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Price Theory

Lecture 9: Quantity Competition

Topics for today’s lecture . . .

1. Cournot competition

2. Costs & market power

3. Competition & welfare

4. Free entry & exit

Definition: Oligopoly

A market in which a small number of firms compete.

Firms in an oligopoly market typically enjoy market power (in the manner of a monopolist),

but face additional strategic considerations that can be modelled using game theory.

The problem with oligopoly

Monopolyone firm

Perfect competitionmany firms

few firms Oligopoly

In perfect competition a firm selects its

quantity to maximise its profits, given the

market price.

A monopolist determines its own price

through its choice of quantity.

In oligopoly, market outcomes are

determined by the strategic interaction of

the firms, and depend on,

• the number of firms.

• the substitutability of products.

• the nature of the strategic interaction.

Definition: Homogeneous goods

Products that are identical in the eyes of consumers (perfect substitutes).

When the products in a market are homogeneous, a consumer will not purchase from a firm if

the same good is available from a second firm at a lower price.

Quiz 1

Suppose that Nature Inc. produces a ‘certified organic’ sunflower oil, and that Chem Co.

produces sunflower oil from crops farmed with chemical pesticides. The two products are

chemically identical (laboratory tests cannot distinguish between them), and there are no

other firms in the sunflower oil market. Which of the following statements is most correct?

(a) Nature Inc. and Chem Co. are both monopolists.

(b) Nature Inc. alone is a monopolist because some consumers prefer ‘organic’ products.

use either ‘organic’ or ‘non-organic’ sunflower oil.

(d) The products of the two firms are homogeneous because they are chemically identical.

Cournot competition

Definition: Cournot competition

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

quantities they will produce.

The Cournot model of competition tends to work well for industries in which firms have to

make production (or capacity) decisions in advance of the market.

Competition in the market for widgets

300

0 Q

Alpha Foundry and Beta Forges engage in

Cournot competition in the market for

widgets, a homogeneous good.

Inverse demand in the market is,

P = 30− 0.1Q.

Using QA for Alpha’s quantity, and QB is

Beta’s quantity, inverse demand can be

restated as,

P = 30− 0.1(QA + QB)

= 30− 0.1QA − 0.1QB

Residual demand

300QB

0 Q

The market price is jointly determined by

the two firms’ choice of quantities.

From Alpha’s perspective, Beta ‘uses up’

a portion of the market demand when it

produces the quantity QB .

Alpha maximises its profit by acting as a

monopolist on its residual demand.

• The demand that is ‘left over’ once Beta’s sales have been accounted for.

Alpha’s problem

QA + QB

MRA

6 MC

300QB

0 Q

Alpha’s total revenue in this market is,

TRA = QA(30− 0.1QA − 0.1QB).

Marginal revenue is the partial derivative

of TRA with respect to QA.

Suppose that both firms face a constant

marginal cost of $6 per widget.

Alpha’s best response to QB is the

quantity that equates marginal revenue

with marginal cost.

Partial derivatives

A partial derivative is a derivative of a multivariate function in which all variables, other than

the variable in question, are treated as constants.

The partial derivative of a function is that rate at which the function changes as the given

variable increases.

Some examples:

• ∂

∂QA (QA + QB) = 1

• ∂

∂QA (QAQB) = QB

• ∂

∂QA

( 2Q2A − 2QAQB

) = 4QA − 2QB

• ∂

∂QB

( 2Q2A − 2QAQB

) = −2QA

Deriving Alpha’s best-response function

First, write the profit function for the firm (in this case Alpha),

ΠA(QA,QB) = QA(30− 0.1QA − 0.1QB)− 6QA = 24QA − 0.1Q2A − 0.1QAQB .

Next, take the partial derivative of the profit function with respect to the firm’s own quantity,

∂ΠA(QA,QB)

∂QA = 24− 0.1× 2Q2−1A − 0.1QB = 24− 0.2QA − 0.1QB .

Finally, set the derivative equal to zero and solve for the firm’s own quantity,

24− 0.2QA − 0.1QB = 0 ⇒ 0.2QA = 24− 0.1QB ⇒ QA = 120− 0.5QB .

Alpha’s best-response function

240 QA = 120− 0.5QB

120

0 QA

We can plot Alpha’s best-response

function in QA-QB space.

• Alpha’s best-response function intersects the QA axis at its monopoly

quantity.

• The QB axis intercept represents the point at which Alpha would withdraw

from the market.

Notice that for each additional widget

Beta produces, Alpha wants to reduce its

production by half a widget.

Exercise: Best-response function

Beta faces the inverse demand function P = 30− 0.1QA − 0.1QB , and the constant marginal cost MCB = $6.

1. Write an expression for Beta’s profit function.

2. Find Beta’s best-response function.

3. Plot Beta’s best-response function in QA-QB space.

4. What is Beta’s monopoly quantity? (The quantity it would choose if it were not

competing against Alpha.)

Exercise solutions

1. Beta’s profit function is,

ΠB(QA,QB) = QB(30− 0.1QA − 0.1QB)− 6QB = 24QB − 0.1QAQB − 0.1Q2B .

