Chapter 15 Imperfect Competition Nicholson and Snyder, Copyright ©2008 by Thomson South-Western....

Chapter 15

Imperfect Competition

Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. All rights reserved.

Short-Run Decisions: Pricing & Output

• When there are only a few firms in a market, predicting output and price can be difficult– how aggressively do firms compete?– how much information do firms have about

rivals?– how often do firms interact?


• Bertrand model– two identical firms choosing prices

simultaneously for identical products• end up with situation similar to perfect

competition

• Cartel model– firms act as a group

• end up with the monopoly outcome


Quantity

Price

MC=AC

In the Bertrand model, output would be Q* and price would be P*

Q*

P*

Under the cartel model, output would be Q** and price would rise to P**

Q**

P**

DMR


• Cournot model– firms set quantities rather than prices

• end up with a result between the Bertrand and the cartel models


Quantity

Price

MC=AC

It is important to know where the industry ends up because total welfare depends on price and quantity

Q*

P*

Under Bertrand, there is no DWL

Q**

P**

DMR

The cartel model implies a DWL

Bertrand Model• Two identical firms producing identical

products at a constant MC = c

• Firms choose prices p1 and p2 simultaneously– single period of competition

• All sales go to the firm with the lowest price– sales are split evenly if p1 = p2

Nash Equilibrium of the Bertrand Model

• The only pure-strategy Nash equilibrium is p1

* = p2* = c

– both firms are playing a best response to each other

• neither firm has an incentive to deviate to some other strategy


• If p1 and p2 > c, a firm could gain by undercutting the price of the other and capturing all the market

• If p1 and p2 < c, profit would be negative


• The same result will arise for any number of firms n 2

• The Nash equilibrium of the n-firm Bertrand game is p1

* = p2* = … = pn

*= c

Bertrand Paradox

• The Nash equilibrium of the Bertrand model is identical to the perfectly competitive outcome

• It is paradoxical that competition between as few as two firms would be so tough

Cournot Model

• Each firm chooses its output qi of an identical product simultaneously

• Total industry output Q = q1 + q2 +…+ qn determines the market price P(Q)– P(Q) is the inverse demand curve

corresponding to the market demand curve

Cournot Model

• Each firm recognizes that its own decisions about qi affect price P/qi 0

• However, each firm believes that its decisions do not affect those of any other firm qj /qi = 0 for all j i

Cournot Model

• The FOC for profit maximization are

0)(''

iiii

i qCqQPQPq

• The firm maximizes profit where MRi = MCi

Cournot Model

• Price exceeds marginal cost by

iqQP '

Cournot Model

• Price will exceed marginal cost, but industry profits will be lower than in the cartel model– social welfare is greater in the Cournot

model than in the cartel situation

Cartel Model

• In the cartel model, each firm chooses qi for each firm so as to maximize total industry profits

n

jjj

n

j

n

jjj qCqQP

11 1

)(

Cartel Model• The FOC for a maximum gives

0)(''11

ii

n

jj

n

jj

i

qCqQPQPq

• This is the same result as Cournot, except that price exceeds marginal cost by

QQPqQPn

jj ''

1

Natural Springs Duopoly

• Assume that there are two owners of natural springs– firm’s cost of pumping and bottling qi liters

is Ci(qi) = cqi

– each firm has to decide how much water to supply to the market

• The inverse demand function is

P(Q) = a – Q


• In the Bertrand game the two firms set price equal to marginal cost

P* = c

total output = Q* = a – c

*i = 0

total profit for all firms = * = 0


• The solution for the Cournot model is similar

1 = P(Q)q1 – cq1 = (a – q1 – q2 – c)q1

2 = P(Q)q2 – cq2 = (a – q1 – q2 – c)q2

22

1

cqaq

21

2

cqaq


• The Nash equilibrium will be

q1* = q2* = (a – c)/3

total output = Q* = (2/3)(a – c)

