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 – fi rm’s information can be partially revealed by its action, – myopic behavior. • Dynamic game (repeated interaction) – fi rm’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 fi rms: fi rm 1 (incumbent), fi rm 2 (potential entrant) Timing: Period 1. – Firm 1 takes a decision (price, advertising, quantity...). – Firm 2 observes fi rm 1’s decision, and takes an action (entry, no entry...). Period 2. If duopoly, fi rms choose they price simultane- ously (Bertrand competition). Period 2 , if entry. • Differentiated products. • Demand curves are symmetric and linear D i ( p i , p j ) = a − bp i + dp j for i, j = 1 , 2 and i 6 = j where 0 < d < b . 2

• The two goods are substitutes ( dD i dp j = d > 0 ) and strategic complements ( d 2 Π i dp i dp j > 0 ). • Marginal cost of fi rm 2 is c 2 , and common knowledge. • Marginal cost of fi rm 1 can take 2 values c 1 ∈ { c H 1 , c L 1 } and is private information . • Firm 2 has only prior beliefs concerning the cost of its rival, x . Thus ( c L with probability x 1 c 1 = with probability (1 − x ) c H 1 • Firm 1’s expected MC from the point of view of 2 is c e 1 = xc L 1 + (1 − x ) c H 1 • Ex post pro fi t is Π i ( p i , p j ) = ( p i − c i )( a − bp i + dp j ) • Firm 1’s program is – if c 1 = c L 1 p 1 ( p 1 − c L 1 )( a − bp 1 + dp ∗ Max 2 ) 3

– If c 1 = c H 1 1 )( a − bp 1 + dp ∗ p 1 ( p 1 − c H 2 ) Max • Firm 2’s program p 2 { x [( p 2 − c 2 )( a − bp 2 + dp L Max 1 )] +(1 − x )[( p 2 − c 2 )( a − bp 2 + dp H 1 )] } which is equivalent to p 2 { ( p 2 − c 2 )( a − bp 2 ) + ( p 2 − c 2 ) p e Max 1 } where p e 1 = xp L 1 + (1 − x ) p H 1 • Best response functions are 1 = a + bc L 1 + dp 2 p L = R L 1 ( p 2 ) 2 b 1 = a + bc H 1 + dp 2 p H = R H 1 ( p 2 ) 2 b p 2 = a + bc 2 + dp e 1 = R 2 ( p e 1 ) 2 b • Graph 4

• Solution of the system of 3 equations gives 2 = 2 ab + ad + 2 b 2 c 2 + dbc e p ∗ 1 4 b 2 − d 2 • where ∂ p ∗ ∂ p ∗ 1 > 0 and ∂ (1 − x ) > 0 2 2 ∂ c e • Then you plug p ∗ 2 in R L 1 ( p 2 ) and R H 1 ( p 2 ) to fi nd the solution p L 1 and p H 1 . • Under asymmetric information, everything is “as if” fi rm 1 has an average reaction curve R e 1 ( p 2 ) = xR L 1 ( p 2 ) + (1 − x ) R H 1 ( p 2 ) = a + bc e 1 + dp 2 2 b • 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 ), fi rms want to appear inoffensive so as to induce its rival to raise its price. • Thus, in fi rst 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 fi rm wants to signal a low cost. • Thus, in fi rst 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 fi rm may rise its price to signal high cost and soften the behavior of its rival. • Riordan (1985)’s model • 2 fi rms Timing: Period A. Price competition Period B. Price competition • Marginal cost is 0. • Firm i’s demand is q i = a − p i + p j • The demand intercept is unknown to both fi rms, and has a mean a e . • In a one-period version of the game, program of fi rm i { E ( a − p i + p j ) p i = ( a e − p i + p j ) p i } Max p i • thus p i = a e + p j , 2 7

• and by symmetry, the Static Bertrand equilibrium is p 1 = p 2 = a e . • 2 period version with same a for each period, and each fi rm observes the realization of its own demand. • In the symmetric equilibrium, – each fi rm sets p A 1 = p A 2 = α in the fi rst period. – Thus, each fi rm learns perfectly a as D A i = a − α + α = a – and the second-period is of complete information, and the program of fi rm i ( a − p B i + p B j ) p B Max i p B i • thus = a + p B j p B , i 2 8

