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Industrial Organization or Imperfect Competition Limit Pricing

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### Industrial Organization or Imperfect CompetitionLimit Pricing

Univ. Prof. dr. Maarten Janssen

University of Vienna

Summer semester 2009

Week 10 (May 18,19)

Limit Pricing – an asymmetric information story about entry deterrence

- Incumbent has private information about cost
- Cost can be either high or low
- Potential entrant does not know cost structure incumbent, but has some beliefs about it
- If incumbent’s cost is really low, entrant cannot compete and make positive profits; if cost is high, entrant does make profit by entering
- Can incumbent prevent entry in all or some cases?

Limit Pricing – basic idea

- A high-cost incumbent has an incentive to pretend to be low-cost if by so doing it can deter entry
- The entrant recognizes this incentive to masquerade as a low cost firm
- What can the entrant infer from observing low price knowing this may be a deception?
- In turn this depends on the probability that observing a low-price means that the incumbent is a low-cost firm
- First, need to develop game theory with private information

TOY GAME

- Student considers applying for a PhD program
- Student knows his own qualities (good, bad)
- University has to decide whether or not to accept student if (s)he applies
- University does not know quality of student, but believes that probability is 50-50
- Pay-offs student, university depend on whether student is good or not in case student is accepted.

Game Structure Toy Game I

2, 2

Accept

U

Apply

-1 , 0

Reject

S

Don’t Apply

0,0

GOOD

½

-2, -3

Accept

N

U

½

Apply

-1 , 0

Reject

S

BAD

Don’t Apply

0,0

What is a strategy?

- Rule that tells a player what to choose given the information (s)he has
- Action to be taken conditional on information
- Student’s strategy
- Example: Apply if, and only if, I am a good student
- Four possible strategies (always apply, never apply, apply iff good, apply iff bad)
- University’s strategy
- Just two possible: accept, reject

Equilibrium definition

- General Nash: one strategy for each player such that no player has incentive to deviate given strategies of other players
- Here, updating of information possible
- Players may learn more about information other players have based on actions they take
- University may learn information about quality student given whether student applies.
- Update information using Bayes’ rule – whenever possible
- P(A/B) = P(B/A)P(A) : P(B)

Equilibria in Toy Game I

- Suppose student chooses: apply iff good, what is optimal reaction of university
- Bayesian updating: P(good/apply) = P(apply/good)P(good)/P(apply) = 1.½ / ½ = 1
- Thus, P(bad/apply) = 0
- University will rationally accept
- Given university will accept, above proposed strategy student is optimal
- Separating (or revealing) equilibrium
- Different types of students choose different actions
- Is there another (pooling) equilibrium?
- Student never applies, university rejects

Limit Pricing-asymmetric info

- An alternative approach to predation: information structure
- suppose that an entrant does not have perfect information about the incumbent’s costs
- if the incumbent is low cost do not enter
- if the incumbent is high-cost enter
- does a high-cost incumbent have an incentive to pretend to be low-cost - to prevent entry, for example by pricing as a low-cost firm?

A (Simple) Example

- Incumbent has a monopoly in period 1
- Threat of entry in period 2
- Market closes at the end of period 2
- Entrant observes incumbent’s price choice at t=1
- Entrant’s choice dependent on price incumbent
- Incumbent is expected to be high-cost or low-cost
- no direct information on incumbent’s costs
- entrant knows that there is a probability that the incumbent is low-cost

Incumbent: 6 + 2 = 8 Entrant: 2

Enter

High Price

Incumbent: 6 + 6 = 12 Entrant: 0

E3

Stay Out

High-Cost

Incumbent: 4 + 2 = 6 Entrant: 2

I1

Enter

Low Price

Nature

E4

Incumbent: 4 + 6 = 10 Entrant: 0

Stay Out

Low-Cost

I2

Enter

Incumbent: 10 + 5 = 15 Entrant: -2

Low Price

E5

Incumbent: 10 + 10 = 20 Entrant: 0

Stay Out

The Example (cont.)With no uncertainty

the entrant enters if the

incumbent is high-cost

With uncertainty and

a low price the entrant

does not know if

he is at E4 or E5

The Example (cont)

- A high-cost incumbent has an incentive to pretend to be low-cost if by so doing it can deter entry
- The entrant recognizes this incentive to masquerade as a low cost firm
- The issue is what can the entrant infer from observing a low price knowing that this may be a deception
- In turn this depends on the probability that observing a low-price means that the incumbent is a low-cost firm

The Example (cont.)

