复旦大学：《产业经济学 Industrial Economics》教学课件_Industrial Organization 3 noncooperative oligopoly

Lecture 3 Models of non-cooperative oligopoly with homogenous products Cooperative oligopoly same as cartels Main assumption in the models discussed here: products are homogeneous: World without brands e Consumers as before assumed to be price takers e Two kinds of market structure o Competition in quantities o Competition in prices Competition in quantities o Cournot model(simultaneous move game)-solve for Nash equilibrium o Stackelberg model(sequential move game)-solve for subgame perfect Nash equilibrium Competition in prices o Bertrand model Model of cournot duopoly e Two firms

Lecture 3 Models of non-cooperative oligopoly with homogenous products • Cooperative oligopoly same as cartels. • Main assumption in the models discussed here: products are homogeneous: World without brands. • Consumers as before assumed to be price takers. • Two kinds of market structure: oCompetition in quantities oCompetition in prices • Competition in quantities: oCournot model (simultaneous move game)- solve for Nash equilibrium o Stackelberg model (sequential move game)- solve for subgame perfect Nash equilibrium • Competition in prices oBertrand model Model of Cournot Duopoly • Two firms

●7(q1)=c;q Inverse demand equation: p(Q=a-bQ ·Q= total output=q+q2 Profit for firm 1 1=P1-c1q1 o Through the demand equation quantity produced by firm 2 affects the profit of firm 1: strategic interaction alI a-bg-c BR=-=0→q 26 alI BR =0→q bq 26 Cournot Nash equlibrium o Mutual best response: Plug equation 2 into equation I or other way round. Cournot Nash equlibrium: 2c;+ 2C+ q1 36 36 Reaction functions plot here

• iii)( = qcqTC • Inverse demand equation: p(Q)=a-bQ • Q = total output = 21 + qq Profit for firm 1 −=Π qcpq 1111 • Through the demand equation quantity produced by firm 2 affects the profit of firm 1: strategic interaction. • )1( 2 0 12 1 1 1 1 − − − =⇒= ∂ Π∂ = b cbqa q q BR • )2( 2 0 21 2 2 2 2 − − − =⇒= ∂ Π∂ = b cbqa q q BR • Cournot Nash equlibrium: oMutual best response: Plug equation 2 into equation 1 or other way round. Cournot Nash equlibrium: • b cca q b cca q 3 2 , 3 2 * 12 1 * 21 1 − + = +− = • Reaction functions plot here

Graphical lllustration of asymmetric Cournot Nash equilibrium in the q1 a-c2 Reaction function of firm2 2b eaction function of firm 1 Cournot N ash eam a-C2/2b a-c1b

1 Graphical Illustration of asymmetric Cournot Nash equilibrium in the q1 q2

Homogeneous firms Cournot duopoly: CI=C2 Output produced by each firm in a symmetric Cournot duopoly d-c equilibrium q 36 e Total output produced in a-c equilibrium g 36 Equilibrium price p a-c ab+26c a+2c a-62 36 36 o Knowing the price in the market and quantity produced by each of the firms: can calculate the profit of each firm d-c 丌1 2 96

2 Homogeneous firms Cournot duopoly: 21 = cc • Output produced by each firm in a symmetric Cournot duopoly equilibrium * q = b ca 3 − • Total output produced in equilibrium * Q = b ca 3 2 − • Equilibrium price * p = 3 2 3 2 3 2 ca b bcab b ca ba + = + = − − • Knowing the price in the market and quantity produced by each of the firms: can calculate the profit of each firm • b ca 9 )( 2 * 2 * 1 − ππ ==

Homogeneous firms Cournot duopoly: Continued o Knowing the price: we can calculate the consumer surplus (CS) Welfare under a homogeneous firms cournot du uop (a-c =CS+兀1+兀2=CS+2 96 o Example: Demand curve Q=1000-1000p c=28cents

3 Homogeneous firms Cournot duopoly: Continued • Knowing the price: we can calculate the consumer surplus (CS) • Welfare under a homogeneous firms Cournot duopoly b ca CS CS 9 )( 2 2 * 2 * 1 − ππ +=++= • Example: Demand curve Q = −10001000 p = 28centsc

