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Industrial Organization or Imperfect Competition Limit 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
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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
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
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
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
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