Introduction; Monopoly
Economics 302 - Microeconomic Theory II: Strategic Behavior
Instructor: Songzi Du
compiled by Shih En Lu
Simon Fraser University
January 6, 2016
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What’s this class about?
Game theory
“game”: interaction of people.
In life you have to interact with others, therefore you have to know
how to play games.
Applying game theory to study markets
Economics of information
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Why is this class hard?
Abstract reasoning
Abstraction: focusing on one or a few aspects of a complex situation.
Using math.
You need to know differential calculus.
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This Course
Read the syllabus!
My office hours: Tuesday 10:30-12:20 in WMC 4657 (but not this
week).
TAs’ OHs: 2 hours/week
Textbooks: none required. See syllabus for recommended text.
Slides will be posted at http://www.sfu.ca/~songzid/302. Take
your own notes!
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Homework
7 assignments. Discuss problems with each other - most of you will
learn a lot from each other - but write your own solutions.
Due at the end of lecture on Tuesday
20% of your grade
Each problem is graded for completeness.
Use them as practices for exams.
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My Expectations
Never fall behind. The class material is cumulative. You should
understand everything in a week’s lecture slides by the end of the
week.
Everybody should really do the assignments and practice exams. You
cannot learn game theory by listening to lecture and reading
lecture slides.
The final grade distribution will be set according to the Economics
department guidelines. You will not be penalized for low class
averages on exams.
Half of the quiz and midterm points can be transferred to the final.
Read the syllabus and course policies carefully and never ask for
an exception: the answer is NO.
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Derivatives and Maximization
6
5
f HxL
4
f ' H1L
3
2
1
1
2
3
4
5
f 0 (x) is the slope of the tangent line at the point (x, f (x)).
At maximum of f (x), f 0 (x) = 0.
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Derivatives and Maximization
(f (x) + g (x))0 = f 0 (x) + g 0 (x)
Product rule: (f (x)g (x))0 = f (x)g 0 (x) + f 0 (x)g (x).
f (x) = ax k , where a and k are constants. Then f 0 (x) = ak · x k−1 .
f (x) = ln(x), then f 0 (x) = 1/x.
Review: quotient rule and chain rule.
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What Is a Monopoly?
A firm that is the only one in the market, i.e., no firm produces a
close substitute.
As a result, a monopoly does not lose all its demand when it raises
price above marginal cost: it has market power.
A monopoly picks a point on the market demand curve.
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Sources of Monopoly
Government policy
1
2
Directly owned/regulated by the state (ICBC)
Patents (drugs), copyrights (music), trademarks (brand names),
licenses (nightclubs), etc.
Large efficient scale
1
2
Natural monopoly: decreasing average cost over the relevant range of
quantities (utilities)
Network externalities on the demand side (Microsoft Office)
Firm actions
1
2
Control of essential input (DeBeers)
Being more efficient than other firms and/or preventing entry
(Walmart) - this is different
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Profit Maximization
Today, we assume that the monopoly charges the same price to every
customer. No price discrimination.
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Profit Maximization
Today, we assume that the monopoly charges the same price to every
customer. No price discrimination.
Notation: P = price per unit, q = quantity.
Demand curve D(P) — the number of people willing to buy at price
P; their willingness-to-pay is greater than P.
Inverse demand curve: P(q), firm’s cost function: C (q)
Profit: π(q) = P(q)q − C (q) (Total Revenue - Total Cost)
First-order condition:
π 0 (q) = P(q) + P 0 (q)q − C 0 (q) = MR(q) − MC (q) = 0
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Profit Maximization (II)
FOC: Marginal Revenue = P(q) + P 0 (q)q = C 0 (q) = Marginal Cost
This pins down the monopoly quantity qm . Read the price off the
demand curve: Pm = P(qm ).
Note: Because P 0 (q) < 0, we have MR < P whenever q > 0.
In words: the monopolist chooses the quantity where marginal
revenue equals marginal cost and charges the maximum price that
bears that quantity.
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Profit Maximization: Some Caveats
We have assumed that the demand and cost functions are
differentiable, and the good is perfectly divisible.