Homogeneous firm Cournot oligopoly .identical firms: (per unit cost of each firm constant =c o Pick up any firm say firm 1 and look at the profit maximization 丌1=pq1-cq1=[a-b(q1+q2+…+qx)q1-cq1 e Firm's problem is to maximize profits by choosing q1 First order condition Best response function an=0=a-2-24-=0 ● Same costs→ symmetric equilibrium: q1=q2 qN=q (a-c) (N+1)b

4 N homogeneous firm Cournot oligopoly • N identical firms: (per unit cost of each firm constant = c • Pick up any firm say firm 1 and look at the profit maximization • 111 21 11 ([ qqbacqpq )]........ cqqq π −= = − + + + N − • Firm’s problem is to maximize profits by choosing 1 q • First order condition: Best response function: 20 0 2 1 1 1 =−−−⇒= ∂ ∂ ∑= cqbbqa q N i i π • Same costs ⇒symmetric equilibrium: * ** 2 * 1 qq ......... qqN ==== • bN ca q )1( )( * + − =

n firm Cournot oligopoly: Continued (a-c) Q=N (N+1)b (a+ Nc P (N+1) AsN→∞:p→p,Q→>Q p, price under perfect competition 2, quantity produced under perfect competition Meaning of large number of firms e Welfare under n firm cournot oligopoly: W=CS+ Nz(a-c)N+2N 2bN2+2N+1 o For very large number of firms no deadweight loss

5 N firm Cournot oligopoly: Continued • bN ca NQ )1( )( * + − = • )1( )( * + + = N Nca p • As c c →→∞→ QQppN * * ,: • , c p price under perfect competition, , c Q quantity produced under perfect competition • Meaning of large number of firms. • Welfare under N firm Cournot oligopoly: ) 12 2 ( 2 )( 2 22 * + + − + =+= NN NN b ca NCSW π • For very large number of firms no deadweight loss

Sequential move game with quantity competition: Stackelberg model: Leader-follower structure ● Case of duopoly o Other things same as the cournot duopoly model- Difference Leader follower structure: firm 1 Leader firm 2-Follower o Solve for the subgame perfect Nash equilibrium: (recall terrorist pilot game) o Solve the game backwards: start from the followers problem o Follower's profit maximization problem gives the best response of firm 2 2 6 ● Leader’ s problen: incorporate the best response of the follower in its problem

6 Sequential move game with quantity competition: Stackelberg model: Leader-follower structure. • Case of duopoly • Other things same as the Cournot duopoly model- Difference: Leader follower structure: Firm 1- Leader, Firm 2- Follower • Solve for the subgame perfect Nash equilibrium: (recall terrorist pilot game) • Solve the game backwards: start from the follower’s problem • Follower’s profit maximization problem gives the best response of firm 2: • 22 1 2 q b ca q − − = • Leader’s problem: incorporate the best response of the follower in its problem

Stackelberg model continued d-c max,=pq -cq=a-b(g,+ 2b2 )qu-cqr o Quantity produced by the leader d-c 26 o Quantity produced by the d-c follower: 92 46 Total output higher under sequential move game Intuition: Price expected(by one of the firms)to fall faster under Cournot than Stackelberg g

7 Stackelberg model continued: • 11 1 111 1 ) 22 max ( cqq q b ca qbacqpq −− − π +−=−= • Quantity produced by the leader: b ca qS 2 1 − = • Quantity produced by the follower: b ca qS 4 2 − = • Total output higher under sequential move game: • Intuition: Price expected (by one of the firms) to fall faster under Cournot than Stackelberg

Changing the nature of competition: Price competition Bertrand duopoly model Main result: Bertrand paradox- Even two firms competing in prices and acting non- cooperatively result in an outcome price= marginal cost o Bertrand nash equilibrium: PI=p2 =c o Important assumptions o No capacity constraints o Homogeneous products o One shot interaction o No information problems Resolution of Bertrand paradox o Capacity constraints o Product differentiation o Repeated interaction o Incomplete information

8 Changing the nature of competition: Price competition￾Bertrand duopoly model • Main result: Bertrand paradox￾Even two firms competing in prices and acting non￾cooperatively result in an outcome price = marginal cost • Bertrand Nash equilibrium: == cpp * 2 * 1 • Important assumptions: o No capacity constraints o Homogeneous products o One shot interaction o No information problems • Resolution of Bertrand paradox o Capacity constraints o Product differentiation o Repeated interaction o Incomplete information