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Price Theory

Lecture 7: Monopoly

Topics for today’s lecture . . .

1. Revenue & market power

2. Profit maximisation

3. Measuring market power

4. Comparative statics

5. Multiple markets

Revenue & market power

Definition: Inverse demand

The highest price at which a firm can sell a given quantity of output.

The inverse demand function is the equation for the demand curve, rearranged such that

price is stated as a function of quantity.

Inverse demand

750

1250

2500

0 Q

Suppose a monopolist, Greg’s florist,

faces the inverse demand function,

P (Q ) = 50 − 0.02Q.

As the only supplier in a market, Greg’s

florist can choose the point on the

demand curve at which it will operate.

• If Greg produces 750 bunches of flowers, the price is $35.

• If Greg produces 1250 bunches of flowers, the price is $25.

Marginal revenue

The total revenue of a firm that sells its product at a uniform price, is given by the function,

TR (Q ) = P (Q ) × Q.

Marginal revenue is the rate at which a firm’s total revenue changes as its output increases;

the derivative of total revenue with respect to Q ,

MR (Q ) = d TR (Q )

dQ = P (Q ) + Q ×

dP (Q )

dQ .

The first term in this equation is the revenue the monopolist receives from the marginal sale.

The second term is the rate at which the monopolist loses revenue from its preexisting sales.

The marginal revenue curve

MR D

0 Q

The position of the marginal revenue

curve, relative to the demand curve, is

described by the function,

MR (Q ) = P (Q ) + Q × dP (Q )

dQ .

At Q = 0, marginal revenue is equal to

(inverse) demand.

Marginal revenue is less than inverse

demand where Q > 0, as demand is

downward sloping (dP (Q )/dQ < 0).

Marginal revenue and elasticity

At any given point on the demand curve, the price elasticity of demand is given by the

function,

εQ,P = dQ

dP ×

Q .

We can use the elasticity to restate marginal revenue as,

MR (Q ) = P (Q ) + Q × dP (Q )

dQ = P (Q ) ×

( 1 +

P (Q ) ×

dP (Q )

) = P (Q )

( 1 +

εQ,P

) .

Where demand is elastic (−∞<εQ,P <−1), marginal revenue is positive.

Where demand is unit elastic (εQ,P = −1), marginal revenue is zero.

Where demand is inelastic (−1 <εQ,P < 0), marginal revenue is negative.

Marginal revenue along a linear demand curve

elastic inelastic

MR D

0 Q

If the monopoly sells nothing (Q = 0), it

has no revenue (TR = 0).

As the quantity increases, total revenue

increases at a decreasing rate.

The maximum total revenue occurs where

marginal revenue is equal to zero.

Thereafter total revenue declines with

quantity, as marginal revenue is negative.

If the monopoly gives its product away

(P = 0), it has no revenue (TR = 0).

Quiz 1

Greg estimates that at his current price, the price elasticity of demand for his flowers is

εQ,P = −0.8. If Greg increases his price by a small amount then,

(a) his total revenue will increase.

(b) his total revenue will not change.

(d) the direction of the change in total revenue will depend on whether or not Greg faces a

linear demand function.

Profit maximisation

Firm profit

00 TC

TRΠ

MR D

0 Q

A firm’s profit is its total revenue minus

its total cost.

Π(Q ) = TR (Q ) − TC (Q ).

For any given quantity, the firm’s profit

can be illustrated as the vertical distance

between the TR and TC curves.

The firm’s problem is to identify the

quantity that maximises the profit

function.

Firm profit

00 TC

TRΠ

MR D

0 Q

A firm’s total revenue is given by:

TR (Q ) = P (Q ) × Q

A firm’s total costs are its marginal costs

plus its fixed costs:

TC (Q ) = MC (Q ) × Q + FC

Marginal costs vary with quantity; fixed

costs do not.

Definition: Marginal cost

The rate at which total cost changes as output increases.

A firm’s marginal cost plays an important role in determining a firm’s profit maximising

behaviour in a market.

The profit maximising condition

At the profit maximising quantity, the derivative (slope) of the profit function is zero,

d Π(Q )

dQ =

d TR (Q )

dQ −

d TC (Q )

dQ = MR (Q ) − MC (Q ) = 0.

