The Optimal Mark-Up
and Price Discrimination
Outline
•The optimal mark-up over cost
•What is price discrimination?
•Examples of price discrimination
•When is price discrimination feasible?
•First, second, and third degree price
discrimination
•Multinational pricing of autos
•Interdependent demand
Price as the decision variable
•Thus far we have assumed that quantity
was the relevant decision-variable.
•In reality, most firms establish a price for
their product and then try to satisfy demand
for their product at that price.
•The price established by management is
generally based on costs plus a mark-up.
The trade-off between price and profit
The firm’s contribution can be written as:
Contribution = (P – MC)Q
We assume that marginal cost (MC)
is constant.
Issue: How far above MC should the firm
raise P to maximize its contribution (and
hence profits)?
It depends on elasticity (EP)
We can show that the
optimal mark-up over
MC is inversely
proportional to elasticity
of demand (EP)
The markup rule
The size of the firm’s mark-up (above marginal cost
expressed as a percentage of price) depends inversely on the
price elasticity of demand for a good or service.
That is, the optimal markup is given by:
P  MC
1

P
 EP
[3.17]
Rearranging [3.1] we obtain:
 EP 
P
 MC
 1  EP 
[3.18]
Elasticities and optimal prices
Elasticity
 EP 


 1  EP 
-1.5
3.0
MC
100
Price
300
-2.0
2.0
100
200
-3.0
1.5
100
150
-5.0
1.25
100
125
-11.0
1.1
100
110
-
1.0
100
100
Students at
Sherwood High
in Sandy
Springs,
Maryland talk
about things that
bother them
What is price discrimination?
Price discrimination is
the practice of selling
the same product to
different buyers (or
groups of buyers) at
different prices.
Examples of price discrimination
•Airlines charge full fares to business travelers,
whereas they offer discount fares to vacationers.
•“Sizing up their income” pricing by dentists,
plumbers, and auto mechanics.
•Publishers of academic journals charge higher
prices for library as compared to individual
subscriptions.
•Senior citizen discounts.
•Discounts for new buyers—e.g., magazine
subscriptions.
•Theater ticket pricing
When is price
discrimination feasible?
1. The seller must be capable of identifying
market segments that differ based on
willingness to pay, or elasticity of demand.
2. The seller must be capable of “enforcing” the
different prices charged to different market
segments—that is, the seller must be able to
prevent “arbitrage.”
1st degree price
discrimination
•Sometimes called “perfect” price
discrimination, the seller charges each buyer
their “reservation price” for every unit
purchased.
•Reservation price is the maximum price a
buyer is willing to pay rather than go
without the last unit of the good.
Auctions
Auctions are
designed to force
buyers nearer to
their reservations
prices.
The Cigarette Czar
•Suppose an individual
gained monopoly control of
the supply of cigarettes in a
particular geographic
location.
•The cigarette czar could
practice 1st degree price
discrimination by holding out
until smokers paid their
reservation price for each
smoke.
3rd degree price
discrimination
This is the practice
of charging
different prices in
different market
segments
Examples of market
“segments”
•Business travelers versus tourists.
•Kids versus adults
•Those covered by health insurance
and those not covered.
•Senior citizens versus everyone else.
•Mercedes Benz owners versus
Chevrolet owners.
•Domestic versus foreign buyers
Multinational pricing
of autos
The problem for a car
manufacturer is to establish
profit-maximizing prices on
cars sold domestically and in
the foreign market segment
The Demand Functions
The inverse demand equation for the home (H)
market is given by:
PH  30,000  500 H
Where PH is the price charge in the home market
and H is the quantity sold in the home market
The inverse demand equation for the foreign (F)
market is given by:
PF  25,000  700 F
The demand for cars
30,000
25,000
Foreign
Home
0
35.7
Quantity
60
Profit maximization in
the Home segment
30,000
20,000
To maximize profits in
the Home segment, set
MRH = MCH
10,000
MCH
MRH
0
20
30
Quantity (000s)
DH
60
Profit-maximization in
the foreign market segment
To maximize
profits in the
Foreign segment,
set MRF = MCF
25,000
18,000
MCF
11,000
MRF
0
10
DF
35.7
Quantity (000s)
60
Summary
Notice that the price is
higher in the Home
market where the
manufacturer faces a
less elastic demand
curve
Interdependent
demand
Consider a
microbrewery that
brews lager and
pilsner. The price of
the lager will likely
affect the demand for
pilsner.
Example
Let A denote lager and B is pilsner. Let the
profit function be given by:
  RA(QA, QB )  RB (QA, QB )  CA(QA)  CB (QB )
Note: We assume that there are no
interdependencies or complementarities
in production
Determining the
optimal quantity
Produce up to the point in which the extra total revenue
(MTR) from the sale of product A is equal to the
marginal cost of A, and similarly for B.
That is:
dRA dRB
MTRA 

 MCA
dQA dQA
And:
dRA dRB
MTRB 

 MCB
dQB dQB
Numerical example
Let MCA = $80; MCB = $40
PA = 280 – 2QA
PB = 180 – QB – 2QA
Notice that increased sales of A adversely
affect sales of B, but not vice versa.
Thus we have:
TR = RA + RB
= (280QA – 2QA2) + (180QB – QB2-2QAQB)
Therefore:
MTRA= 280 – 4QA – 2QB
And:
MTRB= 180 – 2QB – 2QA
So set MTRA = MCA and MTRB = MCB
and solve for QA and QB
The result is a
280 – 4QA – 2QB = 80
linear equation
180 – 2QB – 2QA = 40
system with two
equations and two
unknowns
The solutions
Solving the equation system yields:
QA = 30 and QB = 40
Substituting into the price (or inverse
demand) equations yields:
PA = $220 and PB = $80
Contrast this outcome to the
case where the brewery
ignored the cross effect of A
and B and simply tried to
maximize profits from A.