Monopoly
Price Discrimination
Double Marginalization
Microeconomic Theory: Lecture 4
Monopoly
Parikshit Ghosh
Delhi School of Economics
Summer Semester, 2014
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Non-Discriminating Monopolist
The Monopolist’s Problem
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The monopolist realizes his quantity choice a¤ects market
price through the demand function.
Monopolist’s problem can be described either as one of
choosing optimal price or optimal quantity.
Inverse demand function p (q ) and cost function c (q ).
Maximizing pro…ts:
max p (q )q
q
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Parikshit Ghosh
Monopoly
c (q )
First order condition:
p (q m ) + qp 0 (q m ) = c 0 (q m )
|
{z
}
| {z }
marginal revenue = marginal cost
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Non-Discriminating Monopolist
Monopolist’s Optimum: Another Look
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The FOC can be rewritten as
pm 1 +
or, pm
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Parikshit Ghosh
Monopoly
= c 0 (qm )
= c 0 (qm )
e (q ) is the elasticity of demand at quantity q. The monopolist
never operates on the inelastic part of the demand curve.
Yet another way to write it:
pm
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1
qm dp
.
pm dq
1
e (qm )
c 0 (qm )
1
=
pm
e (qm )
The mark-up is equal to the inverse of demand elasticity.
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Non-Discriminating Monopolist
Comparison With Competitive Markets
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Competitive output level:
p (q ) = c 0 (q )
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The monopolist produces less than the competitive market.
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Suppose q m
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Subtracting the f.o.c’s yields a contradiction:
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Parikshit Ghosh
Monopoly
q ) p (q m )
p (q ) and c 0 (q m )
c 0 (q ).
p (q m ) p (q ) + qp 0 (q m ) = c 0 (q m ) c 0 (q )
|
{z
} | {z }
|
{z
}
=
+
Unlike a competitive …rm, when a monopolist raises output,
he earns lower revenue on previous units.
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Non-Discriminating Monopolist
Monopoly in Pictures
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Non-Discriminating Monopolist
Monopoly in Pictures
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Non-Discriminating Monopolist
Monopoly in Pictures
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Non-Discriminating Monopolist
Monopoly in Pictures
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Price Discrimination: First Degree
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Each consumer buys 0 or 1 unit and is willing to pay up to v .
There is a continuum of consumers whose v follow a
distribution with c.d.f F (v ).
The monopolist can charge each consumer his personal v .
Must choose a cuto¤ v above which to sell:
max
v
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Z ∞
vf (v )dv
Monopoly
F (v ))
First-order condition (using Leibnitz Rule) implies absence of
ine¢ ciency:
v f (v ) + c 0 (1
Parikshit Ghosh
c (1
v
F (v ))f (v ) = 0
v
= c 0 (1 F (v ))
|{z}
|
{z
}
price to marginal customer = marginal cost
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Price Discrimination: Second and Third Degree
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Second degree price discrimination arises when the monopolist
can charge di¤erent prices for di¤erent quantities.
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E.g., bulk discounts, multi-packs, frequent-‡yer miles,
buy-one-get-50%-o¤ on next purchase, etc.
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A way to extract consumer’s surplus from a single consumer.
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Third degree price discrimination arises when observably
di¤erent groups are charged di¤erent prices.
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E.g., student/senior citizen discounts, country speci…c prices.
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A cruder form of …rst degree price discrimination— using
group identity as a predictor of individual traits.
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A common instrument of price discrimination: screening.
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Second Degree Price Discrimination
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Second Degree Price Discrimination
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Second Degree Price Discrimination
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Second Degree Price Discrimination
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Second Degree Price Discrimination
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Second Degree Price Discrimination
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Second Degree Price Discrimination
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Two Part Tari¤
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Two Part Tari¤
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Two Part Tari¤
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Suppose a single consumer with income y has quasi-linear
utility: u (q, m ) = φ(q ) + m.
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The monopolist can charge an entry fee (f ) and a price (p)
per unit of consumption.
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Consumer’s optimum quantity choice (if she subscribes):
max φ(q ) + y
q
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f
Monopoly
1
(p )
Consumer subscribes if (participation constraint):
φ(q (p )) + y
Parikshit Ghosh
pq ) q (p ) = φ0
f
pq (p )
y
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Types of Price Discrimination
Two Part Tari¤
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The monopolist’s problem:
max pq (p ) + f
f ,p
subject to φ(q (p )) + y
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f
pq (p )
y
Participation constraint must be binding at optimum
(otherwise simply increase f ).
max φ(q (p ))
p
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c (q (p ))
c (q (p )) ) φ0 (q
b) = c 0 (q
b)
If the market were competitive (price taking behaviour):
φ 0 (q ) = c 0 (q )
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Parikshit Ghosh
Monopoly
Again, two-part tari¤s remove the monopolistic distortion.
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Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Airfares: More Than 2 Weeks
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Airfares: Less Than 2 Weeks
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Hardcover vs. Paperback: Prices
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Hardcover vs. Paperback: Date and Sales
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Cash Back Coupons
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Rationing
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Rationing
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Price Discrimination by Screening
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Two types of consumers:
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Consumer’s valuation is private information.
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Cost of production is 0.
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Uniform pricing strategy:
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Parikshit Ghosh
Monopoly
high value (value = vH , proportion = α).
low value (value = vL , proportion = 1 α).
charge p = vH if αvH
vL .
charge p = vL ifαvH < vL .