From the final equality it follows that the profit maximising condition can be written as,

MR (Q ) = MC (Q ).

This condition does not depend on the nature of competition in the market.

The profit maximising condition

Q∗

P∗

TRΠ

0 Q

A profit maximising firm selects its

quantity of output such that marginal

revenue equals marginal cost.

The profit maximising (monopoly)

quantity is denoted Q∗.

At the monopoly quantity, the total cost

curve and total revenue curve have the

same slope.

The monopoly price P∗ is inverse demand

at Q∗.

The power rule (and other useful derivatives)

For any non-zero constants a and n, the derivative of the function f (Q ) = aQn with respect

to Q is, df (Q )

dQ = anQn−1.

For any two functions f (Q ) and g(Q ),

( f (Q ) + g(Q )

) =

df (Q )

dQ +

dg(Q )

dQ ,

and, d

( f (Q ) − g(Q )

) =

df (Q )

dQ −

dg(Q )

dQ .

The power rule: examples

• d

dQ (73) = 0

(The derivative of a constant is zero.)

• d

( Q5

) = 5Q4

• d

( 10Q2

) = 20Q

• d

( Q2 − 5Q + 12

) = 2Q − 5

(Note that Q1−1 = Q0 = 1.)

Exercise: Monopoly price & quantity

Greg’s florist has a monopoly over the sale of flowers. Greg faces the inverse demand

function P (Q ) = 50 − 0.02Q . Greg’s total cost function is TC (Q ) = 0.005Q2.

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

2. Find the derivative of Greg’s profit function with respect to Q .

3. Set the derivative equal to zero and solve for Q (this is the monopoly quantity).

4. Substitute the monopoly quantity into inverse demand to find the monopoly price.

Exercise solutions

1. Greg’s florist’s profit function is,

Π(Q ) = TR (Q ) − TC (Q ) = Q (50 − 0.02Q ) − 0.005Q2 = 50Q − 0.025Q2.

2. Using the power rule, the derivative of Greg’s profit function with respect to Q is,

dQ (50Q − 0.025Q2) = 50 − 0.025 × 2Q2−1 = 50 − 0.05Q.

Exercise solutions

3. At the profit maximising quantity the derivative of the profit function is zero,

dQ (50Q − 0.025Q2) = 50 − 0.05Q = 0.

This is known as the first-order condition. Solving the first-order condition for Q yields

the monopoly quantity,

50 − 0.05Q = 0 ⇒ 0.05Q = 50 ⇒ Q∗ = 1000.

4. Substituting the monopoly quantity into the inverse demand function gives us the

monopoly price,

P∗ = 50 − 0.02 × 1000 = $30.

Measuring market power

The monopoly markup

2.5k

0 Q

The monopoly’s market power allows it to

enjoy a markup over its marginal cost.

• Greg’s florist earns a profit of $20 on the marginal sale.

• By contrast, firms in a competitive market sell their marginal units at

marginal cost.

Note: Positive marginal revenue implies

that a monopoly operates on the elastic

portion of its demand curve.

The inverse elasticity pricing rule

The profit maximising condition (MR = MC ) can be rewritten in terms of the price elasticity

of demand,

P∗ (

1 + 1

εQ,P

) = MC (Q∗).

Rearranging yields the inverse elasticity pricing rule,

P∗ − MC (Q∗) P∗

= − 1

εQ,P .

The rule states that a monopolist facing a more inelastic demand, enjoys a higher marginal

profit.

Measuring market power

2.5k

0 Q

The proportional markup of a firm’s price

over its marginal cost is known as the

Lerner index,

L = P∗ − MC (Q∗)

P∗ =

30 − 10 30

= 2

3 .

The Lerner index measures a firm’s

market power:

• Lerner indices range between 0 and 1.

• An index of 0 corresponds to a firm in a competitive market.

Exercise: The market power of cellphone manufacturers

• Apple’s iPhone is estimated to have a marginal cost of US$224, and retails for US$649.

• Samsung’s Galaxy S is estimated to have a marginal cost of US$255, and retails for US$599.

1. Calculate the Lerner index of market power for each firm.

2. Calculate the price elasticity of each firm’s demand. Which firm is face the more inelastic

demand?

