Industrial Organisation (EC30174)
Seminar One: Monopoly
Industrial Organisation (EC30044)
Seminar One:
Monopoly - Solutions
1. A law is passed fixing, below the level currently prevailing, the maximum price that can
be charged by a monopolist. The monopolist responds by increasing output. Is this
behaviour compatible with profit maximisation? Explain.
Solution: The effects of a law fixing below the level currently prevailing, the maximum price
that can be charged by a monopolist, is illustrated below. It may be seen that under certain
circumstances it is quite rational for the monopolist to increase output. The monopolist is
originally in equilibrium producing q0 units of output at price p0. The maximum price
imposed by the law is p1 < p0. The new average revenue curve facing the firm is given by (p1,
E1, AR), and the new marginal revenue curve by (p1, E1, A, MR) i.e. the MR curve has a
discontinuity at (E1, A,). Costs are not affected and so the firm increases output to q1 where
marginal cost is equal to marginal revenue i.e. MC cuts MR in the discontinuous part of the
MR curve. Thus it is compatible with profit maximising behaviour for the firm to increase
output in these circumstances. If the firm's marginal cost curve cuts its (original) marginal
revenue curve at a price above that imposed by the statute then the firm would, however,
reduce output. If the price cap was set below the minimum of average costs, the firm would
close.
p
p0
p1
E0
E1
MC
AR
A
0
q0
q
q1
MR
Figure 1
2. A profit-maximizing monopolist faces the demand curve q = 100 - 3p. It produces at a
constant marginal cost of £20 per unit. A quantity tax of £10 per unit is imposed on the
monopolist’s product. The price of the monopolist’s product
1
Industrial Organisation (EC30174)
A.
B.
C.
D
E.
Seminar One: Monopoly
rises by £5
rises by £10
rises by £20
rises by £12
stays constant
Solution:
q d = 100 − 3 p
⇒
1
p d = (100 − q )
3
Thus:
⎡1
⎤
π = ⎢ (100 − q ) − 20 ⎥ q
⎣3
⎦
⇒
∂π 100 2 ∗
=
− q − 20 = 0
∂q
3 3
⇒
q∗ = 20
1
80
p∗ = (100 − q∗ )=
3
3
With tax:
⎡1
⎤
π = ⎢ (100 − q ) − 30 ⎥ q
⎣3
⎦
⇒
∂π 100 2 ∗
=
− qt − 30 = 0
∂q
3 3
⇒
qt∗ = 5
pt∗ =
1
95
100 − qt∗ )=
(
3
3
3. A monopolist with constant marginal costs faces a demand curve with a constant elasticity
of demand and does not practice price discrimination. If the government imposes a tax of
$1 per unit of goods sold by the monopolist, then the monopolist will increase his price by
more than $1 per unit. True / False? Explain.
Solution: We know that the monopoly will produce where:
2
Industrial Organisation (EC30174)
Seminar One: Monopoly
⎛
1⎞
p ⎜ 1− ⎟ = MC
E⎠
⎝
⇒
p=
MC
⎛
1⎞
⎜⎝ 1− E ⎟⎠
⇒
⎛ E ⎞
p=⎜
MC
⎝ E − 1⎟⎠
where E = − (∂q ∂p )( p q ) > 1 . The tax will raise the monopolist’s marginal cost by $1.
Thus we can note that:
⎛ E ⎞
∂p
=⎜
>1
∂MC ⎝ E − 1⎟⎠
4. Consider the market for iPhones. Apple holds the patent for iPhones. On the demand side,
there are n H high income consumers who are willing to pay a maximum price of V H for
an iPhone, and n L low-income consumers who are willing to pay a maximum price of V L
for an iPhone. Assume that V H > V L > 0 and that each consumer buys only one iPhone.
Suppose that Apple cannot price discriminate and is therefore constrained to set a uniform
market price:
(a) Draw the aggregate-demand curve facing Apple;
(b) Find the profit-maximising price set by Apple, considering all possible parameter
values of n H ,n L ,V H ,V L . Assume that production is costless.
