Question

There are 3 firms in a market with differentiated products. The marginal cost of production for...

There are 3 firms in a market with differentiated products. The marginal cost of production for each firm is c=20. There are no fixed costs. The system of inverse demands in this market is given by:

P1=120-q1-0.5(q2+q3)

P2=120-q2-0.5(q1+q3)

P3=120-q3-0.5(q1+q2)

And the corresponding demand system is

q1=60-1.5P1+0.5(P2+P3)

q2=60-1.5P2+0.5(P1+P3)

q3=60-1.5P3+0.5(P1+P2)

a. Suppose the 3 firms operate independently, and choose prices simultaneously. Find the best response function of each firm to the prices of its two rivals.

b. Find the equilibrium prices, and the profit each firm makes.

c. Suppose firms 1 and 2 merge. Each firm still sells the same differentiated product, hence the demand system remains the same after merger, but the merged firm can choose prices p1 and p2 together to maximize the join profit of the merged firm. Write the profit maximization problem of the merged firm (i.e., maximize sum of profits over a choice of the two prices). Write the 2 first order conditions (one with respect to each price).

d. Solve for the equilibrium market prices after merger (the system you need to solve had the two first order conditions from part c and the best response of firm 3 which is unchanged). How to after merger prices compare to pre-merger prices?

e. Find the profit of the merged firm. Compare it to the total profit of these two firms before merger. Was merger profitable?

f. Was merger profitable for the non-merging firm? Intuitively explain.

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