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Chapter 8: Problem 3

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?

Short Answer

Expert verified
Answer: Both strategies have an average of 6 unbroken eggs. However, preferences depend on an individual's utility function and their attitudes towards risk. Risk-averse individuals may prefer Strategy 2 (two trips with 6 eggs each), while risk-seeking individuals may prefer Strategy 1 (one trip with 12 eggs).

Step by step solution

01

Strategy 1: One Trip with 12 Eggs

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% (\(0.5\)).
02

Strategy 2: Two Trips with 6 Eggs Each

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% (\(0.5 \times 0.5 = 0.25\)).
03

Average Number of Unbroken Eggs

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decision Making Under Uncertainty
In situations like the egg-carrying exercise, individuals face uncertainty in their decision-making. This uncertainty stems from not knowing which possible outcomes will actually occur. In our case, the egg carrier does not know in advance whether the eggs will break on a given trip.
The process of decision making under uncertainty involves evaluating different strategies by considering possible outcomes and their probabilities. By calculating expected results, individuals can make decisions that maximize their expected utility.
For example, both strategies in the exercise lead to an average of 6 unbroken eggs. However, the choice between them depends on the decision maker鈥檚 attitude towards risk. Effective decision-making under uncertainty requires weighing potential risks against rewards, while considering personal preferences and possible repercussions.
Risk Aversion
Risk aversion describes a person's reluctance to take chances, preferring a safer outcome even if it has a lower potential reward. In scenarios like the egg-carrying problem, risk aversion becomes a key component of decision-making.
If the egg carrier is risk-averse, they might choose Strategy 2. This strategy offers a more conservative result because, although the expected number of unbroken eggs is the same in both strategies, Strategy 2 spreads the risk across two trips.
Risk aversion means minimizing potential negative outcomes. A risk-averse individual favors scenarios with more predictable outcomes, often preferring to avoid scenarios that present the same expected result but with higher variability. In economic terms, this behavior aligns with the idea of maximizing utility given imperfect information.
Probability Analysis
Probability analysis involves calculating the likelihoods of specific outcomes occurring. It is crucial for making informed decisions, particularly under conditions involving uncertainty and risk.
In the egg-carrying example, each trip's probability of breaking all eggs is 0.5. By using probability calculations, different strategies are compared to estimate their expected results.
  • In Strategy 1, there are two potential outcomes with equal probabilities: all eggs unbroken (50%) or all eggs broken (50%).
  • Strategy 2, however, yields four potential outcomes, each with a 25% probability, due to the trips' interdependency.
Analyzing probabilities help in identifying the most likely results, assisting individuals in making rational decisions based on potential risk and reward. Through probability analysis, individuals create a clearer vision for their future actions, reducing the element of surprise in decision outcomes.

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Most popular questions from this chapter

A farmer believes there is a \(50-50\) chance that the next growing season will be abnormally rainy. His expected utility function has the form \\[ \begin{array}{c} \mathbf{1} \\ \text { expected utility }=-\boldsymbol{I} \boldsymbol{n} \boldsymbol{Y}_{N R}+-\boldsymbol{I} \boldsymbol{n} \boldsymbol{Y}_{R} \end{array} \\] where \(Y_{N R}\) and \(Y_{R}\) represent the farmer's income in the states of "normal rain" and "rainy," respectively. a. Suppose the farmer must choose between two crops that promise the following income prospects: $$\begin{array}{lcr} \text { Crop } & Y_{H} & Y_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & 19,000 & 15,000 \end{array}$$ Which of the crops will he plant? b. Suppose the farmer can plant half his field with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance, available to farmers who grow only wheat, which costs \(\$ 4000\) and pays off \(\$ 8000\) in the event of a rainy growing season, cause this farmer to change what he plants?

In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is \(p\) and that the fine for receiving the ticket is / Suppose that all individuals are risk averse (that is, \(U^{\prime \prime}(W)<0\), where Wis the individual's wealth) Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking? \([\text {Hint}\) : Use the Taylor \(\text { series approximation }\left.U(W-f)=U(W)-f U^{\prime}(W)+-U^{\prime \prime}(W) .\right]\)

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 \(1978]: 185-200\).

Investment in risky assets can be examined in the state-preference framework by assuming that \(W^{*}\) dollars invested in an asset with a certain return, \(r,\) will yield \(\mathrm{W}^{*}(\mathrm{l}+r)\) in both states of the world, whereas investment in a risky asset will yield \(\mathrm{W}^{*}\left(1+r_{g}\right)\) in good times and \(\left.\mathrm{W}^{*}\left(\mathrm{l}+r_{b}\right) \text { in bad times (where } r_{g}>r>r_{b}\right)\) a. Graph the outcomes from the two investments. b. Show how a "mxied portfolio" containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals' attitudes toward risk will determine the mix of risk- free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual's utility takes the constant relative risk aversion form (Equation 8.62 ), ex plain why this person will not change the fraction of risky asset held as his or her wealth increases.

Suppose Molly Jock wishes to purchase a high-definition television to watch the Olympic Greco-Roman wrestling competition. Her current income is \(\$ 20,000,\) and she knows where she can buy the television she wants for \(\$ 2,000\). She has heard the rumor that the same set can be bought at Crazy Eddie's (recently out of bankruptcy) for \(\$ 1,700,\) but is unsure if the rumor is true. Suppose this individual's utility is given by \\[\text { utility }=\ln (Y).\\] where Fis her income after buying the television. a. What is Molly's utility if she buys from the location she knows? b. What is Molly's utility if Crazy Eddie's really does offer the lower price? c. Suppose Molly believes there is a \(50-50\) chance that Crazy Eddie does offer the lowerpriced television, but it will cost her \(\$ 100\) to drive to the discount store to find out for sure (the store is far away and has had its phone disconnected). Is it worth it to her to invest the money in the trip?
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