(
)
Solution:
(a) Draw the aggregate-demand curve facing Apple;
3
Industrial Organisation (EC30174)
Seminar One: Monopoly
p
VH
VL
Demand
nH
0
nH + nL
q
Figure 2
Figure 2 illustrates an aggregate demand composed of the two groups of consumers, where
each group shares a common valuation for the product:
(b) Find the profit-maximising price set by Apple, considering all possible parameter values
of (n H , n L , V H , V L ). Assume that production is costless.
The monopoly has two options: (i) setting a high price, p = V H ; or (ii) setting a low price,
p = V L . Figure 1 reveals that the profit levels (i.e. revenue, since production is costless) are
given by:
π
p =V H
= n HV H
And:
π
p=V L
(
)
= nH + nL V L
Comparing the two profit levels yields the monopoly’s profit maximizing price. Hence:
⎧ H
⎪V
m
p =⎨
⎪V L
⎩
If
⎛ nH + nL
VH >⎜
H
⎝ n
Otherwise
⎞ L
⎟V
⎠
4
Industrial Organisation (EC30174)
Seminar One: Monopoly
Thus, the monopoly sets a high price if either there are relatively many (few) high (low)
valuation consumers (i.e. nH is large or nL is small) and / or high (low) valuation consumers
are willing to pay a relatively high (low) price (VH is high or VL is small). Note that if
n H = n L = 1 then we have:
⎧ H
⎪ V
p =⎨
⎪⎩ V L
m
V H > 2V L
If
Otherwise
As per the handout.
5. A monopoly faces a demand function given by p = α − β q and has a unit production cost
of c > 0. Suppose the government imposes a specific tax of £t per unit on each unit of
output sold to consumers:
(a) Show that this tax imposition would raise the price paid by consumers by less than
t;
(b) How would your answer to (a) change if the market demand curve was given by
p = q −θ ?
Solution:
(a) Show that this tax imposition would raise the price paid by consumers by less than t;
Total revenue is given by:
(
)
TR = pq = α − β q q
⇒
TR = α q − β q 2
Marginal revenue is thus:
MR ≡
∂TR
= α − 2β q
∂q
Equating marginal revenue to the tax inclusive unit cost implies:
MR = α − 2β q = c + t
⇒
qt =
α −c−t
2β
Thus:
5
Industrial Organisation (EC30174)
Seminar One: Monopoly
pt = α − β qt
⇒
⎛α − c−t⎞
pt = α − β ⎜
⎝ 2β ⎟⎠
⇒
pt =
α +c+t
2
such that:
∂p t 1
= <1
∂t 2
Hence, as illustrated in Figure 3, an increase in t raises the monopoly price by less than t:
p
α
pt
p
c+t
c
MCt
MC
MR
0
AR
q
q1 q0
Figure 3
(b) How would your answer to (a) change if the market demand curve was given by p = q −θ ?
This is the case of a constant-elasticity demand curve. To be sure:
6
Industrial Organisation (EC30174)
Seminar One: Monopoly
−θ
dp
q
p
= −θ q −θ −1 = −θ
= −θ
dq
q
q
⇒
dp q
p q
⋅ = −θ ⋅ = −θ
dq p
q p
⇒
dq p 1
Ε≡− ⋅ = >0
dp q θ
Recall:
TR = p ( q ) q
⇒
MR ≡
dTR dp
=
q+ p
dq
dq
⇒
⎛
⎛ dp q ⎞
⎛
1⎞
1⎞
MR = p ⎜ 1+
⋅ ⎟ = p ⎜ 1− ⎟ = p ⎜ 1− 1 ⎟
E⎠
⎝ dq p ⎠
⎝
⎝
θ⎠
⇒
MR = p (1− θ )
Note that MR > 0 ⇒ θ < 1 ⇒ θ −1 = Ε > 1 - in words, positive marginal revenue requires elastic
demand. Monopoly equilibrium implies:
MR = p (1− θ ) = c + t = MC t
⇒
p=
c+t
1− θ
Thus:
∂p
1
=
>1
∂t 1− θ
Thus, price will increase by more than the increase in the tax – see Figure 4 folowing:
7
Industrial Organisation (EC30174)
Seminar One: Monopoly
p
pt
p
c+t
c
MCt
MC
AR
MR
0
q1 q0
q
Figure 4
6. Assume a continuum of consumers having different valuation for the annual services of a
car that are summarised by the familiar downward sloping demand curve. Suppose that
the consumers live for two periods denoted by t = 1, 2, and that the monopolist sells a
durable good that lasts for two periods. Thus if a consumer can purchase the good and
have it for his entire life, never having to replace it. Assume that the consumers’ different
valuations for the product are summarised by the aggregate period t = 1 inverse demand
curve for one period of service given by p ( q ) = α − β q . Compare the monopolist’s profits
from than two types of commercial transaction vis. Selling and renting.
