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Lecture 3: Monopoly
Advanced Microeconomics II
Yosuke YASUDA
Osaka University, Department of Economics
yasuda@econ.osakau.ac.jp
December 2, 2014
1 / 12
Monopoly Market
We have so far studied consumer and producer behaviors in the
perfectly competitive market in which each economic agent
optimally makes his/her decision given “market prices.”
This is one polar extreme on a spectrum of possible market
structures ranging from the more to the less competitive, since the
competitive market is so competitive that each agent has no power
to affect the price.
The other extreme is (pure) monopoly where there is a single
seller of a product for which there are no close substitutes in
consumption, and entry into the market is completely blocked by
technological, financial or legal impediments.
2 / 12
Monopoly Problem (1)
A monopoly firm takes the market demand function x(p) as given
and chooses price p or/and quantity q to maximize profit.
Let p(q) be the inverse demand function and c(q) be the cost
function. Then, the monopoly problem is given as:
M P 1 : max p(q)q − c(q).
q≥0
where p(q)q is a revenue and c(q) is a cost when the output is
fixed to q. Let π(q) = p(q)q − c(q) denote the revenue function.
Assume that the firm’s objective function π(q) is convex and
differentiable. Then, the first order condition is:
dπ(q)
=
dq
dp(q)
p(q) +
q −
dq

{z
}
marginal revenue (MR)
dc(q)
dq
 {z }
= 0.
marginal cost (MC)
3 / 12
Monopoly Problem (2)
Since the demand function is downward sloping,
dp(q)
dq
< 0,
dc(q)
dc(q) dp(q)
−
q>
.
dq
dq
dq
⇒ The monopoly price must be higher than the marginal cost.
p(q) =
Similarly, we can consider the monopoly problem where the firm
will choose price instead of quantity:
M P 2 : max px(p) − c(x(p)).
p≥0
Although, M P 2 is essentially identical to M P 1, the first order
condition gives us further insight about the monopoly pricing.
d[px(p) − c(x(p))]
dx(p) dc(x) dx(p)
= x(p) + p
−
= 0.
dp
dp
dx dp
Dividing both sides by
dx(p)
dp ,
we obtain
x(p)
1
dc(x)
=p+
= p(1 −
).
dx
dx(p)/dp
ǫ(p)
(1)
4 / 12
Monopoly Problem (3)
Note that ǫ(p) = − dx(p)/dp
x(p)/p (≥ 0) is called elasticity of demand:
% change in the quantity demanded per % change in price.
Rearranging (1), we can obtain an expression for the percentage
deviation of price from marginal cost:
p−
dc(x)
dx
p
=
1
,
ǫ(p)
which is used as a measure of the monopolist’s market power
called the markup, the pricecost margin, or the Lerner Index.
When market demand is infinitely elastic, i.e. under perfect
competition, ǫ(p) will be infinite and p = dc(x)
dx .
However, when market demand is less than infinitely elastic,
the former (strictly) exceeds the latter.
✞
☎
✝
✆
Fg Figure 4.3 (JR, pp.172)
5 / 12
Price Discrimination (1)
A monopoly firm may further raise profit by charging different
prices across consumers. This exercise is called price
discrimination. The traditional classification of the forms of price
discrimination listed as follows is due to Pigou (1920):
First degree price discrimination (or perfect price discrimination)
The seller charges a different price for each unit of the good
in such a way that the price charged for each unit is equal to
the maximum willingness to pay for that unit.
Since the price is always equal to the marginal revenue, the
monopolist can extract all the surplus from the trade (and
hence total surplus is maximized), and the price (charged to
the consumer with least willingness to pay) is set equal to the
marginal cost.
6 / 12
Price Discrimination (2)
Second degree price discrimination (or nonlinear pricing)
Prices differ depending on the number of units of the good
bought, but not across consumers from the beginning.
Each consumer faces the same price schedule, but it involves
different prices for different amounts of the good purchased.
Quantity discount and two part tariffs (in mobile phone,
taxi, printers etc) are the obvious examples.
Third degree price discrimination
Different purchasers are charged different prices, but each
purchaser pays a constant amount for each unit of the good.
This is perhaps the most common form of price discrimination;
examples are student discounts, or charging different prices
on different regions, countries, days of the week, and so on.
7 / 12
Third Degree Price Discrimination: Practice (1)
Question Consider a monopoly market in which there are two
groups of consumers (named 1 and 2). The demand function for
each group is given as follows:
q1 = 200 − p
q2 = 120 − p
Assume that the marginal cost of the monopoly firm is 20
(constant). Then answer the following questions.
(a) Suppose that price discrimination is prohibiteed and thus the
firm should charge the same price to the different goups. Then,
what are the optimal price and quantities?
(b) Now suppose that price discrimination is possible and the firm
can charge different prices for the two groups. What are the
optimal price and quantity for each market?
