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 six eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, six 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: An individual's preferences for risk and cost would affect the decision process by determining which strategy they would prefer. A risk-averse individual may choose Strategy 2, which guarantees at least 6 unbroken eggs in half of the cases, while a risk-taker may choose Strategy 1 for a chance to get all 12 unbroken eggs. The introduction of additional costs for more trips would introduce a trade-off between minimizing broken eggs and minimizing the cost, further impacting the individual's choice depending on their preferences for risk and cost.

Step-by-step solution

Strategy 1: 12 eggs in one trip

The individual carries all 12 eggs in one trip. There is a 50% chance that all 12 eggs will break. Therefore, the possible outcomes are: - All 12 eggs unbroken (50% probability) - All 12 eggs broken (50% probability) Expected value: (12 unbroken eggs * 0.5) + (0 unbroken eggs * 0.5) = 6 unbroken eggs

Strategy 2: 6 eggs in each of the two trips

The individual carries 6 eggs per trip in two separate trips. Since the 50% chance of breaking applies independently to each trip, we have the following possible outcomes: - 6 eggs unbroken in the first trip, 6 eggs unbroken in the second trip (25% probability) - 6 eggs unbroken in the first trip, all eggs broken in the second trip (25% probability) - All eggs broken in the first trip, 6 eggs unbroken in the second trip (25% probability) - All eggs broken in the first trip, all eggs broken in the second trip (25% probability) Expected value: (12 unbroken eggs * 0.25) + (6 unbroken eggs * 0.25) + (6 unbroken eggs * 0.25) + (0 unbroken eggs * 0.25) = 6 unbroken eggs #b. Utility graph and preferred strategy#

Utility graph

To develop the utility graph, we will plot the utility against the quantity of unbroken eggs for both strategies. The utility is a measure of satisfaction derived from the number of unbroken eggs. A person might consider a higher quantity of unbroken eggs to provide higher utility. Depending on personal preferences, the individual might have different utility functions for the two strategies.

Preferred strategy

Which strategy will be preferable depends on the individual's preference for taking risks, their assessment of the value of having more unbroken eggs, and the shape of their utility function. A risk-averse individual may prefer Strategy 2, as it guarantees at least 6 unbroken eggs in half of the cases, while a risk-taker may prefer Strategy 1 for a chance to get 12 unbroken eggs. The utility function will ultimately determine the preferred strategy. #c. Effect of more trips and additional costs on utility#

More trips

To determine if taking more than two trips would increase utility, we can analyze similar strategies - like dividing the dozen into 3 trips of 4 eggs each or 4 trips of 3 eggs each – and calculating probabilities and expected values. However, with any number of trips, the probability of all eggs breaking during at least one trip increases, which may negatively affect the utility.

Additional costs

If additional trips were costly, the individual would have to weigh the benefits of reducing the probability of breaking the eggs against the increasing cost of making more trips. This would introduce a trade-off between minimizing the number of broken eggs and minimizing the cost, affecting the preferred strategy and potentially reducing the overall utility. The choice would then depend on the individual's preferences for risk and cost.

Key concepts

Expected Value

In microeconomic decision-making, the notion of expected value is paramount, especially when outcomes are uncertain. It is a calculated average value that represents the likelihood of various outcomes, weighted by the probabilities of each outcome occurring. Take the egg-carrying dilemma presented: carrying the entire dozen in one trip provides a clear 50% chance of keeping all the eggs intact. Conversely, splitting the trips introduces more possible outcomes, each with its respective probabilities. Although the setup differs, the expected value calculation for both strategies yields six unbroken eggs, demonstrating that, on average, the expected outcome is the same.

This concept is pivotal because it encapsulates the idea that decision-making can lean on mathematical precision, rather than just intuition. Yet, for those grappling with such exercises, remember that expected value doesn't predict individual events. Rather, it gives an average over a large number of instances. So, if our individual makes this choice repeatedly over time, they would average six unbroken eggs per dozen, but each individual trip might yield a different result.

Utility Theory

Delving into utility theory, we find ourselves at the crossroads of economics and psychology. Here, utility refers to the satisfaction or benefit derived from consuming goods or services. When the textbook solution proposes a utility graph, it's suggesting a visual representation of the joy our individual gets from each unbroken egg.

The graph plots 'utility' on one axis against 'number of unbroken eggs' on the other. It's assumed that more unbroken eggs typically offer more utility, but the rate of this increase is not constant for everyone. This is where 'marginal utility' comes into play, the idea that the utility gained from each additional unbroken egg might decrease after reaching a certain point. Discussing utility is essential to understand why the preferred strategy may differ among individuals. For example, if the utility curve is steep, one big loss might be less preferable even if the expected value remains unaltered—hence, the individual might pick the strategy with lesser variability, like splitting the eggs into multiple trips.

Risk Aversion

Last but not least, the concept of risk aversion nudges its way into every decision involving uncertainty. It describes a person's tendency to prefer a sure outcome over a gamble with a potentially higher, but uncertain, payoff. It essentially measures one's discomfort with uncertainty and potential loss.

In our example, the risk-averse individual may shy away from attempting to carry all 12 eggs in a single go, despite the expected outcome being similar for both strategies. Instead, they may opt for multiple trips, even if it ensures only six unbroken eggs, to avoid the risk of losing them all. Improving the understanding of this concept can often be done by juxtaposing extreme scenarios, where the outcomes would either be very favorable or very unfavorable, and observing the choices made by the individual. It's worth noting that real-life factors, such as the potential cost of additional trips, can complicate the assessment of one's risk aversion. When faced with the potential depletion of resources—be they time, effort, or money—even the moderately risk-averse might reconsider their strategy, searching for that delicate balance between expected outcomes and the risks and costs they are willing to bear.