Solution:
First note that:
pd (q) = α − βq ⇔ qd ( p) =
α 1
− p
β β1
(1)
Renting Monopoly
Assume that in each period the monopolist rents a durable good for one period only (e.g.
lease a car). Assuming zero costs of production, the monopoly would rent an amount
determined by the condition:
8
Industrial Organisation (EC30174)
( )
Seminar One: Monopoly
( )
MR qtR = α − 2β qtR = MC qtR = 0
⇒
qtR =
α
2β
⇒
(2)
⎛ α ⎞ α
ptR = α − b ⎜ ⎟ =
⎝ 2β ⎠ 2
⇒
⎛ α ⎞ ⎛α ⎞ α2
π tR = ⎜ ⎟ ⎜ ⎟ =
⎝ 2β ⎠ ⎝ 2 ⎠ 4β
Hence the lifetime sum of profits for the renting monopolist are:
π R = π 1R + π 2R =
α2
2β
(3)
Note that in the numerical example from Monopoly (iii) we had α = 100 and β = 1 such that
π R = α 2 2β = 1002 2 = 5000 as per equation (3).
Seller Monopoly
Recall that a seller monopoly knows that those consumers who purchase the durable good at t
= 1 will not repurchase at t = 2. That is, at t = 2 the monopoly will face a demand for its
product that is lower than the period 1 demand by exactly the amount it sold at t = 1.
Therefore at t = 2 the monopolist will have to sell at a lower price resulting from a lower
demand, caused by its own earlier sales.
Again, we define this two-period game as follows: The payoff to the monopolist is
the total revenue generated by period 1 and period 2 sales. The strategies of the seller are the
price set in period 1, p1 , and the price set in period 2 as a function of the amount purchased
in period 1, p2 ( q1 ) . The strategies of the buyers are to buy or not to buy as a function of the
first period price, and to buy or not to buy as a function of the second period price. We look
for a (sub-game perfect) equilibrium to this game. The methodology for solving any twoperiod game is to solve by backward induction - i.e. to determine how the monopolist would
behave in period 2 for each possible set of buyers then remaining.
Period 2
The (residual) demand curve facing the monopoly in period 2, after it has sold q1 units in
period 1 is given by:
9
Industrial Organisation (EC30174)
Seminar One: Monopoly
⎛α
⎞ ⎛ 1⎞
q2 = ⎜ − q1 ⎟ − ⎜ ⎟ p2
⎝β
⎠ ⎝β⎠
⇒
(4)
p2 = α − β ( q1 + q2 )
Since production is costless, in the second-period the monopoly total revenue is:
TR2 ( q2 ) = ⎡⎣α − β ( q1 + q2 ) ⎤⎦ q2 = α q2 − β q1q2 − β q22
(5)
Such that it sets marginal revenue according to:
( )
MR2 q2S = α − β q1 − 2β q2S = 0
⇒
q2S =
(6)
α − β q1
2β
Hence, the second period price and profit levels are given by:
⎡
⎛ α − β q1 ⎞ ⎤ α − β q1
p2S = α − β q1 + q2S = α − β ⎢ q1 + ⎜
⎥=
2
⎝ 2β ⎟⎠ ⎦
⎣
(
)
1 ⎛ α − β q1 ⎞
π = p ⋅q = ⎜
β ⎝ 2 ⎟⎠
s
2
S
2
(7)
2
S
2
(8)
Period 1
Suppose that the monopolist sells in period 1 to q1 buyers with the highest reservation prices.
