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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 9 months ago
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