Question

In: Economics

Consider the following industry where the inverse market demand is given by the function: p=180-Y where...

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?.

Solutions

Expert Solution

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

Rahul Sunny answered 1 month ago

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