2. The partial derivative of the profit function with respect to Beta’s quantity is,

∂ΠB(QA,QB)

∂QB = 24− 0.1QA − 0.2QB .

Setting the derivative equal to zero and solving for Beta’s quantity,

24− 0.1QA − 0.2QB = 0 ⇒ 0.2QB = 24− 0.1QA ⇒ QB = 120− 0.5QA.

Exercise solutions

120

240

QB = 120− 0.5QA

0 QA

3. Beta’s best-response function is

illustrated in this figure.

4. If QA = 0, leaving Beta as a

monopolist, then Beta’s best response

is to produce 120 widgets.

The Nash equilibrium in Cournot competition

240 QA = 120− 0.5QB

120

240

QB = 120− 0.5QA

0 QA

Nash equilibrium

In a Nash equilibrium each firm’s choice

of quantity is a best-response to its rival.

Substituting for QB into Alpha’s

best-response function,

QA = 120− 0.5(120− 0.5QA)

= 60 + 0.25QA.

Collecting like terms 0.75QA = 60 or

Q∗A = 80 widgets.

Substituting for QA into Beta’s

best-response function Q∗B = 80 widgets.

The equilibrium price

16080

6 MC

300

0 Q

PSA PSB

In equilibrium both firms produce 80

widgets.

The equilibrium price of widgets is,

P∗ = 30− 0.1Q∗A − 0.1Q∗B = 30− 0.1× 80− 0.1× 80 = $14.

Each firm has a producer surplus of $640.

Each firm produces less than the

monopoly quantity, while total output is

greater than the monopoly quantity.

Costs & market power

Discussion: A fall in Alpha’s marginal cost of production

Suppose that Alpha Foundry installs new, more efficient, capital equipment, which reduces its

marginal cost to MCA = $3.

1. How do you think that this change will affect Alpha’s behaviour and producer surplus?

2. How do you think that this change will affect Beta’s behaviour and producer surplus?

Keep your answers in mind as we work through this example.

A fall in Alpha’s marginal cost of production

The change in marginal cost alters Alpha’s profit function,

ΠA(QA,QB) = QA(30− 0.1QA − 0.1QB)− 3QA = 27QA − 0.1Q2A − 0.1QAQB .

The partial derivative of this profit function with respect to QA is,

∂ΠA(QA,QB)

∂QA = 27− 0.1× 2Q2−1A − 0.1QB = 27− 0.2QA − 0.1QB .

Setting the derivative equal to zero and solving for QA,

27− 0.2QA − 0.1QB = 0 ⇒ 0.2QA = 27− 0.1QB ⇒ QA = 135− 0.5QB .

Beta’s best-response function remains QB = 120− 0.5QA, as neither its marginal revenue, nor marginal cost, have changed.

The new Nash equilibrium

100

270 QA = 135− 0.5QB

135

120

240

QB = 120− 0.5QA

0 QA

New Nash equilibrium

Alpha’s best-response function shifts

outward.

Substituting for QB into Alpha’s

best-response function,

QA = 135− 0.5(120− 0.5QA)

= 75 + 0.25QA.

Collecting like terms 0.75QA = 75 or

Q∗A = 100 widgets.

Substituting for QA into Beta’s

best-response function Q∗B = 70 widgets.

Price and producer surpluses

170100

300

0 Q

PSA PSB

The new equilibrium price of widgets is,

P∗ = 30− 0.1Q∗A − 0.1Q∗B = 30− 0.1× 100− 0.1× 70 = $13,

down from $14. (The total quantity is

170 widgets, up from 160 widgets.)

Alpha’s producer surplus is $1000 (up

from $640).

Beta’s producer surplus is $490 (down

from $640).

Quiz 2

Using the two equilibria previously derived in this lecture, answer the following question:

How much would Alpha Foundry be willing to pay for new capital equipment that would

reduce its marginal cost from $6 to $3?

(a) $360.

(b) $490.

(d) $1000.

Market power & marginal cost

170100

300

0 Q

PSA PSB

Alpha’s Lerner index of market power is,

LA = P −MCA

P =

13− 3 13

≈ 0.77.

Beta’s Lerner index is,

LB = P −MCB

P =

13− 6 13

≈ 0.54.

In Cournot competition, the firm with the

lowest marginal cost,

• produces more than its rival.

• enjoys higher margins on each sale.

Competition & welfare

The welfare economics of monopoly

PS DWL

240

120

6 MC

300

0 Q

A monopolist operating in this market

would produce 120 widgets, and sell them

at a price of $18.

The consumer surplus is,

CS = 0.5× 120× (30− 18) = $720.

The producer surplus is,

PS = 120× (18− 6) = $1440.

And the deadweight loss is,

DWL = 0.5×(240−120)×(18−6) = $720.

The welfare economics of duopoly

PS DWL

240

160

6 MC

300

0 Q

In a duopoly, total production rises to 160

widgets, and the price falls to $14.

The consumer surplus is,

CS = 0.5× 160× (30− 14) = $1280.

The total producer surplus is,

PS = 160× (14− 6) = $1280.

