Cournot and Stackleberg PDF

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Kent Dolezal...

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Cournot and Stackleberg. Firms in oligopoly. Few firms, interdependent. Cournot is a simultaneous decision while Stackleberg is a leader-follower situation Firms in oligopoly have three general strategies they can follow in regard to the other firms or its rivals. 1. Ignore – Doesn’t take the other firm’s actions into consideration at all. 2. Collusion – Agrees to fully cooperate with the other firms, forming a monopoly to supply the market 3. Compete – The firms can compete either through quantity or price.

Collusion – Relatively easy to solve. Assume the when the firms agree to fully cooperate, they share technology giving them all the same cost function.

P=100−Q whereQ =q1 +q 2 cos t 1=cos t 2=10 q

Since the firms are now a monopoly, we simply solve the monopoly problem.

TR= PQ = ( 100−Q ) Q=100 Q −Q 2

MR=

dTR =100−2 Q dQ

Profit is maximized where MR=MC ¿

¿

100−2 Q=10 Q=45 q1=q2 =22.5 π 1=π 2=22.5 ( 55 )−10 ( 22.5 ) =$ 1012.50

¿

P =100−45=$ 55

Competition: There are two “classic” models we can study. In the first: Cournot oligopoly (Core no) In this model, firms take the market price as given and compete on the quantity supplied to the market. The second: Bertrand. In this model, firms compete on price and let the market determine the quantity.

Let’s start with Cournot where the firms make their choices at the same time. The market inverse demand curve is Firm 1’s costs are

P=a−bQ , where Q=q 1+q2 .

c 1 q 1 and firm 2’s costs are

c 2 q2 .

The question we are trying to answer is how much each firm will produce. Example:

P=100−Q whereQ =q1 +q 2 cos t 1 :10 q 1 cos t 2 :20 q 2 Firm 1 takes Firm 2’s output as a given (it has no control over what the other firm produces). Firm 1’s profits:

π 1=P q1−c1 =( 100−q1−q 2 ) q1−10 q 1

2

¿ 90 q1 −q1 −q1 q2

Maximization problem for firm 1:

∂ π1 =90−2 q1−q2=0 ∂ q1

q1 =

2 q1=90−q 2

90−q 2 =R 1 2

Firm 2’s profits:

π 1=P q1−c1 =( 100−q1−q 2 ) q2−20 q 1 Maximization problem for firm 1:

∂ π1 =80−2 q2−q 1=0 ∂ q2

q 2=

80−q 1 =R 2 2

2 q2=80−q 1

2

¿ 80 q2 −q2−q1 q2

Finding the

90− q1 =

¿

¿

q1∧q2 , we substitute equation for q2 into the equation for q1 and solve. 80−q1 2 2

2 q1=90−

80−q1 2

q1 =45 −20+

q1 4

3 q =25 4 1

¿

q1=

Putting our value for q1 into the equation for q2 and solving.

100 100 70 3 = =40− 3 6 2

80− q¿2=

Equilibrium price ¿

P =100−Q=100−

100 70 130 ≈ $ 43.33 − = 3 3 3

Finding the profit for each firm.

π 1= π 1=

130 100 100 13000 3000 10000 ≈ $ 1111.11 −10 = − = 9 3 3 3 9 9

70 9100 4200 4900 130 70 ≈ $ 544.44 = − −20 = 9 9 9 3 3 3

Summarizing the profit maximizing equilibrium, we have ¿

q1=

100 ¿ 70 ¿ 170 q= Q= 3 2 3 3

100 3

¿

P =$ 43.33 π 1 =$ 1111.11 π 2 =$ 544.44 Π =$ 1655.55

Stackleberg Firm 1: Leader Firm 2: Follower.

P=100−Q whereQ =q1 +q 2 c 1=10 q1 ∧c 2=20 q2

To complete a Stackleberg game, we need the Cournot reaction function for the following firm. The Cournot game yields

R1 ( q2 ) :q 1=

90−q 2 2

R2 ( q1 ) :q 2=

80−q 1 2

@ the Cournot Nash:

q1¿ =33.3 q¿2=23.3 ¿

P =$ 43.33

π¿1=$ 1122

π¿2=$ 552

Firm 1 is the leader and makes its choice of quantity first. However, Firm 1 cannot discount the reaction from Firm 2. Firm 2 can do no better than using its best reply from the Cournot game. Firm 1 will take Firm 2’s response as given. Firm 1’s problem is then

π 1=( 100−q1−q2 ) q1 −10 q1

π 1=P q1−c1

Substituting Firm’s 2 reaction into the profit equation

(

π 1= 100−q 1−

[

])

80−q1 q 1−10 q1 2

1 2 2 π 1=100 q1− q1−40 q1 + q 1 −10 q1 2

1 2 π 1=50 q1− q 2 1 Firm 1’s maximization

∂ π1 =50− q1= 0 ∂ q1 q1¿ =50

¿

q 2=

80−q 1 30 = =15 2 2

¿

P =100−q 1−q 2=$ 35 π¿1=50 ( 35)−10 ( 50 )=$ 1250

π¿2=15 ( 35)−20 ( 15 )=$ 225

Compared with Cournot, Firm 1 is better off while firm 2 is worse off. The cases of collusion, Cournot, and Stackelberg, graphically. (Collusion, Cournot, Stackelberg)

We can see the differences in the quantities by firm during the different situations. Collusion has the lowest output, then Cournot, and Stackelberg has the highest output. Price works in the reverse manner....