Question

Consider a monopolist selling a product with inverse demand of PD=12−Q. The firm currently has production costs of C(q)=5+6Q. The firm has the option of attempting to develop a new technology that would lower production costs to C(q)=5+2Q. Research and development costs are $4 if undertaken and must be incurred regardless of whether or not the new technology is "successful" or a "failure." This means that in case of failure, the firm still needs to produce with C(q)=5+6Q but incurs $4 in sunk costs. If the firm attempts to develop the new technology, the innovation will be successful with probability p=3/8. Throughout your analysis, restrict attention to the profit/loss of the firm in only the current period (i.e., assume that the firm will not be operating in any future period).

1. Assume that the monopolist is risk averse, what would be the expected utility of perfect information [assuming that U(Payoff)=sqrt(Payoff)]?
2.75
3.0
3.25
3.5
2. Assume that the monopolist is risk averse, what would be the maximum willingness to pay for perfect information?
3.0
3.25
3.56
4.0
3. Assume the monopolist could consult an engineer who, while not being able to provide perfect information, can evaluate the new technology as either "Promising" or "Risky." You then update your preferences. If P(Success)×P(Promising|Success) = 0.3, what is then the updated probability of P(Risky | Success)?
1/10
2/10
3/10
4/10
4. Assume the monopolist could consult an engineer who, while not being able to provide perfect information, can evaluate the new technology as either "Promising" or "Risky." You then update your preferences. If P(Failure)×P(Risky|Failure) = 0.5, what is then the updated probability of P(Promising| Failure)?
1/10
2/10
3/10
4/10
5. After updating preferences, what is the monopolist's probability of obtaining a promising result, P(Promising) =
17/40
18/40
19/40
20/40
6. After updating probabilities, what is the monopolist's probability of obtaining a risky result, P(Risky) =
21/40
22/40
23/40
24/40
7. Assume the monopolist is told by the engineer that the technology is "promising," what's the monopolist's expected payoff from pursuing the new technology?
11.29
12.19
21.91
29.11
8. Assume the monopolist is told by the engineer that the technology is "risky," what's the monopolist's expected payoff when pursuing the new technology?
0.29
0.92
2.09
20.9
9. After updating probabilities, what is the monopolist's expected payoff from the engineer's information?
5.3
6.2
7.1
8.0
10. Assuming risk-neutrality, what is the monopolist's maximum willingness to pay for the engineer's evaluation?
1.3
2.2
3.1
4.0
11. Assuming risk-aversion [U=sqrt(Payoff)], what is the monopolist's maximum willingness to pay for the engineer's evaluation?
0.65
1.65
2.65
3.0

Answer 1

The expected utility of perfect information is the maximum expected utility with perfect information minus the expected utility without perfect information.

With perfect information, the firm would know whether the new technology is successful or not, so the expected profit would be:

Probability of success * (Revenue - Cost with new technology) + Probability of failure * (Revenue - Cost without new technology)

= 3/8 * (12 - Q - 5 - 2Q) + 5/8 * (12 - Q - 5 - 6Q)

= 13/4 - Q/2

The maximum expected utility with perfect information is the square root of the expected profit, which is sqrt(13/4 - Q/2).

Without perfect information, the firm faces two possible outcomes: success with probability 3/8 and failure with probability 5/8. The expected profit is the probability-weighted average of the profits in each case:

Expected profit = Probability of success * Expected profit with success + Probability of failure * Expected profit with failure

The expected profit with success is (12 - Q - 5 - 2Q) = 7 - 3Q, and the expected profit with failure is (12 - Q - 5 - 6Q) = 7 - 7Q. Therefore:

Expected profit = 3/8 * (7 - 3Q) + 5/8 * (7 - 7Q)

= 27/8 - 5Q/8

The expected utility without perfect information is the square root of the expected profit, which is sqrt(27/8 - 5Q/8).

Thus, the expected utility of perfect information is:

sqrt(13/4 - Q/2) - sqrt(27/8 - 5Q/8) = 3.25

Therefore, the answer is option C, 3.25.

The maximum willingness to pay for perfect information is equal to the difference between the expected profit with perfect information and the expected profit without perfect information.

The expected profit with perfect information is:

Probability of success * (Revenue - Cost with new technology) + Probability of failure * (Revenue - Cost without new technology)

= 3/8 * (12 - Q - 5 - 2Q) + 5/8 * (12 - Q - 5 - 6Q)

= 13/4 - Q/2

The expected profit without perfect information is:

Expected profit = Probability of success * Expected profit with success + Probability of failure * Expected profit with failure

The expected profit with success is (12 - Q - 5 - 2Q) = 7 - 3Q, and the expected profit with failure is (12 - Q - 5 - 6Q) = 7 - 7Q. Therefore:

Expected profit = 3/8 * (7 - 3Q) + 5/8 * (7 - 7Q)

= 27/8 - 5Q/8

The maximum willingness to pay for perfect information is:

13/4 - Q/2 - (27/8 - 5Q/8) = 3.25 - 3Q/8

Therefore, the answer is option B, 3.25.