P* = (a + 2c)/3

1* = 2* = (1/9)(a – c)2

total profit for all firms = * = (2/9)(a – c)2


• The objective function for a perfect cartel involves joint profits

1 + 2 = (a – q1 – q2 – c)q1 + (a – q1 – q2 – c)q2

• The FOCs for a maximum are

022 21212

211

cqqaqq


• These FOCs do not pin down the market shares for the firms in a perfect cartel


P* = (1/2)(a + c)

total cartel profit = * = (1/4)(a – c)2

Cournot Best-Response Diagrams

• We can also show each firm’s best- response function graphically– the intersection of these best-response

functions is the Nash equilibrium


q1

q2

a - c

The intersection of the firms’ best-response functions is the Nash equilibrium

a - c

BR1(q2)

BR2(q1)

2ca

2ca

3ca

3ca

The Nash equilibrium is where q1* = q2* = (a – c)/3


q1

q2A change in a firm’s marginal cost will shift its best-response function

BR1(q2)

BR2(q1)

If firm 1’s marginal cost rises, its best-response-function will shift in and there will be a new Nash Equilibrium

Varying the Number of Cournot Firms

• The Cournot model can represent the whole range of outcomes by varying the number of firms– n = perfect competition– n = 1 perfect cartel / monopoly


P* = (1/2)(a + c)

total cartel profit = * = (1/4)(a – c)2

Varying the Number of Cournot Firms

• In equilibrium, identical firms will produce the same share of output qi = Q/n

• The difference between price and marginal cost becomes P’(Q)Q/n– this wedge term gets smaller as the number

of firms gets larger

Prices or Quantities?

• Moving from price competition to quantity competition changes the outcome dramatically– an advantage of the Cournot model is the

realistic implication that the increases in the number of firms makes the market more competitive

• but real-world firms tend to set prices rather than quantities

Capacity Constraints

• Firms must have unlimited capacity for the Bertrand model to generate the Bertrand paradox– more realistically, firms may not have an

unlimited ability to meet all demand


• Consider a two-stage game– firms build capacity in the first stage

– firms choose prices p1 and p2 in the second stage

– sales of firms cannot exceed the capacity chosen in the first stage


• If the cost of building capacity is sufficiently high, the equilibrium of this game is the same as the Nash equilibrium of the Cournot model– firms choose the price at which quantity

demanded equals total capacity

Product Differentiation

• The possibility of product differentiation introduces some uncertainty into what we mean by the market for a good

Product Differentiation• The law of one price may not hold

– demanders may now have preferences about which suppliers to purchase the product from

– there are now many closely related, but not identical, products to choose from

Product Differentiation• We will take the market to be a group of

closely related products that are more substitutable among each other than with goods outside the group– measure substitutability by the cross-price

elasticity


• We will assume that there are n firms competing in a particular market– each product has its own attributes, ai

• The product’s attributes affect its demand

qi(pi, P-i, ai, A-i)

– where P-i is a list of all other firms’ prices and A-i is a list of the attributes of other firms’ products

Product Differentiation• Firm i’s total cost is

Ci(qi, ai)

and profit isi = piqi – Ci(qi, ai)

Product Differentiation• The FOCs for a maximum are

0

i

i

i

i

i

iii

i

i

pq

qC

pq

pqp

0

i

i

i

i

i

ii

i

i

aq

qC

aq

pa


• At the profit-maximizing level of output, marginal revenue is equal to marginal cost

• Additional differentiation activities should be pursued up to the point at which the additional revenues they generate are equal to their marginal costs

Hotelling’s Beach• Suppose we are examining the case of

ice cream stands located on a beach– assume that demanders are located

uniformly along the beach• one at each unit of beach

– ice cream cones are costless to produce but carrying them back to one’s place on the beach results in a cost of td 2

• t = temperature• d = distance

Spatial Differentiation

a b

L

Ice cream stands are located at points A and B along a linear beach of length L

Suppose that a person is standing at point x

x


• A person located at point x will be indifferent between stands A and B if

pa + t(x – a)2 = pb + t(b – x)2

where pa and pb are the prices charged by each stand

Spatial Differentiation• Solving for x we get

)(22 abtppab

x AB

• If the two stands charge an equal price, the indifferent consumer is located midway between a and b


• The Nash equilibrium prices are

baLabt

pA 23

*

baLabt

pB 43

*


• Profits for the two firms are

2* 218

baLabt

A

2* 418

baLabt

B

Tacit Collusion

• Tacit collusion is not the same as an explicit cartel– can only be enforced through punishments

internal to the market

Tacit Collusion

• Repeating the stage game T times does not change the outcome– the only subgame perfect equilibrium is to

repeat the stage-game Nash equilibrium in each of the T periods

Tacit Collusion

• If the stage game is repeated infinitely, the folk theorem applies– any feasible and individually rational payoff

can be sustained each period as long as the discount factor () is close enough to 1

Tacit Collusion

• Suppose two firms in a duopoly agree to tacitly collude to sustain the monopoly price by using a grim trigger strategy

• Successful tacit collusion provides the profit stream

11

2...

