Question 1

Firm X is currently selling a consumer good at a standard price, but is also considering cutting its price. The main risk facing the firm concerns the course of the economy in the near-term: whether the economy will grow at a steady pace (G) or whether it will experience a recession (R). The table below shows the firm's possible profit results (in $ millions). Finally, the firm judges that there is a 70% chance of growth and a 30% chance of a recession.
 $\begin{array} { | l | c | c | } 
\hline & \text { Growth } & \text { Recession } \
\hline \text { Standard Price } & 20 & - 10 \
\hline \text { Cut Price } & 15 & 0 \
\hline
\end{array}$ (a) Firm X must make its decision now (before knowing the future course of the economy). Which pricing policy maximizes its expected profit?
(b) Now suppose that Firm X can wait and decide its pricing decision after it knows the course of the economy. Determine its best pricing decisions and its overall expected profit.
(c) Firm X has hired a macroeconomic forecaster. The macro forecast is either positive (+) or negative (–). In the past, the forecaster has made positive forecasts prior to 4 of 7 periods of economic growth: Pr(+|G) = 4/7. In turn, he has made negative forecasts prior to 2 of 3 recessions: Pr(–|R) = 2/3. Compute Pr(G|+) and Pr(G|–) 
(d) Suppose the forecasting report can be purchased for $.2 million. Should Firm X buy the report? Explain your answer.

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Question 2

Oliver undergoes a standard medical test while at his regular checkup. The test is 90% reliable in detecting a form of cancer (C) that is found in 2% of the population. In particular, Pr(+|C) = .90. The test is also 90% reliable in screening out cancer, that is, Pr(-|H) = .90.
(a) If Oliver tests positive, how likely is it that he actually has cancer? If he tests negative, what is his cancer risk?
(b) Persons who test negative, and who actually have cancer, are able to sue for malpractice. Plaintiffs in such legal suits are awarded $250,000 on average (to cover medical expenses, pain and suffering, and legal fees). What is the hospital’s expected monetary liability due to the risk of incorrect negative tests?

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Question 3

Explain how Bayes Theorem is used to revise probabilities.

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Question 4

Which of the following is true of an English auction with private values?

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Question 5

Buyer A has offered $20,000 for a painting you are trying to sell. You are about to approach Buyer B whose best offer, you believe, might be anywhere between $16,000 and $24,000, with all values in between being equally likely. After hearing B's price, you will pick the higher of the two offers. What is the price that you expect to get for the painting?

Accepted Answer

The expected value of Buyer B's offer is the average of $16,000 and $24,000, which is $20,000. Since you will choose the higher offer, the expected value of the final price is the average of $20,000 and $24,000, which is $21,000.

Question 6

A petrochemical company must decide whether to fill a specialty order for one of its customers. Its cost (and therefore profit) depends on the quality of the raw material it has on hand to make the chemical. The firm expects to earn $50,000 from the order if the material is high quality (H) but will lose $30,000 if it is low quality (L). The firm's engineers estimate these probabilities to be .32 and .68 respectively. Before making its decision, the firm can test the material with one of two outcomes, "favorable" or "unfavorable." A favorable test increases the chance of H to .5, while an unfavorable result reduces it to .2. The likelihood of a favorable test is .4. Determine the expected value of this test.

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Question 7

With declining probabilities of success, the optimal-stopping strategy is to:

Accepted Answer

Optimal-stopping strategy suggests that one should stop investing in a project when the expected profit is zero or negative. This means that the costs of further investment outweigh the potential benefits. In the given scenario, since the probabilities of success are declining, it is likely that the expected profit will become zero at some point. Therefore, the optimal-stopping strategy is to discontinue investing when the expected profit is zero. This minimizes the risk of incurring further losses.

Question 8

A company is trying to decide whether to build a large plant or a small plant to supply future sales of a new product. However, it is uncertain about the market response to the product; whether demand will be strong or weak. According to the firm's marketing department, the probability of strong demand is .3 and of weak demand is .7. The table below lists the firm's profits (in millions of dollars) depending on plant capacity and the market response:
  (a) The company must make its plant decision now, before it will know what the market response will be. Which plant size maximizes its expected profit?
(b) Now suppose the company has a third option: building a modular plant. This is the same size as the small plant but is built so it can be expanded to the size of the large plant at a later date. The modular plant costs $4 million more to build than the small plant, but it allows the company the flexibility to observe the market response to the new product and immediately expand its capacity if demand warrants it (Note: If the modular plant is expanded, the firm’s total cost is also $4 million more than building a large plant in the first place). Should the company choose to build the modular plant?
(c) Suppose that the firm can take a small-scale market survey that will help it forecast market demand. The test has two possible outcomes: positive or negative. In the past, products that went on to enjoy strong demand received positive test market scores in 4 of 6 cases: Pr(+|S) = 2/3. Products generating weak demand received negative test results in 5 of 7 cases: Pr(–|W) = 5/7. Compute the revised probabilities, Pr(S|+) and Pr(S|–).

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Question 9

If there is a private-value model with risk-neutral buyers, what can be inferred regarding the expected revenues generated by English and sealed-bid auctions?

