Sarah has the utility function U(x) = 1 - 1/x, where x is the present value of her lifetime income. Sarah is trying to select a career. If she goes into teaching, she will make x=5 with certainty. If she pursues acting, she will make x = 400 if successful or x = 2 if unsuccessful (and therefore ends up waiting tables). The chance of succeeding in acting is 1% if she pursues acting.

Question

a. Determine which career Sarah will choose. Is she choosing the career with the higher expected value? Explain.
b. An acting career expert charges 0.01 to determine if a person will succeed at acting. By going to an expert, Sarah can choose the best career according to her skills. Assuming that the expert is able to correctly determine if Sarah will be a successful actor, will she pay for this service?

Answer 1

To solve Sarah's career choice problem, we will need to calculate the expected utility for both options and then analyze her decision-making process.

Part a: Determine which career Sarah will choose.

Teaching Career:

• Income (x) = 5
• Utility: \( U(5) = 1 - \frac{1}{5} = 1 - 0.2 = 0.8 \)

Acting Career:

• Income if successful (with probability 1%) = 400
• Income if unsuccessful (with probability 99%) = 2

Expected Income for Acting: \[ E(x) = 0.01 \times 400 + 0.99 \times 2 = 4 + 1.98 = 5.98 \]

Expected Utility for Acting: First, we need to calculate the utility for the expected income: \[ U(5.98) = 1 - \frac{1}{5.98} \] Calculating \( U(5.98) \): \[ U(5.98) \approx 1 - 0.167 = 0.833 \]

Comparison:

• Utility from teaching: \( U(5) = 0.8 \)
• Expected utility from acting: \( U(5.98) \approx 0.833 \)

Conclusion for Part a: Sarah would prefer to pursue acting since the expected utility from acting (\(0.833\)) is higher than the utility from teaching (\(0.8\)).

Is she choosing the career with the higher expected value?

While Sarah's expected income from acting (5.98) is indeed higher than the certain income of 5 from teaching, since she chooses based on expected utility rather than expected value, she is not necessarily choosing the career with the higher expected value (which is 5.98 vs. 5), but rather the one with higher expected utility.

Part b: Should Sarah pay the expert?

Assuming the expert can accurately predict whether Sarah will succeed in acting, we can analyze her decision:

1. If the expert predicts success (successful income of 400):

   • Utility: \[ U(400) = 1 - \frac{1}{400} \approx 1 - 0.0025 = 0.9975 \]

2. If the expert predicts failure (failing income of 2):

   • Utility: \[ U(2) = 1 - \frac{1}{2} = 1 - 0.5 = 0.5 \]

Expected Utility if Sarah pays for the prediction (Price = 0.01): The expected utility from paying the expert can be computed as follows: \[ E(U) = 0.01 \times U(400) + 0.99 \times U(2) - 0.01 \] Substituting the utility values: \[ E(U) = 0.01 \times 0.9975 + 0.99 \times 0.5 - 0.01 \] Calculating this: \[ E(U) \approx 0.009975 + 0.495 - 0.01 \approx 0.495975 \]

Now, let's compare this to the expected utility without the expert:

• Expected utility from acting (without the expert): \( \approx 0.833 \)

Conclusion for Part b:

Since the expected utility of going through the expert's service (about \(0.496\)) is significantly lower than the expected utility of just choosing acting outright (about \(0.833\)), Sarah should not pay for the expert. Therefore, she would still choose acting without paying the expert, as the expected utility from acting is higher than the expected utility after consulting the expert.