- If entrant observes a low-price in period 1, it cannot tell whether it is at node E4 or E5
- As a result, the entrant must rely on the prior probability the incumbent is low-cost
- with probability , profit of –2
- with probability 1-, a profit of 2
- So the expected profit is 2(1 -) - 2 = 2 - 4
- This is negative if > ½. If, there is a “sufficiently high” probability that the incumbent is low cost, an incumbent can deter entry by setting a low price in period 1

Accept

U

Apply

-1 , 0

Reject

S

Don’t Apply

0,0

GOOD

½

-2, 1

Accept

N

½

Apply

-1 , 0

Reject

S

BAD

Don’t Apply

0,0

Game Structure Toy Game IIU

Equilibria in Toy Game II

- Separating equilibrium unaffected
- Student chooses: apply iff good
- University chooses: accept
- Pooling equilibrium unaffected
- Student never applies, university rejects
- But, it is strange now
- University is always better off to accept students
- Reject is what seems to be an incredible threat
- How to get rid of this incredible threat? Subgame perfection?
- What is value of P(good/apply)?
- Out-of-equilibrium beliefs (when certain information sets are not on the equilibrium path, Bayes’ rule cannot be applied)
- Perfect Bayes-Nash equilibrium: impose as an additional restriction that given certain arbitrary off-the-equilibrium beliefs, strategies should be optimal

Perfect Bayes-Nash Equilibria in Toy Game II

- Separating equilibrium unaffected
- There are no out-of-equilibrium beliefs
- Pooling equilibrium affected
- P(good/apply) should be defined. Let us say it equals μ, with 0 < μ < 1.
- For any value of μ, accept is the optimal strategy
- Therefore, in any perfect Bayes-Nash equilibrium, where students do not apply, university should accept.
- Given university accepts, good student will apply.
- No Pooling equilibrium exists

Accept

U

Apply

-1 , 0

Reject

S

Don’t Apply

0,0

GOOD

½

-2, -3

Accept

N

½

Apply

-1 , 0

Reject

S

BAD

Don’t Apply

0,0

Game Structure Toy Game IIIU

Equilibria in Toy Game III

- Separating equilibrium unaffected
- Is a perfect Bayes-Nash equilibrium
- Pooling equilibrium Is a perfect Bayes-Nash equilibrium
- Student never applies, university rejects
- for values of μ with 2μ – 3(1-μ) ≤ 0, P(good/apply) ≤ 3/5
- But, it (again) is strange
- Bad student is always better off not applying
- Why should the university be afraid of bad students applying?
- Domination-based beliefs: if one type of player never has incentives to apply, other type may have an incentive to deviate (for certain reactions of opponent), then out-of-equilibrium beliefs should be such that all probability mass is given to player who may have incentive to deviate.
- Domination-based beliefs restrict the set of reasonable out-of-equilibrium beliefs
- Here, domination-based beliefs requires that μ = 1.
- Given μ = 1, university will accept
- Pooling equilibrium does not satisfy domination-based beliefs requirement

Fuller Model (based on Milgrom, Roberts)

- Same basic information structure as before
- Incumbent’s cost either cL or cH with prob and 1- , resp.
- Entrant’s production cost is cE with cL < cE < cH and fixed cost of entry is f
- Demand is 1 - p
- Monopoly price in absence of entry ½ (1+ci), i = L, H
- (cH - cE)(1- cH) > f entrant can make profit if incumbent has high cost
- cH < ½ (1+cL), high cost incumbent makes profit at low cost monopoly price
- Note that this implies that cE < ½ (1+cL), i.e., if entrant enters low cost incumbent will have to lower its price

Analysis – equilibrium I:pooling on low cost monopoly price to deter entry

- Incumbent: always choose ½ (1+cL), no matter what cost are.
- Entrant stays out if negative expected profits: (1- )(cH - cE)(1- cH) < f
- Equilibrium profits incumbent
- Low cost type: ¼ (1-cL)2 + ¼ (1-cL)2
- High cost type: (½+½cL-cH) (½-½cL)+ ¼ (1-cH)2
- Deviating for both types not profitable
- Low type: this is the highest profit it can get
- High type: if entrant stays out at higher prices, then it can get higher profits by deviating, i.e., must have μ(p)(cH - cE)(1- cH) > f, for all p > ½ (1+cL), where μ(p) is out-of-eq belief after price p
- Discontinuous beliefs
- Moreover, even if entrant enters at higher prices, we should have that profits of setting a lower price in period 1 and monopoly profits in period 2 is larger than just monopoly profits in a period, i.e., ½(1+cL)> cH (which is assumed)

