## Two firms produce homogeneous products and choose quantities simultaneously.

### economics

##### Description

1. Two firms produce homogeneous products and choose quantities simultaneously. The inverse market demand function is given by P = 100−Q. Firm 1 has a total cost function given by C1(q1) = 10q1, while firm 2 has a total cost function given by C2(q2) = 0.5(q2) 2 .

(a) Write out the profit function for each firm, and derive the best response function for each firm. Illustrate these best response functions on a graph.

(b) Solve for the Nash equilibrium. Locate the equilibrium on your graph of best response functions. What are profits and price at the Nash equilibrium?

(c) Suppose now that firm 2’s cost function is changed to C2(q2) = 0.5 α (q2) 2 , where α > 1. Explain how the best response functions change, and illustrate your answer using a graph of best response functions. How do you expect the Nash equilibrium to change? How will the equilibrium price and profits change? Note: an intuitive and graphical explanation will suffice you do not need to solve for the new equilibrium. However, you are welcome to solve for it if you want to.

2. An industry producing a homogeneous commodity is comprised of N(≥ 2) firms. Assume that each firm faces a marginal cost of 1 and no other costs. The industry inverse demand function is P(Q) = 11 − Q, where Q is industry output.

(a) Assuming that the firms choose quantities simultaneously, derive the profits of each firm in equilibrium.

(b) Two of the firms are considering a merger. A merger simply means that these two firms become one firm, with the same marginal cost of 1. Retaining the assumption of Cournot-Nash behaviour in the post-merger equilibrium, derive the condition under which a merger would be profitable; that is under which the equilibrium profits of the new merged firm are greater than the combined profits of the two merging firms pre-merger. What is the largest value of N for which a merger between the two firms would be undertaken? Given this observation, would the merger ever be undertaken?