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Problem Set 1 ANSWER KEY

Due in Class, Thursday 9/22

Be sure to show all of your work and to answer all parts of the questions.

1.   Consider the market for Caribbean cruises. In the wake of the COVID-19 pandemic, the cruise       companies take the following actions. First, they implement testing and tracing programs to limit the potential of COVID outbreaks on ships to occur and to spread. Second, they engage in              widespread advertisement of their new and state of the art safety precautions. Using a generic    model of supply and demand, show and describe how you would expect these initiatives would   affect the market for cruises. Be sure to explain why which curves are shifting, and discuss             changes to market equilibrium.

 

There are two changes to the market. The testing and tracing programs implemented are costly, and  would raise the marginal costs of providing the cruises – as such, we would expect the supply to           decrease and the supply curve would shift left. Second, the press on COVID, it’s effects, and the safety issues would likely alter consumers’ tastes and preferences against going on cruises. Concern over       COVID would likely reduce demand, causing a leftward shift in the demand curve.

Compared to the initial equilibrium (E1), at the new equilibrium (E2) the quantity of cruises sold will      definitely fall. The effects on price would be indeterminate – it depends upon the relative magnitude of the shifts whether or not price would rise or fall. The graph should look something like this:

P              E2                 

E1

Q

2.   Calculate the expected profits for each of the following scenarios:

a.   There is a 0.25 probability that you have profits of 10, a 0.40 probability that you have profits of 15, and a 0.35 probability that your profits are 20.

Just apply the expression for expected value, where we weight each possible level of profits by the probability that those profits occurs:

E() = 0.25 ∗ 10 + 0.40 ∗ 15 + 0.35 ∗ 20 = 15.5

b.   You produce 100 units at a cost of $20 each, but you are uncertain about the price you will receive for your goods. The price you will receive is either $15 or $30, each having equal    probability.

Here we use the same expression for expected value, except that we now have to calculate profits in   each state of the world (profits are total revenue – total cost). Because there are only two prices, their probabilities sum to 1, and they are equal, the probabilities of each possible price is equal to 0.5:

E( ) = 0.5 ∗ (15 ∗ 100 − 20 ∗ 100) + 0.5 ∗ (30 ∗ 100 − 20 ∗ 100) = − 250 + 500 = 250

c.    You win a contract for $5,000, but you are unsure how much time it will actually take you to fulfill the terms of the contract. Each hour you spend on the project costs you $50. There is a 30% chance that the project goes smoothly and you can complete the contract in 50 hours.   There is a 50% chance that the project takes 80 hours. Finally, there is a 20% chance that the project takes much more time than you thought, and it takes you 110 hours to complete.

Here we use the same expression for expected value, except that we now have to calculate profits in       each state of the world but instead of revenues being uncertain our costs are (profits are total revenue – total cost):

E( ) = 0.3 ∗ (5000 − 50 ∗ 50) + 0.5 ∗ (5000 − 50 ∗ 80) + 0.2 ∗ (5000 − 50 ∗ 110) E( ) = 750 + 500 − 100 = 1150

3.   Calculate the profit-maximizing level of output for a perfectly competitive firm in each of the following circumstances:

a.   The firm’s marginal cost function is given by MC(Q) =  Q, and the price = $10 with probability ½, and equals $30 otherwise.

Well, we use the same three-step profit-maximization procedure, only instead of maximizing profits we maximize expected profits. So, we write an expression for expected profits, which are the summation of profits in each state of the world weighted by the probability they occur:

E( ) = pr(P = 10)  10 + pr(P = 30)  30

Now, if there are only two possibilities, price is 10 or 30, and price is 10 with probability ½, then it must be that probability of price being 30 is ½ as well. So, we use our condition for finding a level of output   that maximizes the expected profit:

E(MR) = E(MC)

we have:

 ∗ 10 +  ∗ 30 =  Q

 

We get Q*  = 40 .

b.   The firm’s cost function is given by MC(Q) = Q, and the price equals $6, $9, or $15, each with equal probability.

Recognize that, if each potential price has equal probability and there are three of them, then      each price happens with probability 1/3. Write a similar expression as in (a) above, although with three possible levels of profits, and we get:

E(MR) = E(MC)

we have:

 * 6 +  * 9 +  * 15 = Q

Q*  = 10

4.   On behalf of your company, you are preparing a price bid to supply a fixed quantity of a good to a potential buyer. You are aware that a number of competitors also are eager to obtain the               contract. The buyer will select the lowest bid. Your cost is $100,000. If yours is the winning bid,     your profit is the difference between your bid and your cost. If not, your profit is zero. You are      considering three possible bids:

•    Bid $110,000, giving you a probability of winning of 0.9

•    Bid $130,000, giving you a probability of winning of 0.5

•    Bid $160,000, giving you a probability of winning of 0.2

a)   Assuming your companys aim is to maximize its expected profit, which bid should you submit?

