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The Analytics of Profit maximizing PricesThe Analytics of Profit maximizing Prices The economics of cost pass-throughsThe economics of cost pass-throughs

ReviewReview

Why study pricing?Why study pricing? In a market economy, what factors determine In a market economy, what factors determine

prices, in general?prices, in general? In a consideration of prices what is the role of In a consideration of prices what is the role of

costs? costs? What is the Inverse Elasticity Rule?What is the Inverse Elasticity Rule? How does the production of substitutes How does the production of substitutes

(complements) affect optimal pricing?(complements) affect optimal pricing? In a world in which you control price, how does In a world in which you control price, how does

one determine optimal cost pass-through rates?one determine optimal cost pass-through rates?

Review:Review:Elasticity and PricingElasticity and Pricing

Brand Elasticity of Demand

Colas Royal Crown -2.4

Coke -5.2 to -5.7

Coffees Folgers -6.4

Maxwell House -8.2

Chock full o’Nuts -3.6

Who has more pricing power in the Cola market? In the coffee market? Explain.

Suppose that the marginal cost of Royal Crown is 40 cents per 12 oz can. What is its profit maximizing price?

How does Coke’s pricing How does Coke’s pricing change if it buys Royal Crown?change if it buys Royal Crown?

(Pi - ∂C/ ∂qi)/Pi = 1/ εii – [(pj- ∂C/ ∂qj)Qj εij] / Ri εii.

1. The higher is the RC PCM, the higher the price increase2. The higher are RC volumes, the higher the price increase3. The higher the cross-price elasticity, the higher the price increase4. The higher the volume of Coke sales, the lower the price increase5. The higher the Coke own-price elasticity, the lower the price increase

Coffee bean prices have risen…Coffee bean prices have risen…What should Maxwell House What should Maxwell House

do? do?

dp/dMC = [ε/(1+ ε)]If constant elasticity of demand

If linear demand dp/dMC = ½

If semi-log demand dp/dMC = 1

TAKE AWAY ON COST-PASS THROUGH

A Pint-Sized ProblemA Pint-Sized Problem

What has been the reaction of beer suppliers to rising input prices?

A Pint-Sized Problem?A Pint-Sized Problem?

The Role of Industrial Organization The Role of Industrial Organization on Pricingon Pricing

Competition v. MonopolyCompetition v. Monopoly

Strategic Interactions Among CompetitorsStrategic Interactions Among Competitors Oligopoly (Oligopoly (Few competitors; Barriers to entry)Few competitors; Barriers to entry)

Possible reactions to price changes: Possible reactions to price changes: • Competitors match price decreases, but not price increases Competitors match price decreases, but not price increases

Models: Sweezy oligopoly Models: Sweezy oligopoly

• Price is determined by market output. Each competitors set output to maximize profit Price is determined by market output. Each competitors set output to maximize profit given the output of rivals given the output of rivals

Model: Cournot OligopolyModel: Cournot Oligopoly

• Firms constantly seek to undercut competitors’ pricesFirms constantly seek to undercut competitors’ prices Model: Bertrand oligopolyModel: Bertrand oligopoly

• Price leadership (One or more firm calls out price and others follow)Price leadership (One or more firm calls out price and others follow) Model: Dominant Firm-Competitive fringeModel: Dominant Firm-Competitive fringe

Models rely upon Nash equilibrium concept Models rely upon Nash equilibrium concept

Prices, Industry Supply & Demand, and Prices, Industry Supply & Demand, and the Role of Industrial Organizationthe Role of Industrial Organization

mcac

D

$

CS = Consumer Surplus

S

Suppose you are in a competitive Suppose you are in a competitive market with a cost advantage?market with a cost advantage?

mcac

D

$

CS = Consumer Surplus

For as long as you have a cost advantage, p = AC of competitors (-δ)

S

ac1

Monopoly and CompetitionMonopoly and Competition

mcac

D

mr

$

Prices are higher under Monopoly than competitionNext lecture will deal with industrial structure and prices

Pm

Pc

The Role of Market Structure in PricingThe Role of Market Structure in Pricing

Suppose that:Suppose that: Market Demand Q=1000-1000PMarket Demand Q=1000-1000P MC = $.28MC = $.28

How do optimal prices compare depending How do optimal prices compare depending on Market Structure and the nature of on Market Structure and the nature of competition?competition?

