Pricing in Monopoly
Goal:
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In the previous lecture, we considered pricing in a market with
many …rms.
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Now we consider a single …rm setting its price in a market.
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Interpretations: true monopoly
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One large …rm with a competitive fringe of small …rms
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natural monopoly
legal monopoly (patent, copyright, etc.)
Small …rms’reactions can be interpreted as part of the
demand curve
No game theory needed this time
We cover linear prices, price discrimination and non-linear
pricing.
Key Ingredient: Demand Curve
Optimal Pricing
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Trade can only occur if a surplus can be realized between the
trading partners.
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Here surplus in monetary value of utility to buyer net of
production cost.
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Hence pricing is based on both valuations and costs.
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It is not a good idea to base pricing on production costs only
as in cost plus pricing.
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Simplest cost plus rule: calculate average cost of production
and add a …xed percentage.
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Maybe the seller knows its production cost.
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How does the monopolist learn the demand curve?
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How does an outside observer (statistician) learn the demand
curve?
Optimal Linear Price
Market description:
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Large number of buyers represented by demand curve
q = d (p ) ,
where d 0 (p ) < 0.
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A single seller produces the good with cost function c (q ) for
producing q units of the good.
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Monopolist chooses the price, and quantity is read o¤ the
demand curve.
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Prices are linear so that revenue is pq.
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The monopolist chooses p to maximize revenue net of cost.
Optimal Linear Price
Monopolist’s Problem
c (q )
max pq
p,q 0
subject to q = d (p ) .
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Substituting the constraint into the objective function gives:
max pd (p )
p
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c (d (p )) .
Notice that this objective function is not always concave.
Hence you should check all points at which the …rst-order
condition holds and also the point where p is high enough to
make q = 0 and pick the point that results in the highest
pro…t.
First-order condition:
pd 0 (p ) + d (p )
c 0 (d (p )) d 0 (p ) = 0.
Optimal Linear Price
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Dividing through by pd 0 (p ) , and rearranging yields:
p
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c 0 (d (p ))
=
p
pd 0 (p )
Writing εp =
for the price elasticity of demand and
d (p )
q = d (p ) for the amount demanded, we have:
p
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d (p )
.
pd 0 (p )
1
c 0 (q )
= .
p
εp
In words, the percentage markup of the optimal monopoly
price over marginal cost is the inverse of the elasticity of
demand in the market.
Alternatively:
p c 0 (p )
1
=
.
0
c (q )
εP 1
In both formulas, less elastic demand leads to higher markups.
What are examples of markets with inelastic demand?
Implications for multi-product …rms?
Optimal Linear Price
Comparison of Optimal Pricing and Cost-Plus Pricing
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For concreteness, assume a cost function with a …xed cost
F = 2000 and variable cost 60q. Hence c (q ) = 2000 + 60q if
q > 0 and c (0) = 0.
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The demand function is given by
q = 200
p.
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Exercise: show that optimal monopoly price p here is 130 and
optimal monopoly sales q = 70, and optimal pro…t is 2900.
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Suppose next that the …rm considers cost-plus pricing.
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This form of pricing is widely used across many industries.
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Its supposed virtue is that pricing decision are based on hard
engineering data from production side rather than softer
demand estimates.
Optimal Linear Price
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Suppose the capacity at the plant is at 100.
Cost plus pricing takes the form of setting a margin
requirement for the sales price.
Suppose that the goal is to have a 100% margin on
production costs.
How to set the price? How to allocate the …xed cost?
Variable cost is 60, at full capacity, average …xed cost is 20
per unit.
Hence cost-plus pricing demands a price of
(1 + α) (60 + 20) = 160
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since at the required 100% margin, α = 1.
Realized demand is then 40 and total pro…t is 2000.
But now the production department reports that the true …xed
cost per unit sold is 50 since production is well below capacity.
If the price is raised in response to this report to 220, the
demand disappears altogether and the …rm makes a loss of
2000.
Optimal Linear Price
Moral of the Story:
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Firms do not generally have very accurate information about
the elasticity of demand for their product.
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Nevertheless, they can perform thought experiments along the
lines:
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Suppose we increase price by x per cent.
What is the maximal drop in sales volume that makes this
increase still pro…table?
Do we consider a price response of that magnitude is likely?
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Room for consumer surveys.
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Room for experimenting with di¤erent prices to …nd out more
about the demand function.
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Models on how to price when also learning about the demand
curve are beyond the scope of this course.
Personalized Prices or First-Degree Price Discrimination
Disaggregating the Demand Side:
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Recall that the market demand is obtained by summing
together all individual demand functions:
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d (p ) =
∑ di (p ) ,
i =1
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where di (p ) is the individual demand function of buyer i.
Suppose now that the seller knows all the di (p ) and can set
individual prices pi for each buyer.
Let εP ,i be the price elasticity of the individual demand of
buyer i.
Optimal pricing is given by:
pi
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c 0 (q )
1
=
.
pi
εP ,i
Notice that the marginal cost depends on the aggregate
demand.
Personalized Prices or First-Degree Price Discrimination
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Special case: Unit Demands
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Good sold in discrete units.