3. Which of the two products do you think has the closer substitutes? Explain your answer.

Exercise solutions

1. For Apple, the Lerner index is,

LA = P∗ − MC (Q∗)

P∗ =

649 − 224 649

≈ 0.655.

For Samsung, the Lerner index is,

LS = P∗ − MC (Q∗)

P∗ =

599 − 255 599

≈ 0.574.

Exercise solutions

2. The inverse elasticity pricing rules states,

P∗ − MC (Q∗) P∗

= − 1

εQ,P .

The left-hand side of this equation is the Lerner index. It follows that the price elasticity

of Apple’s demand is,

εQ,P = − 1

LA = −

0.655 ≈−1.527.

The price elasticity of Samsung’s demand is,

εQ,P = − 1

LS = −

0.574 ≈−1.742.

Apple’s demand is the more inelastic (closer to zero).

Exercise solutions

3. Demand for a product tends to be more elastic, the closer are the available substitutes.

Given that the demand faced by Samsung is more elastic than the demand faced by

Apple, it is likely that the Galaxy S has the closer substitutes.

Comparative statics

An increase in the marginal cost of production

0.7k

MC1

MC2

2.5k

0 Q

The new intersection of marginal revenue

and marginal cost occurs above and to

the left of the original intersection.

There is an increase in the monopoly

price, and a fall in the monopoly quantity.

Note: If MC is flat or upward-sloping, the

increase in the monopoly price is less than

the increase in marginal cost.

The impact of a cost increase on total reveue

30k

25k

0.7k

MC1

MC2

0 Q

An increase in marginal cost must result

in a decrease in the monopolist’s total

revenue.

• The monopoly operates on the elastic portion of its demand curve.

• Any reduction in the quantity produced must result in a movement

down along the total revenue curve.

It follows that an increase in marginal

cost must result in a fall in profits.

An increase in demand

P∗1

Q∗1 Q ∗ 2

P∗2

D1 D2

MR1

MR2 MC

0 Q

A rightward shift of the demand curve

causes a rightward shift of the marginal

revenue curve.

The intersection of marginal revenue and

marginal cost occurs to the right of the

original monopoly quantity.

• The monopoly quantity increases.

• In this example, the monopoly price also increases.

An increase in demand causing a fall in price

P∗1

Q∗1 Q ∗ 2

P∗2

D1 D2MR1 MR2

0 Q

It is important to note that an increase in

demand can results in a fall in the

monopoly price.

This can only occur if marginal cost is

downward sloping over some range.

Lowering the price is profitable in this

example, as the monopolist achieves

considerable cost advantages by

expanding production.

Multiple markets

A monopolist with multiple markets

180

0 Q

Tiger’s Door entertainment has a

monopoly over the sale of movies in two

countries:

• Demand for movies in the ‘Alpha Islands’ is given by the function

QA(P ) = 180 − P .

• The “choke price” for Alpha Islands occurs when QA(P ) = 0, at a price of

P = 180.

A monopolist with multiple markets

180

120

0 Q

Tiger’s Door entertainment has a

monopoly over the sale of movies in two

countries:

• Demand for movies in ‘Beta Peninsula’ is given by the function

QB(P ) = 120 − P .

• The “choke price” for Beta Peninsula occurs when QB(P ) = 0, at a price of

P = 120.

Combining the demands of the two markets

180

120

300

0 Q

Suppose that Tiger’s Door entertainment

must charge a uniform price across both

markets.

The total demand for the monopolist’s

movies is the sum of the demand in the

two markets:

• If 120 < P < 180 then Q = QA = 180 − P .

• If P ≤ 120 then Q = QA + QB = 300 − 2P .

Inverse demand & marginal revenue

180

120

30 MC

180

300

120

0 Q

The corresponding inverse demand is:

• P = 180 − Q if 120 < P < 180.

• P = 150 − Q/2 if P < 120.

The firm’s cost functions are:

• TC (Q ) = 30Q .

• MC (Q ) = d TC (Q )

dQ = 30.

Profit maximisation with a uniform price

Let’s begin by assuming that 120 < P < 180.

The firm’s profit function is,

Π(Q ) = TR (Q ) − TC (Q ) = Q (180 − Q ) − 30Q = 150Q − Q2.