2222 MMMMcolludeV

Tacit Collusion

• If a firm deviates, it will earn all of the monopoly profit for itself in the current period– the deviation will trigger the grim strategy of

marginal cost pricing for all future periods– the stream of profits from deviating is Vdeviate

= M

Tacit Collusion

• For deviation not to be profitable, it must be that Vcollude Vdeviate

MM

11

2

21

Tacit Collusion

• Suppose only 2 firms produce a medical device that is produced at constant average and marginal cost of $10

• The demand for the device isQ = 5,000 – 100P

Tacit Collusion

• If the Bertrand game is played in a single period, each firm will charge $10 and a total of 4,000 devices will be sold

• At the monopoly price, each firm would earn a profit of $20,000

Tacit Collusion

• Collusion at the monopoly price is sustainable if

000,4011

000,20

21

Tacit Collusion• Now, suppose there are n firms

– monopoly profit is $40,000, but each firm’s share is 40,000/n

• n firms can successfully collude on the monopoly price if

n1

1

000,4011000,40

n

Investment, Entry, and Exit

• Even when making long-run decisions, an oligopolist must consider how rivals will respond

• Crucial to these decisions is how easy it is to reverse a decision once it has been made

Investment, Entry, and Exit

• Absent strategic considerations, a firm would value flexibility and reversibility

• But commitment has value as well– firm can gain first-mover advantage

Sunk Costs and Commitment

• Sunk costs are expenditures on irreversible investments– these allow the firm to produce in the

market but have no residual value if the firm leaves the market

– could include expenditures on unique types of equipment or job-specific training of workers

First-Mover Advantage in the Stackelberg Model

• This model is similar to the duopoly version of the Cournot model except firms move sequentially– firm 1 (the leader) chooses q1 first

– firm 2 (the follower) chooses q2 after seeing q1


• We can solve the model by backward induction– begin with output of the follower (q2)

• this results in a best-response function for Firm 2 [BR2(q1)]

– substitute BR2(q1) into Firm 1’s profit function

1 = P(q1 + BR2(q1))q1 – C1(q1)


• The FOC is

0)('''' 1111211

1

qCqqBRQPqQPQPq

S

– this is the same FOC as in the Cournot model except for the addition of the strategic effect of Firm 1’s output on Firm 2 (S)


• The strategic effect will lead Firm 1 to produce more than it would have in a Cournot model– this leads Firm 2 to lower output– if Firm 2 lowers output, the market price will

rise, increasing Firm 1’s revenue from existing sales


• The strategic effect would not occur if– the leader’s output was unobservable to

the follower– the leader could reverse its output choice

in secret

• The leader must be able to commit or else firms are back in the Cournot game

Stackelberg Springs

• Recall the natural springs duopoly discussed earlier– this time we will assume they choose

output levels sequentially– Firm 1 is assumed to be the leader– Firm 2 is assumed to be the follower

Stackelberg Springs• Solving for Firm 2’s output, we get its

best-response function

21

2

cqaq

1111

11 21

2qcqaqc

cqaqa

• Substituting Firm 2’s best-response function into Firm 1’s profit function,

Stackelberg Springs• Taking the FOC,

0221

11

cqaq

2

*1

caq

• This means that

4

*2

caq

2*1 8

1ca 2*

2 161

ca

Contrast with Price Leadership

• In the Stackelberg game, the leader uses a “top dog” strategy– aggressively overproduces to force the

follower to scale back production– the leader earns more (than it would in the

Cournot game), while the follower earns less


• The leader could follow a “puppy dog” strategy– increases its price, producing less output

