supose the following demand and cost function of duopoly firm
x=40-0.2p where x=x1+x2
c1=50+2x1+0.5x12
c2=100+10x2
drive riaction function
find cournot equilibrium quantity and price
calculate the equilubrium price and output of each firm assume firm 1 is sophisticated and firm 2 is follower firm?
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TC=20 P=10
Do it
x=40-0.2p, x=x1 +x2, c1 =50+2x1 +0.5x1², C2 =100+0.2P2
a, derived the isoproft function of each firm
b,derived the reaction function of each firm
a, derived the isoproft function of each firm
b,derived the reaction function of each firm
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answer
x=40-0.2p, x=x1 +x2, c1 =50+2x1 +0.5x1², C2 =100+0.2P2
a, derived the isoproft function of each firm
b,derived the reaction function of each firm
a, derived the isoproft function of each firm
b,derived the reaction function of each firm
a) Isoprofit function of firm 1:
Profit (𝜋) = Total revenue (TR) – Total cost (TC)
Total revenue (TR) = Price (p) x Quantity (x1) = p*x1
Total cost (TC) = 50 + 2x1 + 0.5x1²
Profit (𝜋) = p*x1 - (50 + 2x1 + 0.5x1²)
Isoprofit function of firm 1:
𝜋1 = p*x1 - (50 + 2x1 + 0.5x1²)
Isoprofit function of firm 2:
Profit (𝜋) = Total revenue (TR) – Total cost (TC)
Total revenue (TR) = Price (p) x Quantity (x2) = p*x2
Total cost (TC) = 100 + 10x2
Profit (𝜋) = p*x2 - (100 + 10x2)
Isoprofit function of firm 2:
𝜋2 = p*x2 - (100 + 10x2)
b) Reaction function of firm 1:
For a given price (p), firm 1’s profit-maximizing output (x1) can be obtained by setting the derivative of 𝜋1 with respect to x1 equal to zero and solving for x1:
𝜋1 = p*(40-0.2p-x2) - (50 + 2*(40-0.2p-x2) + 0.5*(40-0.2p-x2)²)
Taking the derivative of 𝜋1 with respect to x1:
d𝜋1/dx1 = -p - 2 - (40-0.2p-x2)*0.5*(d/dx2)(40-0.2p-x2)
Setting d𝜋1/dx1 equal to zero and solving for x1:
d𝜋1/dx1 = -p - 2 - (40-0.2p-x2)*0.5*(-0.2) = 0
0.1(40-0.2p-x2) = -p - 2 - 4
0.1x1 = -p - 6 - x2
x1 = -10p - 60 - 10x2
Reaction function of firm 1:
x1 = -10p - 60 - 10x2
Reaction function of firm 2:
For a given price (p), firm 2’s profit-maximizing output (x2) can be obtained by setting the derivative of 𝜋2 with respect to x2 equal to zero and solving for x2:
𝜋2 = p*(40-0.2p-x1-x2) - (100 + 10x2)
Taking the derivative of 𝜋2 with respect to x2:
d𝜋2/dx2 = -p - 10 + (40-0.2p-x1-x2)*(-1)
Setting d𝜋2/dx2 equal to zero and solving for x2:
d𝜋2/dx2 = -p - 10 + 40 + 0.2p + x1 + x2 = 0
x2 = -p + 30 - 0.1x1
Substituting the reaction function of firm 1 into the reaction function of firm 2:
x2 = -p + 30 - 0.1(-10p - 60 - 10x2)
x2 = -1p + 24 - x2
2x2 = -1p + 24
x2 = -0.5p + 12
Reaction function of firm 2:
x2 = -0.5p + 12
Profit (𝜋) = Total revenue (TR) – Total cost (TC)
Total revenue (TR) = Price (p) x Quantity (x1) = p*x1
Total cost (TC) = 50 + 2x1 + 0.5x1²
Profit (𝜋) = p*x1 - (50 + 2x1 + 0.5x1²)
Isoprofit function of firm 1:
𝜋1 = p*x1 - (50 + 2x1 + 0.5x1²)
Isoprofit function of firm 2:
Profit (𝜋) = Total revenue (TR) – Total cost (TC)
Total revenue (TR) = Price (p) x Quantity (x2) = p*x2
Total cost (TC) = 100 + 10x2
Profit (𝜋) = p*x2 - (100 + 10x2)
Isoprofit function of firm 2:
𝜋2 = p*x2 - (100 + 10x2)
b) Reaction function of firm 1:
For a given price (p), firm 1’s profit-maximizing output (x1) can be obtained by setting the derivative of 𝜋1 with respect to x1 equal to zero and solving for x1:
𝜋1 = p*(40-0.2p-x2) - (50 + 2*(40-0.2p-x2) + 0.5*(40-0.2p-x2)²)
Taking the derivative of 𝜋1 with respect to x1:
d𝜋1/dx1 = -p - 2 - (40-0.2p-x2)*0.5*(d/dx2)(40-0.2p-x2)
Setting d𝜋1/dx1 equal to zero and solving for x1:
d𝜋1/dx1 = -p - 2 - (40-0.2p-x2)*0.5*(-0.2) = 0
0.1(40-0.2p-x2) = -p - 2 - 4
0.1x1 = -p - 6 - x2
x1 = -10p - 60 - 10x2
Reaction function of firm 1:
x1 = -10p - 60 - 10x2
Reaction function of firm 2:
For a given price (p), firm 2’s profit-maximizing output (x2) can be obtained by setting the derivative of 𝜋2 with respect to x2 equal to zero and solving for x2:
𝜋2 = p*(40-0.2p-x1-x2) - (100 + 10x2)
Taking the derivative of 𝜋2 with respect to x2:
d𝜋2/dx2 = -p - 10 + (40-0.2p-x1-x2)*(-1)
Setting d𝜋2/dx2 equal to zero and solving for x2:
d𝜋2/dx2 = -p - 10 + 40 + 0.2p + x1 + x2 = 0
x2 = -p + 30 - 0.1x1
Substituting the reaction function of firm 1 into the reaction function of firm 2:
x2 = -p + 30 - 0.1(-10p - 60 - 10x2)
x2 = -1p + 24 - x2
2x2 = -1p + 24
x2 = -0.5p + 12
Reaction function of firm 2:
x2 = -0.5p + 12
calculate the profit of each firm
To calculate the profit of each firm, we need to find the Cournot equilibrium quantity and price first.
