Monopoly, setting quantity

X

Inverse demand function

X

p

X

)(Xp

p

)( pX

demand function

inverse demand function

Revenue, costs and profit

Revenue: Costs: Profit:

XXpXR )(

XCXRX XC

Marginal revenue with respect to quantity

goes up by p, but goes down by dp/dX (the quantity increase diminishes the

price) multiplied by X

When a firm increases the quantity by one unit, revenue

XdX

XdpXpMR

dX

dRX

)()(

pX

X

p

dXXdp

XpX

XpMR

,

11

)()(

1)(

Amoroso-Robinson relation:

Exercise (marginal revenue)

State three cases where the marginal revenue equals the price.

First order condition

Notation:

0

!

XX MC

dX

XdC

MRdX

XpdXXp

dX

Xd

X

X

MCMC

MRMR

:

:

Linear demand curve in a monopoly Demand:

Revenue:

Marginal revenue:

bXaXp )(

2)( bXaXXR

bXaMR 2

Exercise (Depicting the linear demand curve)

Slope of demand curve: ....

Slope of marginal revenue curve: ....

The .... has the same vertical intercept as the demand

curve.

Economically,

– the vertical intercept is ...,

– the horizontal intercept is ... .

Depicting demand and marginal revenue

a

a/ba/(2b)

MR

)(Xp

X

p

b

1

2b

1

Depicting the Cournot monopoly

Cournot point

MX

Mp

MR

)(Xp

MC

X

p

Profit in a monopoly

MX

Mp

MR

)(Xp

MC

X

p

MX

Mp

X

p

AC

Marginal point of view: Average point of view:

MC

MR

)(Xp

Exercise (monopoly)

Consider a monopoly facing the inverse demand function p(X)=40-X2. Assume that the cost function is given by C(X)=13X+19.

Find the profit-maximizing price and calculate the profit.

Price discrimination

First degree price discrimination:

Second degree price discrimination:

Third degree price discrimination:

Every consumer pays a different price which is equal to his or her willingness to pay.

Prices differ according to the quantity demanded and sold (quantity rebate).

Consumer groups (students, children, ...) are treated differently.

Inverse elasticities rule for third degree price discrimination

Supplying a good X to two markets results in the inverse demand

functions p1(x1) and p2(x2).

21

22

222

11

11121, xxC

xR

xxp

xR

xxpxx Profit function:

First order conditions: 0

!,2111

1

21

xxMCxMRx

xx

0!,

21222

21

xxMCxMRx

xx

Equating the marginal revenues (using the Amoroso-Robinson relation) leads to:

2222

1111

11

11

xxp

xxp

)()()()( 22112211 xpxpxx

Exercise I(price discriminating monopoly)A monopoly sells in two markets:

p1(x1)=100-x1 and p2(x2)=80-x2.

The cost function is given by C(X)=X2.

a) Calculate the maximizing quantities and the profit at these quantities.

b) Suppose now that the monopoly plant is decomposed into two plants, where each plant sells in one market independently (profit center).

Calculate the sum of profits.

Exercise II(price discriminating monopoly)

c) Assume now that the cost function is given by C(X)=10X. Repeat the comparison.d) What happens if price discrimination between both markets will not be possible anymore? Find the profit-maximizing quantity and price. Consider the cost function C(X)=10X.(Hint: Differentiate between quantities below and above 20.)

Oz Shy; Industrial Organization

The deadweight loss of a monopoly

Without price discrimination a monopoly realizes a deadweight loss.

MX

Mp

MR)(Xp

MC

X

p

X

p

MCMR

MCp

Exercise (deadweight loss)

Consider a monopoly where the demand is given by p(X)=-2X+12. Suppose that the marginal costs are given by MC=2X.

Calculate the deadweight loss.

Exercises (price cap in a monopoly)

How does a price cap influence the demand and the marginal revenue curves?

X

p

)(Xp

MR

MC

MX

capp

Mp

Right or wrong? Why?