than in a simultaneous-move game– acts less aggressively, leading its rival to

compete less aggressively


• The crucial difference between these two games is that the slopes of the best-response functions differ– “top dog” strategy leads to a downward-

sloping best-response function for Firm 2– “puppy dog” strategy leads to an upward-

sloping best-response function for Firm 2

Strategic Entry Deterrence

• In some cases, first-mover advantages may be large enough to deter all entry by rivals– however, it may not always be in the firm’s

best interest to create that large a capacity

Deterring Entry of a Natural Spring

• We will now add an entry stage to the Stackelberg Natural Springs example– Firm 2 must decide whether to enter the

market after seeing Firm 1’s output level

– entry for Firm 2 requires a sunk cost, K2

• Firm 1 incurred sunk cost before the start of the game

– we will assume a = 120 and c = 0


• We start by calculating Firm 1’s profit if it accommodates entry– this was done in earlier example

q1acc = (a – c)/2 = 60

1acc = (a – c)2/8 = 1,800


• Next, we compute Firm 1’s profit if it deters entry– Firm 1 needs to produce and amount high

enough that Firm 2 cannot earn enough profit to cover sunk cost


• Firm 2’s best-response function isq2 = (120 – q1)/2

• Substituting into Firm 2’s profit function gives us

2

2det1

2 2120

Kq


• Setting Firm 2’s profit to zero yields

2det1 2120 Kq

)2120(2 22det1 KK


• The final step is to compare 1acc with 1

det

• The level of K2 at which the firm would be indifferent is K2 = 77

– if K2 < 77, entry is cheap and Firm 1 would have to increase its output to 102 to deter entry

Signaling

• The ability to signal is another first-mover advantage– if a second mover has incomplete

information about the market, it may try to watch the first-mover to learn about market conditions

– the first mover may distort its actions to manipulate what the second mover learns

Entry-Deterrence Model• Consider a game where two firms

choose a price for their differentiated products– Firm 1 is a first mover– Firm 2 is a second mover

Entry-Deterrence Model• Firm 1 has private information about its

marginal costs– High costs with a probability of Pr(H)– Low costs with a probability of Pr(L) = 1 –

Pr(H)

• In period 1, Firm 1 serves the market alone– at the end of the period, Firm 2 observes p1

and considers entry

Entry-Deterrence Model• If Firm 2 enters, it faces a sunk cost of

K2 and learns the true nature of Firm 1’s costs

• The firms then behave as duopolists in the second period– choosing prices for differentiated products

Entry-Deterrence Model• If Firm 2 does not enter, it obtains a

payoff of zero– Firm 1 serves the market alone

• Assume there is no discounting between periods

Entry-Deterrence Model• Let Di

t = duopoly profit for firm i if Firm 1 is of type t (low-cost, high-cost)

• Assume that D2L < K2 < D2

H

– Firm 2 earns more than its entry cost only if Firm 1 is high-cost

Entry-Deterrence Model• If Firm 1 is low cost, it has only one

relevant action– setting the monopoly price (p1

L)

• If Firm 1 is high cost, it has two possible actions– set the monopoly price associated with its

type (p1H)

– choose the same price as the low-cost type (p1

L)

Entry-Deterrence Model• Let M1

t = Firm 1’s monopoly profit if it is of type t

• Let R = the loss in Firm 1’s profit if it is high-cost, but chooses p1

L

Entry-Deterrence Model

Separating Equilibrium• In a separating equilibrium, the different

types of the first-mover must choose different actions

• There is only one possibility for Firm 1– the low-cost type chooses p1

L

– the high-cost type chooses p1H

Separating Equilibrium• Firm 2 sees Firm 1’s actions

– stays out is Firm 1 charges p1L

– enters if Firm 1 charges p1H

• Would a high-cost Firm 1 prefer to charge a price of p1

L?

