Chapter 9: Information and Strategic Behavior Asymmetric - - PDF document

Chapter 9: Information and Strategic Behavior

• Asymmetric information.
• Firms may have better (private) information on

– their own costs, – the state of the demand...

• Static game

– firm’s information can be partially revealed by its action, – myopic behavior.

• Dynamic game (repeated interaction)

– firm’s information can be partially revealed, – can be exploited by rivals later, – and thus manipulation of information.

• Accommodation
• entry deterrence (Limit Pricing model, Milgrom-

Roberts (1982)) 1

1 Static competition under Asym- metric Information

• 2 period model
• 2 risk-neutral firms: firm 1 (incumbent), firm 2

(potential entrant) Timing: Period 1. – Firm 1 takes a decision (price, advertising, quantity...). – Firm 2 observes firm 1’s decision, and takes an action (entry, no entry...). Period 2. If duopoly, firms choose they price simultane-

• usly (Bertrand competition).

Period 2, if entry.

• Differentiated products.
• Demand curves are symmetric and linear

Di(pi, pj) = a − bpi + dpj

for i, j = 1, 2 and i 6= j where 0 < d < b. 2

• The two goods are substitutes (dDi

dpj = d > 0) and

strategic complements ( d2Πi

dpidpj > 0).

• Marginal cost of firm 2 is c2, and common knowledge.
• Marginal cost of firm 1 can take 2 values c1 ∈ {cH

1 , cL 1}

and is private information.

• Firm 2 has only prior beliefs concerning the cost of its

rival, x. Thus

c1 = ( cL

1

with probability x

cH

1

with probability (1 − x)

• Firm 1’s expected MC from the point of view of 2 is

ce

1 = xcL 1 + (1 − x)cH 1

• Ex post profit is

Πi(pi, pj) = (pi − ci)(a − bpi + dpj)

• Firm 1’s program is

– if c1 = cL

1

Max

p1 (p1 − cL 1)(a − bp1 + dp∗ 2)

3

– If c1 = cH

1

Max

p1 (p1 − cH 1 )(a − bp1 + dp∗ 2)

• Firm 2’s program

Max

p2 {x[(p2 − c2)(a − bp2 + dpL 1)]

+(1 − x)[(p2 − c2)(a − bp2 + dpH

1 )]}

which is equivalent to

Max

p2 {(p2 − c2)(a − bp2) + (p2 − c2)pe 1}

where

pe

1 = xpL 1 + (1 − x)pH 1

• Best response functions are

pL

1 = a + bcL 1 + dp2

2b = RL

1 (p2)

pH

1 = a + bcH 1 + dp2

2b = RH

1 (p2)

p2 = a + bc2 + dpe

1

2b = R2(pe

1)

• Graph

4

• Solution of the system of 3 equations gives

p∗

2 = 2ab + ad + 2b2c2 + dbce 1

4b2 − d2

• where ∂p∗

2

∂ce

1 > 0 and

∂p∗

2

∂(1−x) > 0

• Then you plug p∗

2 in RL 1 (p2) and RH 1 (p2) to find the

solution pL

1 and pH 1 .

• Under asymmetric information, everything is “as if”

firm 1 has an average reaction curve

Re

1(p2) = xRL 1 (p2) + (1 − x)RH 1 (p2)

= a + bce

1 + dp2

2b

• Firm 1 has an incentive to prove that it has a high cost

before engaging in price competition. 5

2 Dynamic Game

• Assume that direct disclosure is impossible.

Timing: Period 1. Price competition Period 2. Price competition

• If entry is not an issue (accommodate), firms want to

appear inoffensive so as to induce its rival to raise its price.

• Thus, in first period: high price to signal high cost.
• Thus, accommodation calls for puppy dog strategy

(be small to look inoffensive).

• If deterrence is at stake, more aggressive behavior: the

firm wants to signal a low cost.

• Thus, in first period, low price to induce its rival to

doubt about the viability of the market (limit pricing model).

• Thus, deterrence calls for top dog strategy.

6

3 Accommodation

• A firm may rise its price to signal high cost and soften

the behavior of its rival.

• Riordan (1985)’s model
• 2 firms

Timing: Period A. Price competition Period B. Price competition

• Marginal cost is 0.
• Firm i’s demand is

qi = a − pi + pj

• The demand intercept is unknown to both firms, and

has a mean ae.

• In a one-period version of the game, program of firm i

Max

pi

{E(a − pi + pj)pi = (ae − pi + pj)pi}

• thus

pi = ae + pj 2 ,

7

• and by symmetry, the Static Bertrand equilibrium is

p1 = p2 = ae.

