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An industry consists of three firms with identical costs C(q)18q +q2. Market demand is Q = 150 – p. I. What is the industry equilibrium (price, output and profits) if the firms have Cournot beliefs?
II. Would it pay for Firm 1 and Firm 2 to merge, if, after the merger,
the remaining firm plays Cournot? (Hint: carefully consider if the
merged firm would produce using both original firms’ plants and just
those of one firm.) III. What happens if their costs are C(q) =18q instead?
Solution: I)For firm 1: Profit= PQ-cost =q1[150-(q1+q2+q3)] (18q1+q1^2) =150 q1 q1^2 q1 q2 q1q3 -18q1 -q1^2 Partially differentiating both sides with respect to q1: Profit/ q1 =150 2q1 q2 q3 -18 -2 q1 The FOC for firm 1 gives us: Profit/ q1=0 q1 = 1/4[132- q2- q3]..(1) For firm 2: Profit= PQ-cost =q2[150-(q1+q2+q3)] (18q2+q2^2) =150 q2 q1q2 q2^2 q2q3 -18q2 -q2^2 Partially differentiating both sides with respect to q2: Profit/ q2 =150 q1 2q2 q3 -18 -2 q2 The FOC for firm 2 gives us: Profit/ q1=0 q2 = 1/4[132- q1- q3] ..(2) Similarly,( By symmetry,): q3 =1/4[132- q1- q2] ..(3) Adding 1, 2 and 3 we get: q 1+q2+q3= [3*132 -2q1- 2q2 2q3] or, 4(q 1+q2+q3)= 396 -2 (q 1+q2+q3) or, q1+q2+q3= 396/6 =66 or, q1= 66-q2-q3 Substituting the above in equation 2: q2=[132 (66-q2- q3) q3] or, 4q2 =66+q2 or, 3q2 =66/3 =22 By symmetry, q1=q2=q3 =22 Therefore P= 150- 66 = 84 Each firms Profit= PQ-cost =22[150-66] (18*22+22^2) =22*84 (396+484) =1848-880 =968 II)If firm 1 and 2 merge, in effect there would now be two firms in the industry. Proceeding as before, For firm 1( comprising of firm 1 and 2): Profit= PQ-cost =q1[150-(q1+q2)] (18q1+q1^2) =150 q1 q1^2 q1 q2 -18q1 -q1^2 Partially differentiating both…

es with respect to q1: Profit/ q1 =150 2q1 q2 -18 -2 q1 The FOC for firm 1 gives us: Profit/ q1=0 q1 = 1/4[132- q2]..(1) For firm 2 (initially firm 3): By symmetry, q2 = 1/4[132- q1]..(2) Solving the above equations, we get, q1= q2 = 132/5 =26.4 and industry output = 26.4*2 = 52.8 Thus we find that the industry output has decreased from 66 to 52.8. Also before merger, firm 1 and firm 2 together produced 22+22 =44 units and after merger they together produce 26.4 units. However, after merger, Firm 3 (after merger firm 2) produces higher amount i.e its production increased from 22 to 26.4. Thus it would NOT pay firm 1 and firm 2 to merge since they would produce lower output and thus earn lower profits. III) If now the cost is C(q) =18q for the three firms: For firm 1: Profit= PQ-cost =q1[150-(q1+q2+q3)] 18q1 =150 q1 q1^2 q1 q2 q1q3 -18q1 Partially differentiating both sides with respect to q1: Profit/ q1 =150 2q1 q2 q3 -18 The FOC for firm 1 gives us: Profit/ q1=0 q1 = 1/3[132- q2- q3]..(1) For firm 2: Profit= PQ-cost =q2[150-(q1+q2+q3)] (18q2) =150 q2 q1q2 q2^2 q2q3 -18q2 Partially differentiating both sides with respect to q2: Profit/ q2 =150 q1 2q2 q3 -18 The FOC for firm 2 gives us: Profit/ q1=0 q2 = 1/3[132- q1- q3] ..(2) Similarly,( By symmetry,): q3 =1/3[132- q1- q2] ..(3) Adding 1, 2 and 3 we get: q 1+q2+q3= 1/3[3*132 -2q1- 2q2 2q3] or, 3(q 1+q2+q3)= 396 -2 (q 1+q2+q3) or, q1+q2+q3= 396/4 =99 each firm produces =33 units Here, price P= 150-99 =51 And Profit for each firm = 33[150-99] (18*33) = 1089 For the case where the firms merge, the situation would be as follows: For firm 1( comprising of firm 1 and 2): Profit= PQ-cost =q1[150-(q1+q2)] (18q1) =150 q1 q1^2 q1 q2 -18q1 Partially differentiating both sides with respect to q1: Profit/ q1 =150 2q1 q2 -18 The FOC for firm 1 gives us: Profit/ q1=0 q1 = 1/3[132- q2]..(1) For firm 2 (initially firm 3): By symmetry, q2 = 1/3[132- q1]..(2) Solving the above equations, we get, q1= q2 = 132/4 =33 and industry output = 33*2 = 66 Thus, clearly when cost becomes c(q)=18q, the industry as well the firm output increases for both cases i.e when there are 3 firms and also when firm 1 and 2 merge. But when 2 firms merge the industry output falls.

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