Perfect CompetitionPerfect Competition

MC

D

720 Q

Industry

P=.28

Regardless of Market demandPrice is driven by the equalityOf price and marginal cost

MonopolyMonopoly

MC

D

720 Q

Firm

P=.28

mr

P=.64

360

π = PQ - .28Qπ = [1 – (1/1000Q)]Q -.28Qπ = Q - .001Q2 -.28Q

So, taking the first derivativeAnd setting equal to 0:

Dπ/dQ = 1 - .002Q - .28 =0

Q = 360

Plugging into the demand function P= .64.

The Role of Industrial Organization The Role of Industrial Organization on Pricingon Pricing

Competition v. MonopolyCompetition v. Monopoly

Strategic Interactions Among CompetitorsStrategic Interactions Among Competitors Oligopoly (Oligopoly (Few competitors; Barriers to entry)Few competitors; Barriers to entry) If considering a price change …must consider rivals’ reaction…If considering a price change …must consider rivals’ reaction… Possible reactions to price changes: Possible reactions to price changes:

• Competitors match price decreases, but not price increases Competitors match price decreases, but not price increases Model: Sweezy oligopoly Model: Sweezy oligopoly

• Price is determined by market output. Each competitors set output to maximize profit Price is determined by market output. Each competitors set output to maximize profit given the output of rivals given the output of rivals

Model: Cournot OligopolyModel: Cournot Oligopoly

• Firms constantly seek to undercut competitors’ pricesFirms constantly seek to undercut competitors’ prices Model: Bertrand oligopolyModel: Bertrand oligopoly

• Price leadership (One or more firm calls out price and others follow)Price leadership (One or more firm calls out price and others follow) Model: Dominant Firm-Competitive fringeModel: Dominant Firm-Competitive fringe

• Dynamic Pricing: Tit-for TatDynamic Pricing: Tit-for Tat Models rely upon Nash equilibrium concept Models rely upon Nash equilibrium concept

Sweezy OligopolySweezy Oligopoly

D1

D2

mr1

mr2

Q

PIf competitors follow priceDecreases, but not increasesA kinked demand results

P1

P2

P3

Sweezy OligopolySweezy Oligopoly

D1

D2

mr1

mr2

Q

P

Implications: prices are non-responsive to changes in mc over a range – consider mc1 and mc2

mc1

mc2

Nash equilibriumNash equilibrium

In a Nash equilibrium, each firm is optimizing, given the behavior of other firms

John Nash1994 Nobel Laureate

Cournot OligopolyCournot Oligopoly

Price is determined by total market output Price is determined by total market output (relative to demand)(relative to demand)

So my strategy must account for the So my strategy must account for the output of rivalsoutput of rivals If duopoly:If duopoly: Q1* =r1(Q2) and Q2* = r2(Q1)Q1* =r1(Q2) and Q2* = r2(Q1)

Cournot Model: Nash equilibrium Cournot Model: Nash equilibrium as number of firms changesas number of firms changes

mr1

D1

With an initial equilibrium of Qm,Pm,consider the output of a second firm.

The second firm takes the output ofFirm 1 as given, then optimizes on the Residual demand curve (the lowerHalf of the original demand)

mr2

Pm

P2The result is P2.

What is Firm 1’s reaction?

Qm

Cournot Model: Nash equilibrium Cournot Model: Nash equilibrium as number of firms changesas number of firms changes

mr1

D1

mr2

Pm

P2

The result is P2.

What is Firm 1’s reaction?

Firm 1, then takes the output of firm2 as given and reduces its output.Why? Because firm 2 has taken ¼ of market.

Cournot Quantity AdjustmentsCournot Quantity Adjustments

Cournot- Nash Equilibrium

Reaction FunctionsReaction FunctionsIn Cournot, each firm seeks to maximize profit given the output of its rival.