And the deadweight loss is,

DWL = 0.5×(240−160)×(14−6) = $320.

Cournot competition with N identical firms

Suppose now that N identical firms, including Alpha Foundry, compete in the market for

widgets.

• Each firm has a marginal cost of MC = $6.

From Alpha’s perspective, the total quantity in the market can be written Q = QA + X ,

where X is the combined production of all other firms.

• This allows inverse demand to be written as P = 30− 0.1QA − 0.1X .

• And Alpha’s profit function to be written,

ΠA = QA(30− 0.1QA − 0.1X )− 6QA = 24QA − 0.1Q2A − 0.1QAX .

The Nash equilibrium with N firms

The corresponding first-order condition is,

∂ΠA ∂QA

= 24− 0.2QA − 0.1X = 0.

From which it follows that Alpha’s best response function is QA = 120− 0.5X .

Given that all firms are identical, each firm will produce the same quantity in equilibrium.

If we replace X with (N − 1)QA in Alpha’s best-response function,

QA = 120− (N − 1)QA

2 or (N + 1)QA = 240,

implying that each firm produces Q∗A = 240/(N + 1) widgets in equilibrium.

Equilibrium price & quantity

PS DWL

240

180

6 MC

300

0 Q

In equilibrium, the total quantity produced

by the firms in the market is,

Q∗ = NQ∗A = 240 N

N + 1 .

(The figure illustrates the N = 3 case.)

The equilibrium price at this quantity is,

P∗ = 30− 0.1Q∗ = 30− 24 N

N + 1 .

As N increases, both the equilibrium price

and quantity approach competitive levels.

Exercise: Welfare with N identical firms

240

6 MC

300

0 Q

Five firms compete in the market for

widgets. Inverse demand in the market is,

P = 30− 0.1Q.

Each firm’s marginal cost is MC = $6.

Calculate the following:

1. P∗ and Q∗ (use the expressions from

the previous slide).

2. Consumer surplus.

3. Total producer surplus.

4. Deadweight loss.

Exercise solutions

240

200

6 MC

300

0 Q

1. With N = 5 the equilibrium quantity

is,

Q∗ = 240 N

N + 1

= 240× 5

6 = 200 widgets.

The equilibrium price is,

P∗ = 30− 24 N

N + 1

= 30− 24× 5

6 = $10.

Exercise solutions

PS DWL

240

200

6 MC

300

0 Q

2. Consumer surplus is,

CS = 0.5× 200× (30− 10)

= $2000.

3. Producer surplus is,

PS = 200× (10− 6) = $800.

4. Deadweight loss is,

DWL = 0.5× (240− 200)× (10− 6)

= $80.

Firm producer surplus with N firms

240

200

6 MC

300

0 Q

As the number of firms contesting the

market increases, the total producer

surplus decreases.

The producer surplus of each individual

firm is,

PSfirm = Q ∗ firm(P

∗ −MC )

= 240

N + 1

( 30− 24

N + 1 − 6

) =

5760

(N + 1)2 .

Free entry & exit

Definition: Free entry and exit

A condition under which firms can freely establish or dismantle a presence in a market. (The

absence of barriers to entry and exit.)

If a market has free entry and exit, firms will enter the market if they will earn a profit by

doing so, and firms will exit the market if they are incurring losses.

Cournot competition with free entry

With free entry and exit, a firm will enter and remain in the market so long as its resulting

producer surplus is sufficient to cover its fixed costs.

A firm engaged in Cournot competition earns a profit of,

Πfirm = PSfirm − FC firm,

where FC firm is the firm’s fixed cost.

FC firm is a constant (does not vary with Qfirm), therefore FC firm has no effect on a firm’s

first-order condition or best-response function.

As fixed costs increase, the number of firms that can profitably contest the market falls.

Recovering fixed costs

240

6 MC

300

0 Q

Suppose that each firm in the market for

widgets faces fixed costs of $200.

• If N = 4 firms compete in the market each firm will earn a profit of,

Πfirm = 5760

(N + 1)2 − 200 = $30.40.

• If N = 5 firms compete then Πfirm = −$40.

Thus free entry and exit will result in 4

firms competing in the market.

Quiz 3

Suppose that the equilibrium producer surplus of a firm operating in the market for widgets is

given by the equation,

PSfirm = 5760

(N + 1)2 ,

where N represents the number of firms in the market. With free entry and exit, how many

firms will contest the market if all firms have fixed costs of $360?

(a) N = 3.

(b) N = 4.

(d) N = 1439.

Quiz 4

Suppose that the equilibrium producer surplus of a firm operating in the market for widgets is

given by the equation,

PSfirm = 5760

(N + 1)2 ,

where N represents the number of firms in the market. With free entry and exit, what is the

highest level of fixed costs at which 11 firms would compete?

(a) FC = $21.91.

(b) FC = $40.

(d) FC = $480.

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:

• Perfect competition

• Oligopoly

• Homogeneous goods

• Cournot (quantity) competition

• Residual demand

• Best-response function

• Business stealing

• Welfare

• Deadweight loss

• Free entry & exit

• Excess capacity

• Credible commitment