• and the symmetric equilibrium of second period is p B 1 = p B 2 = a. • Consider a strategic behavior in period A: fi rm i deviates and chooses p A i 6 = α • Firm j observes a demand of D A j = a − α + p A i • Firm j has a wrong perception of a , and has a perception e a , a − α + p A i = e a − α + α = e a and thus a ( p A i ) = a − α + p A e i • In the second period, j believes it is playing a game of perfect information, with intercept e a ( p A i ) , so it charges p B a ( p A i ) = a − α + p A j = e i 9

and thus ∂ p B j = 1 ∂ p A i • A unit increase in the fi rst 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 { Π B = ( a − p B a ( p A i )) p B i + e Max i } i p B i • Thus a ( p A = a + p A i = a + e i ) i − α p B 2 2 • The derivative of the second period pro fi t with respect to p A i is d Π B = ∂ Π B ∂ p B + ∂ Π B i i i i dp A ∂ p B ∂ p A ∂ p A i i i i a ( p A ∂ e i ) = p B i ∂ p A i = p B i 10

• Firm i maximizes its expected present discounted pro fi t, thus the FOC is Ed Π A + δ Ed Π B i i = 0 dp A dp A i i • where δ is the discount factor. • Thus, it is equivalent to i + α + δ ( a e + p A i − α a e − 2 p A ) = 0 2 • In equilibrium p A i = α , thus α = a e (1 + δ ) > a e • In a dynamic model, a fi rm may induce its rival to raise its price. 11

4 The Milgrom-Roberts (1982) Model of Limit Pricing • Asymmetric information drives fi rms to cut their price in fi rst period. • 2 risk-neutral fi rms: fi rm 1 (incumbent), fi rm 2 (potential entrant) • Asymmetric information on fi rm 1’s costs. Firm 2 has only prior beliefs concerning the cost of its rival, x . Thus ( c L with probability x 1 c 1 = c H with probability (1 − x ) 1 Timing: Period 1. • Firm 1 chooses a fi rst period price p 1 . – Firm 2 observes p 1 and decides whether to enter { e, ne } . Period 2. If fi rm 2 enters: price competition. If not, monopoly. 12

• Firm 2 learns 1’s cost immediately after entering. • The incumbent’s pro fi t when price is p 1 is M t 1 ( p 1 ) = ( p 1 − c t 1 ) Q ( p 1 ) where t = H, L. (strictly concave function in p 1 ) – Thus p L 1 , p H 1 are the monopoly prices charged by the incumbent, p L 1 < p H 1 . • Duopoly’s payoffs are D t i for t = H, L and i = 1 , 2 . • Assume D H 2 > 0 > D L 2 : if low cost, no room for 2 fi rms, if high cost, room for duopoly. • δ Discount factor. • To simplify : only 2 prices p L 1 , p H 1 and not a continuum of 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 fi rm 2 to enter? – if low cost: does not enter, – if high cost: enters. • Decision of fi rm 1? – if low cost, fi rm 1 chooses a low price if M L 1 ( p L 1 ) + δ M L 1 ( p L 1 ) > M L 1 ( p H 1 ) + δ M L 1 ( p L 1 ) ⇒ M L 1 ( p L 1 ) > M L 1 ( p H 1 ) which is always satis fi ed. – if high cost, fi rm 1 chooses a high price if M H 1 ( p H 1 ) + δ D H 1 > M H 1 ( p L 1 ) + δ D H 1 ⇒ M H 1 ( p H 1 ) > M H 1 ( p L 1 ) Result 1. Under symmetric information ♦ If c = c L 1 , ( p L 1 , ne ) is a Perfect Nash Equilibrium ♦ If c = c H 1 , ( p H 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 fi rst period price is independent of the cost level. Separating equilibrium • Only one possible kind of separating: – If c = c L 1 , ne – If c = c H 1 , e • Is it an equilibrium? and under what kind of circum- stances? • It is an equilibrium if none of the fi rms deviate. – If c = c L 1 M L 1 ( p L 1 ) + δ M L 1 ( p L 1 ) > M L 1 ( p H 1 ) + δ D L 1 (1) ⇒ M L 1 ( p L 1 ) − M L 1 ( p H 1 ) > δ ( D L 1 − M L 1 ( p L 1 )) 15

– If c = c H 1 M H 1 ( p H 1 ) + δ D H 1 > M H 1 ( p L 1 ) + δ M H 1 ( p H 1 ) (2) ⇒ M H 1 ( p H 1 ) − M H 1 ( p L 1 ) > δ ( M H 1 ( p H 1 ) − D H 1 ) – The equation (1) is always satis fi ed, whereas (2) must be satis fi ed. Result 2. If (2) is satis fi ed, there exists a separating equilibrium such that ♦ the incumbent chooses p L 1 and fi rm 2 does not enter ( ne ) if c = c L 1 , ♦ the incumbent chooses p H 1 and fi rm 2 enters ( e ) if c = c H 1 . Pooling equilibrium • Two possible kinds of pooling: P1. the incumbent always chooses p L 1 , whatever the cost, P2. the incumbent always chooses p H 1 , whatever the cost. • Updated beliefs equal to prior beliefs. 16