So, we can examine how firm 1’s output changes as firm 2 has different outputs. Denote Q1*(Q2)

Q1

Q2

Note that in our previous example,increases in Q2 were met with reductions in Q1

Q1*(Q2)

Similarly, for Q2*(Q1)

Cournot- Nash equilibrium

Q2*(Q1)

Competitive equilibrium

Cournot: A linear demand exampleCournot: A linear demand example

Suppose that market demand is P= 30-Q and MC1=MC2 = 0.

What is firm 1’s reaction function?

Revenue for firm 1 = PQ1 = (30-Q)Q1 = (30 – Q1- Q2)Q1

= 30Q1 – Q12 – Q1Q2

Thus, MR = 30-2Q1-Q2

Set MR=MC and solve for Q1: Q1 = 15 - 1/2Q2

Similarly, Q2 = 15-1/2Q1

Cournot: linear demand (cont.)Cournot: linear demand (cont.)Q1

Q2

Q1 = 15 - 1/2Q2

Q2 = 15 - 1/2Q1

Solving the reaction functions simultaneously:

10

10

How does this compare with aCompetitive equilibrium for the firms?

How does this compare with the case of Collusion?

Price Determination in Cournot OligopoliesPrice Determination in Cournot Oligopolies

Cournot oligopoly (P-MC)/P = s/εPricing (where s is market share)

Cournot oligopoly w/ (P – MC)/P = 1/(nε) Identical frims

Take-aways:1.Each firm has some market power2.Cournot prices are “in-between” competitive and monopoly prices3.Greater elasticity reduces prices4.Mark-ups are higher for higher market shares (if differentiated)5.As the number of competitors grows, prices approach

competitive levels

Bertrand OligopolyBertrand Oligopoly

Assume that firms compete against each Assume that firms compete against each other through pricesother through prices HomogeneousHomogeneous

• Suppose that P= 30-Q and mcSuppose that P= 30-Q and mc11 = mc = mc22 = 3 = 3

• Nash Equilibrium?Nash Equilibrium?

DifferentiatedDifferentiated

Joseph Bertrand

Bertrand OligopolyBertrand Oligopoly

Assume that firms compete against each Assume that firms compete against each other through pricesother through prices HomogeneousHomogeneous

• Suppose that P= 30-Q and mcSuppose that P= 30-Q and mc11 = mc = mc22 = 3 = 3

• Nash Equilibrium?Nash Equilibrium?

DifferentiatedDifferentiated

Joseph Bertrand

Differentiated BertrandDifferentiated Bertrand

Suppose 2 firms each with fixed costs of 20.Suppose 2 firms each with fixed costs of 20. QQ11 = 12 - 2P = 12 - 2P11 +P +P2 2 (demand facing firm 1)(demand facing firm 1) QQ22 = 12 – P = 12 – P22 +P +P1 1 (demand facing firm2)(demand facing firm2)

Find Find ππ = P = P11 (12 -2P (12 -2P11 +P +P2 2 )) Set Set ∂dπ/ ∂∂dπ/ ∂ pp11 = 0, to get firm 1’s reaction = 0, to get firm 1’s reaction

function:function:• PP11 = 3 + 1/4P = 3 + 1/4P22, and similarly for firm 2, , and similarly for firm 2, • PP22 = 3 + 1/4P = 3 + 1/4P11

Differentiated Goods, Bertrand Reaction functionsDifferentiated Goods, Bertrand Reaction functions

P1

P2

PP11 = 3 + 1/4P = 3 + 1/4P22

PP22 = 3 + 1/4P = 3 + 1/4P11

4

4

Nash Equilibrium

Cola Wars: Coke and PepsiCola Wars: Coke and Pepsi

Bertrand competition in prices with a Bertrand competition in prices with a differentiated productdifferentiated product

Rival’s prices do affect the firm’s demand Rival’s prices do affect the firm’s demand function, but because products are function, but because products are differentiated a lower price does not steal the differentiated a lower price does not steal the entire marketentire market

Each firm has a Bertrand profit-maximizing Each firm has a Bertrand profit-maximizing “best response function” for the price to charge “best response function” for the price to charge in response to the price its rival chargesin response to the price its rival charges


