Consider the given problem here the market demand function is given by, “P = 90 - 5*Q”, where “Q = Q1 + Q2”. Now, the cost function of two firms are given by, “C1(Q1) = 10 + 5*Q1” and “C2(Q2) = 5 + 10*Q1”, => the marginal cost functions are given by.
=> MC1 = 5 and MC2 = 10, where “MC1 < MC2”. Now, since marginal cost of “firm1” is less than “firm2”, => under the cooperation only “firm1” will produce and “firm2” will not produce ay output. So, here the “MR” is given by.
=> MR = 90 - 10*Q and “MC1 = 5”, => at the equilibrium “MR = MC1”, => 90 - 10*Q = 5, => Q = 85/10 = 8.5. So, here the profit maximizing total output is given by “Q = 8.5”, where “Q1=8.5” and “Q2=0”.
Combined, the firms will produce “Q=8.5 units” of output of which “firm1” will produce “Q1=8.5 units” and “firm2” will produce “Q2 = 0 units”.
Now, under the Cournot model the profit function of the individual firms are given by.
=> A1 = P*Q1 - C1 = (90 – 5*Q1 – 5*Q2)*Q1 - (10 + 5*Q1) = 90*Q1 – 5*Q1^2 – 5*Q2*Q1 - 10 - 5*Q1.
=> A1 = 85*Q1 – 5*Q1^2 – 5*Q2*Q1 - 10.
Similarly, the profit function of “firm2” is given by.
=> A2 = P*Q2 – C2 = (90 – 5*Q1 – 5*Q2)*Q2 - (5 + 10*Q2) = 90*Q2 – 5*Q1*Q2 – 5*Q2^2 - 5 - 10*Q2.
=> A2 = 80*Q2 – 5*Q1*Q2 – 5*Q2^2 - 5.
So, the FOC require “dA1/dQ1 = dA2/dQ2 = 0”.
=> dA1/dQ1 = 85 - 10*Q1 - 5*Q2 = 0, => 10*Q1 = 85 - 5*Q2, => Q1 = 8.5 - 0.5*Q2, be the reaction function of “firm1”.
Now, “dA2/dQ2 = 0”, => 80 - 5*Q1 - 10*Q2 = 0, => 10*Q2 = 80 - 5*Q1, => Q2 = 8 - 0.5*Q1, be the reaction function of “firm2”.
Now, by simultaneously solving those equations we get the optimum solutions, => “Q1 = 6 units” and “Q2 = 5 units”, => “Q = Q1 + Q2 = 11 units”.
So, if the firms compete, then “firm1” will produce “6 units” of output and “firm2” will produce “5 units” of output.
Consider two firms facing the demand curve P=90-SQ. 17 where Q Q1+02. The firms' cost functions...
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