Man 503                                  Homework IX                                     Fall 2009

 

1)  Elizabeth Airlines (EA) flies only one route: Chicago-Honolulu.  The demand for each flight on this route is Q = 500 - P.  Elizabeth’s cost of running each flight is $30,000 plus $100 per passenger.

a.         What is the profit-maximizing price EA will charge?  How many people will be on each flight?  What is EA’s profit for each flight?

To find the profit-maximizing price, first find the demand curve in inverse form:

P = 500 - Q.

We know that the marginal revenue curve for a linear demand curve will have twice the slope, or

MR = 500 - 2Q.

The marginal cost of carrying one more passenger is $100, so MC = 100. Setting marginal revenue equal to marginal cost to determine the profit-maximizing quantity, we have:

500 - 2Q = 100, or Q = 200 people per flight.

Substituting Q equals 200 into the demand equation to find the profit-maximizing price for each ticket,

P = 500 - 200, or P = $300.

Profit equals total revenue minus total costs,

p = (300)(200) - {30,000 + (200)(100)} = $10,000.

Therefore, profit is $10,000 per flight.

b.         Elizabeth learns that the fixed costs per flight are in fact $41,000 instead of $30,000.  Will she stay in this business long?  Illustrate your answer using a graph of the demand curve that EA faces, EA’s average cost curve when fixed costs are $30,000, and EA’s average cost curve when fixed costs are $41,000.

An increase in fixed costs will not change the profit-maximizing price and quantity. If the fixed cost per flight is $41,000, EA will lose $1,000 on each flight.  The revenue generated, $60,000, would now be less than total cost, $61,000.  Elizabeth would shut down as soon as the fixed cost of $41,000 came due. 

Figure 11.6.b

c.         Wait!  EA finds out that two different types of people fly to Honolulu.  Type A is business people with a demand of QA = 260 - 0.4P. Type B is students whose total demand is QB = 240 - 0.6P.  The students are easy to spot, so EA decides to charge them different prices.  Graph each of these demand curves and their horizontal sum.  What price does EA charge the students?  What price does EA charge other customers?  How many of each type are on each flight?

Writing the demand curves in inverse form, we find the following for the two markets:

PA = 650 - 2.5QA    and

       PB = 400 - 1.67QB.

Using the fact that the marginal revenue curves have twice the slope of a linear demand curve, we have:

MRA = 650 - 5QA    and

           MRB = 400 - 3.34QB.

To determine the profit-maximizing quantities, set marginal revenue equal to marginal cost in each market:

650 - 5QA = 100, or QA = 110  and

         400 - 3.34QB = 100, or QB = 90.

Substitute the profit-maximizing quantities into the respective demand curve to determine the appropriate price in each sub-market:

PA = 650 - (2.5)(110) = $375  and

      PB = 400 - (1.67)(90) = $250.

When she is able to distinguish the two groups, Elizabeth finds it profit-maximizing to charge a higher price to the Type A travelers, i.e., those who have a less elastic demand at any price.

Figure 11.6.c

d.         What would EA’s profit be for each flight?  Would she stay in business? Calculate the consumer surplus of each consumer group.  What is the total consumer surplus?

With price discrimination, total revenue is

(90)(250) + (110)(375) = $63,750.

Total cost is

41,000 + (90 + 110)(100) = $61,000.

Profits per flight are

p = 63,750 - 61,000 = $2,750.

 

Consumer surplus for Type A travelers is

(0.5)(650 - 375)(110) = $15,125.

Consumer surplus for Type B travelers is

(0.5)(400 - 250)(90) = $6,750

Total consumer surplus is $21,875.

e.         Before EA started price discriminating, how much consumer surplus was the Type A demand getting from air travel to Honolulu?  Type B?  Why did total surplus decline with price discrimination, even though the total quantity sold was unchanged?

When price was $300, Type A travelers demanded 140 seats; consumer surplus was

(0.5)(650 - 300)(140) = $24,500.

Type B travelers demanded 60 seats at P = $300; consumer surplus was

(0.5)(400 - 300)(60) = $3,000.

Consumer surplus was therefore $27,500, which is greater than consumer surplus of $21,875 with price discrimination.  Although the total quantity is unchanged by price discrimination, price discrimination has allowed EA to extract consumer surplus from those passengers who value the travel most.

