, such as taking three trips with 4 eggs each or four trips with 3 eggs each. We must compute the possible outcomes and probabilities for these new strategies and calculate their average number of unbroken eggs. If additional trips were costly, an individual's utility function would need to take the cost into account. The utility for each strategy would diminish due to the increased cost, shifting the preferences. The individual may prefer taking fewer trips to save costs, even if it means a higher chance of broken eggs. In this case, the individual would need to weigh the additional cost against the benefits of potentially having more unbroken eggs.