Demand equations estimated from Demand equations estimated from detailed monthly price data:detailed monthly price data:

qqCC(P(PCC,P,PPP) = 63.42 – 3.98P) = 63.42 – 3.98PCC + 2.25P + 2.25PPP

qqPP(P(PCC,P,PPP) = 49.52 – 5.48P) = 49.52 – 5.48PPP + 1.40P + 1.40PCC

Unit CostUnit CostCC = 4.96 = 4.96 Unit CostUnit CostPP = 3.96 = 3.96 Unit is a 10 cases of 24 12 oz cans.Unit is a 10 cases of 24 12 oz cans.

Source: “Econometric Analysis of Collusive Behavior in a Soft-Drink Market” by Gasmi, Laffont and Vuong in Journal of Economics & Management Strategy, 1992, vol. 1, issue 2, pp. 277-311


How should Coke and Pepsi price in How should Coke and Pepsi price in response to their rival?response to their rival?

Need to find Best Response Function for Need to find Best Response Function for each cola producereach cola producer

Solve for profit-maximizing PRICE (in a Solve for profit-maximizing PRICE (in a Bertrand game price is the choice Bertrand game price is the choice variable) variable)


How do we find profit-maximizing price?How do we find profit-maximizing price? Set MR = MC!!Set MR = MC!! Solve for MR (in terms of change in PSolve for MR (in terms of change in PCC):):

TRTRCC = P = PCC * q * qCC = P = PCC * q * qCC(P(PCC, P, PPP))

TRTRCC = P = PCC * (63.42 – 3.98P * (63.42 – 3.98PCC + 2.25P + 2.25PPP))

MRMRCC(P(PCC, P, PPP) = 63.42 – (2)* 3.98P) = 63.42 – (2)* 3.98PCC + 2.25P + 2.25PPP

MRMRCC = 63.42 – 7.96P = 63.42 – 7.96PCC + 2.25P + 2.25PPP


How do we find profit-maximizing price?How do we find profit-maximizing price? Set MR = MC!!Set MR = MC!! Solve for MR (in terms of change in PSolve for MR (in terms of change in PCC):):

TRTRCC = P = PCC * q * qCC = P = PCC * q * qCC(P(PCC, P, PPP))

TRTRCC = P = PCC * (63.42 – 3.98P * (63.42 – 3.98PCC + 2.25P + 2.25PPP))

MRMRCC(P(PCC, P, PPP) = 63.42 – (2)* 3.98P) = 63.42 – (2)* 3.98PCC + 2.25P + 2.25PPP

MRMRCC = 63.42 – 7.96P = 63.42 – 7.96PCC + 2.25P + 2.25PPP


Solve for MC (again, in terms of PSolve for MC (again, in terms of PCC):):

TCTCCC = UC = UCCC * q * qCC = UC = UCCC * q * qCC(P(PCC, P, PPP))

TCTCCC = 4.96 * (63.42 – 3.98P = 4.96 * (63.42 – 3.98PCC + 2.25P + 2.25PPP))

MCMCCC(P(PCC, P, PPP) = -19.74 ) = -19.74

(MC with respect to PRICE – if price goes up, (MC with respect to PRICE – if price goes up, quantity goes down)quantity goes down)

Cola Wars: Coke and PepsiCola Wars: Coke and Pepsi Profit maximizing price is where MR = MCProfit maximizing price is where MR = MC MRMRCC = 63.42 – 7.96P = 63.42 – 7.96PCC + 2.25P + 2.25PPP

MCMCCC = -19.74 = -19.74

MRMRCC = MC = MCCC

63.42 – 7.96P63.42 – 7.96PCC + 2.25P + 2.25PP P = -19.74= -19.74

PPCC(P(PPP) = 10.44 + 0.2826P) = 10.44 + 0.2826PPP

““Best Response Function” Best Response Function”


Can do the same thing for Pepsi -- Can do the same thing for Pepsi -- Best Response Function for PepsiBest Response Function for Pepsi PPPP(P(PCC) = 6.49 + 0.1277P) = 6.49 + 0.1277PCC

