The Optimal Mark-Up and Price Discrimination

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The Optimal Mark-Upand 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

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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.

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The trade-off between price and profit
The firms 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)?
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It depends on elasticity (EP)
We can show that the optimal mark-up over MC is
inversely proportional to elasticity of demand
(EP)
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The markup rule
The size of the firms 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
3.17
Rearranging 3.1 we obtain
3.18
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Elasticities and optimal prices
Elasticity MC Price
-1.5 3.0 100 300
-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
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Students at Sherwood High in Sandy Springs,
Maryland talk about things that bother them
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(No Transcript)
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What is price discrimination?
Price discrimination is the practice of selling
the same product to different buyers (or groups
of buyers) at different prices.
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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 buyerse.g., magazine
  subscriptions.
• Theater ticket pricing

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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
   segmentsthat is, the seller must be able to
   prevent arbitrage.

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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.

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Auctions
Auctions are designed to force buyers nearer to
their reservations prices.
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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.

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3rd degree price discrimination
This is the practice of charging different prices
in different market segments
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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

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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
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The Demand Functions
The inverse demand equation for the home (H)
market is given by
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
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The demand for cars
30,000
Foreign
25,000
Price
Home
60
0
35.7
Quantity
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Profit maximization in the Home segment
30,000
To maximize profits in the Home segment, set MRH
MCH
20,000
Price
MCH
10,000
DH
MRH
60
0
30
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Quantity (000s)
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Profit-maximization in the foreign market segment
25,000
To maximize profits in the Foreign segment, set
MRF MCF
Price
18,000
MCF
11,000
DF
MRF
60
0
35.7
10
Quantity (000s)
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Summary
Notice that the price is higher in the Home
market where the manufacturer faces a less
elastic demand curve
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Interdependent demand
Consider a microbrewery that brews lager and
pilsner. The price of the lager will likely
affect the demand for pilsner.
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Example
Let A denote lager and B is pilsner. Let the
profit function be given by
Note We assume that there are no
interdependencies or complementarities in
production
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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
And
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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.
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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 linear equation system with two
equations and two unknowns
280 4QA 2QB 80180 2QB 2QA 40
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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.