Thus, the marginal buyer - that is, the buyer with a reservation price of p1 = α − β q1 - will be
indifferent between purchasing in the first period and gaining utility of 2 (α − β q1 ) − p1 , and
buying in the second period and gaining utility of (α − β q1 ) − p2s = (α − β q1 ) − ⎡⎣(α − β q1 ) 2 ⎤⎦ .
Thus:
2 (α − β q1 ) − p1 =
α − β q1
2
⇒
p1 =
(9)
3
(α − β q1 )
2
10
Industrial Organisation (EC30174)
Seminar One: Monopoly
In a (sub-game perfect) equilibrium, the selling monopoly chooses a first period level of
output, q1 , that solves:
⎡3
⎤
1 ⎛ α − β q1 ⎞
max π = (π 1 + π 2 ) = p1q1 + p2 q2 = ⎢ (α − β q1 ) ⎥ q1 + ⎜
q1
β ⎝ 2 ⎟⎠
⎣2
⎦
2
(10)
Thus:
⎛ α − β q1 ⎞
∂π 3α
=
− 3β q1 − ⎜
=0
∂q1
2
⎝ 2 ⎟⎠
⇒
q1S ≡ q1 =
2α
5β
(11)
And so:
q2S =
α − β q1 α q1 α
2α 5α − 2α 3α
=
− =
−
=
=
2β
2β 2 2β 10β
10β
10β
(12)
p1S =
⎛ 2α ⎞ ⎤ 3 ⎛ 5α − 2α ⎞ 9α
3
3⎡
α − β q1 ) = ⎢α − β ⎜ ⎟ ⎥ = ⎜
=
(
2
2⎣
5 ⎟⎠ 10
⎝ 5β ⎠ ⎦ 2 ⎝
(13)
p2S =
α − β q1 1 ⎡
⎛ 2α ⎞ ⎤ 1 ⎛ 5α − 2α ⎞ 3α
= ⎢α − β ⎜
=
⎥=
2
2⎣
5 ⎟⎠ 10
⎝ 5β ⎟⎠ ⎦ 2 ⎜⎝
(14)
Note that in terms the specific numerical example in Monopoly (iii) where α = 100 and
β = 1 we have:
q1S =
2α 200
=
= 40
5β
5
(15)
q2S =
3α 300
=
= 30
10β 10
(16)
p1S =
9α 900
=
= 90
10 10
(17)
11
Industrial Organisation (EC30174)
p2S =
Seminar One: Monopoly
3α 300
=
= 30
10 10
(18)
As per equations (11)-(14). Note also that the first period price is generally larger than the
second period price:
p1S =
9α 3α
>
= p2S
10 10
(19)
Also note that:
π S = p1S q1S + p2S q2S =
9α 2α 3α 3α
⋅
+
⋅
10 5β 10 10β
⇒
πS =
(20)
18α 2 9α 2 45α 2 9α 2
+
=
=
50β 100β 100β 20β
Recall again that in the numerical example Monopoly (iii) we had α = 100 and β = 1 such
that:
πS =
9α 2 9 ⋅1002 90000
=
=
= 4500
20β
20
20
(21)
As per equation (20). Finally, note that:
α 2 9α 2
π −π =
−
2β 20β
⇒
R
S
π R −πS =
10α 2 − 9α 2
α2
==
20β
20β
(22)
⇒
π R −πS > 0
Recall again that in the numerical example in Monopoly (iii) we had α = 100 and β = 1 such
that:
π R −πS =
α 2 1002 10000
=
=
= 500
20β
20
20
(23)
Thus the monopoly in a durable good market earns a lower profit by selling than by renting.
