Man
503 Homework
IX Fall
2009
1)
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.
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.
Figure 11.6.b
c. Wait!
EA finds out that two different types of people
fly to
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,
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
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