Question

three identical firms Cournot output merge

5. Suppose the market for a good consists of three identical firms, who each have (25 marks total) a. What is the Cournot output of each of the three firms and the market total costs of TG = 2091; 1 = 1,2,3· The market inverse demand is P 260-0 price? What profit does each firm make? (8 marks) Firm 1 and 2 decide to merge into a new firm. Let's refer to this merged firm as firm 4. The new merged firm has the same cost structure, where TCs-20q4 Firms 3 and 4 now compete against each other as Coumot duopolists. b. Calculate the Coumot equilibrium price and output for firm 3 and the merged firm (firm 4). What is the profit for the merged firm? (6 marks) c. If the profit of the merged firm (firm 4) are distributed equally back to the original pre-merged firms 1 and 2, discuss whether or not the merger (3 marks) is profitable to the firms. d. Calculate the profit for the unmerged firm (firm 3) post-merger. Explain why firm 3 benefits from the merger. Is this intuitive? (8 marks)

Answer 1

Solution:

Market inverse demand function: P = 260 - Q

Also, Q = q1 + q2 + q3

Given the total cost function as TCi = 20*qi, i = 1, 2, 3, we find the marginal cost for each firm, MCi = $\partial TCi/\partial qi$ = 20

a) Solving for Cournot equilibrium:

Profit = Total revenue - total cost

Total revenue = Price*Quantity = P*qi

So, first finding the best response curves for each firm:

Profit for firm i, say Zi = P*qi - TCi

Zi = (260 - qi - sum(qj))*qi - 20*qi ; where sum(qj) is the summation of quantity produced by firms other than firm i, so, summation with j not equal to i

Zi = (260 - 20)*qi - qi² - qi*(sum(qj))

Best response function is found by solving for the first order conditions (FOCs) for the three firms: $\partial Zi/\partial qi$ = 0

$\partial Zi/\partial qi$ = 240 - 2*qi - sum(qj)

So, $\partial Zi/\partial qi$ = 0 gives us 240 - 2*qi - sum(qj) = 0

Thus, best response function for firm i is qi* = (240 - sum(qj))/2

Or in similar terms, best response function for firm 1 is q1* = (240 - (q2 + q3))/2

best response function for firm 2 is q2* = (240 - (q1 + q3))/2

best response function for firm 3 is q3* = (240 - (q1 + q2))/2

Solving the above three equations simultaneously gives us the Cournot equilibrium levels of output for the three firms. Looking at the symmetry of equations, on solving them we will get condition q1 = q2 = q3

Then, substituting this in any of the three equations, q1 = (240 - 2*q1)/2

q1 = 60 units

So, q2 = q3 = 60 units

Q = q1 + q2 + q3 = 60 + 60 + 60 = 180

P = 260 - 180 = $80

Profits for each firm are also same ,i.e, Zi = 80*60 - 20*60 = $3,600

Z1 = Z2 = Z3 = $3,600

(b) Since, the cost for each firm is symmetric, and the market demand share for each firm is similar here, there exists a shortcut for the equilibrium quantity for each firm (you may also verify this for part (a), as the conclusion or final result of the shortcut solution is driven from the longer method itself):

For Inverse demand function: P = a - bQ, and Marginal cost, c, qi* = (a - c)/((n+1)b); where n is the number of firms in the market

In the given question, a = 260, b = 1, c = 20, n = 2 (the two firms are q3 and q4)

So, qi* = (260 - 20)/(1*(2+1)) = 240/3 = 80 units

So, equilibrium total quantity = q3 + q4 = 80 + 80 = 160 units

Equilibrium price, P = 260 - 160 = $100

Profits for each firm = 100*80 - 20*80 = $6,400

Thus, profit of the merged firm = $6,400

c) If the profits earned after merger by firm 1 and firm 2 (as obtained in part (b)) are distributed equally, profit with each firm individually = profits after merger/2

So, Z1' = Z2' = 6400/2 = $3,200

Without merger, from part (a) we saw that both, firm 1 and 2 earned profit of $3,600, which was higher than after merger individual profit of the firms, i.e., $3,200. So, the merger isn't profitable for the two firms.

d) As already seen in part (b), profit of the un-merged firm, firm 3 (post-merger) = $6,400. Initially, that is before merger however, firm 3 earned profits of $3,600. Clearly, the un-merged firm has benefited from the merger off remaining two firms ($6,400 > $3,600).

Firm 3 benefits because initially it had competition from 2 more firms, however after the merger, competition is from only 1 firm, thereby firm 3 getting a bigger piece of market share than it was initially getting, as the firms still compete in Cournot environment. With the same reasoning, it makes sense why profits for firm 1 and 2 have decreased post-merger (though the competition for them as well has reduced, they share a smaller market share than before, taken together).