Custom «The Computation of Economic Equilibrium» Essay Paper Sample

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:

 Y₂

f₁y_{2 (reaction function for firm 2)}                        .

Cournot Nash equilibrium

f₂y_{1 (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).

Assumptions:

Let q_i= y₁₊ y_2.

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 =0.5-0.5y_1; 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=0.5-0.5y_{1 by} direct substitution y₂=0.375

Total industrial output = 0.25+0.375=0.625units.

Price= 0.625(P)

P=1-(y1+y2)

0.625[1-(y₁+y₂)] =0.703175.

In the case of three firms:

Π = [1-(y₁+y₂)] (y₁+y₂)]-0

= y₁₊ y₂- y₁ ² +y₂²-y₁y₂

∂π/∂y₂ =1+2y₂-y₁                             y_2=. 0.5y_1-0.5

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          

2(-1.25)

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 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.