We have assumed that the second-order condition (π 00 (q) < 0) holds,
i.e. that MR intersects MC from above. We will usually deal with
decreasing MR and increasing MC , in which case this automatically
holds.
Just like for perfectly competitive firms, you need to check that the
monopoly wouldn’t prefer shutting down.
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Example
Demand: D(P) = 12 − 12 P
Monopolist’s cost: C (q) = q 2
Find the monopoly price and quantity.
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Example
Demand: D(P) = 12 − 12 P
Monopolist’s cost: C (q) = q 2
Find the monopoly price and quantity.
Monopolist solves:
max q · 2(12 − q) − q 2
q
FOC: 2(12 − q) − 2q − 2q = 0.
Monopoly quantity qm = 4, monopoly price Pm = 16.
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Welfare
20
15
P
10
5
0
0
2
4
6
8
10
12
q
Continue with our example: MC (q) = 2q and P = 24 − 2q
⇒ qm = 4.
Set equal the marginal social cost and benefit (willingness-to-pay)
⇒ 2q = 24 − 2q ⇒ q = 6.
Units between 4 and 6 yield a benefit between 16 and 12, but only
cost between 8 and 12.
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Welfare
Both the consumers and the monopolist would be better off if:
two additional units were produced by the monopolist and sold at a
price of 12 to consumers whose willingness-to-pay is between 16 and 12
the monopolist keep the “regular” price of Pm = 16, which is for
consumers whose whose willingness-to-pay is above 16
this improvement assumes that the monopolist is able to identify
consumers who have lower willingness-to-pay. For example: price
discounts offered to students (student ID), senior citizens (driver’s
licence), etc.
The monopoly outcome is not Pareto efficient: everyone
(consumers and the monopolist) could be better off!
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Welfare
20
15
P
10
5
0
0
2
4
6
8
10
12
q
The deadweight loss (inefficiency) is the area between the demand
and marginal cost curves, from the actual quantity to the efficient
quantity, = 12 × (6 − 4) × (16 − 8) = 8 in this example.
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Price, Marginal Cost and Elasticity at the Monopolist’s
Optimum
Remember FOC: P(q) + P 0 (q)q = C 0 (q)
P − MC = − dP
dq q
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Price, Marginal Cost and Elasticity at the Monopolist’s
Optimum
Remember FOC: P(q) + P 0 (q)q = C 0 (q)
P − MC = − dP
dq q
P−MC
P
q
1
= − dP
dq P = , where is the price elasticity of demand:
=−
dq/q
dP/P
Equivalently,
markup =
ECON 302 (SFU)
P − MC
1
=
.
MC
−1
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Price, Marginal Cost and Elasticity at the Monopolist’s
Optimum
Remember FOC: P(q) + P 0 (q)q = C 0 (q)
P − MC = − dP
dq q
P−MC
P
q
1
= − dP
dq P = , where is the price elasticity of demand:
=−
dq/q
dP/P
Equivalently,
markup =
P − MC
1
=
.
MC
−1
Markup over marginal cost gets lower as elasticity increases: the
firm’s market power decreases.
What happens when → ∞?
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Example Continued
We found qm = 4 and Pm = 16.
Marginal cost is C 0 (q) =
is 100%.
Thus =
P
P−MC
ECON 302 (SFU)
=
16
16−8
d
2
dq (q )
= 2q, so it is 8 at qm . The markup
= 2.
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Example Continued
We found qm = 4 and Pm = 16.
Marginal cost is C 0 (q) =
is 100%.
Thus =
P
P−MC
=
16
16−8
d
2
dq (q )
= 2q, so it is 8 at qm . The markup
= 2.
Now lets find directly using the demand D(P) = 12 − 12 P.
=
ECON 302 (SFU)
dq
dP
− q
P
=−
Lecture 1
− 12
4
16
= 2.