Each buyer gets a utility vi from the …rst unit, no additional
unit from further units.
Without loss of generality, rename the buyers so that
v1 v2 ... vI .
If each unit costs c to produce, sell to all buyers with vi
c.
If n units cost c (n) to produce, then sell to the …rst n buyers,
where
n = maxfn : vn c (n) c (n 1)g.
Interpretation?
For i n , set pi = vi .
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With unit demands, monopolist extracts all consumer surplus
in the market.
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With more general demands, the consumers do get some
consumer surplus with linear individual prices.
Personalized Prices or First-Degree Price Discrimination
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A peek ahead:
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Suppose the monopolist can use two-part tari¤s:
pi (qi )
pi (0)
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= fi + pi qi if qi > 0,
= 0 if qi = 0.
fi is the …xed purchase fee of i.
pi is the linear individual price for i.
Suppose for ease of exposition that c (q ) = cq, i.e. there is a
constant marginal cost.
Can the monopolist extract all surplus in the market? How
should the fi and pi be set?
Does this conclusion really depend on constant marginal costs?
Personalized Prices or First-Degree Price Discrimination
Features of the Model:
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With two-part tari¤s, the Pareto-e¢ cient market outcome is
obtained.
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Caution: This does not necessarily hold in markets with free
entry of sellers.
Extreme distributional asymmetry. Sellers get all, buyers
nothing.
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This is Pareto-e¢ cient, but is this a good societal outcome?
Relies on the seller’s perfect knowledge of the preferences of
the buyers.
Is this realistic?
Personalized Prices or First-Degree Price Discrimination
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What about arbitrage, e.g. resale between buyers?
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Always a question for models of price discrimination.
Technological progress might make the model more relevant.
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Collect information on individual buyers through loyalty cards,
social media etc.
Tailor price o¤ers available only on their loyalty card.
You can even experiment relatively cheaply by issuing coupons
(price discounts) and observing the demand reactions.
Combined with statistical analysis of all data in the database
of the selling …rm, this is a potentially successful pricing tool.
Third-Degree Price Discrimination or Pricing to Market
Segments
Di¤erent Prices for Di¤erent Identi…able Groups:
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Student/Pensioner/Disabled/Unemployed/Military Service
discounts.
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Di¤erential fees for municipal services for single parents.
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What about di¤erential insurance premiums based on sex/age
etc.?
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Key assumption: membership in a market segment cannot be
manipulated.
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A linear price for each market segment.
Third-Degree Price Discrimination or Pricing to Market
Segments
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N market segments.
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Each with a demand curve dn (p ) .
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Since the markets are separate, optimal pricing formula is as
before:
1
pn c 0 (q )
=
.
pn
εP ,n
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Implications are then clear: set higher prices for the segments
with less elastic demand,
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What does this mean in terms of the examples listed above?
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What are the welfare consequences of this model?
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What do we mean by welfare?
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Sum of consumer surpluses and pro…t.
Third-Degree Price Discrimination or Pricing to Market
Segments
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Assume for simplicity that marginal cost is constant at c and
there are no …xed costs.
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The comparison is to the case where pn = p for all
n 2 f1, ..., N g and hence there is no price discrimination.
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Let qn = dn (pn ) .
Consumer surplus in market segment n at prices pn is denoted
by:
Z ∞
CSn (pn ) =
dn (z ) dz..
pn
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Notice that this is the area under the usual demand curve up
to demand level qn net of the payment pn qn .
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Total consumer surplus is then Σn CSn (pn ) .
Third-Degree Price Discrimination or Pricing to Market
Segments
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If the monopolist had to set a single price for all markets that
would be p .
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Let qn = Dn (p ) .
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Pro…t is then Σn (p
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Let ∆qn = qn
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Di¤erence in total surplus between the discrimination and no
discrimination cases is:
c ) qn . Consumer surplus is Σn CSn (p ).
qn .
∆W = (Σn CSn (pn )
Σn CSn (p )) + (Σn (pn
c ) qn
Σn (p
c)
Third-Degree Price Discrimination or Pricing to Market
Segments
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How can we get bounds for ∆W ?
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CSn (pn ) is a convex function of pn .
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But then CSn (pn )
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Since
CSn0
CS (p )
(p ) =
Similarly: CSn (p )
∆W
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p ).
qn , we have:
∆W
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CSn0 (p ) (pn
Σn (p
Σn (pn
CS (pn )
c ) ∆qn .
CSn0 (pn ) (p
c ) ∆qn = (p
pn ) and thus:
c ) (Σn ∆qn ) .
Hence an increase in the aggregate supply is a necessary but
not su¢ cient condition for an increase in the total welfare.
Third-Degree Price Discrimination or Pricing to Market
Segments
Conclusion
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Both of the above models rely on the seller’s ability to assign
buyers to right categories.
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In the …rst-degree case, individual identi…cation.
In pricing to market segments, identi…cation at the level of the
segment.
What about the case when the buyer can manipulate this
classi…cation?
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Second-degree price discrimination.
Information economics: adverse selection, mechanism design.