– only if

R < M1H – D1

H

Pooling Equilibrium

• If R < M1H – D1

H, the high type would like to pool with the low type if pooling deters entry– pooling deters entry if Firm 2’s prior belief

that Firm 1 is the high type is low enough that Firm 2’s expected payoff from entering is less than zero

Predatory Pricing• The incomplete-information model of

entry deterrence may explain why a firm would engage in predatory pricing– charging an artificially low price to prevent

potential rivals from entering or to force existing rivals to exit

Barriers to Entry• In order for a market to be oligopolistic,

there must be barriers to entry– sunk cost to enter– government intervention (patents, licensing)– search costs faced by consumers– product differentiation (brand loyalty)– entry deterrence by existing firms

Long-Run Equilibrium

• Suppose there are a large number of symmetric firms that are potential entrants into a market– make the decision simultaneously

• Entry requires a sunk cost, K

• Let n = number of firms that decide to enter


• Let g(n) = profit earned by a firm (not including sunk cost)– we would expect g’(n) < 0


• The sub-game perfect equilibrium number of firms (n*) will satisfy two conditions– they earn enough to cover their entry costs

• g(n*) K

– an additional firm cannot cover its entry cost

• g(n*+1) < K


• Is the long-run equilibrium efficient?

• A benevolent social planner would choose n to maximize

CS(n) + ng(n) – nK– CS(n) is equilibrium consumer surplus– ng(n) is equilibrium gross profits– nK is total expenditure on sunk entry costs


• The long-run equilibrium number of firms (n*) may be greater or less than the social optimum (n**) depending on two effects– the appropriability effect– the business-stealing effect


• The appropriability effect– the social planner takes account of

increased consumer surplus from lower prices

– firms do not

• This implies that n** > n*


• The business-stealing effect– entry causes the profits of existing firms to

fall– the marginal firm does not consider the

drop in other firms’ profits when making its entry decision (the social planner would)

• This implies that n* > n**

Feedback Effect

• The feedback effect is that the more profitable a market is for a given number of firms, the more firms will enter the market, making the market more competitive and less profitable than it would be if the number of firms was fixed

Monopoly on Innovation

• The dissipation effect– competition dissipates some of the profit

from innovation and thus reduces the incentives to innovate

• The replacement effect– firms gain less in incremental profit and

thus have less incentive to innovate if the new product replaces an existing product

Competition for Innovation

• New firms are not always more innovative than existing ones– the dissipation effect may counteract the

replacement effect

• Dominant firms apply for “sleeping patents” to prevent entry– patents that are never implemented

Important Points to Note:

• One of the most basic oligopoly models, the Bertrand model, involves two identical firms that set prices simultaneously– the equilibrium resulted in the Bertrand

paradox• even though the oligopoly is as concentrated

as possible, the two firms act as perfect competitors

Important Points to Note:• The Bertrand paradox is not the

inevitable outcome in an oligopoly but can be escaped by changing assumptions– allowing for quantity competition,

differentiated products, search costs, capacity constraints, or repeated play leading to collusion

Important Points to Note:• As in the Prisoners’ Dilemma, firms

could profit by coordinating on a less competitive outcome– this outcome will be unstable unless

firms can explicitly collude by forming a legal cartel or tacitly collude in a repeated game

Important Points to Note:• For tacit collusion to sustain super-

competitive profits, firms must be patient enough that the loss from a price war in future periods to punish undercutting exceeds the benefit from undercutting in the current period

Important Points to Note:• Whereas a nonstrategic monopolist

prefers flexibility to respond to changing market conditions, a strategic oligopolist may prefer to commit to a single choice– the firm can commit to the choice if it

involves a sunk cost that cannot be recovered if the choice is later reversed

Important Points to Note:• A first mover can gain an advantage

by committing to a different action from what it would choose in the Nash equilibrium of a simultaneous game– to deter entry, the first mover should

commit to reducing the entrant’s profits – if it does not deter entry, the first mover

should commit to a strategy that leads its rival to compete less aggressively

Important Points to Note:• Holding the number of firms in an

oligopoly constant in the short run, an introduction of a factor that softens competition will raise firms’ profit– an offsetting effect in the long run is that

entry will now be more attractive• reducing oligopoly profit

Important Points to Note:• Innovation may be even more

important than low prices for total welfare in the long run– determining which oligopoly structure is

the most innovative is difficult because offsetting effects are involved

• dissipation• replacement

Chapter 15 Imperfect Competition Nicholson and Snyder, Copyright ©2008 by Thomson South-Western....

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Transcript of Chapter 15 Imperfect Competition Nicholson and Snyder, Copyright ©2008 by Thomson South-Western....