Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
Industrial Organisation (ES30044)
Topic One:
Outline: 1.
2.
3.
4.
1.
Monopoly (iii) – Durable Goods
Introduction
The Coase Conjecture
A Durable Good Monopoly Facing a Continuous Demand Curve
A Durable Goods Monopolist Facing a Discrete Demand Curve
Introduction
Now assume that monopolist sells durable rather than flow goods. Flow goods are purchased
repeatedly and perish after usage (e.g. food). Durable goods are bought infrequently (once?)
and last a long time (e.g. house, car, land). Clearly, with the exception of land, all goods
ultimately perish and so the two concepts are relative to a certain time horizon that is relevant
to the consumer.
2.
The Coase Conjecture (1972)
In his pioneering analysis, Coase (1972) conjectured that a monopoly selling a durable good
would behave differently to one selling a flow good. Essentially, such a monopolist creates
his own competition. By selling a durable good today, he reduces demand for the good
tomorrow. In order to sell to the residual demand, the monopolist lowers the prices
tomorrow. Consumer’s, however, anticipate this and hold back on the purchases today. These
rational expectations hurt the monopolist because strategic buyers, anticipating that the price
would drop in the ‘twinkling of an eye’, will refuse to buy as long as the price remained
above the competitive level.
To illustrate, consider the extreme case of someone who owns all the land in the
world. The demand curve for land is:
pd (q) = α − βq
⇔
qd ( p) =
(1)
α 1
− q
β β
Thus we assume that α β is the total amount of land in the world, being the demand that
would occur if the seller ‘gave it away for nothing’. Assume for simplicity the ‘cost’ of land
to the seller is zero. If we regarded land as a flow good then the seller would maximise
profits (i.e. revenues) vis:
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
max π ( q ) = p ( q ) ⋅ q = (α − β q ) ⋅ q = α q − β q 2
q
⇒
∂TR ( q )
∂q
= α − 2β q = 0
(2)
⇒
q=
α
2β
And:
⎛ α ⎞ α
pm qm = a − βqm = α − β ⎜ ⎟ =
⎝ 2β ⎠ 2
( )
(3)
That is, the seller would sell half of the total amount of land in the world, for half of the
maximum price consumers would be ever willing to pay.
p
pd = α - βq
α
pm = α/2
0
E1
qm = α/2β
α/β
q
Figure 1
Note that at E1:
Ε=−
∂q p
1 (α 2 )
⋅ =− ⋅
=1
∂p q
β (α 2β )
(4)
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
Intuitively, the seller maximises revenue by supplying the level of output at which the
elasticity of demand is unitary, since at this point a one per cent fall (rise) in the price leads to
a one percent rise (fall) in demand.
Now what will happen next year? The monopoly still owns half of the world’s land,
and there is no reason to assume that it will not offer that land for sale next year.
p
pd = α - βq
α
E1
p1 = α/2
E2
p2 = α/4
0
q1 = α/2β!0
q2!α/4β
α/β!α/2β
q
Figure 2
In Figure 2, the initial equilibrium of the firm (point E in Figure 1) is now defined as E1, and
the associated year 1 equilibrium price and quantity as p1 = α 2 and q1 = α 2β . Thus in year
2 we can subtract the land already supplied by the monopolist - i.e. q1 = a 2b - from the
demand curve, thereby envisaging a year 2 origin at q1 = α 2β = 0 . Repeating the year 1
analysis, we locate the year 2 optimal price and output for the monopolist at p2 = a 4 and
q2 = α 4β respectively.
Note that the monopolist price in year 2 is less than the monopolist price in year 1
vis. p1 = α 2 > α 4 = p2 . Thus, those consumers who do not discount time too heavily will
postpone buying land until year 2. Hence, the current demand facing the monopolist falls,
implying that the monopoly will charge a lower price than that which a monopolist selling a
perishable flow good would charge.
3.
A Durable Good Monopoly Facing a Continuous Demand Curve
Numerical Example
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
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
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 ) = 100 − q
(5)
Now compare the monopolist's profits from than two types of commercial transaction vis.
selling and renting. Note:
(1) By selling a product to a consumer for a price of pS the firm transfers all rights of ownership for
using the product and getting the product back from the consumer from the time of purchase
extended indefinitely;
(2) By renting a product to a consumer for a price of pR the renter maintains ownership of the product
but contracts with the consumer to allow the consumer to derive services from the product for a given
period specified in the renting contract.
