–
Either you say “yes”,
–
Or you say “no” and explain the difference(s) between the contexts.
p
D
(
p
)
Pictures that have boxes representing revenue/profit underneath the demand
curve are common in market segmentation and volume discounting.
Each
box indicates some more revenue.

Page
18
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/~metin
Calculating Differentiated Prices
by Eliminating Arbitrage
Arbitrage happens when a lower priced product i is
transport
ed to be sold as a higher priced product j
transportation cost
a
ij
upgrade
d to be sold as a higher priced product j
upgrade cost
a
ij
stock
ed to be sold as a higher priced product j
inventory holding cost
a
ij
Arbitrage can be eliminated with constraints
𝑝
?
≤ 𝑝
?
+ 𝑎
??
and
𝑝
?
≤ 𝑝
?
+ 𝑎
??
on the price decision variables.
Constraints are called
“
arbitrage elimination constraint
”.
The constraint makes the cost of using arbitrage
p
i
+
a
ij
higher than the price
p
i
.
i
j
𝑎
??
𝑎
??

Page
19
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/~metin
Motivating Example of Chips
Intel chips are sold both in US (country 1) and in Brazil (country 2).
The
demand functions are given by
p
p
b
a
p
d
p
p
b
a
p
d
10000
48600
)
(
and
100000
508000
)
(
2
2
2
1
1
1
Since the constraint is not satisfied, an arbitrageur can buy chips in Brazil and
transport them to US to sell them in US.
We need to jointly optimize prices in US and Brazil to eliminate arbitrage rather
than separately as we have done above.
The transportation cost per chip is $0.08 between these countries.
With the constant slope demand curves (lines), the revenue maximizing prices
are given as
21
0
2
0
1
2
2
0
2
1
1
0
1
51
.
2
54
.
2
but
43
.
2
2
and
54
.
2
2
a
p
p
b
a
p
b
a
p

Page
20
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Quadratic Program Formulation to
Eliminate Arbitrage
When we have linear demands
d
i
(
p
i
), the profit (p
i
-c
i
)
d
i
(
p
) is a quadratic
function (it is a polynomial of degree 2).
Summing these profits over N
regions, we still have a quadratic objective for the joint optimization problem:
N
i
p
N
j
i
a
p
p
p
d
c
p
i
ij
i
j
N
i
i
i
i
i
p
p
N
1
for
0
1
for
subject to
)
(
)
(
max
1
...
1
This is a quadratic programming problem and it can be solved efficiently with
algorithms similar to those used to solve linear programming problems.