Pro…t = maxfαvH , vL g. Assume αvH < vL .
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Price Discrimination by Screening
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Parikshit Ghosh
Monopoly
Monopolist can impose a burden B on consumers (delay,
coupons, uncertainty, etc.).
Cost of the burden is cH for high value types and cL for low
value types (cH > cL ).
The burden reduces the willingness to pay of all customers.
Its direct impact on pro…ts is negative. Why will the
monopolist hurt his own interests?
By imposing the burden, the monopolist can gather valuable
market information which allows him to price discriminate.
This indirect bene…t may compensate for the direct loss of
lower reservation prices.
The screening technique is useful in other contexts (employers
seeking dedicated workers, governments targeting anti-poverty
programmes at the poor).
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Price Discrimination by Screening
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Monopolist o¤ers a menu (pH , 0) and (pL , B ), satisfying:
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Self Selection Constraint: H-type chooses (pH , 0), L-type
chooses (pL , B ).
pL + cH
pH
pH
pL + cL
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Participation Constraint: Both types want to buy.
vL
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Parikshit Ghosh
Monopoly
(IC-H)
(IC-L)
vH
pL
pH
cL
0
0
Monopolist solves: maxpH ,pL αpH + (1
constraints.
(PC-H)
(PC-L)
α)pL subject to these
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Price Discrimination by Screening
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Assumption A: Relative to the L-type, the H-type gains
more from the good than she su¤ers from the burden:
vH
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vL )
or, vH
vH
Monopoly
cH
cL
Step 1: PC-H is implied by the other constraints and can be
dropped.
vL + (vH
Parikshit Ghosh
vL
(pL + cL ) + (cH
pL + cH
cH
cL )
(combining PC-L and
(using IC-H)
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Price Discrimination by Screening
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Step 2: PC-L must be binding at the optimum.
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Step 3: IC-H must be binding at the optimum.
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Parikshit Ghosh
Monopoly
If not, increase both pH and pL by e > 0.
IC’s continue to hold, pro…ts have increased.
Otherwise, increase pH by e > 0.
The other remaining constraints continue to be satis…ed.
Step 4: Binding IC-H implies IC-L, so it can be dropped.
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Monopolistic Screening
Price Discrimination by Screening
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Optimal solution:
pL
pH
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Assume vH > vL + cH
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Pro…t from price discrimination:
π
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Parikshit Ghosh
Monopoly
= vL cL
= vL + cH
cL
cL .
= α(vL + cH cL ) + (1
= vL + αcH cL
α)(vL
cL )
Price discrimination is better than uniform pricing if
αcH cL > 0.
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Vertical Mergers
Horizontal vs. Vertical Mergers
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Downstream product X (furniture), upstream product Y
(wood). Fixed coe¢ cient technology (1:1).
Each supplied by a separate monopolist, with marginal costs
cx and cy .
Inverse demand function for the …nal good: p = p (x ). Let
R (x ) = p (x )x be downward sloping.
Let the price charged by the upstream monopolist be q.
Downstream problem:
max R (x )
x
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( cx + q ) x
First order condition:
R 0 ( x ) = cx + q
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Vertical Mergers
Upstream Problem
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The upstream monopolist takes into account the downstream
demand function:
max yq
y
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cy y
max y R 0 (y )
y
cx
cy y
First order condition (replacing y by x ):
R 0 (x ) + x R 00 (x ) = cx + cy
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Suppose the two monopolies merged. Then the …rm will solve:
max R (x )
x
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( cx + cy ) x
First order condition shows vertical merger increases e¢ ciency:
R 0 ( x ) = cx + cy ) x > x
Parikshit Ghosh
Monopoly
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Tax Harmonization
Centre and State Taxes
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Let q = a
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Supply is horizontal at some price p
b (net of taxes).
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The Centre …rst chooses a tax tc . Then the State chooses its
own tax ts .
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Both governments aim to maximize tax revenue.
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Centre and State do not coordinate when choosing their tax
policies. They do not maximize total government revenue.
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This can be ine¢ cient. Cumulative tax rates are too high.
Parikshit Ghosh
Monopoly
bp be the demand function.
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Tax Harmonization
State’s Problem
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The State solves:
max ts [a
ts
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State’s FOC de…nes its reaction function ts (tc ):
a
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b (p
b + tc + ts )]
Di¤erentiating w.r.t tc :
b (p
b + tc ) = 2bts
ts0 (tc ) =
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Parikshit Ghosh
Monopoly
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If Central taxes are higher, State will lower its own taxes to
some degree but not completely.
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Tax Harmonization
Centre’s Problem
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Centre chooses tc to solve:
max tc [a
tc
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Replacing State’s reaction function:
tc
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b (p
b + tc + ts (tc ))]
1
= arg max tc (a
tc 2
a bb
p
=
2b
Monopoly
btc )
Plugging back:
ts =
Parikshit Ghosh
bb
p
a
bb
p
4b
Delhi School of Economics
Monopoly
Price Discrimination
Double Marginalization
Tax Harmonization
Tax Harmonization
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Consider a harmonized single tax rate to maximize tax
collection for Centre and State. It solves
b (p
b + t )]
max t [a
t
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FOC is gives the optimal tax rate
t =
bb
p
2b
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Since ts = t , we have ts + tc > t .
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A harmonized single tax will
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Parikshit Ghosh
Monopoly
a
Decrease tax burden and increase consumer’s surplus
Increase tax collection which can be suitably shared.
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