• 2 period version with same a for each period, and each

firm observes the realization of its own demand.

• In the symmetric equilibrium,

– each firm sets

pA

1 = pA 2 = α

in the first period. – Thus, each firm learns perfectly a as

DA

i = a − α + α = a

– and the second-period is of complete information, and the program of firm i

Max

pB

i

(a − pB

i + pB j )pB i

• thus

pB

i

= a + pB

j

2 ,

8

• and the symmetric equilibrium of second period is

pB

1 = pB 2 = a.

• Consider a strategic behavior in period A: firm i

deviates and chooses

pA

i 6= α

• Firm j observes a demand of

DA

j = a − α + pA i

• Firm j has a wrong perception of a, and has a

perception e

a, a − α + pA

i = e

a − α + α = e a

and thus

e a(pA

i ) = a − α + pA i

• In the second period, j believes it is playing a game of

perfect information, with intercept e

a(pA

i ), so it charges

pB

j = e

a(pA

i ) = a − α + pA i

9

and thus

∂pB

j

∂pA

i

= 1

• A unit increase in the first period triggers a unit increase

in the rival’s second period price.

• However i knows the intercept is not the right one, and

the program of i in the second period is

Max

pB

i

{ΠB

i

= (a − pB

i + e

a(pA

i ))pB i }

• Thus

pB

i = a + e

a(pA

i )

2 = a + pA

i − α

2

• The derivative of the second period profit with respect

to pA

i is

dΠB

i

dpA

i

= ∂ΠB

i

∂pB

i

∂pB

i

∂pA

i

+ ∂ΠB

i

∂pA

i

= pB

i

∂e a(pA

i )

∂pA

i

= pB

i

10

• Firm i maximizes its expected present discounted

profit, thus the FOC is

EdΠA

i

dpA

i

+ δEdΠB

i

dpA

i

= 0

• where δ is the discount factor.
• Thus, it is equivalent to

ae − 2pA

i + α + δ(ae + pA i − α

2 ) = 0

• In equilibrium pA

i = α, thus

α = ae(1 + δ) > ae

• In a dynamic model, a firm may induce its rival to raise

its price. 11

4 The Milgrom-Roberts (1982) Model of Limit Pricing

• Asymmetric information drives firms to cut their price

in first period.

• 2 risk-neutral firms: firm 1 (incumbent), firm 2

(potential entrant)

• Asymmetric information on firm 1’s costs. Firm 2 has
• nly prior beliefs concerning the cost of its rival, x.

Thus

c1 = ( cL

1

with probability x

cH

1

with probability (1 − x) Timing: Period 1.

• Firm 1 chooses a first period price p1.

– Firm 2 observes p1 and decides whether to enter

{e, ne}.

Period 2. If firm 2 enters: price competition. If not, monopoly. 12

• Firm 2 learns 1’s cost immediately after entering.
• The incumbent’s profit when price is p1 is

Mt

1(p1) = (p1 − ct 1)Q(p1)

where t = H, L. (strictly concave function in p1) – Thus pL

1, pH 1 are the monopoly prices charged by the

incumbent, pL

1 < pH 1 .

• Duopoly’s payoffs are Dt

i for t = H, L and i = 1, 2.

• Assume DH

2 > 0 > DL 2 : if low cost, no room for 2

firms, if high cost, room for duopoly.

• δ Discount factor.
• To simplify: only 2 prices pL

1, pH 1 and not a continuum

• f prices.
• Perfect Bayesian Equilibrium concept.
• See tree of the game

13

Benchmark case: symmetric information

• Cost is low with probability x = 1
• Cost is high with probability x = 0.
• Decisions of firm 2 to enter?

– if low cost: does not enter, – if high cost: enters.

• Decision of firm 1?

– if low cost, firm 1 chooses a low price if

ML

1 (pL 1) + δML 1 (pL 1) > ML 1 (pH 1 ) + δML 1 (pL 1)

⇒ ML

1 (pL 1) > ML 1 (pH 1 )

which is always satisfied. – if high cost, firm 1 chooses a high price if

MH

1 (pH 1 ) + δDH 1 > MH 1 (pL 1) + δDH 1

⇒ MH

1 (pH 1 ) > MH 1 (pL 1)

Result 1. Under symmetric information

♦ If c = cL

1, (pL 1, ne) is a Perfect Nash Equilibrium

♦ If c = cH

1 , (pH 1 , e) is a Perfect Nash Equilibrium

14

Asymmetric Information

• Separating equilibrium?