Draw a decision tree to help think through the problem. The first three branches should             represent each of the potential bids that you can make (the actions you can take), and then the potential outcomes will branch off from there:

Lose 10%

Win 50%

 

Lose 50%

Win 20%

 

Lose 80%

w  = $110K − $100K

 

 l  = 0

w  = $130K − $100K

 

 l  = 0

w  = $160K − $100K

 

 l  = 0


We calculate the expected profit from each possible bid, and then choose the bid that gives us the highest expected profits! The expression for expected profits is:

E() = pT(win)几w + pT(lose)l

Our decision tree becomes:

Win 90%

E(几) = 0.9 ∗ 10 = $9K

Bid $110K

 

 

E() = 0.5 ∗ 30 = $15K

Lose 50%

 

Bid $160K

E() = 0.2 ∗ 60 = $12K

Lose 80%

So the choice that maximizes our expected profits is to bid $130K!

b)   In part (a), your cost is $100,000 for certain. Now suppose it is uncertain: either $80,000 or  $120,000, with each cost equally likely. Will this fact change your bidding behavior? Explain.

No, because your expected costs are still $100,000, so your expected profits would remain the same from each bid!

c)    Building upon part (b), suppose it is possible to gain information about the cost so that you will   know exactly what the cost will be (either $80,000 or $120,000) before submitting the bid. What would be the value of this information to the firm?

Okay, now we have two decision points in our decision tree. First, we decide on whether or not to       purchase the perfect forecast (i.e. gain the information). If we don’t, then we get the expected profits that we calculated in part (a) and (b), where we had no information:  E(几) = $15K.

However, if we purchase the information we get to see what the cost would be, and then we decide our optimal bid. The decision tree becomes (recall that profits are 0 when you lose the bid) :

w = $110K − $80K

 

E() = $15K

Bid $110K

Bid $130K

几w = $160K − $80K

$80K, 50%

Bid $160K

Lose 80%

 

$120K, 50%

Bid $130K

w = $130K − $120K

Bid $160K    

 

w = $160K − $120K

Lose 80%

So, you have three actions you might take if there is a low cost, and then three if there is a high cost…     calculate the expected profit from each, and then choose the maximum expected profits in each state of the world (high cost or low cost). You should see that if the cost is low, you maximize expected profit by  choosing a low bid, giving you an expected profit of $27K. Similarly, if there is a high cost you maximize   your expected profit with a high bid, yielding expected profit of $8K. So, the red lines indicate that these are the choices that we should make in either state of the world. Now, each state of the world (high cost or low cost) occurs with 50% probability, so the expected profit if we pay for the information is now:

E(几 |info) =  ($27K) +  ($8K) = $17.5

Because the expected profits with information is greater than without, we should choose to purchase   the information! How much would be willing to pay? Well, the difference between our expected profits with information and the expected profits without:

Value of information = E(几 |info) − E(几 |no info) = $2.5K


5.   Consider a perfectly competitive firm choosing a profit-maximizing level of output. However,        there is uncertainty surrounding the market price, P, of their good, generating some uncertainty. The firms cost function is given by C(Q) = 10 + 2Q + 4Q2. There is 60% chance that the market price is $42  otherwise a relatively low price of $22 is realized. (Hint: the firm’s marginal cost is   given by: MC(Q) = 2 + 8Q)

a)   Calculate the firm’s profit maximizing level of output under this scenario. What are the firm’s expected profits? Will that level of profits ever be realized?

First, recognize that the uncertainty is stemming from the realized market price. There are two   states of the world – a high price and a low price. So, if our firm is an expected- profit maximizer, then their problem is:

Q(ma)x E(几(Q)) = () (42Q − 10 − 2− 4Q2) + ()(22Q − 10 − 2− 4Q2)

Using our expected profit-maximization condition, we have:

E(MR) = E(MC)

() ∗ 42 + () ∗ 22 = 2 + 8Q

Solving the condition we get the expected-profit maximizing level of output, QE" = 4. If the firm chooses this level of output ex ante, then when the price is high they receive profits of 86 and     when price is low they realize profits of 6. Their expected profits are then:

E() = () (86) + () 6 = 54

although this level of profits is never actually realized.

b)   Suppose the firm can hire an outside consulting firm to provide an improved understanding of the perspective prices. They provide a perfect forecast. How much would our firm be willing to pay for the consulting firm’s forecast, i.e. for their information on the future market price?

If the firm hires the consulting firm to get a perfect forecast, then once the future price is         declared the firm will tailor output to produce maximum profits under each possible price. We can solve for the firm’s supply function generally, where output is a function of P:

max (Q) = PQ − 10 − 2− 4Q2

Q

In other words, and using our profit maximization condition, we have:

P = 2 + 8Q

Implying that Q(P) = (P − 2)/8. So, when P=42 the optimal quantity is QP=42 = 5, yielding  profits of 90, and when P=22 the optimal quantity is QP=22 = 2.5, yielding profits of 15. Thus:

E((perfect information)) = ( ) (90) + () 15 = 60

The value of information is the difference between expected profits with the information minus expected profits without it! This tells us how much the firm would be willing to pay for the          information before deciding how much to produce. The expected profits without information    was found in part (a), so:

Value of Info = E(几(perfect information)) − E(几(imperfect information)) = 6