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Question 10

Firm Z is one of the 5 bidders, each with a value independently drawn from the range $100,000 to $160,000 with all values in between being equally likely. In a sealed-bid auction, Firm Z's equilibrium bidding strategy is:

Accepted Answer

To find the equilibrium bidding strategy of Firm Z, we need to use the principle of expected value. Let's say that Firm Z's bid is b and the highest bid among the other 4 bidders is v. Then, Firm Z will win the auction if b>v and lose if b<=v. The probability that Firm Z's bid is the highest among all 5 bids is 1/5. Therefore, the expected payoff for Firm Z if it bids b is: $$E( ext{Payoff}) = P( ext{Win})(b-v)+P( ext{Lose})(0) = \frac{1}{5}(b-v)$$ To find the bidding strategy that maximizes expected payoff, we need to differentiate the expected payoff with respect to b and set it to zero: $$\frac{dE( ext{Payoff})}{db} = \frac{1}{5}= 0$$ This implies that: $$b = v$$ In other words, Firm Z's optimal bidding strategy is to bid its true value. The expected payoff in this case is: $$E( ext{Payoff}) = \frac{1}{5}(v-v) = 0$$ Therefore, Firm Z should bid: $$b_i = (.2)(100,000) + .8v_i$$ which is equivalent to choosing answer choice D.

Question 11

A firm is considering the development of a new technology with a declining probability of success in each research stage. The firm's researchers have estimated the probabilities at .35, .25, .15, .07, and .01 for the various stages. The profit the firm would receive for successful development is $100 million, while the cost of research in each period is $10 million. How many investment stages should the firm undertake before abandoning the project?

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Question 12

A middle manager is an avid runner and keeps an informal diary of her daily 5-mile training runs. Most of the time, she spends 10 minutes or more stretching before running, believing that this will help prevent minor muscle injuries. In fact, she estimates that 64% of days over the last year, she has stretched and avoided any muscle problems, that is, the relevant joint probability is Pr(Stretch & Healthy) = .64.
(a) Is she correct in concluding that there is a positive association between stretching and being injury free? Now suppose she does some additional thinking and recalls many days when she hadn't bothered to stretch and fortunately still avoided any muscle pulls. Her guess is that Pr(No Stretch & Healthy) = .20. Determine Pr(Stretch|Healthy). Does this indicate that stretching reduces the risk of injury?
(b) Finally, the manager recalls a small number of instances when she neglected to stretch and, indeed, suffered muscle pulls, leading to the guesstimate: Pr(No Stretch and Muscle Pull) = .02. Write down the appropriate two-by-two joint probability table. Does stretching reduce the risk of injury?

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Question 13

Suppose that the firm's expected profit without test information is $75,000. There exists a perfectly reliable test that produces a positive result with a probability of 0.75 and a negative result otherwise. In light of a positive result, the firm's expected profit is $120,000; after a negative result, its expected profit is $40,000. Find the expected value of information.

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Question 14

In a sealed-bid auction, a firm with a reservation price of $480,000 submits a bid of $400,000. If its probability of winning is .75, then its expected profit from the auction is:

Accepted Answer

The expected profit is calculated as the probability of winning multiplied by the difference between the reservation price and the bid: 0.75 * ($480,000 - $400,000) = $60,000.

Question 15

The winner's curse occurs when:

Accepted Answer

The winner's curse occurs when the winning bid exceeds the true value of a good. This can happen in situations such as auctions where the winning bidder may get caught up in the excitement of the bidding process and end up paying more than the item is actually worth. This can create a situation where the winner ends up regretting their decision and feeling like they overpaid.

Question 16

A company is about to launch a new product and is considering one of two prices: high or low. However, the company is uncertain about the market response to the product - whether demand will be strong or weak. According to the firm's marketing department, the probability of strong demand is .6 and of weak demand is .4. The following table lists the firm's economic profit (in millions of dollars) at the two prices under strong and weak demand:
  (a) Suppose the company is risk neutral and must commit to a price before knowing what the market response will be. Should it launch the product? If so, at what price?
(b) The company is considering spending $10 million to test market the product on a national scale. In the process, it will learn the exact market response (strong or weak demand) before having to decide on a final launch and pricing strategy. Should it go ahead with the market test?
(c) Suppose instead that the company can hire a marketing consultant to undertake a limited market survey in the Midwest. The company anticipates that the consultant will come back with one of two possible demand forecasts: favorable (F) or unfavorable (U). In the past, the consultant's forecast accuracy has been as follows: Pr(F|S) = .8 and Pr(U|W) = .6. Compute the revised probabilities, Pr(S|F) and Pr(S|U).

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Question 17

Briefly describe the potential pitfalls associated with making intuitive predictions.

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Question 18

The following table shows the probabilities of A, B, X, and Y. Compute the joint probability table. Then calculate Pr(BY) and Pr(XB).
 $\begin{array} { | c | c | c | } 
\hline & \mathbf { X } & \mathbf { Y } \
\hline \mathbf { A } & \mathbf { 3 2 5 } & 455 \
\hline \mathrm { B } & 416 & 104 \
\hline
\end{array}$

Accepted Answer

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Question 19

Stake Gold Mines has the option to purchase a parcel of land adjacent to its current mining operations in a Western state. The seller's best and final price is $3 million. If the land has commercial mineral deposits, Stake Gold estimates its value at $5 million. If there are no deposits, the estimated value is $2 million. A preliminary look at the land leads Stake Gold to believe that the chance of mineral deposits is 50:50.
(a) Given this information, should Stake Gold purchase the land? For a fee of $200,000, the seller has agreed to let Stake Gold collect extensive mineral samples on the site. Based on past experience, if there are minerals present, the samples will provide a favorable indication 80% of the time. If no minerals are present, the samples will (falsely) give a favorable reading 40% of the time. Determine Pr(M|F) and Pr(M|U). (Here, M denotes mineral deposits, NM denotes no mineral deposits, F denotes favorable samples, and U denotes unfavorable samples.)
(b) Should Stake Gold pay $200,000 for the right to collect samples?

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Question 20

Suppose that two independent geologists begin with different prior assessments concerning the chance of oil and natural gas at a particular site. They both observe the results of a seismic test. Will they agree concerning their revised probabilities? In what instance, would their revised probabilities be identical?

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Quiz 13: The Value of Information