Industrial Organization or Imperfect Competition Limit Pricing

1. Industrial Organization or Imperfect CompetitionLimit Pricing Univ. Prof. dr. Maarten Janssen University of Vienna Summer semester 2009 Week 10 (May 18,19)

2. 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?

3. 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

4. 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.

5. 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

6. 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

7. 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)

8. 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

9. 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?

10. 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

11. 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

12. 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

13. 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

14. 2, 2 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 II U

15. 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

16. 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

17. 2, 2 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 III U

18. 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

19. 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

20. 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)

21. 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

22. 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

23. 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

24. 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?

25. 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

26. 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