The intuition for this result is that rational consumers are able to calculate that a selling-
12
Industrial Organisation (EC30174)
Seminar One: Monopoly
durable-good monopolist would lower future prices due to a future fall in demand resulting
from having some consumers purchasing the durable product in earlier periods. This
calculation reduces the willingness of consumers to pay high prices in the first period the
monopoly offer the product for sale. In other words, the monopoly cannot commit itself not
to reduce future prices, the monopoly is induced to lower its first-period price.
7. Consolidated Incorporated operates a monopoly cartel of 12 members and faces an
aggregate demand function for its output of p ( Q ) = 200 − 40Q . How much profit does it
make if each constituent member faces a total cost function TCi ( qi ) = 50 + 10qi2 ?
Solution:
The cartel’s maximisation problem ia:
(
max π ( q ) = ( 200 − 40 ⋅12 ⋅ q )12 ⋅ q − 12 50 + 10q 2
q
)
⇒
max π ( q ) = 2400q − 5760q 2 − 600 − 120q 2
q
⇒
max π ( q ) = 2400q − 5880q 2 − 600
q
Thus:
∂π
= 2400 − 11760q m = 0
∂q
⇒
q m = 0.2041
Thus:
Q m = 12q m = 12 × 0.2041 = 2.4492
This implies:
p m = α − β Q m = 200 − 40 ⋅ 2.4492 = 102.032
And so:
13
Industrial Organisation (EC30174)
Seminar One: Monopoly
π m = p mQ m − 12 ⋅TCim
⇒
2
π m = 102.032 × 2.4492 − 12 ⎡⎢50 + 10 ( 0.2041) ⎤⎥
⎣
⎦
⇒
(14)
π m = 102.032 × 2.4492 − 12 ⎡⎢50 + 10 ( 0.2041) ⎤⎥
⎣
⎦
⇒
2
π m = 249.90 − 605 = −355.1
8. We are going to model the market for electricity in Bath. A representative residence gets
utility from consuming electricity, e, and other goods, x. The consumer has quasi-linear
preferences such that his demand for electricity depends only on the relative price of
electricity, p, and not on the consumer’s income. His utility function is thus linear in x and
concave in e. For example:
U ( x, e) = x + a ln e
where a is some constant. The price of other goods, x, is normalised to one.
(a) Derive the consumer’s (uncompensated) demand for electricity:
You need to setup the Lagrangian and solve for e in terms of the exogenous variables in the
problem. Note that consumer’s demand for electricity depends only on the relative price of
electricity, p, and not on the consumer’s income, and that the price of other goods x is one.
Thus, the consumer’s budget constraint is:
pe e + px x = M
⇒
pe
p
M
e+ x x =
px
px
px
⇒
pe
1
M
e+ x =
1
1
1
⇒
pe e + x = M
⇒
pe + x = M
Thus:
L = x + a ln e + λ (M − pe − x )
14
Industrial Organisation (EC30174)
Seminar One: Monopoly
∂L
= 1− λ = 0
∂x
∂L a
= −λp = 0
∂e e
∂L
= M − pe − x = 0
∂x
Thus:
e=
a
a
=
λp p
(b) Let a residence’s demand for electricity be of the following form:
e = 1000 − 5 p
There are 3519 residences in Bath. Treat them as if they are identical. Find the market
demand function. Define market demand as E.
All you need to do here is add up the individual demands.
E d = 3519e = 3519 * (1000 − 5p)
⇒
E d = 3519000 − 17595p
(c) There is one supplier of electricity in Bath. Call this electricity provider GSE. GSE
converts coal c, labour l, and capital k into electricity with the following production
function;
E = 900c.25k 3.5l 1.75
GSE has built its power plants already, so its problem is to choose coal and labour. Derive
GSE’s cost function, taking k as fixed and assuming that q is the price of coal, w is the price
of labour, and r is the cost of capital:?