)(Xp

MR

MC

cappMp

MX X

p

Additional deadweight loss due to quantity tax

consumer’ surplus: ABCA

producer’s surplus: TEF EB

T

additional deadweight loss

A

E F

B C

)(XpMR

MC

X

p

XTX

pTp

tMC

Taxes on profits

MR

X

p

MX

Mp

)1)(( tX

)(X

)(Xp

)(XC

MC

)(XR

Exercise (quantitiy taxes)

A monopoly is facing a demand curve given by p(X)=a-X. The monopoly’s unit production cost is given by c>0. Now, suppose that the government imposes a specific tax of t dollars per unit sold.

a) Show that this tax would raise the price paid by consumers by less than t.b) Would your answer change if the market inverse demand curve is given by p(X)=-ln(X)+5.

c) If the demand curve is given by p(X)=X-1/2, what is the influence on price?

Oz Shy; Industrial Organization

Illustrating the solutions

a) b)

MR

c

tc

p

X

)(XpMR

)( tcpM

)(cpM

c

tc

p

X

)(Xp

)( tcpM

)(cpM

Lerner index of monopoly power

First order condition:

Lerner index:

)()()(!

XMCdXdp

XXpXMR

pX

pX

p

pp

pMCp

,

, 1

11

Monopoly profits and monopoly power

Cournot point

MX

Mp

MR)(Xp

MC

X

p

AC

Comparison: monopoly and monopsony

Monopolist Monopsonist

= the only supplier = the only demander

First order condition(Output):

First order condition (input) of factor labor (L):

0

!

MCdX

XCd

MR

Xd

XRd

Xd

d 0

!,,,

LL MC

L

KLC

MR

L

KLR

L

KL

XCXRX KLCKLRKL ,,,

The monopsonist’s profit function Production function: X=X(L,K) Profit function:

)(

2)(1

)(

)),((

),()),((),(

Kc

Kr

Lc

LLw

KLXR

KLXKLXpKL

Demand for labor

Cost:

Supply elasticity:

Marginal costs:

Amoroso-Robinsonrelation:

)()( LwLLC

L

w

dw

dL

wdwL

dL

wL ,

dL

dw

w

Lw

dL

wdLwMC

dL

LCdMC

L

L

1

wL

w,

11

First order condition (concerning labor)

0!)()()),()),(((),(

L

dL

LdwLwL

KLXKLXp

L

KL

LL

L

MCMPMR

MCL

XXpX

dX

dp

LdL

LdwLw

L

XXpX

L

X

dX

dp

)(

)()()(

MRL

MCL

First order condition for the factor input

Product market Factor market

111 dx

dwxwMC

Special case: price taker

i.e. p = const. :

Special case: price taker

i.e. :01

1dx

dw

111 MVPMPpMR wMC 1

1

111

MP

x

X

MR

dX

dR

x

RMR

Marginal revenue productof factor 1

Marginal costs of factor 1

Value of the marginal product

1

!

1 MCMR

Depicting the labor market

LMC

LMR

)(Lw

w

L0L

0w

Monopsonist:

Exercise (Equal wages for the same work?)

employs men and women, equal productivity M men wage rate: AM, A>0 F women wage rate: BFc, B,

c>0

Find A, B and c such that the monopsonist pays lower wages to women than to men.

Exercise(Minumum wages in a monopsony)

How does a minimum wage

change the supply of labor

and the marginal costs of

labor?

LMC

LMR

)(Lwmw

w

L0L

0w

Executive summary A profit-maximizing monopolist always sets the quantity in the

elastic region of the demand curve. The lower the marginal cost and the higher the demand at each

price, the higher the profit and the monopoly quantity. Distinction between monopolistic power and monopoly profits:

– Monopolistic power: price will be set above the marginal cost by a profit maximizer.

– If the demand curve is tangent to the average cost curve, the profit-maximizing price is set above marginal cost and equal to the average cost monopolistic power, zero profits.