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, – canbeexploitedbyrivalslater, – 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- ously (Bertrand competition).
Period 2,ifentry. Differentiated products. • Demand curves are symmetric and linear • Di(pi,pj)=a bpi + dpj − for i, j =1, 2 and i = j where 0 2 The two goods are substitutes (dDi = d>0)and dpj • 2 i strategic complements ( d Π > 0). dpidpj Marginal cost of firm 2 is c2, and common knowledge. • H L Marginal cost of firm 1 can take 2 values c1 c1 ,c1 • and is private information. ∈ { } Firm 2 has only prior beliefs concerning the cost of its • rival, x. Thus cL with probability x c = 1 1 H ( c1 with probability (1 x) − Firm 1’s expected MC from the point of view of 2 is • ce = xcL +(1 x)cH 1 1 − 1 Ex post profitis • i Π (pi,pj)=(pi ci)(a bpi + dpj) − − Firm 1’s program is • L – if c1 = c1 L Max(p1 c1 )(a bp1 + dp2∗) p1 − − 3 H – If c1 = c1 H Max(p1 c1 )(a bp1 + dp2∗) p1 − − Firm 2’s program • L Max x[(p2 c2)(a bp2 + dp1 )] p2 { − − H +(1 x)[(p2 c2)(a bp2 + dp )] − − − 1 } which is equivalent to e Max (p2 c2)(a bp2)+(p2 c2)p1 p2 { − − − } where pe = xpL +(1 x)pH 1 1 − 1 Best response functions are • a + bcL + dp pL = 1 2 = RL(p ) 1 2b 1 2 a + bcH + dp pH = 1 2 = RH(p ) 1 2b 1 2 a + bc + dpe p = 2 1 = R (pe) 2 2b 2 1 Graph • 4 Solution of the system of 3 equations gives • 2ab + ad +2b2c + dbce p = 2 1 2∗ 4b2 d2 − ∂p2∗ ∂p2∗ where ∂ce > 0 and ∂(1 x) > 0 • 1 − L H Then you plug p2∗ in R1 (p2) and R1 (p2) to find the • L H solution p1 and p1 . Under asymmetric information, everything is “as if” • firm 1 has an average reaction curve e L H R (p2)=xR (p2)+(1 x)R (p2) 1 1 − 1 a + bce + dp = 1 2 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 • e Max E(a pi + pj)pi =(a pi + pj)pi pi { − − } thus • ae + p p = j , i 2 7 and by symmetry, the Static Bertrand equilibrium is • e p1 = p2 = a . 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 A A p1 = p2 = α in the first period. – Thus, each firm learns perfectly a as DA = a α + α = a i − – and the second-period is of complete information, and the program of firm i Max(a pB + pB)pB B i j i pi − thus • a + pB pB = j , i 2 8 and the symmetric equilibrium of second period is • B B p1 = p2 = a. Consider a strategic behavior in period A: firm i • deviates and chooses pA = α i 6 Firm j observes a demand of • DA = a α + pA j − i Firm j has a wrong perception of a, and has a • perception a, a α + pA = a α + α = a − i − and thus e a(pA)=a α + pA i e− i e In the second period, j believes it is playing a game of • A perfect information,e with intercept a(pi ),soitcharges pB = a(pA)=a α e+ pA j i − i e 9 and thus B ∂pj A =1 ∂pi 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 ΠB =(a pB + a(pA))pB B i i i i pi { − } Thus • a + a(pA) epA α pB = i = a + i − i 2 2 The derivative of thee second period profit with respect • A to pi is B B B B dΠi ∂Πi ∂pi ∂Πi A = B A + A dpi ∂pi ∂pi ∂pi A B ∂a(pi ) = pi A ∂pi B e = pi 10 Firm i maximizes its expected present discounted • profit, thus the FOC is A B dΠi dΠi E A + δE A =0 dpi dpi where δ is the discount