The Optimal Mark-Up and Price Discrimination

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.