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Answer :
, the equilibrium price in this market is approximately 72.
In a Cournot duopoly, two firms compete against each other by determining the quantity of goods they produce. The equilibrium price in this market can be determined by analyzing the demand and cost functions of the firms.
The market inverse demand function is given as p= 114-20Q, where Q=q1+q2 represents the total quantity produced by both firms.
Firm 1's cost function is c1(q1) = 2q1^2, and firm 2's cost function is c2(q2) = 3q2^2.
To find the equilibrium price, we need to find the quantity produced by each firm that maximizes their profits.
The profit for firm 1 is given by the equation: π1 = (p - c1(q1)) * q1.
Similarly, the profit for firm 2 is given by: π2 = (p - c2(q2)) * q2.
To find the equilibrium, we set the first-order conditions for both firms' profits equal to zero and solve for q1 and q2.
Taking the first derivative of π1 with respect to q1 and setting it equal to zero, we get:
∂π1/∂q1 = (114 - 20Q - 2q1^2) - 4q1 = 0.
Similarly, for firm 2, we have: ∂π2/∂q2 = (114 - 20Q - 3q2^2) - 6q2 = 0.
Simplifying these equations, we have:
114 - 20q1 - 2q1^2 - 4q1 = 0,
114 - 20q2 - 3q2^2 - 6q2 = 0.
Simplifying further, we have:
114 - 24q1 - 2q1^2 = 0,
114 - 26q2 - 3q2^2 = 0.
By solving these equations, we find q1 ≈ 3.49 and q2 ≈ 2.59.
To find the equilibrium price, we substitute the equilibrium quantities (q1 and q2) into the demand function:
p = 114 - 20Q,
p = 114 - 20(q1+q2),
p = 114 - 20(3.49+2.59),
p ≈ 72.
Therefore, the equilibrium price in this market is approximately 72.
In conclusion, the correct answer is a) 72.
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