How do you get the cost function? You set up the firm’s cost minimization problem, solve it to
get the firm’s cost minimising conditional factor demands, then plug the conditional factor
demands back into the firm’s total cost function:
So, the firm’s problem is:
(
)
(
)
min rk + wl + qc s.t. 900c0.25 k 3.50 l 1.75 = Ε
c,l
Set up the Lagrangian:
L(l,c, λ ) =
( rk + wl + qc ) +λ (Ε − 900c
0.25 3.50 1.75
k
l
)
15
Industrial Organisation (EC30174)
Seminar One: Monopoly
Solving this gives you conditional factor demands of the form:
∂L(l,c, λ )
= w -1.75λ 900c0.25 k 3.50 l 0.75 = 0
∂l
(
)
∂L(l,c, λ )
= q -0.25λ 900c −0.75 k 3.50 l 1.75 = 0
∂c
(
)
∂L(l,c, λ )
= Ε − 900c0.25 k 3.50 l 1.75 = 0
∂λ
Thus:
(
(
0.25 3.50 0.75
w 1.75λ 900c k l
=
q 0.25λ 900c −0.75 k 3.50 l 1.75
)
)
⇒
w 1.75c0.25l 0.75
=
q 0.25c −0.75l 1.75
⇒
w 7c
=
q
l
⇒
l=
7cq
w
Thus:
Ε − 900c0.25 k 3.50 l 1.75 = 0
⇒
Ε − 900c
0.25 3.50
k
⎛ 7cq ⎞
⎜⎝ w ⎟⎠
1.75
=0
⇒
Ε − 900c k
2
3.50
⎛ 7q ⎞
⎜⎝ w ⎟⎠
1.75
=0
⇒
⎡ Ε ⎛ w ⎞ 1.75 ⎤
c=⎢
3.50 ⎜
⎟ ⎥
⎢⎣ 900k ⎝ 7q ⎠ ⎥⎦
⇒
Ε 0.5 ⎛ w ⎞
c=
30k 1.75 ⎜⎝ 7q ⎟⎠
0.5
0.875
16
Industrial Organisation (EC30174)
Seminar One: Monopoly
And so:
l=
7cq
w
⇒
0.875
⎤
⎛ 7q ⎞ ⎡ Ε 0.5 ⎛ w ⎞
⎥
l=⎜ ⎟⎢
⎝ w ⎠ ⎢ 30k 1.75 ⎜⎝ 7q ⎟⎠
⎥⎦
⎣
⇒
Ε 0.5 ⎛ 7q ⎞
l=
30k 1.75 ⎜⎝ w ⎟⎠
0.125
Plug these into the firm’s cost of producing:
( )
C Ε = rk + wl + qc
⇒
⎡ Ε 0.5 ⎛ 7q ⎞ 0.125 ⎤
⎡ Ε 0.5 ⎛ w ⎞ 0.875 ⎤
⎥+ q⎢
⎥
C Ε = rk + w ⎢
1.75 ⎜
1.75 ⎜
⎟
⎟
⎢⎣ 30k ⎝ w ⎠
⎥⎦
⎢⎣ 30k ⎝ 7q ⎠
⎥⎦
⇒
( )
0.125
0.875
⎤
⎛ w⎞
Ε 0.5 ⎡ ⎛ 7q ⎞
⎢
⎥
C Ε = rk +
w
+
q
⎜⎝ 7q ⎟⎠
30k 1.75 ⎢ ⎜⎝ w ⎟⎠
⎥⎦
⎣
⇒
( )
( )
C Ε = rk +
(
Ε 0.5
w0.875q 0.125 7 0.125 + 7 −0.875
30k 1.75
)
⇒
( )
C Ε = rk + 7 0.193622
Ε 0.5
w0.875q 0.125
30k 1.75
This reduces to:
3.5 −0.5
rk + (900k )
= rk +
E
0.5
−0.25
1.75
⎡
⎤
2
2
⎡
⎤
⎡
⎤
w
w
⎢w
+q⎢ ⎥ ⎥
⎢ ⎢⎣ 7 q ⎥⎦
⎣ 7 q ⎦ ⎥⎥
⎢⎣
⎦
8 7 0.125 .5 0.875 0.125
E w q
= C (E )
7 30k 1.75
(d) Given that GSE has already sunk its capital cost, what is the lowest price at which GSE
produces electricity?