Thus ‘selling’ means charging a single price for an indefinite period, whereas ‘renting’
means charging a price for using the product for a specific limited time period.1
A Renting Monopoly
Assume that in each period the monopolist rents a durable good for one period only (e.g.
lease a car). In what follows we will show that leasing yields a higher profit than selling.
Assume that in each of the two periods the monopolist faces the demand curve shown in
Figure 3.
Assuming zero costs of production, the monopoly would rent an amount determined
by the condition:
( )
( )
MR qtR = 100 − 2qtR = MC qtR = 0
⇒
(p
R
t
)
,qtR = (50,50 )
(6)
⇒
π tR = 2500
Hence the lifetime sum of profits for the renting monopolist are:
π R = π 1R + π 2R = 5000
(7)
A Seller Monopoly
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.
1
Note that Definition 1 does not imply that by selling, the manufacturer always transfers all rights to the
products sold. For example, even when a product is sold (rather than rented) the new owner does not have the
rights to produce identical or similar products if ht product is under patent protection.
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
Formally, 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.
p
pd = 100 - q
100
pt = 50
Et
0
qt = 50
100
q
Figure 3
Period 2
Figure 4 illustrates the (residual) demand curve facing the monopoly in period 2, after it has
sold q1 units in period 1, given by:
q2 = (100 − q1 ) − p2
⇒
(8)
p2 = 100 − ( q1 + q2 )
Thus total revenue in period two is given by:
TR2 ≡ p2 q2 = 100q2 − ( q1 + q2 ) q2
⇒
(9)
TR2 = 100q2 − q1q2 − q
2
2
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
Since production is assumed to be costless, in the second-period the monopoly sets:
MR2 ≡
∂TR2
= 100 − q1 − 2q2S = 0
∂q2
⇒
(10)
q2S = 50 −
q1
2
Hence, the second period price and profit levels are given by:
⎡
⎛
q ⎞⎤
q
p2S = 100 − q1 + q2S = 100 − ⎢ q1 + ⎜ 50 − 1 ⎟ ⎥ = 50 − 1
2 ⎠⎦
2
⎝
⎣
(
)
⎛
q⎞
π = p ⋅ q = ⎜ 50 − 1 ⎟
2⎠
⎝
S
2
S
2
(11a)
2
S
2
(11b)
p
p2 = 100 − (q1 + q2 )
100 − q1
E2
p2
D2
0
q2
100 − q1
MR2
q
Figure 4
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 100 − q1 - will be
indifferent between purchasing in the first period and gaining a net utility of 2 (100 − q1 ) − p1 ,
and
buying
in
the
second
period
and
gaining
a
net
utility
of
(100 − q1 ) − p2 = (100 − q1 ) − (50 − q1 2) . Thus:
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
⎛
q⎞
2 (100 − q1 ) − p1 = (100 − q1 ) − ⎜ 50 − 1 ⎟
2⎠
⎝
⇒
p1 = 150 −
(12)
3q1
2
Note that (12) can also be derived by observing that the period 1 price should include the
period 2 price in addition to pricing the period 1 service, because buying in the first period
yields services for two periods, hence, the product can be resold in the second period for a
price of p2. Thus:
p1 = 100 − q1 + p2 = 100 − q1 + 50 −
q1
3q
= 150 − 1
2
2
(13)
which is identical to (12).
