The Optimal Mark-UpThe Optimal Mark-Upand Price Discriminationand 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 variablePrice 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 The trade-off between price and profitprofit
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 (EIt depends on elasticity (EPP))
We can show that the optimal mark-up over
MC is inversely proportional to elasticity
of demand (EP)
The markup ruleThe 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:
PEP
MCP
1[3.17]
Rearranging [3.1] we obtain:
MCE
EP
P
P
1
[3.18]
Elasticities and optimal pricesElasticities and optimal prices
ElasticitElasticityy
MCMC PricePrice
-1.5-1.5 3.03.0 100100 300300
-2.0-2.0 2.02.0 100100 200200
-3.0-3.0 1.51.5 100100 150150
-5.0-5.0 1.251.25 100100 125125
-11.0-11.0 1.11.1 100100 110110
-- 1.01.0 100100 100100
P
P
E
E
1
Students at Sherwood High in Sandy Springs, Maryland talk about things that bother them
What is price discrimination?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 Examples of price discriminationdiscrimination
•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 When is price discrimination feasible?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.”
11stst degree price degree price discrimination 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.
AuctionsAuctions
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.
33rdrd degree price degree price discriminationdiscrimination
This is the practice of charging
different prices in different market
segments
Examples of market Examples of market “segments”“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 pricingMultinational pricing of autos 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 FunctionsThe Demand Functions
The inverse demand equation for the home (H) market is given by:
HHP 500000,30
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:
FFP 700000,25
30,000
600Quantity
Pri
ce
25,000
Home
Foreign
35.7
The demand for carsThe demand for cars
30,000
600Quantity (000s)
Pri
ce
DH
30
Profit maximization in Profit maximization in the Home segmentthe Home segment
MRH
MCH10,000
20,000
20
To maximize profits in the Home segment, set MRH = MCH
18,000
600Quantity (000s)
Pri
ce
25,000
DF
35.7
Profit-maximization in Profit-maximization in the foreign market segmentthe foreign market segment
MRF
MCF11,000
10
To maximize profits in the Foreign segment, set MRF = MCF
SummarySummary
Notice that the price is higher in the Home market where the
manufacturer faces a less elastic demand
curve
Interdependent Interdependent demanddemand
Consider a microbrewery that brews lager and
pilsner. The price of the lager will likely
affect the demand for pilsner.
ExampleExample
Let A denote lager and B is pilsner. Let the profit function be given by:
)()(),(),( BBAABABBAA QCQCQQRQQR
Note: We assume that there are no interdependencies or complementarities in production
Determining the Determining the optimal quantityoptimal 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:
AA
B
A
AA MC
dQ
dR
dQ
dRMTR
And:
BB
B
B
AB MC
dQ
dR
dQ
dRMTR
Numerical exampleNumerical 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.
TR = RA + RB
= (280QA – 2QA2) + (180QB – QB
2-2QAQB)
Thus we have:
Therefore:
MTRA= 280 – 4QA – 2QB
And:
MTRB= 180 – 2QB – 2QA
So set MTRA = MCA and MTRB = MCB
and solve for QA and QB
280 – 4QA – 2QB = 80180 – 2QB – 2QA = 40
The result is a linear equation system with two equations and two unknowns
The solutionsThe solutionsSolving 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.