The incumbent does not choose the same price when its cost is high or low.

• Pooling equilibrium?

The first period price is independent of the cost level. Separating equilibrium

• Only one possible kind of separating:

– If c = cL

1, ne

– If c = cH

1 , e

• Is it an equilibrium? and under what kind of circum-

stances?

• It is an equilibrium if none of the firms deviate.

– If c = cL

1

ML

1 (pL 1) + δML 1 (pL 1) > ML 1 (pH 1 ) + δDL 1

⇒ ML

1 (pL 1) − ML 1 (pH 1 ) > δ(DL 1 − ML 1 (pL 1))

(1) 15

– If c = cH

1

MH

1 (pH 1 ) + δDH 1 > MH 1 (pL 1) + δMH 1 (pH 1 )

⇒ MH

1 (pH 1 ) − MH 1 (pL 1) > δ(MH 1 (pH 1 ) − DH 1 )

(2) – The equation (1) is always satisfied, whereas (2) must be satisfied. Result 2. If (2) is satisfied, there exists a separating equilibrium such that

♦ the incumbent chooses pL

1 and firm 2 does not enter

(ne) if c = cL

1,

♦ the incumbent chooses pH

1 and firm 2 enters (e) if

c = cH

1 .

Pooling equilibrium

• Two possible kinds of pooling:
• P1. the incumbent always chooses pL

1, whatever the cost,

• P2. the incumbent always chooses pH

1 , whatever the cost.

• Updated beliefs equal to prior beliefs.

16

• P1. (pL

1) Player 2 stays out if

0 > xδDL

2 + (1 − x)δDH 2

⇒ x > e x = DH

2

DH

2 − DL 2

• e

x ∈ [0, 1]?

• e

x > 0 if DH

2 > DL 2 ,

• e

x < 1 if DL

2 < 0.

• Thus, for x > e

x firm 2 prefers to stay out.

• Can firm 1 do better?

– If c = cL

1

ML

1 (pL 1) + δML 1 (pL 1) > ML 1 (pH 1 ) + δDL 1

OK and ML

1 (pL 1) + δML 1 (pL 1) > ML 1 (pH 1 ) + δML 1 (pL 1)

OK 17

– If c = cH

1

MH

1 (pL 1) + δMH 1 (pH 1 ) > MH 1 (pH 1 ) + δDH 1

OK and MH

1 (pL 1) + δMH 1 (pH 1 ) > MH 1 (pH 1 ) + δMH 1 (pH 1 )

NO

• Thus, with an out-of-equilibrium prob(e/pH

1 ) = 1,

there exists a pooling. Result 3. If (2) is not satisfied, there exists a pooling equilibrium such that

♦ the incumbent always chooses pL

1,

♦ and firm 2 does not enter (ne) ♦ with an out-of-equilibrium probability prob(e/pH

1 ) = 1.

18

• P2. (pH

1 ) Player 2 enters if

xδDL

2 + (1 − x)δDH 2 > 0

⇒ x < e x = DH

2

DL

2 − DH 2

• Then for x < e

x firm 2 will enter.

• Can firm 1 do better?

– If c = cL

1

ML

1 (pH 1 ) + δDL 1 > ML 1 (pH 1 ) + δML 1 (pL 1)

NO and ML

1 (pH 1 ) + δML 1 (pL 1) > ML 1 (pL 1) + δML 1 (pL 1)

NO – Thus firm 1 will always deviate.

• There is no pooling P2.
• If (2) is not satisfied, the incumbent manipulates the

price such that its action does not reveal any cost information. 19

• In continuous p ∈ [0, ∞[, same results except that

prices are different.

• Single-crossing condition

∂2[(p1 − c1)Qm

1 (p1)]

∂p1∂c1 = −∂Qm

1

∂p1 > 0

• It is more costly to the high type to charge low price.

Separating equilibrium

• – if c = cH

1 , pH 1 = pH m

– if c = cL

1, pL 1 ∈ [e

e p1, e p1] where e p1 < pL

• m. Low cost

type makes pooling very costly to the high cost type.

• There exists a reasonable separating equilibrium where

– if c = cH

1 , pH 1 = pH m and entry occurs,

– if c = cL

1, pL 1 = e

p1and no entry.

• The incumbent does not fool the entrant
• But, there exists a limit pricing.

20

Pooling equilibrium

• The incumbent chooses pL

m.

• The incumbent manipulates its price.
• Less entry occurs than under symmetric information.
• High cost type is engaged in limit pricing.

21