8 / 12
Third Degree Price Discrimination: Practice (2)
Answer This is a standard monopoly problem. Note that the
profit in (b) is higher than that in (a). Here are the answers.
(a) p = 90, q1 = 110, q2 = 30.
(b) p1 = 110, q1 = 90, p2 = 70, q2 = 50.
✞ ☎
✝Rm ✆How about the effect to the social welfare (total surplus)?
The producer’s surplus (profit) will always increase.
The consumers’ surplus may increase or decrease.
⇒ In general, the total effect is indecisive.
In our practice, T S becomes 16300 in (a), while 15900 in (b).
⇒ Price discrimination turns out to decrease social welfare.
9 / 12
Limitation of Price Discrimination
Arbitrage (Resell) Arbitrage must be impossible or costly so
that price discrimination is possible.
Hidden Information The monopoly firm usually does not know
the willingness to pay of each consumer.
1
To (literally) implement perfect price discrimination is
impossible, since there is cost to extract such information.
2
The second degree tries to extract the private information by
asking consumers to selfselect menus.
3
The third degree makes use of rough (observable) group
information, instead of the detailed personalized information.
Limited Commitment Power Firms may need to commit price
schedule in dynamic pricing, especially in durable goods markets.
10 / 12
Monopolistic Competition
In monopolistic competition, a relatively large group of firms sell
differentiated products that buyers view as close, though not
perfect, substitutes for one another.
Each firm therefore enjoys a limited degree of monopoly
power in the market for its particular product variant, though
the markets for different variants are closely related.
In a monopolistically competitive group, entry occurs when a
new firm introduces a nonexistent variant of the product.
In the short run, a fixed finite number of active firms choose price
to maximize profit, given the prices chosen by the others. In a
long run equilibrium, entry and exit decisions can also be made.
✞
☎
✝
✆
Fg Figure 4.4 (JR, pp. 178)
11 / 12
Imperfect Market Competition
Many markets display a blend of monopoly and (perfect)
competition simultaneously.
Firms become more interdependent the smaller the number of
firms in the industry, the easier entry, and the close the
substitute goods available to consumers.
When firms perceive their “interdependence,” then have an
incentive to take account of their rivals’ actions and to
formulate their own plan “strategically.”
In Advanced Microeconomics II, we study a mathematical tool
that can deal with these strategic situations, game theory.
Oligopoly competition can well be analyzed by game theory.
Collusive behaviors (cartels) can also be analyzed by dynamic
game theory, called the theory of repeated games.
12 / 12
Advanced Microeconomics II
Yosuke YASUDA
Osaka University, Department of Economics
yasuda@econ.osakau.ac.jp
December 2, 2014
1 / 12
Monopoly Market
We have so far studied consumer and producer behaviors in the
perfectly competitive market in which each economic agent
optimally makes his/her decision given “market prices.”
This is one polar extreme on a spectrum of possible market
structures ranging from the more to the less competitive, since the
competitive market is so competitive that each agent has no power
to affect the price.
The other extreme is (pure) monopoly where there is a single
seller of a product for which there are no close substitutes in
consumption, and entry into the market is completely blocked by
technological, financial or legal impediments.
2 / 12
Monopoly Problem (1)
A monopoly firm takes the market demand function x(p) as given
and chooses price p or/and quantity q to maximize profit.
Let p(q) be the inverse demand function and c(q) be the cost
function. Then, the monopoly problem is given as:
M P 1 : max p(q)q − c(q).
q≥0
where p(q)q is a revenue and c(q) is a cost when the output is
fixed to q. Let π(q) = p(q)q − c(q) denote the revenue function.
Assume that the firm’s objective function π(q) is convex and
differentiable. Then, the first order condition is:
dπ(q)
=
dq
dp(q)
p(q) +
q −
dq

{z
}
marginal revenue (MR)
dc(q)
dq
 {z }
= 0.
marginal cost (MC)
3 / 12
Monopoly Problem (2)
Since the demand function is downward sloping,
dp(q)
dq
< 0,
dc(q)
dc(q) dp(q)
−
q>
.
dq
dq
dq
⇒ The monopoly price must be higher than the marginal cost.
p(q) =
Similarly, we can consider the monopoly problem where the firm
will choose price instead of quantity:
M P 2 : max px(p) − c(x(p)).
p≥0
Although, M P 2 is essentially identical to M P 1, the first order
condition gives us further insight about the monopoly pricing.
d[px(p) − c(x(p))]
dx(p) dc(x) dx(p)
= x(p) + p
−
= 0.
dp
dp
dx dp
Dividing both sides by
dx(p)
dp ,
we obtain
x(p)
1
dc(x)
=p+
= p(1 −
).
dx
dx(p)/dp
ǫ(p)
(1)
4 / 12
Monopoly Problem (3)
Note that ǫ(p) = − dx(p)/dp
x(p)/p (≥ 0) is called elasticity of demand:
% change in the quantity demanded per % change in price.
