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Price Theory

Lecture 10: Price Competition

Topics for today’s lecture . . .

1. Bertrand competition

2. Collusion

3. Hotelling competition

Bertrand competition

Definition: Bertrand competition

A form of oligopoly competition in which the firms in a market simultaneously select the

prices they will charge.

The Bertrand model of competition tends to work well for industries in which firms can easily

adjust the quantities they produce in response to consumer demand.

The market for widgets

6 MC

300

0 Q

Alpha Foundry and Beta Forges engage in

Bertrand competition in the market for

widgets; a homogeneous good.

Demand in the market is,

Q = 300 − 10P.

Note: This is the same demand curve as

we used with Cournot competition in the

previous lecture.

Both firms have a constant marginal cost

of MCA = MCB = $6.

Residual demand in Bertrand competition

6 MC

300

0 Q

Given that widgets are a homogeneous

good, consumers purchase exclusively

from the firm with the lowest price.

• Consumers split their purchases between firms if firms choose the

same price.

For any price PB chosen by Beta, demand

for Alpha’s widgets will be,

QA =

 

0 PA > PB

150 − 5PA PA = PB 300 − 10PA PA < PB

Nash equilibrium with discontinuous demand

6 MC

300

0 Q

In a Nash equilibrium, each firm’s price is

a best-response to the price of its rival.

The discontinuity in the residual demands

means that we cannot construct

best-response functions for firms engaged

in Bertrand competition.

Instead, we will employ a process of

elimination, excluding cases in which a

firm can increase its profit by unilaterally

altering its price.

Step 1: Neither firm will choose a price below marginal cost

PSA < 0 PA

6 MC

300

0 Q

Suppose that Alpha chooses a price that

lies below marginal cost.

• Alpha’s producer surplus is negative as it incurs a loss on each sale.

If, instead, Alpha set its price equal to

marginal cost, its producer surplus would

be zero.

• It follows that pricing at marginal cost cannot be a best-response.

Step 2: Both firms choose the same price

PSA

6 MC

300

0 Q

Suppose that Alpha chooses a price that

is greater than PB.

• Alpha makes no sales and receives a producer surplus of zero.

If, instead, Alpha matches Beta’s price, it

sells to half the market and receives a

positive surplus.

• It follows that selecting a higher price than the rival firm cannot be a

best-response.

Step 3: Firms do not choose a price above marginal cost

PSA gain

loss

6 MC

300

0 Q

Suppose that Alpha and Beta choose the

same price, in excess of marginal cost.

• Alpha sells to half the market and receive a positive surplus.

If, instead, Alpha chooses a price that is

just below PB, it captures the entire

market.

• The surplus lost from existing sales is arbitrarily small, relative to the gain

from more than doubling sales.

Nash equilibrium prices

120 240

6 MC

300

0 Q

In a Nash equilibrium, both firms set their

price equal to marginal cost.

• Each firm sells 120 widgets, half market demand at the price $6.

• The producer surplus of each firm is zero.

• The outcome is efficient; there is no deadweight loss.

Neither firm can gain by unilaterally

altering its price. (You should check this.)

Quiz 1

Suppose that natural gas is a homogeneous good, and that demand for gas is given by the

function,

Q = 75 − 2P.

The marginal cost of producing gas is $7.50. If two firms (A and B ) engage in Bertrand

competition, in the Nash equilibrium,

(a) Q∗A = 15 and Q ∗ B = 15.

(b) Q∗A = 30 and Q ∗ B = 30.

(d) Q∗A = 60 and Q ∗ B = 60.

Quiz 2

Suppose that natural gas is a homogeneous good, and that demand for gas is given by the

function,

Q = 75 − 2P.

The marginal cost of producing gas is $7.50. If two firms (A and B ) engage in Bertrand

competition, which of the following statements is false?

(a) The Nash equilibrium prices are P∗A = $7.50 and P ∗ B = $7.50.

(b) The Nash equilibrium consumer surplus is larger than under monopoly.

than under perfect competition.

(d) The entry of a third firm would not alter the efficiency of the market.

