There are 3 firms producing a homogeneous product. Let qi be the output level of

Question

There are 3 firms producing a homogeneous product. Let qi be the output level of firm i, and let Q be the aggregate output level. The market demand curve is given by p = 50 - Q. Firms have identical constant marginal costs, which we normalize to zero. Solve for the Cournot equilibrium output and profit level of each firm. Now suppose that firms 2 and 3 merge into a single firm (without any change in costs). Call this merged entity firm X. Calculate the profit level of the merged firm under Cournot competition. Do firms 2 and 3 benefit from the merger, compared to their status quo profit levels? Now suppose that firm 1 merges with firm X. Do firms X and 1 benefit from this merger? Explain why the first and second mergers yield different results regarding the profitability of the mergers. Explain how these results would change if firms were competing in prices with differentiated products (just give the intuition, no math).

Explanation / Answer

a. Q=q1+q2+q3

so p=50 - q1-q2-q3

All the firms are working to maximize profit

MC=MR

For Firm 1

R1=p*q1

R1=50q1-q1^2-q1q2-q1q3

so MR1 = dR1/dq1

MR1 = 50-2q1-q2-q3

And MC=0

so 50-2q1-q2-q3=0 eq1

For Firm 2

R2=p*q2

R2=50q2-q2^2-q1q2-q2q3

so MR2 = dR2/dq2

MR2 = 50-2q2-q1-q3

And MC=0

so 50-2q2-q1-q3=0 eq2

For Firm 3

R3=p*q3

R3=50q3-q3^2-q1q3-q2q3

so MR3 = dR3/dq3

MR3 = 50-2q3-q2-q1

And MC=0

so 50-2q3-q2-q1=0 eq3

solving eq1, eq2 and eq3 we get q1=q2=q3

so 50-4q1=0

so q1=12.5=q2=q3

p = 50-q1-q2-q3 = 12.5

So profit for each firm = 12.5*12.5 = 156.25

b. When firm 2 and 3 will form firm x then it will act as duopoly

Q=q1+qx

so p=50 - q1-qx

All the firms are working to maximize profit

MC=MR

For Firm 1

R1=p*q1

R1=50q1-q1^2-q1qx

so MR1 = dR1/dq1

MR1 = 50-2q1-qx

And MC=0

so 50-2q1-qx=0 eq1

For Firm x

Rx=p*qx

Rx=50qx-qx^2-q1q2

so MRx = dRx/dqx

MRx = 50-2qx-q1

And MC=0

so 50-2qx-q1=0 eq2

solving eq1 and eq2 we get q1=qx

so 50-3q1=0

so q1=16.67=qx

p = 50-q1-qx = 16.67

So profit for each firm = 16.67*16.67 = 277.89

So firm 2 and 3 will get =277.89 = 138.94 each so basically firm 1 is at greater profit while firm 2 and 3 had lowered their profits

c. Now the complete system will act as a monopoly as all the firms have merged.

p=50-Q

R=p*Q

R=50Q-Q^2

MR=dR/dQ = 50-2Q

MC=0

and to maximize profit MC=MR

50-2Q=0

Q=25

p=50-25=25

Profit = 25*25=625

So profit for each firm = $625/2 = $312.5 so both X and 1 have benefited from the merger.

d. The first merger made it a duopoly which has a better control over markket so overall profit has reduced but as firm 1 was at special benefit of being alone in comarision to x so reaped more profits. In second merger it became a monopoly giving more control in production. If differentiated products were produced in these firms then merger will not make much of difference as they were already monopolistic competitive firms.

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