Market Structure and Competition

1a. Derive the reaction functions of each firm. Show your working.

P1 = 300 – 6q1 - 6q2                        MC1 = 6

TR= P1Q1 = (300 – 6q1 – 6q2) Q1

               = 300Q1 – 6q12 – 6q2 Q1

MR1= dTR = 300 – 12q1 – 6q2

          dQ1 

MR1 = MC1

300 – 12q1 – 6q2  = 6

300 – 6 – 12q1 – 6q2  = 0

294 – 6q2  = 12q1 

294/12 – 6q2 /12 = 12q1/12

24.5 – 0.5q2  = q1   reaction function for firm 1

FIRM 2

P2 = 300 – 6q1 - 6q2                        MC2 = 6

TR= P2Q2 = (300 – 6q1 – 6q2) Q2

               = 300Q2 – 6q1Q2 – 6q22

MR2= dTR = 300 – 6q1 – 12q2

          dQ2 

MR2 = MC2

300 – 6q1 – 12q2  = 6

300 – 6 – 6q1 – 12q2  = 0

294 – 6q1  = 12q2 

294/12 – 6q1 /12 = 12q2/12

24.5 – 0.5q1  = q2   reaction function for firm 2

1b. Solve for the Cournot equilibrium levels of firms’ outputs, price and firms’ profits. Show your working.

Put equation q2 into q1 to get quantity

q1 = 24.5 – 0.5 (24.5 – 0.5q1 )

q1 = 24.5 – 12.25 - 0.25q1

q1 = 12.25 - 0.25q1

0.75q1 = 12.25

q1 = 16.33

Put equation q1into q2 to get quantity

q2 = 24.5 – 0.5 (24.5 – 0.5q2 )

q2 = 24.5 – 12.25 + 0.25q2

q2 = 12.25 + 0.25q2

0.75q2 = 12.25

q2 = 16.33

To get the price of each firm, substitute quantities of each firm into the demand curve equations for firm 1 or firm 2.

P1 = 300 – 6q1 - 6q2                       

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P1 = 300 – 6 x 16.33 – 6 x 16.33

P1 = 300- 97.98 – 97.98

P1 = 104.04

To get profit, it will be price x quantity so,

1 = P1Q1 = 16.33 X 104.04 = 1698.97

2 = P2Q2 = 16.33 X 104.04 = 1698.97

Total profit is  ∏ = ∏1 + ∏2 = 3397.95  

1c. Draw the reaction functions on a diagram and illustrate the Cournot equilibrium.

Reaction function and quantity setting for ...

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