Econ 302, Spring 2013
Assignment 1 – Solution
Problem 1
a) Price elasticity of demand
ò( p ) = −
−2 p p
dD ( p ) p
= −100( 4 )
=2
dp D( p )
p 100
p2
b) Monopolist quantity and price.
From the demand function, derive the inverted demand function knowing that in
equilibrium the quantity demanded should be equal to the quantity supplied by the
monopolist. That is,
D( p) = q ⇒
100
= q ⇒ P (q ) = ±10 / q
p2
Since p ≥ 0 , we retain P (q ) = 10 / q .
The monopolist’s profit function is then given by:
π (q ) = P(q ) * q − c(q ) = 10q −1/2 q − 2q = 10q1/2 − 2q
The first order conditions:
∂π
= 0 ⇔ 5q −1/2 − 2 = 0
∂q
25
⇔ qm =
4
Next, substitute this quantity into the inverted demand function to get the price:
pm = P (
25
10
)=
=4
4
5/2
c) Monopolist price and quantity when sales taxes is 12%
The new profit function becomes:
π=
P(q) × q
10q1/ 2
− c (q ) =
− 2q = 8.93q1/ 2 − 2q
1+ t
1.12
Taking the FOC:
4.46q −1/2 − 2 = 0 ⇒ q = 2.232 = 4.98 , which is the monopolist’s quantity.
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The consumer’s price is
p = 10 / 2.23 = 4.48 , and the monopolist’s price is
4.48/1.12 = 4.
d) Price elasticity of demand if D ( p ) =
ò( p ) = −
C
Pò
 −ε p ε −1  p
dD ( p ) p
= −C 
=ε

2ε
dp D ( p )
 p
 C
pε
Problem 2
D ( p ) = 10 − p for p ∈ [ 0,10]
The inverted demand curve is : P(q)=10-q
a) Price elasticity of Demand
ò( p ) = −
dD ( p ) p
p
p
= − ( −1)
=
dp D ( p )
10 − p 10 − p
b) Monopolist’s quantity and price
Profit function
π (q) = (10 − q)q − 2q
Taking the FOC:
∂π
= 0 ⇔ 10 − 2q − 2 = 0
∂q
And pm = 6
⇔ qm = 4
c) Effect of an unexpected decrease of price from p1 to p2
Number of people willing to buy the good at p1 or higher:
D( p1 ) = 10 − p1
Number of people willing to buy the good at a price between p1 and p2
D( p2 ) − D( p1 ) = 10 − p2 − (10 − p1 ) = p1 − p2
The total quantity produce by the firm will be
q = 10 − p1 + p1 − p2 = 10 − p2
Thus the new profit function is :
2
π ( p1 , p2 ) = (10 − p1 ) p1 + ( p1 − p2 ) p2 − 2(10 − p2 )
Taking the FOC with respect to p1 and p2 :
 ∂π ( p1, p2 )
22

=0
p1 =
 ∂p

10 − 2 p1 + p2 = 0


3
1
⇔
⇔

2
2
0
p
−
p
+
=
π
∂
(
p
p
)
14
 1
2
1, 2

p =
=0
 2 3
 ∂p2
The total quantity is therefore:
q = 10 − p2 = 10 −
14 16
=
3
3
c) Comparaisons
p2 < pm < p1
This makes sense because with two prices, the monopolist can afford to have a
high initial price, to reap profits from the high-value customers first – then he posts a
discounted price to cover the low-value customers. If the monopolist can only post one
price, then he is forced to hit the middle ground.
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Problem 3
The monopolist’s profit at quantity q is
π(q) = qP (q) − C(q),
where P (q) is the inverse demand function, and C(q) is the cost function. Therefore, the
monopolist’s marginal profit at quantity q is
π 0 (q) = P (q) + qP 0 (q) − C 0 (q).
(1)
Since the elasticity at quantity q is
(q) = −
P (q)/q
P (q)/q
dq/q
=−
=− 0
≤ 1,
dP (q)/P (q)
dP (q)/dq
P (q)
we have P (q) ≤ −qP 0 (q) since −P 0 (q) ≥ 0 and q ≥ 0. Plug P (q) ≤ −qP 0 (q) into the
marginal profit in (1) gives (since the marginal cost C 0 (q) is negative):
π 0 (q) = P (q) + qP 0 (q) − C 0 (q) ≤ −qP 0 (q) + qP 0 (q) − C 0 (q) < 0,
i.e., the marginal profit is negative at quantity q. Since q is arbitrary, monopolist’s marginal
profit is negative at every quantity. Therefore, it is best for the monopolist to produce
nothing: q = 0.
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