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Given the inverse demand function above with zero costs (fixed and variable), where n is the number of firms. Assume firms engage in Cournot competition. Find the equilibrium price and profits per firm if there are two firms. Repeat for the case of three firms.
In the case of cournot equilibrium, it is a case where there are homogenous products and different levels of output.
In the case of two firms:
f1y2 (reaction function for firm 2) .
Cournot Nash equilibrium
f2y1 (reaction curve for firm 1) Y1
Firm 1’s profit maximization problem is max y1π =p (y1+y2) y1-0 (Ramusen, 100)
The profits of firm 1 depend on the economic decisions made by firm 2 thus in order to make an informed decision then firm 1 must forecast firm2’s output decision. In order to obtain the equilibrium output (y1*, y2*) in which each firm is maximizing its outputs we must obtain the first order equations (Scarf, 93).
Let qi= y1+ y2.
Firm 1- leader; firm 2 –follower
The above equation becomes p=1- (y2+y2)
Profits= [1-(y1+y2)] y2-0
Therefore π= y2- y1 y2-y22
First order condition y2= (∂π/∂y2) = 1-y1-2y2
y2 =0.5-0.5y1; this is the reaction function for firm two.
Y2= f (y1)
Therefore: π 1 = [1-(y1+y2)] y1
Π= y1-y21-y1y2 but from the above equation we already obtained the value of y2, thus by substitution the equation becomes y1-y12 –y1 (0.5-0.5y1)
Π =y1-y1 2 -0.5y1+0.5y2
∂π /∂y1= 1-2y1-0.5
y1= 0.25 and y2=0.5-0.5y1 by direct substitution y2=0.375
Total industrial output = 0.25+0.375=0.625units.
In the case of three firms:
Π = [1-(y1+y2)] (y1+y2)]-0
= y1+ y2- y1 2 +y22-y1y2
∂π/∂y2 =1+2y2-y1 y2=. 0.5y1-0.5
Substituting directly we get:
y1+ (0.5y1-0.5)-y2+ (0.5y1-0.5)2-y1 (0.5y1-0.5) = 1.5y1-0.25-1.25y12
The above solution is a quadratic equation thus to solve quadratically we use the quadratic equation, which is given as: x=-b±b2-4ac2a
Therefore y= -1.5-√ (-1.5)2 +4(-1.25) (-0.25) = 1.348
The quadratic solution gives y1 two possible values but one is negative and since y1 represents output which cannot be negative so we take the positive solution of y1whichis given as 1.348.
Given y1 as 1.348 then y2 =0.174
Thus price will be given as 1-(0.174+1.348) = -0.522 but since it is an inverse of the demand curve then we take that absolute values of ‘p’ which is 0.522.