Please explain all steps Two firms compete as duopolists, producing identical go
ID: 1211510 • Letter: P
Question
Please explain all steps
Two firms compete as duopolists, producing identical goods in quantities qt and q2, respectively. Assume the inverse market demand function for their product is P = 420 - q_1 - q_2. Firm #1 and firm #2 have the same total cost function TC_1 = 0.25(qi)^2 [i.e., MC_1 = MC_2 = 0.5(q_i)]. When viewed as a single "multi-plant" firm, the two firms have the combined profit function 420(q_1) + 420(q_2) - (q_1)(q_2) - 0.75(q_1)^2 - 0.75(q_2)^2 Find the profit maximizing output for each "plant" when they act as a Cartel.Explanation / Answer
Given the profit function, find the profit maximizing level of output for each firm by setting the partial derivatives of the profit function equal to zero:
= 420q1 +420q2 - q1q2 - 0.75(q1)2 - 0.75(q2)2
Partial Derivatives are:
(q1) = 0
420 - q2 - 1.5q1 = 0
q2 + 1.5q1 = 420
(q2) = 0
420 - q1 - 1.5q2= 0
q1 + 1.5q2 = 420
Solving these two equations give:
q1 + 1.5q2 = q2 + 1.5q1
q1 = q2
This suggets that the joint profit is maximum when both firms produce equal level of output. Since this equation is true, substituting them in the partial derivatives equation gives:
q1 + 1.5q2 = 420
q1 + 1.5q1 = 420
2.5q1 = 450
q1 = q2 = 168 units.
This the profit maximizing combination where both firms produce 168 units each