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assignment four
the monopoly model (I): standard pricing
optimal production ………….1
optimal allocation (I) ………….2
optimal allocation (II) ………….5
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
assignment 4
the monopoly model (I): standard pricing
the analytics of constrained optimal
decisions
optimal production / profit maximization
140
130
► Banana Republic’s customer have the
following demand:
P = 140 – 6Q
► From the demand function we get
immediately the marginal revenue
MR = 140 – 12Q
monopoly price
Pm = 80
MR
110
► Item are produced at a constant
marginal cost:
MC = 20
 marginal revenue
demand
120
100
90
Pm
80
70
monopoly quantity
Qm = 10
60
50
40
 profit maximization
► Condition:
30
MC = 20
20
MR = MC
we get 140 – 12Q = 20 with solution:
Qm = 10 , Pm = 140 – 6∙10 = 80
 2016 Kellogg School of Management
10
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Qm
11.67
assignment 4
23.34
page | 1
microeconomics
assignment 4
the monopoly model (I): standard pricing
the analytics of constrained optimal
decisions
optimal allocation (I) / profit maximization
140
130
► Banana Republic has a stock of 12
items already produced. While it faces
exactly the same demand as before (P =
140 – 6Q) the marginal cost is now zero
(up to maximum capacity) because any
production cost is sunk.
► The problem is now one of allocation
of available units (since they are already
produced) rather then how many units to
produce.
► Another important issue here is to
clearly understand what are the allocation
alternatives
and their corresponding
payoffs.
► The two alternatives are:
(A): sell through the store – marginal
revenue is MR(A) = 140 – 12Q
(B): burn units – marginal revenue is in
this case MR(B) = 0
“max “ capacity
demand
120
110
100
MR(A)
90
Pm
80
70
60
50
40
30
20
10
MR(B)
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
11.67
 2016 Kellogg School of Management
assignment 4
23.34
page | 2
microeconomics
assignment 4
the monopoly model (I): standard pricing
the analytics of constrained optimal
decisions
optimal allocation (I) / profit maximization
140
► The problem can be restated as:
… we have 12 units to allocate
between two alternatives that give
different payoffs …
► We are given the marginal revenue
that we obtain for each unit allocated to a
certain alternative...
► … which gives a very easy way to
decide on how to allocate each unit...
► Start with the first unit…allocate it to
the alternative that gives the highest
marginal revenue, then the second … and
so on … until the last unit is allocated.
130
“max “ capacity
demand
120
110
100
MR(A)
90
80
allocation to (A)
Q(A) = 11.67
70
60
50
40
30
► It’s easy to see that the marginal
revenue from alternative (A) is larger than
the marginal revenue from alternative (B)
for the first 11.67 units obtained from
setting MR(A) = MR(B).
20
10
MR(B)
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
11.67
 2016 Kellogg School of Management
assignment 4
23.34
page | 3
microeconomics
assignment 4
the monopoly model (I): standard pricing
the analytics of constrained optimal
decisions
optimal allocation (I) / profit maximization
140
► Since 12 units are sold through the
stores the price will be
P(A) = 140 – 6∙11.67 = 70
► The remaining 0.33 units are burned…
thus Q(B) = 0.33
130
120
100
► No doubt this is puzzling… why burn
those units when you can actually still get
a fairly high price for them (slightly below
P(A)
$70/unit)?
► If you choose to still sell these 0.33
units you would actually decrease the
price on all previous 11.67 units thus you
sell more units at a lower price…
60
► The marginal revenue measures
exactly that: it tells you how the total
revenue changes with changes in units …
however MR(A) is negative beyond 11.67
burning these 0.33 units gives
a higher marginal revenue
than from selling through the
chain stores
110
90
► How do you know whether this is
better or worse than burning the units?
“max “ capacity
demand
MR(A)
80
allocation to (A)
Q(A) = 11.67
70
50
40
30
20
10
MR(B)
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
11.67
 2016 Kellogg School of Management
assignment 4
23.34
page | 4
microeconomics
assignment 4
the monopoly model (I): standard pricing
the analytics of constrained optimal
decisions
optimal allocation (II) / profit maximization
140
► There are three alternatives now:
(A) sell through chain stores MR(A)
(B) burn MR(B) = 0
(C) sell as private label MR(C) = 10
► The problem essentially remains the
same: how should we allocate the 12 units
among the three alternatives given their
payoffs.
► Same logic applies: each unit should
be allocated to the alternative that offers
the highest marginal revenue…
130
120
110
100
MR(A)
90
allocation to (A)
Q(A) = 10.83
80
70
60
50
► The marginal revenue MR(A) is largest
up to the output Q(A) = 10.83 obtained
from the condition
40
MR(A) =MR(C)
20
that is
30
MR(C)
10
140 – 12Q = 10
► What about the remaining units?
“max “ capacity
demand
MR(B)
0
0
1
2
3
4
5
6
7
8
10.83
 2016 Kellogg School of Management
assignment 4
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
11.67
23.34
page | 5
microeconomics
assignment 4
the monopoly model (I): standard pricing
the analytics of constrained optimal
decisions
optimal allocation (II) / profit maximization
140
► The remaining 1.17 units can be
burned, for a marginal revenue of 0, or
sold under private label for a marginal
revenue of 10…
130
► Obviously the 1.17 units are sold
under private label at a price of 10 each,
thus Q(C) = 1.17
100
“max “ capacity
demand
120
selling these 1.17 units gives
a higher marginal revenue
than from selling through the
chain stores or burning them
110
MR(A)
90
allocation to (A)
Q(A) = 10.83
80
► Again, it might be counterintuitive to
sell 1.17 units at $10 per unit under
private label when those units could be
sold at a price (slightly below) $70 through
the chain stores…
► Adding the extra units to the chain
stores sales would decrease the price on
all previous 10.83 units which would result
in an overall lower revenue than selling
them at $10 under the private label …
► No unit is burned, thus Q(B) = 0
70
60
50
40
30
20
MR(C)
10
MR(B)
0
0
1
2
3
4
5
6
7
8
10.83
 2016 Kellogg School of Management
assignment 4
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
11.67
23.34
page | 6