In a (sub-game perfect) equilibrium, the selling monopoly chooses a first period level
of output, q1 , that solves:
⎛
⎛
3q ⎞
q⎞
max π = (π 1 + π 2 ) = p1q1 + p2 q2 = ⎜ 150 − 1 ⎟ q1 + ⎜ 50 − 1 ⎟
q1
2 ⎠
2⎠
⎝
⎝
2
(14)
Thus:
⎛
qS ⎞
5q S
∂π
= 150 − 3q1S − ⎜ 50 − 1 ⎟ = 100 − 1 = 0
∂q1
2⎠
2
⎝
⇒
q1S = 40
40
= 30
2
120
p1S = 150 −
= 90
2
40
p2S = 50 −
= 30
2
(15)
q2S = 50 −
Hence:
π S = p1S q1S + p2S q2S = 90 ⋅ 40 + 30 ⋅30 = 3600 + 900 = 4500 < 5000 = π R
(16)
Comment
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 sellingdurable-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 consumer to pay high prices in the first period. In other
words, the monopoly cannot commit itself not to reduce future prices, the monopoly s
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
induced to lower its first-period price.2 Coase’s conjecture (i.e. the above argument) is not
valid when the number of consumers is finite – see Section IV following.
4.
A Durable Good Monopoly Facing a Discrete Demand Curve
The analysis of Section 3 confined itself to a demand curve with a continuum of non-atomic
buyers. Following Bagnoli et al. (1989), we now provide a simple example that demonstrates
that Coase’s conjecture is false when the number of consumers is finite.3
Consider an economy with two consumers, i = H, L, who live for two periods. Both
consumers desire car services for the two periods of their lives, but they differ in their
willingness to pay for these services. To be sure, the maximum price that Consumer H is
willing to pay for one period of car services is V H and the maximum amount that Consumer
L is willing to pay for one period of car service is V L . We assume that consumers’
willingness to pay per period of car services are substantially different with V H > 2V L > 0 .
Figure 5 below illustrates the aggregate inverse demand function for one period of car
services facing the monopolist each period.
pt
VH
VL
0
1
2
qt
Figure 5: Durable Good Monopoly – One Period (Discrete) Demand
Because the product is durable, consumers buy it only once in their life at either t = 1 or t = 2.
The utility functions for consumers type i = H, L that yield the demand structure illustrated in
Figure 4 are given by:
2
This type of argument has led some economists to argue that monopolies have an incentive to produce less
than the optimal level of durability (e.g. light-bulbs that burn very fast).
3
See Bagnoli, M., S. W. Salant and J. E. Swierzbinski. (1989). ‘Durable-Goods Monopoly with Discrete
Demand.’ The Journal of Political Economy, 97(6), pp. 1459-1478.
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Industrial Organisation (ES30044
⎧ 2V i − p
1
⎪⎪
i
i
U ≡⎨ V −p
2
⎪
0
⎪⎩
Topic One: Monopoly (iii) - Durable Goods
if he buys a car in period 1
if he buys a car in period 2
(17)
if he does not buy a car in any period
Thus, if consumer i buys a car in the first period, he gains a benefit of 2V i since the car
provides services for two periods, and he pays whatever the monopoly charges in t = 1. In
contrast, if consumer i waits and purchases the car at t = 2, he gains only one period of utility,
V i , minus the price charges in period 2.
On the production side, we assume that there is only one firm producing cars and that
they do this at zero cost. Like the consumers, the monopoly lives for two periods and
maximises sales during the two periods – i.e. he chooses p = ( p1 , p2 ) to maximise the sum of
revenues from two periods of sales given by π = p1q1 + p2 q2 . We denote by qt the quantity of
cars produced and by pt the period t price of a car set by the monopoly in period t, t = 1, 2.4
A Renting Monopolist
Assume first that the monopoly does not sell cars but instead rents them for one period only.
Thus, each consumer who rents a car at t = 1 has to return the car at the end of the first period
and rent it again at t = 2. We denote by ptR the rental price for one period of renting in period
t. Since car rentals last for one period only, it is sufficient to calculate ptR for each period
separately. Since the renting firms is a monopoly, it has two options: (i) Setting ptR = V H ,
which, by (17) induces only Consumer H to rent a car in each period, while Consumer L will
not rent; (ii) setting ptR = V L which induces both consumers to rent a car in each period. In
the first case, the two period profit is π R = 2V H and in the second case it is π = π R = 4V L .
Since V H > 2V L by assumption, a renting monopolist would rent cars only to the high
valuation consumer by setting a rental price equal to ptR = V H , t = 1, 2; and it will earn a twoperiod profit of π R = 2V H .