The Bertrand paradox

Market outcomes depend critically on whether it is prices that are determined by firms’

choices of quantities, or quantities that are determined by firms’ choices of prices.

In Cournot competition, small changes in strategy lead to small changes in price and producer

surplus.

• Firms are accommodating, willing to give one another enough room to make a profit.

In Bertrand competition, a small change in price could be the difference between selling

nothing and capturing the entire market.

• Firms are cut-throat, unwilling to yield anything to their rivals.

Collusion

Framing oligopoly as a game

We have examined both Cournot competition and Bertrand competition as games.

• In both games we assume that firms choose their strategies simultaneously.

• Both games have finite-horizons, in that they are played once only.

In many markets firms interact repeatedly, over an indefinite time horizon.

• Repeated interaction over an indefinite time-horizon creates the possibility of cooperation between firms.

• Firms have an incentive to cooperate as competition reduces the total producer surplus in a market.

Definition: Collusion

An agreement between firms to coordinate prices, quantities, or some other aspect of market

behaviour.

By colluding, the firms in a market can mimic the behaviour of a monopolist, increasing

industry profits while creating a (larger) deadweight loss.

Collusion in the market for widgets

DWL

240

PSA PSB

12060

6 MC

300

0 Q

Suppose that Alpha and Beta are engaged

in an infinite-horizon repeated game.

The two firms agree to select the prices

PA = PB = $18 (the monopoly price)

each time they interact.

• Each firm receives a producer surplus of $720.

• Both consumer surplus and deadweight loss are $720.

Note: The welfare cost of the collusive

agreement is the same as for monopoly.

The incentive to cheat

PSB

PSA

60 120

6 MC

300

0 Q

We have already established that the

prices PA = PB = $18 do not constitute a

mutual best-response.

• If Alpha shaves its price to slightly less than $18, Alpha captures the entire

market.

• Alpha’s producer surplus becomes (arbitrarily close to) $1440.

The punishment for cheating

gain

loss

7 2

collude

1 4

4 0

0 t

Suppose that the firms utilise the

grim-trigger strategy.

• Collude by choosing the price $18 so long as the other firm does the same.

• If either firm cheats, choose the price $6 (marginal cost) thereafter, thereby

reducing both firms’ surpluses to $0.

This strategy will sustain collusion so long

as both firms place sufficiently high value

on future profits.

Quiz 3

Which of the following factors would facilitate collusion in a market?

(a) The firms in a market heavily discount future profits, relative to profits earned today.

(b) It is difficult to distinguish between fluctuations in consumer demand, and the effects

of another firm cheating.

the collusive agreement collapses.

(d) A firm can rapidly change its price if a rival is observed to be cheating.

Exercise: Collusion with N firms

Suppose that natural gas is a homogeneous good, and that demand for gas is given by the

function Q = 75 − 2P . The marginal cost of producing gas is $7.50, and firms compete by selecting prices.

1. Find the monopoly price, quantity, and producer surplus, for this market. (Hint: The first

step is to derive inverse demand.)

2. If N firms collude by selecting the monopoly price, what is each firm’s producer surplus?

3. If a firm cheats on the agreement by shaving its price, what producer surplus would it

receive?

4. Suppose that the colluding firms employ the grim-trigger strategy to punish cheating.

How does the number of colluding firms affect their ability to cooperate?

Exercise solutions

1. The first step is to rearrange the demand function to derive inverse demand,

P = 37.5 − 0.5Q.

The profit function of a monopolist in this market would be,

Π(Q ) = Q (37.5 − 0.5Q ) − 7.5Q = 30Q − 0.5Q 2.

The corresponding first-order condition is,

∂Π(Q )

∂Q = 30 − Q = 0 implying Q∗ = 30.

The monopoly price is P∗ = 37.5 − 0.5 × 30 = $22.5, and the corresponding producer surplus is PS = 30 × (22.5 − 7.5) = $450.

Exercise solutions

2. If there are N firms in the market, with all firms selecting the monopoly price, then each

firm will sell a quantity Qfirm = 30/N . Therefore, each firm’s producer surplus is,

PS firm = 30

N × (22.5 − 7.5) =

450

N .

