# Custom «The Computation of Economic Equilibrium» Essay Paper Sample

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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:

Y_{2}

f_{1}y_{2 (reaction function for firm 2)} .

## Cournot Nash equilibrium

f_{2}y_{1 (reaction curve for firm 1)} Y_{1}

Firm 1’s profit maximization problem is max _{y1}π =p (y_{1}+y_{2}) 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 (y_{1}*, y_{2}*) in which each firm is maximizing its outputs we must obtain the first order equations (Scarf, 93).

Assumptions:

Let q_{i= }y_{1+ }y_{2.}

## Firm 1- leader; firm 2 –follower

The above equation becomes p=1- (y2+y_{2})

Profits= [1-(y_{1}+y_{2})] y_{2}-0

Therefore π= y_{2}- y_{1 }y_{2}-y_{2}^{2}

First order condition y_{2= }(∂π/∂y2) = 1-y_{1}-2y_{2}

y_{2} =0.5-0.5y_{1; }this is the reaction function for firm two.

Y_{2}= f (y1)

Therefore: π _{1 }= [1-(y1+y2)] y1

Π= y_{1}-y^{2}_{1}-y_{1}y_{2} but from the above equation we already obtained the value of y_{2}, thus by substitution the equation becomes y_{1}-y_{1}^{2} –y_{1} (0.5-0.5y_{1})

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Π =y_{1}-y_{1 }^{2 }-0.5y_{1}+0.5y_{2 }

∂π /∂y_{1= }1-2y_{1}-0.5

y_{1= }0.25 and y_{2}=0.5-0.5y_{1 by} direct substitution y_{2}=0.375

Total industrial output = 0.25+0.375=0.625units.

Price= 0.625(P)

P=1-(y1+y2)

0.625[1-(y_{1}+y_{2})] =0.703175.

In the case of three firms:

Π = [1-(y_{1}+y_{2})] (y_{1}+y_{2})]-0

= y_{1+ }y_{2}- y_{1 }^{2 }+y_{2}^{2}-y_{1}y_{2}

∂π/∂y_{2 }=1+2y_{2}-y_{1 }y_{2=. }0.5y_{1-}0.5

Substituting directly we get:

y_{1}+ (0.5y_{1}-0.5)-y_{2}+ (0.5y_{1}-0.5)^{2}-y_{1} (0.5y_{1}-0.5) = 1.5y_{1}-0.25-1.25y_{1}^{2}

The above solution is a quadratic equation thus to solve quadratically we use the quadratic equation, which is given as: *x*=-*b*±*b*2-4*ac*2*a*

Therefore y= -1.5-√ (-1.5)^{2 }+4(-1.25) (-0.25) = 1.348

2(-1.25)

The quadratic solution gives y_{1} two possible values but one is negative and since y_{1} represents output which cannot be negative so we take the positive solution of y_{1}whichis given as 1.348.

Given y_{1} as 1.348 then y_{2 =}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.

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