Eco 300
HW 3 Answer Key
11.6
First prove the hint:
In the graph
Q
max
is the quantity demanded when
P
= 0. Since
MR
= 0 at
1
/
2
Q
max
(there
e
=
–1), it is clear that
MR
bisects the distance from the
P
axis to the demand curve. So, if
Q
*
represents quantity demanded when
P
=
MC, MR
=
MC
at
1
/
2
Q
*. Notice also that the
profitmaximizing price is given by 0.5 (
P
max
+
MC
) where
P
max
is the price for which
quantity demanded is zero.
Note:
The results of this hint are used to solve several problems in later chapters.
a.
If
Q
1
= 55 –
P
, then since
MC
= 5,
Q
1
= 55 – 5 = 50
1
Q *
= 25
2
At that output level,
P
1
= 30
π
= (
P
1
– 5)(
Q
1
) = 25
×
25 = 625.
If
Q
2
= 70 – 2
P
2
,
Q
2
*
= 70 – 2(5) = 60 = 70 – 2(5) = 60
*
2
30
2
Q
=
Therefore,
P
2
= 20
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π
= (20 – 5)
×
30 = 450.
Total profits =
π
1
+
π
2
= 1,075.
b.
The problem when transport costs = $5 cannot be solved explicitly without
calculus. In general, one would answer that the prices can in that case only differ
by $5 and that profits would fall relative to the pure separation case.
Using calculus it can be shown that the exact solution to the problem requires that
P
1
= 26
2
3
P
2
= 21
2
3
Q
1
= 28
1
3
Q
2
= 26
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 Fall '09
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 Economics, Consumer Surplus, Supply And Demand, Harshad number

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