The corresponding first-order condition is,

dQ (150Q − Q2) = 150 − 2Q = 0

Solving for Q yields the monopoly quantity Q∗ = 75.

Substituting for Q∗ into inverse demand P∗ = 180 − 75 = $105. This is not consistent with the initial assumption.

Profit maximisation with a uniform price cont. . .

Alternatively, lets assume that P ≤ 120.

The firm’s profit function in this instance is,

Π(Q ) = TR (Q ) − TC (Q ) = Q (

150 − Q

) − 30Q = 120Q −

2 .

The corresponding first-order condition is,

( 120Q −

) = 120 − Q = 0

Solving for Q yields the monopoly quantity Q∗ = 120.

Substituting for Q∗ into inverse demand P∗ = 150 − 0.5 × 120 = $90. This is consistent with the initial assumption.

Producer surplus with uniform pricing

120

180

120

30 MC

180

300

120

0 Q

The monopolist’s producer surplus is the

area between its price and its marginal

cost curve.

With a constant marginal cost, producer

surplus can be calculated as,

PS = Q∗(P∗ − MC )

= 120 × (90 − 30) = $7200.

Note: Profit is equal to producer surplus

minus fixed cost.

Definition: Price discrimination

The practice of charging consumers different prices for the same good or service.

For a firm to engage in price discrimination it must (a) have market power, and (b) be able to

prevent the resale (arbitrage) of its good or service.

Producer surplus with price discrimination

105

MRA DA

180

30 MC

MRB DB

120

30 MC

0 Q

If the Tiger’s Door entertainment can

charge a different price each country,

• Q∗A = 75 movies, P ∗ A = $105 and

PSA = $5625.

• Q∗B = 45 movies, P ∗ B = $75 and

PSB = $2025.

(You should check this.)

The total producer surplus across the two

markets is thus $7650, $450 more than

under uniform pricing.

Price discrimination and elasticity

Uniform monopoly price105

MRA DA

180

30 MC

Uniform monopoly price

MRB DB

120

30 MC

0 Q

In each market, the profit maximising

price satisfies the inverse elasticity pricing

rule.

• The monopoly price is higher in market A because demand is less

elastic in that market.

Compared to uniform pricing, consumers

with,

• inelastic demands are worse off.

• elastic demands are better off.

Methods for separating markets (and market segments)

Geo-blocking: A technology used by digital media companies to prevent consumers in one

country from accessing content in another country.

Screening: Offering consumers who are members of a group with elastic demand a discount,

if they can prove their membership of the group (eg. student discounts with a student ID).

Inter-temporal price discrimination: Altering the price of a product over time. (eg. Flights

tend to be cheaper if purchased in advance because last-minute travellers tend to have less

elastic demand.)

Methods for separating markets (and market segments)

Coupons & rebates: Coupons offer consumers a discount on an item if the coupon is

presented at the point of sale. Rebates return a portion of the purchase price to a consumer

if they complete a form and send it to the manufacturer. Consumers are more likely to cut

out coupons, or mail in rebates, if they are price sensitive.

Versioning: Producing multiple versions of product, with different levels of performance, so

that consumers who derive the highest benefits pay higher prices. (eg. In the early 1990s IBM

produced the LaserPrinter in two versions. The two printers were identical except that the low

priced version had an extra chip that caused the printer to pause, thereby slowing it down.)

Quiz 2

2.5k

0 Q

Using the figure provided, find the

producer surplus for Greg’s florist.

(a) 20,000 bunches of flowers.

(b) $20,000.

(d) $25,000.

Quiz 3

Suppose that a monopolist serves two markets, and that the monopolist can price

discriminate between markets.

• In the first market, at the optimal price εQ,P = −2, and Q∗1 = 500 units.

• In the second market, at the optimal price εQ,P = −2, and Q∗2 = 1000 units.

Which of the following statements is true?

(a) The optimal price in market 1 is higher than the optimal price in market 2.

(b) The optimal price in market 2 is higher than the optimal price in market 1.

(d) We cannot compare the optimal prices without the inverse demand and marginal cost

functions.

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:

• Monopoly

• Inverse demand

• Marginal revenue

• Marginal cost

• Elasticity

• Profit maximisation

• Market power

• Inverse elasticity pricing rule

• Lerner index

• Uniform price

• Price discrimination

• Producer surplus