factor. • Thus, it is equivalent to • pA α ae 2pA + α + δ(ae + i − )=0 − i 2 In equilibrium pA = α, thus • i 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 • only prior beliefs concerning the cost of its rival, x. Thus cL with probability x c = 1 1 H ( c1 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 profitwhenpriceisp1 is • t t M (p1)=(p1 c )Q(p1) 1 − 1 where t = H, L. (strictly concave function in p1) L H – Thus p1 , p1 are the monopoly prices charged by the L H incumbent, p1 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 L L L L L H L L M1 (p1 )+δM1 (p1 ) >M1 (p1 )+δM1 (p1 ) M L(pL) >ML(pH) ⇒ 1 1 1 1 which is always satisfied. – if high cost, firm 1 chooses a high price if H H H H L H M1 (p1 )+δD1 >M1 (p1 )+δD1 M H(pH) >MH(pL) ⇒ 1 1 1 1 Result 1. Under symmetric information L L ♦ If c = c1 , (p1 ,ne) is a Perfect Nash Equilibrium H H ♦ If c = c1 , (p1 ,e) is a Perfect Nash Equilibrium 14 Asymmetric Information Separating equilibrium? • The incumbent does not choose the same price when itscostishighorlow. Pooling equilibrium? • The first period price is independent of the cost level. Separating equilibrium Only one possible kind of separating: • L – If c = c1 , ne H – If c = c1 , e Is it an equilibrium? and under what kind of circum- • stances? It is an equilibrium if none of the firms deviate. • L – If c = c1 L L L L L H L M1 (p1 )+δM1 (p1 ) >M1 (p1 )+δD1 (1) M L(pL) M L(pH) > δ(DL M L(pL)) ⇒ 1 1 − 1 1 1 − 1 1 15 H – If c = c1 H H H H L H H M1 (p1 )+δD1 >M1 (p1 )+δM1 (p1 ) (2) M H(pH) M H(pL) > δ(M H(pH) DH) ⇒ 1 1 − 1 1 1 1 − 1 – The equation (1) is always satisfied, whereas (2) must be satisfied. Result 2. If (2) is satisfied, there exists a separating equilibrium such that L ♦ the incumbent chooses p1 and firm 2 does not enter L (ne)ifc = c1 , H ♦ the incumbent chooses p1 and firm 2 enters (e)if H c = c1 . Pooling equilibrium Two possible kinds of pooling: • L P1. the incumbent always chooses p1 , whatever the cost, H P2. the incumbent always chooses p1 , whatever the cost. Updated beliefs equal to prior beliefs. • 16 L P1.(p1 ) Player 2 stays out if 0 >xδDL +(1 x)δDH 2 − 2 DH x>x = 2 ⇒ DH DL 2 − 2 x [0, 1]? e • ∈ x>0 if DH >DL, • 2 2 x L L L L L H L L and M1 (p1 )+δM1 (p1 ) >M1 (p1 )+δM1 (p1 ) OK 17 H – If c = c1 H L H H H H H M1 (p1 )+δM1 (p1 ) >M1 (p1 )+δD1 OK H L H H H H H H and M1 (p1 )+δM1 (p1 ) >M1 (p1 )+δM1 (p1 ) NO H Thus, with an out-of-equilibrium prob(e/p1 )=1, • there exists a pooling. Result 3. If (2) is not satisfied, there exists a pooling equilibrium such that L ♦ the incumbent always chooses p1 , ♦ and firm 2 does not enter (ne) ♦ with an out-of-equilibrium probability H prob(e/p1 )=1. 18 H P2.(p1 )Player2entersif xδDL +(1 x)δDH > 0 2 − 2 DH x L H L L L L L L and M1 (p1 )+δM1 (p1 ) >M1 (p1 )+δM1 (p1 ) 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 m m ∂ [(p1 c1)Q (p1)] ∂Q − 1 = 1 > 0 ∂p1∂c1 − ∂p1 It is more costly to the high type to charge low price. • Separating equilibrium H H H – if c = c1 , p1 = pm • L L L – if c = c1 , p1 [p1, p1] where p1 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