Question

Consider the following industry where the inverse market demand is given by the function: p=180-Y where Y is the total market output. There are two firms in the market, each has a total cost function: ci (yi)=3(yi)2 where i=1,2 is the label of the firm. Suppose the firms act as Cournot duopolists. What output level will each firm produce in order to maximize profits?.

Answer 1

P = 180 - Y

Y = y₁ + y₂

C₁(y₁) = 3y²₁

C₂(y₂) = 3y²₂  

₁ = Py₁ - C₁(y₁)

= (180 - Y)y₁ -   3y²₁  

= 180y₁ - y₁(y₁ + y₂ ) -    3y²₁  

= 180y₁ - y²₁ - y₁y₂ -   3y²₁  

= 180y₁ - 4y²₁ - y₁y₂

₁/y₁ = 180 - 8y₁ - y₂

put ₁/y₁ = 0

180 - 8y₁ - y₂ = 0

180 - y₂ = 8y₁  

y₁   = 180/8 - y₂/8 (i)   

₂ = Py₂ - C₂(y₂)

= (180 - Y)y₁ -   3y²₂

= 180y₂ - y₂(y₁ + y₂ ) -    3y²₂

= 180y₂  - y²₂ - y₁y₂ -   3y²₂

= 180y₂- 4y²₂- y₁y₂  

₂/y₂ = 180 - y₁ - 8y₂

put ₂/y₂ = 0  

180 - y₁ - 8y₂ = 0  

180 - y₁ = 8y2

y₂ = 180/8 - y₁/8 (ii)  

from (i) and (ii)  

y₂ = 180/8 - y₁/8

= 180/8 - 1/8(180/8 - y₂/8)  

y₂ = 180/8 - 180/64 + y₂/64  

y₂ -   y₂/64 = 180/8( 1 - 1/8)

y₂( 1 - 1/64) = (180/8)(7/8)  

y₂ (63/64) =  1260/64

y₂ = 20

y₂(63/64) =  (180/8)(63/64)  

y₂ = 180/8

y₂ = 22.5

y₁ = 180/8 -  y₂/8  

= 180/8 - 20/8  

= (180 - 20)/8

= 160/8

= 20