Managerial Econ – Required Homework for Module 7 Solution
Problem 1:
Part a: By inspection of the inverse demand function for foreign buyers, we can see that any
price above $16,000 per tractor will result in no sales to foreign buyers.
Part b: To answer this part, we need to sum the demands of the two groups, but we must do so
horizontally. That is, for each price, we need the sum of the quantities from the two groups. To
do this, we need the ordinary demand functions for each group. I left the ordinary demand
functions in fractional form. This produces “nice numbers for answers. If you converted to
decimals, your numbers will be different from mine and depending on how many decimal places
you carried through, your numbers may be significantly different. As long as your method was
sound, there are no deductions for the differences.
1, 250 1
− PD
3
60
1
Foriegn Demand: PF = $16, 000 − $20QF  QF = 800 − PF
20
3, 650 1
Total Demand: QT = QD + QF =
− P
3
15
Domestic Demand: PD = $25, 000 − $60QD  QD =
 P = $18, 250 − $15Q for P  $16, 000 and PD = $25, 000 − $60QD for P  $16, 000
Notice that from the answer to part a, we know that at prices below, $16,000 this market demand
function includes both domestic and foreign buyers. At prices above $16,000, the market demand
is just the demand by the domestic buyers since foreign buyers are priced out of the market
above $16,000. In fact, the market demand curve is kinked as depicted below.
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Combined Demand
$26000
Domestic
Foreign
Combined
Combined MR
$24000
$22000
$20000
Dollars ($)
$18000
$16000
$14000
$12000
MC
$10000
$8000
$6000
$4000
$2000
$0
0
100
200
300
400
500
600
700
800
900
1000
Quantity
Part c: All that is required here is to solve the “profit-max” problem.
P = $18, 250 − $15Q  TR = P = $18, 250Q − $15Q 2  MR =
TC = $1, 000, 000 + $10, 000Q  MC =
dTR
= P = $18, 250 − $30Q
dQ
dTC
= $10, 000
dQ
MC = MR
$10, 000 = $18, 250 − $30Q
Q* = 275
P* = $18, 250 − $30 ( 275 ) = $14,125
Titan will charge a price of $14,125 and sell 275 tractors using this pricing strategy in order to
make maximum profits. Since this price is below $16,000, some foreign buyers will buy. In fact,
the total sales will be divided as follows:
2
1, 250 1
− ( $14,125 ) = 181.25
3
60
1
Foriegn Demand: QF = 800 − ( $14,125 ) = 93.75
20
Domestic Demand: QD =
Part d: All that is required here is to solve the “profit-max” problem for each segment of the
market separately.
PD = $25, 000 − $60QD  TRD = $25, 000QD − $60QD2  MRD =
dTRD
= $25, 000 − $120QD
dQD
MC = MRP
$10, 000 = $25, 000 − $120QD
QP * = 125
PD * = $25, 000 − $60 (125 ) = $17,500
PF = $16, 000 − $20QF  TRS = $16, 000 − $20QF2  MRF =
dTRF
= $16, 000 − $40QD
dQF
MC = MRF
$10, 000 = $16, 000 − $40QD
QF * = 150
PF * = $16, 000 − $20 (150 ) = $13, 000
So, using price discrimination Titan sells its tractors its to the domestic buyers at $17,500 each
and sells 125, while it sells to foreign buyers at $13,000 each and sells 150 units.
Part e: This part requires finding the profits under the two pricing schemes.
Single Price:
Operations = TR − TC
TR = $14,125  275 = $3,884,375
TC = $1, 000, 000 + $10, 000 ( 275 ) = $3, 750, 000
Profit = 134,375
Price Discrimination:
Operations = (TRD + TRF ) − TC
TRD = $17,500 125 = $2,187,500
TRF = $13, 000 150 = $1,950, 000
TC = $1, 000, 000 + $10, 000 ( 275 ) = $3, 750, 000
Profit = $387,500
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As the above calculations show, Titan can drastically increase its profits by segmenting the
market and engaging in third-degree price discrimination.
Problem 2:
a. This is simply a monopoly pricing strategy that can be solved in the standard fashion by
setting marginal revenue equal to marginal cost.
MC ( Q ) = 50 = 150 − 5Q = MR ( Q )
100 = 5Q
Q* = 20
P* = 150 − 2.5  20 = $100
b. The annual profit per typical student (ignoring fixed costs) is found as follows:
 = TR ( Q* ) − VC ( Q* ) = ( P*  Q* ) − VC ( Q* )
= $100  20 − $50  20
= $1, 000
c. Under the two-part pricing scheme books should be priced at the marginal cost per book
($50). This results in the typical student buying 40 books per year.