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Estimated Price Elasticities of Demand in U.S., 1990s
Salt
Matches
Toothpicks
Airline travel, short-run
Gasoline, short-run
Gasoline, long-run
Residential natural gas, short-run
Residential natural gas, long-run
Coffee
Fish (cod) consumed at home
Tobacco products, short-run
Legal services, short-run
Physician services
Taxi, short-run
Automobiles, long-run
0.1
0.1
0.1
0.1
0.2
0.7
0.1
0.5
0.25
0.5
0.45
0.4
0.6
0.6
0.2
Movies
Housing, owner occupied, long-run
Shellfish, consumed at home
Oysters, consumed at home
Private education
Tires, short-run
Tires, long-run
Radio and television receivers
Restaurant meals
Foreign travel, long-run
Airline travel, long-run
Fresh green peas
Automobiles, short-run
Chevrolet automobiles
Fresh tomatoes
0.9
1.2
0.9
1.1
1.1
0.9
1.2
1.2
2.3
4.0
2.4
2.8
1.2 –
4.0
4.6
Source: http:
//scholar.harvard.edu/files/alada/files/price_elasticity_of_demand_handout.pdf
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Why Would a Government Create a Monopoly?
Monopolies come with a deadweight loss. Why create them?
The government might take control of a monopoly that would exist
anyway (e.g. utilities).
The government can generate revenue or reduce expenditures by
selling the right to form a monopoly (e.g. certain toll highways).
The government might want to encourage innovation through patents
and copyright. The alternative would be to subsidize, which increases
the need for revenues and could therefore increase deadweight losses
elsewhere.
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What Can Government Do about Monopolies?
Divestiture: monopolies that control a key resource can be forced to
sell off some of that resource. For example, AT&T had to sell off its
local operations in 1982.
In other cases, such as natural monopolies, the government can still
affect the price:
1
2
through taxes (or subsidies);
directly through regulation.
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Taxes and Monopolies
Does taxing a monopoly improve efficiency?
It does redistribute money from the monopoly (and its customers)
toward the government.
Just like under perfect competition, there is a tax wedge between the
curves determining quantity.
Here, they are the MR and MC curves (rather than the supply and
demand curves).
Once you know the quantity, read the price paid by consumers
(consumer’s price, Pc ) off the demand curve.
The price received by the monopoly (monopolist’s price, Pm ) is Pc
less the tax.
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Taxes and Monopolies: Per-unit Tax
Return to our example: MC (q) = 2q, P(q) = 24 − 2q and
MR(q) = d(P(q)q)
= 24 − 4q.
dq
We found qm = 4 and Pm = 16 with no taxes. Now suppose there’s a
per unit tax of 6.
Find the new quantity, price paid by consumers and price received by
the monopolist.
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Taxes and Monopolies: Per-unit Tax
Return to our example: MC (q) = 2q, P(q) = 24 − 2q and
MR(q) = d(P(q)q)
= 24 − 4q.
dq
We found qm = 4 and Pm = 16 with no taxes. Now suppose there’s a
per unit tax of 6.
Find the new quantity, price paid by consumers and price received by
the monopolist.
Monopolist solves maxq P(q)q − C (q) − 6q.
24 − 4qm = 2qm + 6
With tax: qm = 3, Pc = 18, Pm = 18 − 6 = 12.
Per-unit tax of 6 increases the consumer’s price by only 2.
Does tax leads to the efficiency of P(q) = MC (q)?
No, we are even farther away from market clearance.
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Taxes and Monopolies: Ad Valorem (Sales) Tax
Instead of a per unit tax of 6, in the previous example suppose there’s
a percentage tax (sales tax) of 100% of the price received by the
monopoly (equivalently, 12 of the price paid by consumers).
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Taxes and Monopolies: Ad Valorem (Sales) Tax
Instead of a per unit tax of 6, in the previous example suppose there’s
a percentage tax (sales tax) of 100% of the price received by the
monopoly (equivalently, 12 of the price paid by consumers).
Monopolist solves:
max
q
1
(24 − 2q)q − q 2 ,
1+t
where t = 100%, FOC: 1/2 · (24 − 4qm ) − 2qm = 0, or qm = 3.
Consumer’s price Pc is still read off the demand curve: Pc = 18.
But what does the monopoly get now? Monopolist’s price
1
Pm = 1+t
Pc = 9.
1
In words: total revenue is 1+t
= 12 of what it was without tax at
every q. Thus so is marginal revenue ⇒ MR(q) = 12 − 2q. Setting
MR = MC gives: 12 − 2qm = 2qm , or qm = 3.
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