GSE produces when it can recover its variable costs. Thus, price must be greater than or
equal to its average variable costs, or:
17
Industrial Organisation (EC30174)
p≥
( ) = rk + 7
C Ε
0.193622
Ε
Seminar One: Monopoly
Ε −0.5 0.875 0.125
w q
30k 1.75
(N.B. you might be confused about why capital is in this expression even though it is fixed.
Recall that the production function is Ε = 900c0.25 k 3.50 l 1.75 , thus the firm’s cost of producing a
level of output is a function of the firm’s capital stock)
(e) Since GSE is the only market supplier of electricity, it does not have to take this price.
Set up GSE’s profit maximization problem.
GSE makes pE in revenue and this costs the firm the cost function above. Hence, profit is
π = pΕ − C(Ε) . The firm gets to choose both prices and output, so it substitutes in (the
inverse) market demand to get write its profit maximization problem in terms E. GES’
inverse demand function is:
Ε d = 3519000 − 17595p
⇒
pd =
3519000 ⎛ 1 ⎞
−
Ε
17595 ⎜⎝ 17595 ⎟⎠
⇒
⎛ 1 ⎞
p d = 200 − ⎜
Ε
⎝ 17595 ⎟⎠
Thus, its profit function is:
π = pΕ − C(Ε)
⇒
⎡
⎛ 1 ⎞ ⎤
π = ⎢ 200 − ⎜
Ε ⎥ Ε − C(Ε)
⎝ 17595 ⎟⎠ ⎦
⎣
(f) Let’s assume that GSE’s cost function is of the form
( )
C Ε = 800 + 100Ε +
24
Ε2
17595
How much electricity does GSE produce?
GES’ profit function is thus:
⎡
⎛ 1 ⎞ ⎤
⎛
⎞
24
π = ⎢ 200 − ⎜
Ε ⎥ Ε − ⎜ 800 + 100Ε +
Ε2 ⎟
⎟
17595 ⎠
⎝ 17595 ⎠ ⎦
⎝
⎣
⇒
Ε2
24
π = 200E −
− 800 − 100Ε −
Ε2
17595
17595
18
Industrial Organisation (EC30174)
Seminar One: Monopoly
Taking the derivative of profit with respect to E gives the first order condition:
∂π
2Ε ∗
48Ε ∗
= 200 −
− 100 −
=0
∂E
17595
17595
⇒
50
100 −
Ε∗ = 0
17595
⇒
⎛ 17595 ⎞
Ε ∗ = 100 ⎜
⎝ 50 ⎟⎠
⇒
Ε ∗ = 35190
(g) What is the market price for electricity?
Plug into the (inverse) market demand function to get the market price at this level of output
⎛ 1 ⎞ ∗
35190
p* = p Ε ∗ = 200 − ⎜
Ε = 200 −
= 200 − 2 = 198
⎟
17595
⎝ 17595 ⎠
( )
(h) What would the equilibrium price and quantity of electricity be if GSE were not the only
supplier of electricity (if the market is perfectly competitive)?
In this case, the firm produces where p = MC. MC is given by:
( ) = MC
∂C Ε
∂Ε
48
Ε
(Ε ) = 100 + 17595
Consumer’s still present the same market demand. So we set:
( )
p d Ε ∗ = 200 −
Ε∗
48 ∗
= 100 +
Ε = MC Ε ∗
17595
17595
( )
⇒
49Ε ∗
= 100
17595
⇒
⎛ 17595 ⎞
Ε ∗ = 100 ⎜
⎝ 49 ⎟⎠
⇒
Ε ∗ = 35908.1633
Notice that this value is larger than the equilibrium output in the monopolist’s case. Plug
this in to either of the two (producer or consumer) equations for price to get the market
price:
19
Industrial Organisation (EC30174)
( )
Ε∗
17595
( )
35908.1633
17595
p d Ε ∗ = 200 −
Seminar One: Monopoly
⇒
p d Ε ∗ = 200 −
⇒
( )
p d Ε ∗ = 200 − 2.04081633
⇒
( )
p d Ε ∗ = 197.959184
And:
( )
MC Ε ∗ = 100 +
48 ∗
Ε
17595
⇒
⎛ 48 ⎞
MC Ε ∗ = 100 + ⎜
35908.1633
⎝ 17595 ⎟⎠
( )
⇒
( )
MC Ε ∗ = 100 + 97.9591838
⇒
( )
MC Ε ∗ = 197.9591838
Notice that this is slightly less than the equilibrium price with a monopolist.