Rearranging (1), we can obtain an expression for the percentage
deviation of price from marginal cost:
p−
dc(x)
dx
p
=
1
,
ǫ(p)
which is used as a measure of the monopolist’s market power
called the markup, the pricecost margin, or the Lerner Index.
When market demand is infinitely elastic, i.e. under perfect
competition, ǫ(p) will be infinite and p = dc(x)
dx .
However, when market demand is less than infinitely elastic,
the former (strictly) exceeds the latter.
✞
☎
✝
✆
Fg Figure 4.3 (JR, pp.172)
5 / 12
Price Discrimination (1)
A monopoly firm may further raise profit by charging different
prices across consumers. This exercise is called price
discrimination. The traditional classification of the forms of price
discrimination listed as follows is due to Pigou (1920):
First degree price discrimination (or perfect price discrimination)
The seller charges a different price for each unit of the good
in such a way that the price charged for each unit is equal to
the maximum willingness to pay for that unit.
Since the price is always equal to the marginal revenue, the
monopolist can extract all the surplus from the trade (and
hence total surplus is maximized), and the price (charged to
the consumer with least willingness to pay) is set equal to the
marginal cost.
6 / 12
Price Discrimination (2)
Second degree price discrimination (or nonlinear pricing)
Prices differ depending on the number of units of the good
bought, but not across consumers from the beginning.
Each consumer faces the same price schedule, but it involves
different prices for different amounts of the good purchased.
Quantity discount and two part tariffs (in mobile phone,
taxi, printers etc) are the obvious examples.
Third degree price discrimination
Different purchasers are charged different prices, but each
purchaser pays a constant amount for each unit of the good.
This is perhaps the most common form of price discrimination;
examples are student discounts, or charging different prices
on different regions, countries, days of the week, and so on.
7 / 12
Third Degree Price Discrimination: Practice (1)
Question Consider a monopoly market in which there are two
groups of consumers (named 1 and 2). The demand function for
each group is given as follows:
q1 = 200 − p
q2 = 120 − p
Assume that the marginal cost of the monopoly firm is 20
(constant). Then answer the following questions.
(a) Suppose that price discrimination is prohibiteed and thus the
firm should charge the same price to the different goups. Then,
what are the optimal price and quantities?
(b) Now suppose that price discrimination is possible and the firm
can charge different prices for the two groups. What are the
optimal price and quantity for each market?
8 / 12
Third Degree Price Discrimination: Practice (2)
Answer This is a standard monopoly problem. Note that the
profit in (b) is higher than that in (a). Here are the answers.
(a) p = 90, q1 = 110, q2 = 30.
(b) p1 = 110, q1 = 90, p2 = 70, q2 = 50.
✞ ☎
✝Rm ✆How about the effect to the social welfare (total surplus)?
The producer’s surplus (profit) will always increase.
The consumers’ surplus may increase or decrease.
⇒ In general, the total effect is indecisive.
In our practice, T S becomes 16300 in (a), while 15900 in (b).
⇒ Price discrimination turns out to decrease social welfare.
9 / 12
Limitation of Price Discrimination
Arbitrage (Resell) Arbitrage must be impossible or costly so
that price discrimination is possible.
Hidden Information The monopoly firm usually does not know
the willingness to pay of each consumer.
1
To (literally) implement perfect price discrimination is
impossible, since there is cost to extract such information.
2
The second degree tries to extract the private information by
asking consumers to selfselect menus.
3
The third degree makes use of rough (observable) group
information, instead of the detailed personalized information.
Limited Commitment Power Firms may need to commit price
schedule in dynamic pricing, especially in durable goods markets.
10 / 12
Monopolistic Competition
In monopolistic competition, a relatively large group of firms sell
differentiated products that buyers view as close, though not
perfect, substitutes for one another.
Each firm therefore enjoys a limited degree of monopoly
power in the market for its particular product variant, though
the markets for different variants are closely related.
In a monopolistically competitive group, entry occurs when a
new firm introduces a nonexistent variant of the product.
In the short run, a fixed finite number of active firms choose price
to maximize profit, given the prices chosen by the others. In a
long run equilibrium, entry and exit decisions can also be made.
✞
☎
✝
✆
Fg Figure 4.4 (JR, pp. 178)
11 / 12
Imperfect Market Competition
Many markets display a blend of monopoly and (perfect)
competition simultaneously.
Firms become more interdependent the smaller the number of
firms in the industry, the easier entry, and the close the
substitute goods available to consumers.
When firms perceive their “interdependence,” then have an
incentive to take account of their rivals’ actions and to
formulate their own plan “strategically.”
In Advanced Microeconomics II, we study a mathematical tool
that can deal with these strategic situations, game theory.
Oligopoly competition can well be analyzed by game theory.
Collusive behaviors (cartels) can also be analyzed by dynamic
game theory, called the theory of repeated games.
12 / 12