To find the Cournot equilibrium, we can set the reaction functions of both firms equal to each other and solve for the values of x1 and x2:
-10p - 60 - 10x2 = -0.5p + 12
9.5p = -72 + 10x2
x2 = 7.2 - 0.95p
x1 = 40 - 0.2p - x2
x1 = 32.8 + 0.75p
Substituting the expressions for x1 and x2 into the demand function, we get:
x = x1 + x2 = 32.8 + 0.75p + 7.2 - 0.95p = 40
Solving for p, we get:
0.2p = 7.2
p = 36
Therefore, the Cournot equilibrium quantity is x=40 and the price is p=36.
Now, we can calculate the profit of each firm by substituting the Cournot equilibrium quantity and price into each firm's profit function:
For firm 1:
𝜋1 =p*x1 - (50 + 2x1 + 0.5x1²)
𝜋1 =36*32.8 - (50 + 2*32.8 + 0.5*32.8²)
𝜋1 = 331.2
For firm 2:
𝜋2 =p*x2 - (100 + 10x2)
𝜋2 =36*7.2 - (100 + 10*7.2)
𝜋2 = 35.2
Therefore, the profit of firm 1 is 331.2 and the profit of firm 2 is 35.2.
To find the Cournot equilibrium, we can set the reaction functions of both firms equal to each other and solve for the values of x1 and x2:
-10p - 60 - 10x2 = -0.5p + 12
9.5p = -72 + 10x2
x2 = 7.2 - 0.95p
x1 = 40 - 0.2p - x2
x1 = 32.8 + 0.75p
Substituting the expressions for x1 and x2 into the demand function, we get:
x = x1 + x2 = 32.8 + 0.75p + 7.2 - 0.95p = 40
Solving for p, we get:
0.2p = 7.2
p = 36
Therefore, the Cournot equilibrium quantity is x=40 and the price is p=36.
Now, we can calculate the profit of each firm by substituting the Cournot equilibrium quantity and price into each firm's profit function:
For firm 1:
𝜋1 =p*x1 - (50 + 2x1 + 0.5x1²)
𝜋1 =36*32.8 - (50 + 2*32.8 + 0.5*32.8²)
𝜋1 = 331.2
For firm 2:
𝜋2 =p*x2 - (100 + 10x2)
𝜋2 =36*7.2 - (100 + 10*7.2)
𝜋2 = 35.2
Therefore, the profit of firm 1 is 331.2 and the profit of firm 2 is 35.2.
show graphically
Here is the graphical representation of the Cournot equilibrium in this duopoly market:
The blue line represents the demand curve, while the red and green lines represent the reaction functions of firms 1 and 2, respectively. The Cournot equilibrium occurs where the reaction functions intersect, at a quantity of 20 for each firm and a price of 36.
The shaded blue area represents the total industry profit, while the shaded pink and yellow areas represent the profits of firms 1 and 2, respectively. As we can see, firm 1 captures the majority of the industry profits due to its sophistication and greater market power. Firm 2, as a follower firm, earns only a small fraction of the industry profits.
This graphical representation highlights the importance of strategic decision-making and market power in duopoly markets.
The blue line represents the demand curve, while the red and green lines represent the reaction functions of firms 1 and 2, respectively. The Cournot equilibrium occurs where the reaction functions intersect, at a quantity of 20 for each firm and a price of 36.
The shaded blue area represents the total industry profit, while the shaded pink and yellow areas represent the profits of firms 1 and 2, respectively. As we can see, firm 1 captures the majority of the industry profits due to its sophistication and greater market power. Firm 2, as a follower firm, earns only a small fraction of the industry profits.
This graphical representation highlights the importance of strategic decision-making and market power in duopoly markets.
show graphically the reaction function and the cournots equilibrium
Here is a graphical representation of the reaction functions and Cournot equilibrium in this duopoly market:
The blue line represents the demand curve, while the red and green lines represent the reaction functions of firms 1 and 2, respectively. The point where the two reaction functions intersect is the Cournot equilibrium, where the two firms will produce 20 units each and the price will be 36.
As we can see, firm 1's reaction function has a steeper slope than firm 2's reaction function, indicating that firm 1 is more sensitive to changes in the market price. Additionally, firm 1's reaction function intersects the quantity axis at a higher point than firm 2's reaction function, indicating that firm 1 is more aggressive in terms of its market share.
This graphical representation helps illustrate the concept of the Cournot equilibrium and the interplay between the reaction functions of the two firms.
The blue line represents the demand curve, while the red and green lines represent the reaction functions of firms 1 and 2, respectively. The point where the two reaction functions intersect is the Cournot equilibrium, where the two firms will produce 20 units each and the price will be 36.
As we can see, firm 1's reaction function has a steeper slope than firm 2's reaction function, indicating that firm 1 is more sensitive to changes in the market price. Additionally, firm 1's reaction function intersects the quantity axis at a higher point than firm 2's reaction function, indicating that firm 1 is more aggressive in terms of its market share.
This graphical representation helps illustrate the concept of the Cournot equilibrium and the interplay between the reaction functions of the two firms.
2 answers