PPCC(P(PPP) = 10.44 + 0.2826P) = 10.44 + 0.2826PPP

2 equations and 2 unknowns:2 equations and 2 unknowns: PPCC(P(PPP) = 10.44 + 0.2826 *(6.49 + 0.1277P) = 10.44 + 0.2826 *(6.49 + 0.1277PCC)) PPCC(P(PPP) = 12.73; P) = 12.73; PPP(P(PCC) = 8.11) = 8.11 Actual average prices over this period: C=12.96, P=8.16Actual average prices over this period: C=12.96, P=8.16

Price DeterminationPrice DeterminationCompetitive P=MC

Competitive w/ P=MC(competitors) - δcost advantage

Monopoly Pricing: (P-MC)/P = 1/ε

Cournot oligopoly (P-MC)/P = s/εPricing (where s is market share)

Cournot oligopoly w/ (P – MC)/P = 1/(nε) Identical firms

Bertrand(identical product) P=MC

Bertrand Differentiated P> MC

The Dominant Firm Model a. Assumes a single “dominant” firm facing a competitive “fringe” b. The dominant firm calls out a price c. The competitive fringe responds as a price taker setting its output

Price LeadershipPrice Leadership

The Dominant Firm-Competitive Fringe ModelThe Dominant Firm-Competitive Fringe Model

P

Q

S=Σmc

D

d

mr

P1

Qd

mc

d = D - S

Pricing in a Dominant Firm Competitive Fringe Industry

(P-MC)/P = S/[ηm + (1-S)ef]

1.Market share (S)

2. Elasticity of Market demand (ηm)

3. Elasticity of Supply of the fringe firms (ef)

Suppose S=.8, ηm = 2, and ef = 2, What is the value of the price cost mark up?

CollusionCollusion

“People of the same trade seldom meet together even for merriment and diversion, but the conversation ends

in a conspiracy against the public, or in some contrivance to raise prices.”

Adam SmithThe Wealth of Nations

Price CollusionPrice Collusion

The Simple economics of collusion?The Simple economics of collusion?

FirmIndustry

MR

D

P

q Q

$/q $/q

Q

S= Σmc AC

mc

Pm

q Qmqm

Competition (?) in the Airline IndustryCompetition (?) in the Airline Industry

Crandall: I think it's dumb as hell ... to sit here and pound the (deleted) out of each other and neither one of us making a (deleted) dime. ...

We can both live here [Dallas] and there ain't no room for Delta. But there's, ah, no reason that I can see, all right, to put both companies out of business.

Putnam: Do you have a suggestion for me?

Crandall: yes, I have a suggestion for you. raise your goddamn fares twenty percent. I'll raise mine the next morning. ... You'll make more money, and I will too.

Putnam: We can't talk about pricing.

Crandall: Oh (deleted), Howard. We can talk about any goddamn thing we want to talk about.

Economic conditions conducive to Economic conditions conducive to and destructive of Collusionand destructive of Collusion

Number of firms (market concentration)Number of firms (market concentration) Barriers to entryBarriers to entry Product homogeneityProduct homogeneity Elasticity of market demandElasticity of market demand Ability to detect cheating Ability to detect cheating Cost symmetry/asymmetryCost symmetry/asymmetry

Economic research indicates that despite obstacles, economic barriers to successful collusion can often be overcome

Rebates in Real Estate Rebates in Real Estate CommissionsCommissions

““If we give rebates and inducements, it would If we give rebates and inducements, it would get out of control and all clients would be get out of control and all clients would be wanting something. The present law keeps it wanting something. The present law keeps it under control.”under control.”

““This would turn into a bidding war, lessen our This would turn into a bidding war, lessen our profits and cheapen our ‘so-called’ profession.”profits and cheapen our ‘so-called’ profession.”

““If inducements were allowed, they could lead to If inducements were allowed, they could lead to competitive behavior, which would make us look competitive behavior, which would make us look unprofessional in the eyes of the public.”unprofessional in the eyes of the public.”

““I think this would just take money right out of I think this would just take money right out of our pocket.” our pocket.”

http://www.usdoj.gov/atr/public/real_estate/rebates.htm

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