Analysis – equilibrium I: Summary

- Both types of incumbents choose ½ (1+cL).
- Entrant stays out if price is at or below ½ (1+cL); enter if price is larger than ½ (1+cL)
- Threat to enter at high prices gives incentive to lower prices
- μ(p) is out-of-eq belief that incumbent has high cost after any p other than ½ (1+cL)
- μ(p)> f/(cH - cE)(1- cH), for all p > ½ (1+cL),
- μ(p)< f/(cH - cE)(1- cH), for all p < ½ (1+cL),
- This is required by (weak) perfect Bayes-Nash eq
- Specify out-of-eq belief
- Optimal behaviour given this belief
- Consistent with domination-based beliefs?
- Yes, only restriction that μ(p) = 0 for all p < cH

Other pooling equilibria?

- Where entrant enters?
- No, if entrant enters, high cost firm would like to get maximal profits in period 1 and set its monopoly price
- Pooling on high cost monopoly price and entry cannot be an equilibrium outcome as then low cost incumbent will want to deviate to its own monopoly price
- Entrant enters after pooling strategy if (1- )(cH - cE)(1- cH) > f
- No pooling equilibrium for these parameter restrictions
- Where entrant stays out?
- Yes, potentially
- Construct one

Analysis – other pooling equilibria

- Incumbent: always choose p*, no matter what cost are.
- Entrant: stay out if, and only if, p ≤ p*
- Requires: (1- )(cH - cE)(1- cH) < f
- Equilibrium profits incumbent
- Low cost type: (1-p*)(p*-cL)+ ¼ (1-cL)2
- High cost type: (1-p*)(p*-cH)+ ¼ (1-cH)2
- Deviating for incumbent not profitable
- Low type: p* ≤ ½ (1+cL) as otherwise firm can lower price in the first period, obtain monopoly profit in both periods
- High type: can always get monopoly profit in first period. Hence, cH≤ p*
- If p* < ½ (1+cL) and low type sets ½ (1+cL) in first period, then profit ¼ (1-cL)2 + (1-cE)(cE-cL), which is lower than eq profit as cE <cH≤ p*.
- Deviating for entrant not profitable if appropriate out-of-eq beliefs

What type of equilibrium when (1- )(cH - cE)(1- cE) > f ?

- Potentially, revealing (or separating) equilibrium
- High cost incumbent sets pH; low cost pL
- Entrant enters after p>pL including pH ;stays out after p ≤ pL
- Equilibrium profits incumbent
- Low type: ¼ (1-cL)2 + (1-pL)(pL-cL)
- High type (1-pH)(pH-cH)
- Imitating not profitable
- Out-of-equilibrium beliefs
- What should they be?

Can there be a revealing equilibrium where both types set their monopoly prices?

- Equilibrium profits incumbent
- Low type: ¼ (1-cL)2 + ¼ (1-cL)2
- High type ¼ (1-cH)2
- Imitating not profitable
- Low type will never want to imitate
- But what about high type?
- Imitate low type yields (½+½cL-cH) (½-½cL)+ ¼ (1-cH)2
- As ½+½cL>cH this type of revealing equilibrium not possible, i.e., monopolist must distort somehow his pricing decision for revealing to be part of eq
- Who must distort?
- If high quality is revealed, he better makes the best of period 1: no distortion

Revealing equilibrium

- Incumbent
- High type sets ½ (1+cH) and makes profit ¼ (1-cH)2
- Low type should set price so that imitating is not profitable:
- (1-pL)(pL-cH) + ¼(1-cH)2 ≤ ¼(1-cH)2
- Thus low type should set price at or below high cost, i.e., pL ≤ cH
- If low type wants to deviate best is to set monopoly price in first period yielding ¼ (1-cL)2 + (1-cE)(cE-cL). This is not optimal if cE ≤ pL
- What about out-of-eq beliefs?
- At prices above pL there should be high chance of entry.
- Summary;
- For cE ≤ pL ≤ cH revealing equilibrium exists

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