3. If a firm cheats by shaving its price, it will sell (close to) the monopoly quantity at (close

to) the monopoly price. Therefore, its producer surplus would be (close to) $450.

4. As the number of firms in the market increases, the surplus a firm earns from colluding

decreases, while the payoff to cheating remains the same. It follows that sustaining a

collusive agreement becomes more difficult.

Hotelling competition

Definition: Differentiated products

Products that differ in one or more characteristic that is significant to consumers.

When products are differentiated some consumers may be willing to pay a premium for their

preferred product.

Linear city

Imagine a city in which 300 consumers live on Main Street, which is one kilometre long.

The people of Linear City dislike travelling to visit a store.

• A consumer in Linear City suffers $10 of disutility for each kilometre they travel.

There are only two stores in Linear City, Alice’s Groceries and Brett’s Bargains, located at

either end of Main Street.

• The groceries sold by the two stores are identical.

• The marginal cost of groceries is $15.

• The stores compete by selecting prices.

Consumers in linear city

1km

1 − xx

Alice BrettConsumer

Suppose that the people of Linear City

are evenly distributed along Main Street.

• Each consumer demands one unit of groceries, regardless of price.

Consumers choose a store based on price

and location.

• The effective price of Alice’s groceries to a consumer at point x is PA + 10x .

• The effective price of Brett’s groceries is PB + 10(1 − x ).

Quiz 4

1km

Alice BrettJohn

Paul George

If John buys his groceries from Alice, we

can conclude that,

(a) both Paul and George will also buy

from Alice.

(b) Paul will buy from Alice, and

George will buy from Brett.

George’s choice is uncertain.

(d) George will buy from Brett, and

Paul’s choice is uncertain.

Market share in Hotelling competition

The indifferent consumer is the consumer for whom the cost of purchasing from Alice is

equal to the cost of purchasing from Brett.

• All consumers to the left of the indifferent consumer prefer Alice, while those to the right prefer Brett.

If the indifferent consumer is located at a point x∗ then,

PA + 10x ∗ = PB + 10(1 − x∗) or x∗ =

PB − PA + 10 20

Therefore, demand for Alice’s groceries is the number of consumers in the market (300),

times Alice’s market share x∗. (Brett’s demand is 300(1 − x∗).)

Profit functions in Hotelling competition

Given that Alice’s Groceries faces a constant marginal cost, Alice’s profit function will be,

ΠA(PA,PB) = QAPA − QAMCA = (PA − MCA)QA.

Given that Alice’s market share is x∗, and that the number of consumers in the market is

300, we have QA = 300x ∗ and therefore,

ΠA(PA,PB) = (PA − MCA)300x∗

= 300(PA − 15) PB − PA + 10

= 15(PAPB − P 2A + 10PA − 15PB + 15PA − 150)

= 15(PAPB − P 2A + 25PA − 15PB − 150).

Best-response functions in Hotelling competition

To find Alice’s best-response function we first need to find the partial derivative of the profit

function with respect to PA,

∂

∂PA 15(PAPB − PA2 + 25PA − 15PB − 150) = 15(PB − 2PA + 25).

Alice’s best response to PB satisfies the first-order condition,

∂Π(PA,PB)

∂PA = 15(PB − 2PA + 25) = 0 ⇒ PA =

PB + 25

2 .

Alice’s best-response function

2012.5

PA = PB + 25

0 PA

We can plot Alice’s best-response

function in PA-PB space.

Alice’s location is a source of market

power:

• If Brett prices at marginal cost, Alice’s best-response is $20—a price

in excess of marginal cost.

• A fraction 0.25 of consumers will buy from Alice at this price, to avoid the

cost of travelling to Brett’s.

Exercise: Brett’s Bargains best-response function

Recall that Brett’s Bargains faces a constant marginal cost of $15, there are 300 consumers

in linear city, and that the location of the indifferent consumer is given by the equation,

x∗ = PB − PA + 10

20 .

