chapter 9 information and strategic behavior
play

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 rms information can be partially revealed by


  1. 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

  2. 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

  3. • 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

  4. – 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

  5. • 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

  6. 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

  7. 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

  8. • 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

  9. • 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

  10. 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

  11. • 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

Recommend


More recommend