P ( Q ) = MC ( Q )
150 − 2.5Q** = 50
Q** = 40
d. Under the two-part pricing scheme the annual access fee should be established to extract
the consumer surplus that results from the per-book price being set equal to marginal cost.
Thus, we need to determine the consumer surplus that results from charging $50 per book
and selling 40 books per year. The access fee should be set at $2,000.
1
CS = (150 − 50 )  40 = $2,000
2
e. Your expected profit per typical student under this scheme is just the consumer surplus you
have extracted via the annual access fee. So, the profit per typical student is $2,000.
f. If the annual fixed cost of bookstore operations is $2,000,000 and the profit per typical
student ignoring these costs is $2,000, bookstore needs to sell at least 1,000 memberships
annually to cover all its costs and be willing to operate the course.
Breakeven Memberships =
$2, 000, 000
= 1, 000
$2, 000
The graph on the next page shows these results. The area of the blue triangle is the original
consumer surplus if the single monopoly price is charged, while the area of the green rectangle is
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the profit earned under this pricing scheme ignoring fixed costs. The areas of the orange triangle
plus the green rectangle plus the blue triangle are the profits (equal to total consumer surplus) using
the two-part pricing scheme of charging an access fee equal to this area and a per-unit price that is
equal to marginal cost.
Campus Bookstore Operations
150
Demand
MR
MC
140
130
120
110
Dollars ($)
100
90
80
70
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
45
50
55
60
Books per year
Problem 3:
Part a: All that is required here is to solve the standard profit maximization problem facing the
monopoly firm.
P = $18 − Q  TR = $18Q − Q 2  MR =
VC ( Q ) = Q 2  MC =
dTR
= $18 − 2Q
dQ
dVC
= 2Q
dQ
MC = MR
2Q = $18 − 2Q
Q* = 4.5
P* = $18 − 4.5 = $13.50
5
Part b: Simply compute total revenue and total variable cost. So profits are:
 = TR − TC = ( $13.50  4.5) − ( 4.52 ) = $40.50
Dollars
Bottled Water - Typical Resort Guest
$20
$19
$18
$17
$16
$15
$14
$13
$12
$11
$10
$9
$8
$7
$6
$5
$4
$3
$2
$1
$0
MC
AC
Demand
MR
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Bottles per Week
Part c: The goal here is to extract all the consumer is willing to pay for each bottle until that
value is equal to our marginal cost. We won’t be willing to sell the next bottle at a price less than
marginal cost and the consumer will be unwilling to pay a price for the next bottle that is equal to
that marginal cost because they don’t value the extra bottle that highly. The optimal number of
units in the all-or-nothing package is that output where price equals marginal cost. So solve the
following:
P = $18 − Q = 2Q = MC
3Q = $18
Q* = 6
Part d: The price to charge for this all-or-nothing block of your output is the total value to the
consumer of the 6 bottles. This is just the consumer surplus of the block of 6 units plus what the
consumer is willing to pay for the 6 units. Or, thinking of it in another way, what is the
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maximum amount the consumer would be willing to pay for each bottle purchased one at a time
until 6 bottles are purchased? We could represent this by the area under the demand curve
between 0 and 6 bottles. What makes this problem different from the demonstration problem in
the text is that in this case, marginal cost is not constant but increasing in Q. So here we cannot
simply add the total cost to consumer surplus to determine our price since the consumer is
willing to pay more than that for a block of six bottles. See the graph below.
1
( $18 − $12 )  6  = $18
2
Willingness to pay = $12  6 = $72
PBlock = $18 + $72 = $90
CS =
Dollars
Bottled Water - Typical Resort Guest
$20
$19
$18
$17
$16
$15
$14
$13
$12
$11
$10
$9
$8
$7
$6
$5
$4
$3
$2
$1
$0
MC
AC
Price to charge =
area in red outline
Total cost to
produce 6 units
Demand
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Bottles per Week
Part e: Under the single price strategy, we saw that profits were $40.50.
To determine the profits under the block pricing strategy, we must subtract the cost of production
from the block price we are charging.
 = TR − TC = $90 − ( 62 ) = $54
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We can see that by using the block pricing strategy (in this case an all-of-nothing strategy), your
company is able to increase profits from $40.50 to $54. This comes from extracting all the
consumer surplus from the typical consumer plus their willingness to pay for the block.
Bonus:
Thursday. The only day they both tell the truth is Sunday; but today can’t be Sunday because the
monkey also tells the truth on Saturday (yesterday). Going day by day, the only day one of them
is lying and one of them is telling the truth with those two statements is Thursday.
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