Question

Lisa just inherited a vineyard from a distant relative. In good years (no rain or frost), she earns $10,000 from the vineyard. In bad years, she earns only $2,500. She estimates that the probability of a good year is 60%.  

a) Calculate the expected value of her earnings from the vineyard.

b) Suppose Lisa has a utility function U(w)= pw, where w is her wealth. Assume that she has 0 initial wealth. Ethan, a grape buyer, offers to lease the vineyard from Lisa for $6,500 next year, so that Lisa would get $6,500 regardless of whether it was a good year or a bad year. Will Lisa accept this over?

Answer 1

Answer 2

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Step-by-Step Calculation:

Part (a): Expected Earnings

• Good year (60% chance): $10,000

• Bad year (40% chance): $2,500

Formula:

$$\text{Expected Value}=(\text{Probability of Good Year}\times \text{Good Earnings})+(\text{Probability of Bad Year}\times \text{Bad Earnings})$$$$=(0.6\times \mathrm{\$}10,000)+(0.4\times \mathrm{\$}2,500)=\mathrm{\$}6,000+\mathrm{\$}1,000=\mathrm{\$}7,000$$

Conclusion: Lisa’s average expected earnings are $7,000/year.

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Part (b): Should Lisa Accept Ethan’s Offer?

Lisa’s utility function is $U(w)=\sqrt{w}$. She compares:

1. Risking the vineyard:

   $$=0.6\times \sqrt{\mathrm{\$}10,000}+0.4\times \sqrt{\mathrm{\$}2,500}=0.6\times 100+0.4\times 50=60+20=80$$

• Expected utility = Utility of each outcome × Probability.

2. Leasing to Ethan (guaranteed $6,500):

   $$\text{Utility}=\sqrt{\mathrm{\$}6,500}\approx 80.62$$

Comparison:

• Risking it: Utility = 80

• Leasing: Utility ≈ 80.62

Final Decision: She accepts Ethan’s offer!