Limit Pricing

Limit Pricing

How does a monopolist maintain its dominant position?


•A cost advantage, say from holding a patent


•A cost advantage, say from holding a patent•Is it possible for a firm without a cost advantage to deter entry by competitors?

Limit pricing with a cost advantage

• Simplest model non-strategic.• Incumbent has cost advantage over potential

entrants.• Incumbent who prices above the (constant)

marginal cost of entrants faces gradual erosion of market control. Pricing at the entrant’s cost is called limit pricing.

• Issue examined is timing.

Limit pricing with a cost advantage

• In the first three examples the key simplifying assumption is that the incumbent either continues last period’s production or switching to limit price.

• Assume that time moves in discrete periods and that every period for which price is above the MC of entrants, some fixed amount of entry occurs, shifting inward the demand curve of the incumbent.

• Goal of incumbent is to maximize present discounted profits by choosing a time to switch to limit pricing.

Discounted present value

Recall how to calculate the present value of a stream of profits Π

10 )1(1

)1( −

∞

= + +Π

=Π=Π ∑ iit iPV t

Limit pricing example 1

Profit maximization for a monopolist: constant marginal cost and linear demand

bQaP t −= IcMC =

MRMC = ⇒ Itt cbQa =− Π2

bcaQ

It

t 2−

=Π )(21 I

tt caP +=Π41 )( I

tbMt ca −=Π


The incumbent can forestall entry by pricing at the marginal cost of production for the potential entrants. This yields a per period profit of

where solves

)( IELt

Lt ccQ −=Π

LtQ

ELtt

Lt cbQaQP =−=)(

Limit pricing example 1The incumbent chooses which period to switch by comparing:

Switch in period 0

Switch in period 1

Switch in period 2

M

100

)1( −+Π

=Πii

L

PV

i

LM

PV1

00 Π

+Π=Π

)1()1(20

00

iii

LMM

PV +Π

++Π

+Π=Π


• The later the switch is made, the smaller is the incumbent's demand in the limit price market. Traded off against the profits earned in the periods before entry.

• The higher are interest (discount) rates, the less the future erosion matters, and the longer entry is allowed to continue.


• This very simple story can be slightly modified to allow the incumbent to adjust output as market share erodes.

• As the demand curve shifts inward, the per-period output should be reduced. Then if the incumbent switches to limit pricing at period 2, for example, present discounted profits are

• Same basic story)1()1(

210

0

iii

LMM

PV +Π

++Π

+Π=Π


• It seems plausible that the incumbent can use a cost advantage to forestall entry, but what if all firms have access to the same technology?

• If firms have constant cost, limit price means zero profits, so best to just allow the market to erode.

• We can still tell a limit pricing story. Key is the existence of increasing returns to scale. This might be due to a fixed cost of production, so that unit cost falls as production increases.

Limit pricing example 3P

LRMCLRAC

D

PM

QM MR Q


LRMCLRAC

Entrants residual Demand

D

QQM


LRMCLRAC

D

PL

QL MR Q


LRMCLRAC

Entrants residual Demand

D

QQL

Critique of Game Theorists

• Do these stories make sense? Each has the basic assumption that the monopolist will ignore the entrant

• Consider example 3: why should the potential entrant believe that the incumbent will continue to price at if entry is not successfully blocked. Shouldn’t the two firms engage in an oligopoly pricing game?

LP

An entry game

• Assume that a dominant firm operates in a market and faces a potential entrant.

• Assume there is no commitment in pricing: what I do today does not restrict my choices tomorrow.

• Assume technology and demand are as in example 3.

• This leads to a simple pricing game.

An entry game

PE

DF1

DF2

OUT

IN

Profit max

Profit max

Limit price

Limit price

)0,( MΠ

)0,( LΠ

),( CC ΠΠ

),( lossLΠ

An entry game

• The inability to commit to future prices makes the threat to price low upon entry not credible.

• Milgrom and Roberts show that limit pricing may make sense when potential entrants do not know the exact cost structure of the incumbant (asymmetric information).

• This is a complicated model using a sophisticated equilibrium concept. There can be multiple equilibria.

Milgrom and Roberts

• They show there may be a separating equilibrium where the high cost firm produces the optimal short run quantity and attracts entry, while the low cost firm limit prices.

• There also may be a pooling equilibrium in which both firms produce the short run optimal amount for the low costs firm, and rivals do not enter.

Predatory Pricing

• The core of game theoretician's criticism of the naïve model is that present pricing lacks commitment.

• What about responding to entry? Can a firm be aggressive after entry to encourage exit?

• This is called predatory pricing.• May work if a firm can build a reputation

for being aggressive.

Chain store paradox

• Sequential entry game with markets• Each stage game identical• Consider the final market ( )

N

1

Chain store paradox

PE

DF1

DF2

OUT

IN

Profit max

Profit max

Limit price

Predatory price

)0,( MΠ

)0,( LΠ

),( CC ΠΠ

),( lossPΠ

Chain store paradox

• So entry is not profitable in the last period• What about the next to last?• Reputation has no value• And so on, back to the market.• Kreps and Wilson tell an asymmetric information

story about this situation. Posit the existence of “strong” competitors who always fight entrants. Then “weak” entrants can fight in early rounds to deter future entry. Uses mixed strategies

thN

Evidence?• Models are hard to test• Some evidence exists that firms respond to

potential entry by sacrificing short term profits.• United Shoe Machinery is a clear case where they

were found by US Supreme Court to be abusing a dominant position.

• They had some time to correct their behaviour, and they raised prices and attracted entry

• Predatory pricing has been alleged in airlines and elsewhere.

• Wal-Mart was accused but acquitted on appeal in Arkansas supreme court.

Documents

Limit Pricing