A Seller Monopoly
Now suppose that the monopoly sells the cars to consumers. We denote the selling price
by ptS , t = 1, 2. By paying p1S , the consumer pays for two periods of car-use (as compared
with the renting price ptR that entitles the consumer to one period of car use only). As
always, we solve this game recursively.
The Second Period
The effect of selling in the first period on second period demand is illustrated in Figure 6. If
Consumer H purchases in period 1, then only Consumer L demands a car in period 2. If
Consumer H does not purchase a car in period 1, then the second period demand is the given
(rental) demand curve as per Figure 5.
Figure 7 illustrates the sub-games associated with consumers H’s decisions whether
to purchase a car in the first period. It is apparent from Figure 6 that when Consumer H buys
4
Note that we have implicitly assumed that buyers and the monopoly do not discount future utility and profit,
since assuming otherwise would not have a qualitative effect on our results.
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
in the first period, then the monopoly will maximise period 2 profit by setting p2S = V L and
earning a period 2 profit of π 2 = V L (and a total profit of π = 2V H + V L ) and thereby
extracting all surplus from the consumers. If, however, Consumer H does not buy in the first
period, then in the second period the monopoly faces that entire demand and will therefore ,
by assumption, charge p2S = V H (i.e. selling only to Consumer H since V H > 2V L > 0 )
yielding a second period profit of π 2 = V H .
The First Period
In the first period, the monopoly sets p1S and the consumers decide whether to purchase the
good or not. Figure 7 illustrates the sequence of moves in the two periods.
Since Consumer L knows that the price in the second (i.e. last) period will never fall
below V L , Consumer L will buy in the first period at any price less than or equal to 2V L .
Hence, if the seller sets p1S = 2V L , both consumers would purchase initially. To simplify the
game tree, we report in Figure 7 the payoffs to the three players if the monopoly sets this low
price.
p2
VL
0
q2
1
Figure 6: Durable Good Monopoly – Period 2 Demand Given Period 1 Sales
Clearly, the monopoly will not set p1S > 2V H because the price exceeds the two period sum
of Consumer H’s valuation. Therefore, we now check whether p1S = 2V H is the profitmaximising first period price for the seller monopoly. From the second period analysis, we
conclude that Consumer H earns a utility of zero ( U H = 0 ) irrespective of whether he buys
the product in the first period. Hence, buying the product is an optimal response for
Consumer H to the first period price of p1S = 2V H . Thus, in a sub-game perfect equilibrium
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Industrial Organisation (ES30044
Topic One: Monopoly (iii) - Durable Goods
(SPE), p1S = 2V H , Consumer H buys in the first period, p2S = V L , and Consumer L buys in
the second period. We can therefore conclude that the Coase Conjecture is false under
discrete demand since durable good monopoly facing a discrete demand will:
1. Charge a first period selling price that is equal to the sum of the per period rental
prices vis: p1S = 2V H = 2 ptR
2. Earn a higher profit than a renting monopoly vis: π S = 2V H + V L > 2V H = π S
Thus, in the case of discrete demand, a selling monopoly can extract a higher surplus from
consumers than the renting monopoly. Coase conjectured that the ability of a durable good
monopoly to extract consumer surplus is reduced when the monopoly is forced to sell rather
than to rent because it was unable to commit not to reduce future prices. Here we have
demonstrated that opposite case, that is, selling enables the monopoly to price discriminate
across different consumers by setting prices that induce different consumers to purchase at
different time periods.
S
t = 1: Monopoly sets p1
⎧
π S = 4V L
⎪⎪
UL =0
p = 2V ⎨
⎪ H
H
L
⎪⎩U = 2 (V − V )> 0
S
1
L
p1S = 2V H
t = 1: Consumer H decides whether to purchase
Consumer H Buys
Consumer H Does Not Buy
S
t = 2: Monopoly sets p2
p2S = V L
p2S = V H
p2S = V L
π S = 2V H + V L
π S = 2V H
π S = 2V L
U =U = 0
U =U = 0
U H =V H −V L > 0
H
L
H
L
UL =0
p2S = V H
π S =V H
U H =U L = 0
Figure 7: Two-Period Durable-Good Monopoly Game with Discrete Demand
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