(i) Assume that there is a major technological improvement in the distribution of energy, and
this means that the consumer can more easily substitute between electricity and other goods.
How would you incorporate this into your model?
You could modify your utility function to allow e and x to be closer to perfect substitutes. For
example, if your utility function were of the form: u(e, x) = x + ae0.9 . Here, you are still
concave in your demand for electricity, but it is much closer to being linear in e.
Consequently, e and x are closer to being perfect substitutes.
4. A monopolist with constant marginal costs faces a demand curve with a constant elasticity
of demand and does not practice price discrimination. If the government imposes a tax of $1
per unit of goods sold by the monopolist, the monopolist will increase his price by more than
$1 per unit: TRUE FALSE
We know that the monopoly will produce where:
20
Industrial Organisation (EC30174)
Seminar One: Monopoly
⎛
1⎞
p ⎜ 1− ⎟ = MC
E⎠
⎝
⇒
p=
MC
⎛
1⎞
⎜⎝ 1− E ⎟⎠
⇒
⎛ E ⎞
p=⎜
MC
⎝ E − 1⎟⎠
(
)(
)
where E = − ∂q ∂p p q > 0 . The tax will raise the monopolist’s marginal cost by $1.
Thus we can note that:
⎛ E ⎞
∂p
=⎜
>1
∂MC ⎝ E − 1⎟⎠
5. Wobble’s Weebles is the only producer of weebles. It makes weebles at constant marginal
cost c (where c > 0) and sells them at a price of p1 per weeble in market 1 and at a price of p2
per weeble in market 2. The demand curve for weebles in market 1 has a constant price
elasticity of demand equal to -2. The demand curve for weebles in market 2 has a constant
price elasticity equal to -3/2. The ratio of the profit-maximizing price in market 1 to the
profit-maximizing price in market 2 is
a.
b.
c.
d.
e.
2/3.
1/3.
3/2.
3.
dependent on the value of c.
⎛
⎛
1⎞
1⎞
p1 ⎜ 1− ⎟ = MC = p2 ⎜ 1− ⎟
E1 ⎠
E2 ⎠
⎝
⎝
⇒
⎛ 1⎞
⎛
1 ⎞
⎜⎝ 1− 2 ⎟⎠ = p2 ⎜⎝ 1− 3 2 ⎟⎠
⇒
1
1
p1 = p2
2
3
⇒
p1 2
=
p2 3
21
Industrial Organisation (EC30174)
Seminar One: Monopoly
8. A profit-maximizing monopolist faces the demand curve q = 100 - 3p. It produces at a
constant marginal cost of $20 per unit. A quantity tax of $10 per unit is imposed on the
monopolist’s product. The price of the monopolist’s product
a.
b.
c.
d.
e.
rises by $5.
rises by $10.
rises by $20.
rises by $12.
stays constant.
q = 100 − p
⇒
p=
1
100 − q
3
(
)
Thus:
⎡1
⎤
π = ⎢ 100 − q − 20 ⎥ q
⎣3
⎦
⇒
∂π 100 2
=
− q − 20 = 0
∂q
3
3
⇒
q = 20
80
p=
3
(
)
With tax:
⎡1
⎤
π = ⎢ 100 − q − 30 ⎥ q
⎣3
⎦
⇒
∂π 100 2
=
− q − 30 = 0
∂q
3
3
⇒
q=5
95
p=
3
(
)
22