91Ó°ÊÓ

Perry the picker has stumbled across a piece of pottery in an antique shop. Because of its style, he believes it might be by Frog Woman, a famous Hopi artist. If he buys the pot for \(\$ 3,000\), he will be able to resell it for \(\$ 4,500\), provided it is a genuine Frog Woman pot. If it is not genuine, he will be forced to unload it for only \(\$ 1,000\). Perry estimates that there is a $$2 / 3$$ chance the pot is genuine. a. If Pemy bases his decision on expected monetary value, should he buy the pot? b. Perry's utility depends on his wealth: Specifically, \(U=W^{1 / 4}\) Compute Perry's utility at the possible levels of final wealth he might experience. c. If Perry bases his decision to buy on expected utility, what should he do? d. Suppose Perry spends a little bit too much time in the hotel bar, and buys the pot without doing the math first. At breakfast the next moming, Penny, another picker, notices the pot and decides to make an offer for it. What is the minimum Penny would have to offer to convince Perry to sell. (Remember - Perry still doesn't know whether the pot is genuine; an offer from Penny removes all risk.) e. What is Perry's risk premium?

Step by step solution

01

Calculate Expected Monetary Value (EMV)

Determine the expected monetary value for Perry based on the probabilities and potential resale values provided. The formula is: \( \text{EMV} = (\text{Probability of Genuine}) \times (\text{Resale Value if Genuine}) + (\text{Probability of Not Genuine}) \times (\text{Resale Value if Not Genuine}) \). Substitute the given values: \( \text{EMV} = \left( \frac{2}{3} \right) \times 4500 + \left( \frac{1}{3} \right) \times 1000 \). Calculating gives: \( \text{EMV} = 3000 + 333.33 = 3333.33 \). Since the EMV of 3333.33 is greater than the purchase cost of 3000, Perry should buy the pot based on EMV.

02

Compute Perry's Utility for Different Wealth Levels

Calculate Perry's utility for each possible outcome based on his utility function \( U = W^{1/4} \). For wealth when genuine: \( U = (W_i + 4500 - 3000)^{1/4} \) and when not genuine: \( U = (W_i + 1000 - 3000)^{1/4} \). Without initial wealth \( W_i \), simplify as for genuine \( U = 1500^{1/4} \approx 5.27864 \), and not genuine \( U = -2000^{1/4} \), which is not defined hence stop at zero.

03

Calculate Expected Utility

Find the expected utility \( EU \) using \( EU = \frac{2}{3} \times 1500^{1/4} + \frac{1}{3} \times U_{not genuine} \). Since when not genuine, \( U = 0 \), the expected utility simplifies to \( EU = \frac{2}{3} \times 5.27864 + \frac{1}{3} \times 0 = 3.519 \approx 3.52 \). Compare with utility of keeping the money: \( 3000^{1/4} \approx 8.18 \). Since 3.52 < 8.18, based on expected utility Perry should not buy the pot.

04

Determine Minimum Offer from Penny

Perry should accept an offer that provides utility equal to the expected utility if he kept the money. Thus, solve \( U = W_{min}^{1/4} = 3.52 \). Solving for \( W_{min} = 3.52^4 \), gives approximately \( W_{min} = 153.76 \). Add original purchase loss: \( 3000 + 153.76 = 3153.76 \). Penny should offer a minimum of 3153.76.

05

Calculate Perry's Risk Premium

The risk premium is the difference between the expected monetary value and the certainty equivalent (minimum offer from Penny after adjusted for zero risk). \( RP = 3333.33 - 3153.76 = 179.57 \). The risk premium Perry derives for purchasing the pot is 179.57.

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Key Concepts

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

Risk Premium

The risk premium is an important concept when making decisions under uncertainty, especially within the framework of Expected Utility Theory. It represents the amount of money that a person is willing to forgo in order to avoid risk. In simpler terms, it's the difference between the expected monetary value (EMV) of a risky venture and the amount a person would accept if offered a risk-free alternative.

For Perry, the pot's expected monetary value is calculated to be \(3333.33\) dollars. However, when considering the minimal offer he would accept from another picker to eliminate all risk, it turns out to be \(3153.76\) dollars. This difference of \(179.57\) dollars is Perry's risk premium. It essentially quantifies how much he values certainty over the potential fluctuations of outcome.

In practical terms, understanding the risk premium allows individuals to make informed choices by weighing the benefits of potentially higher, yet uncertain, returns against the appeal of a more predictable, guaranteed payoff.

Expected Monetary Value

Expected Monetary Value (EMV) is a fundamental tool used to evaluate decisions involving uncertainty. It provides a way to compute the average outcome of a decision when there are various possible outcomes each with their associated probabilities.

For Perry's decision on whether to buy the pottery, EMV is determined by considering the possible financial outcome if the pot is a genuine Frog Woman piece and if it's not. The formula used is:

• EMV = (Probability of Genuine) \(\times\) (Resale Value if Genuine) + (Probability of Not Genuine) \(\times\) (Resale Value if Not Genuine)

Plugging in Perry's data:

• EMV = \(\left( \frac{2}{3} \right) \times 4500 + \left( \frac{1}{3} \right) \times 1000 = 3000 + 333.33 = 3333.33\)

Since this EMV is higher than the purchase price of \(3000\) dollars, EMV suggests that Perry should proceed with the purchase if he bases his decision on potential monetary gain. This calculation highlights how EMV objectively informs whether potential financial gains justify the purchase price.

Utility Function

A utility function measures the satisfaction or happiness derived from different levels of wealth or consumption, reflecting an individual's preference under uncertainty. Perry's utility function is given by \(U = W^{1/4}\), implying that as wealth increases, his utility increases at a diminishing rate.

To analyze Perry's decisions using utility, we assess his satisfaction from possible outcomes:

- For the genuine pot sale, his utility becomes \(U = 1500^{1/4} \approx 5.27864\) when considering initial and resale values.
- For a non-genuine pot, however, the utility calculation becomes problematic, but ultimately simplifies to zero because negative values are undefined in this context.

This utility approach helps Perry evaluate his decision based not on just potential monetary gains but also on his personal satisfaction tied to his final wealth. It recognizes that financial decisions are not always linear and highlights how wealth increments can affect personal utility differently.

Decision Making under Uncertainty

Making choices when facing uncertain outcomes is a common and challenging aspect of decision-making in economics. Perry faces this challenge with his potential pottery purchase.

With the use of Expected Utility Theory, Perry evaluates his options by considering both potential monetary outcomes and his personal utility derived from these outcomes. In uncertain environments, decision-making involves balancing potential benefits against risks, often requiring one to decide between maximizing financial gain and ensuring personal satisfaction.

The process involves the use of:

• Expected Monetary Value (EMV): to assess potential financial outcomes.
• Utility Function: to account for personal satisfaction and diminishing returns of increased wealth.
• Risk Premium: to determine how much certainty is valued over potential risk.

Using these elements, Perry realizes that while his EMV suggests buying the pot, the utility approach discourages it due to a lower expected utility compared to holding onto his money. It underscores that decision-making isn’t just about potential profits, but also personal risk tolerance and satisfaction from outcomes.