Question

Consider two firms facing the demand curve P=90-SQ. 17 where Q Q1+02. The firms' cost functions are С, (A) 10 +sal 13 12 and C2(92)#5+10Q2. Suppose that both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? Combined, the firms will produceunits of output, of which Firm 1 will produce units and Firm 2 will produce units. (Enter a numeric response using a real number rounded to two decimal placos.) 123456710 1 12 13 14 15 18 17 1 Firm 2's output What is each firm's equilibrium output and profit if they behave noncooperatively? Use the Cournot model f the frms compete, then Filrm 1 will produce units of output and Firm 2 will produce units of output

Answer 1

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.