Strategic Production Game 1

Transcription

1 Lec5-6.doc Strategic Production Game Consider two firms, which have to make production decisions without knowing what the other is doing. For simplicity we shall suppose that the product is essentially identical. If outputs levels are q and q so that the total supply is q = q + q, the market-clearing price is p = a bq= a b( q + q ) Firm i has a production cost of C firm s profit is = c q. Then if firm produces q, i i i Π ( q, q ) = pq c q = ( a c bq ) q bq Equal-profit curves for firm are depicted in Figure E-. To understand why they must have the indicated shape, first note that as q increases, firm 's profit must decline. Expressing this mathematically, Π < 0 q ext differentiating firm 's profit by q, Π = a c bq bq = b a c bq b q q ( ) [( )/ ] Π Hence > 0if q < ( a c bq)/b q The earliest discussion of this game goes all the way back to Augustin Cournot (848).

2 Lec5-6.doc q c )/ b Π > q 0 Π ( q, q ) = const c )/b q Figure E- Iso-profit curves for firm Π > < q 0if q ( a c bq )/b Best response Π Choose output such that marginal profit q is zero. B = q ( a c bq )/b Similarly for player B = q ( a c bq )/b

3 Lec5-6.doc q c )/b E c )/b c )/ b q Figure E- Best responses For equilibrium, q = c bq)/b= q Similarly for player q = c bq)/b= q Rearranging, and bq+ bq = a c q + bq = a c 4bq + bq = ( a c ) q + bq = a c Subtracting 3bq = c) c) Arguing symmetrically 3 bq = c) c) Dividing by 3 and summing bq ( + q) = c) c) 3 3 Then the equilibrium price p = a b( q + q ) = a+ c + c Class Exercise With 3 firms, and 3 c c c < < under what conditions will there be (i) firm (ii) firms producing a positive output? 3

4 Lec5-6.doc identical firms p = a bq Q= q+ z, z = output of all other firms profit of firm, Π = pq cq = ( p c) q = c b( q+ z)) q = c bz ) q bq Π marginal profit, = a c bz bq. q Then firm s best response q = c bz). b Given the symmetry we seek a solution in which each firm chooses the same output. Then q = Q and z = Q But bq = a c bz Substituting, Q= a c Q. Hence Then c) Q = + b and p = a bq= ( ) a+ ( ) c bq = a c ote that as the number of firms rises, the price falls. In the limit as the number of firms gets large, the equilibrium price approaches the unit cost c. 4

5 Lec5-6.doc Sequential rather than simultaneous play The oligopoly game is a natural one to use to introduce games in which moves are sequential, rather than simultaneous. (The classic childhood sequential game is 0 s and X s or tic-tac-toe) If firms are very similar in size, the simultaneous model is more likely to be appropriate. However, when one firm is larger, or the incumbent, it may be in a position to make the first move, leaving other firms to follow. What should the first mover do? RULE Look ahead and work backwards Once firm has made its move, firm simply gets to maximize profit conditional upon firm s move. That is, it will choose its best response q = c bq )/b (Best Response by firm ) q c )/b E Π ( q, q ) = const q A D A q A ( a c )/ b q Figure F- Leader-Follower equilibrium has the same best response rule as before, firm is not simply responding to firm s choice. Suppose that firm initially chooses the same output as in the simultaneous move game. Then the outcome is the 5

6 Lec5-6.doc point E in the figure. But firm knows that firm will choose a response along the heavy response curve. Thus firm optimizes by picking the most profitable point on this curve. As depicted it is the point A. Example a= 00, b=, c = c = 0. Simultaneous moves q = c bq )/ b= (90 q )/ q = c bq )/ b= (90 q )/ Solving, q = q = 30 Sequential moves q = c bq )/ b= (90 q )/ p = a b( q + q ) = 00 ( q + (90 q ) = 55 q Π = pq cq = ( p c) q = (45 q ) q. Differentiating, Π = 45 q. Thus q q = 45 and q = (90 q ) / =.5 6

7 Lec5-6.doc Entry Game Incumbent monopolist and a potential entrant. Incumbents payoffs indicated first. Entrant Stay out (6,0) Come In Incumbent Fight Share (,-) (,) ash equilibrium Incumbent announces that he will fight if there is entry. Entrant s is to choose OUT. Given that Entrant stays OUT, the strategy FIGHT is a for the Incumbent. ash equilibrium Entrant believes Incumbent will share. Then is COME I. Given that Entrant does come in, Incumbent s is SHARE. Which equilibrium is more plausible? If you look down the tree and ask what each player would actually do at a decision node, one equilibrium is eliminated 7