2)  You are an executive for Super Computer, Inc. (SC), which rents out super computers.  SC receives a fixed rental payment per time period in exchange for the right to unlimited computing at a rate of P cents per second.  SC has two types of potential customers of equal number--10 businesses and 10 academic institutions.  Each business customer has the demand function Q = 10 - P, where Q is in millions of seconds per month; each academic institution has the demand Q = 8 - P.  The marginal cost to SC of additional computing is 2 cents per second, regardless of the volume.

a.         Suppose that you could separate business and academic customers.  What rental fee and usage fee would you charge each group?  What would be your profits?

For academic customers, consumer surplus at a price equal to marginal cost is

(0.5)(8 - 2)(6) = 18 million cents per month or $180,000 per month.

Therefore, charge $180,000 per month in rental fees and two cents per second in usage fees, i.e., the marginal cost.  Each academic customer will yield a profit of $180,000 per month for total profits of $1,800,000 per month.

For business customers, consumer surplus is

(0.5)(10 - 2)(8) = 32 million cents or $320,000 per month.

Therefore, charge $320,000 per month in rental fees and two cents per second in usage fees.  Each business customer will yield a profit of $320,000 per month for total profits of $3,200,000 per month.

Total profits will be $5 million per month minus any fixed costs.

b.         Suppose you were unable to keep the two types of customers separate and charged a zero rental fee.  What usage fee maximizes your profits?  What are your profits?

Total demand for the two types of customers with ten customers per type is

.

Solving for price as a function of quantity:

, which implies

To maximize profits, set marginal revenue equal to marginal cost,

, or Q = 70.

At this quantity, the profit-maximizing price, or usage fee, is 5.5 cents per second.

p = (5.5 - 2)(70) = $2.45 million cents per month, or $24,500.

c.         Suppose you set up one two-part tariff- that is, you set one rental and one usage fee that both business and academic customers pay.  What usage and rental fees would you set?  What would be your profits?  Explain why price would not be equal to marginal cost.

With a two-part tariff and no price discrimination, set the rental fee (RENT) to be equal to the consumer surplus of the academic institution (if the rental fee were set equal to that of business, academic institutions would not purchase any computer time):

RENT = CSA = (0.5)(8 - P*)(8 – P*) = (0.5)(8 - P*)2.

Total revenue and total costs are:

TR = (20)(RENT) + (QA + QB )(P*)

TC = 2(QA + QB ).

Substituting for quantities in the profit equation with total quantity in the demand equation:

p = (20)(RENT) + (QA + QB)(P*) - (2)(QA + QB ), or

p = (10)(8 - P*)2 + (P* - 2)(180 - 20P*).

Differentiating with respect to price and setting it equal to zero:

Solving for price, P* = 3 cent per second.  At this price, the rental fee is

(0.5)(8 - 3)2 = 12.5 million cents or $125,000 per month.

At this price

QA = (10)(8 - 3) = 50

QB = (10)(10 - 3) = 70.

The total quantity is 120 million seconds.  Profits are rental fees plus usage fees minus total cost, i.e., (12.5)(20) plus (120)(3) minus 240, or 370 million cents, or $3.7 million per month.  Price does not equal marginal cost, because SC can make greater profits by charging a rental fee and a higher-than-marginal-cost usage fee.

 

3)  The local zoo has hired you to assist them in setting admission prices.  The zoo’s managers recognize that there are two distinct demand curves for zoo admission.  One curve applies to those 15 and above, while the other is for children.                   

                        PA = 9.6 – 0.08QA,    Demand from adults

                        PC = 4 – 0.05QC,        Demand from children

            Crowding is not a problem at the zoo, so that managers consider marginal cost is zero.

a)         Assuming that the zoo can not price discriminate between its two types of customers, calculate the profit maximizing price and output.

b)         Assuming that the zoo can price discriminate between its two types of customers, calculate the profit maximizing price and output.

a) Total Demand Q = 200 – 32.5P

    MR = 6.15 -0.06Q

    MR = MC

    P = 3.075

    Q = 102.5

b)  Optimal price discrimination requires the zoo to set MRA = MRCS = MC.

                                Setting MRA = 0

9.6 - 0.16QA = 0

9.6 = 0.16QA

QA = 60

 

PA = 9.6 - 0.08(60)

PA = $4.8

MRCS = 4 - 0.10QCS = 0

4 = 0.10QCS

QCS = 40

 

PCS = 4 - 0.05(40) = $2

PCS = $2

 

TRA = PA • QA

TRA = 4.8 • 60 = $288

 

TRCS = PCS • QCS

TRCS = 2 • 40 = $80

TR = 288 + 80 = $368