1. Write an expression for the quantity sold by Brett’s Bargains as a function of firm prices.

2. Write a profit function for the firm.

3. Find Brett’s Bargains’ best response function.

Exercise solutions

1. Brett’s Bargains sells the quantity,

QB = 300(1 − x∗) = 300 (

1 − PB − PA + 10

) = 300

( 20

20 −

PB − PA + 10 20

) = 300

( 20 − PB + PA − 10

) = 300

PA − PB + 10 20

Exercise solutions

2. Brett’s Bargains’ profit function is,

ΠB(PA,PB) = (PB − MCB)QB = 300(PB − 15) PA − PB + 10

= 15(PAPB − P 2B + 10PB − 15PA + 15PB − 150)

= 15(PAPB − P 2B + 25PB − 15PA − 150).

3. Brett’s first-order condition is,

∂ΠB(PA,PB)

∂PB = 15(PA − 2PB + 25) = 0.

Solving for PB yields the best-response function,

PB = PA + 25

2 .

The Nash equilibrium

2512.5

PA = PB + 25

1 2

PB = PA + 25

0 PA

We need to solve the best-response

functions simultaneously.

Substituting for PB into Alice’s

best-response function,

PA = 12.5 + 1

( PA + 25

) = 18.75 +

PA 4 .

Collecting like terms 0.75PA = 18.75 or

P∗A = $25.

Substituting for PA into Brett’s

best-response function P∗B = $25.

Exercise: Equilibrium market shares and profits

There are 300 consumers living in linear city. Each consumer demands a single unit of

groceries, and the indifferent consumer is located at,

x∗ = PB − PA + 10

20 .

The marginal cost of each firm is MC = $15, and the equilibrium prices are P∗A = P ∗ B = $25.

1. Calculate the equilibrium market share of each store.

2. What is each store’s equilibrium profit?

3. Calculate each store’s Lerner index of market power.

Exercise solutions

1. To find the equilibrium market shares, substitute the prices into the equation for the

indifferent consumer,

x∗ = PB − PA + 10

20 =

25 − 25 + 10 20

= 1

2 .

Therefore, each store sells to half the market (150 consumers).

2. Alice’s equilibrium profit is,

ΠA = 150 × (P∗A − MC ) = 150 × (25 − 15) = $1500.

Brett’s equilibrium profit is,

ΠB = 150 × (P∗B − MC ) = 150 × (25 − 15) = $1500.

Exercise solutions

3. Alice’s Lerner index is,

LA = P∗A − MC

P∗A =

25 − 15 25

= 0.4.

Brett’s Lerner index is,

LB = P∗B − MC

P∗B =

25 − 15 25

= 0.4.

Deriving market power from location

strongly

prefer Alice

strongly

prefer Brett

Alice Brett

The Hotelling model illustrates that

differences in location may be sufficient to

negate the Bertrand paradox.

• Each store enjoys market power due to the reluctance of nearby consumers

to travel to the more distant retailer.

While stores do compete for consumers

closer to the centre, this competition is

not sufficient to eliminate all profits.

Other interpretations of spatial product differentiation

The Linear City example treats the concept of distance literally.

An alternative interpretation is that the spatial dimension represents an attribute of a product.

• Motorcars might be classed as either family cars, located at the left end of the line, or sports cars located at the right end.

• Music could be divided into classical at the left end, and popular music at the right end.

The location of a consumer on the line then represent her/his preference over the attribute in

question, while the travel cost represents the intensity of that preference.

The distribution of consumers

less

competition

Alice Brett

Thus far we have assumed that the

consumers in the market are evenly

spread along the line.

Competition is reduced if consumers are

clustered around firm locations.

• There are fewer consumers who are (close to) indifferent between the two

firms.

Conversely, competition is increased if

consumers are clustered at the centre.

Questions?

Key concepts from today’s lecture

You can use these concepts (as search terms) to conduct further research into the topics

covered in today’s lecture:

• Bertrand competition

• Discontinuity in demand

• Price shaving

• Bertrand paradox

• Collusion

• Grim-trigger strategy

• Temporary punishment

• Hotelling competition

• Differentiated products

• Indifferent consumer

• Spatial product differentiation

• Distribution of consumers