Industrial Economics Homework Help
1. Two firms producing a homogeneous product are the only firms on the market. The inverse demand for the product is equal to: ��=680 0,5��, where �� is the total demand on the market, which consists of the demand for the product of Firm 1 (��1) and the demand for the product of Firm 2 (��2) . �� =��1 +��2. The firms have identical cost structures: ��1 =20��1 and ��2 =20��2. The firms compete through output (Cournot).
a) What will the output and price level be in equilibrium, if the two firms compete by setting quantities and choose how much to produce simultaneously? How high are the profits of the firms? (1 point)
Under Cournot competition, the firms choose their output by maximizing their profits. The profit function of Firm 1 is:
In order to maximize the profit using output, we need to take the derivative of the profit function with respect to Firm 1’s output and set it to zero:
From this we can derive the profit-maximizing output of Firm 1 as a function of the output of Firm 2 (i.e. Firm 1’s reaction function):
Since the two firms have symmetric costs, one would expect that the reaction function of Firm 2 is symmetric to that of Firm 1. Hence:
In equilibrium both of these conditions have to be fulfilled, hence:
b) Now assume that Firm 1 has a technological advantage and can produce faster than Firm 2. As such it chooses how much to produce first and Firm 2 observes this before making its own output choice. What are the new equilibrium levels for the output of the two firms? What is the market price? How high are the profits of the two firms? (1 point)
Firm 1 can now be sure that Firm 2 will observe its actions and have time to react. Since Firm 1 knows that Firm 2 will use its reaction function (which depends on the output of Firm 1) to choose its optimal output, Firm 1 can effectively predetermine not only its own output but
also the output of the competitor. Thus its profit function is now only dependent on its choice of q1:
c) Example theory question (DO NOT ANSWER IN SUBMITTED HOMEWORK): How would the equilibrium output of the two companies change if their marginal costs were quadratic (i.e. increasing in output) rather than constant?
2. There are two firms on the market selling differentiated products. They engage in price competition (Bertrand model).
The demand for the product of Firm 1 is equal to ��1 =20 2��1 +��2 , while the demand for the product of its competitor is equal to ��2 =20+��1 2��2 , where ��1 and ��2 are the prices of Firm 1 and Firm 2 respectively and ��1 and ��2 are the demanded quantities. The firms have a symmetric cost structure: ��1 =5��1 and ��2 =5��2.
a) What prices and output quantities will the firms choose if they compete and choose their prices simultaneously? How high will their profits be? (1 point)
The logic is analogous to that with Cournot competition, but now the firm maximizes its profit by choosing the appropriate price, rather than the appropriate output, hence we calculate the profit as a function of the price and then take the derivative.
Since the two firms have symmetric cost and demand functions, one would expect that the reaction function of Firm 2 is symmetric to that of Firm 1. Hence:
b) What prices and output levels would the firms choose if they were to form a cartel and agree to set the same price (��1 =��2 =��������������) and split the market? What profits would they make if both firms stick to the cartel? (1 point)
If the firms form a cartel, they will maximize joint profits, assuming that ��1 =��2 =��. In this case the joint profits are equal to 2 times the individual profits of the firms, given the agreedupon price:
c) Example theory question (DO NOT ANSWER IN SUBMITTED HOMEWORK): Is the cartel stable? How would firms adjust their prices to maximize profits, given that their competitor sticks to cartel pricing?
3. Two companies engage in Bertrand competition. If they collude, both would make a profit of 3. If one colludes while another undercuts substantially, the profits of the undercutting company are equal to 6, while the one that colludes doesn’t make a profit. If both compete, profits are equal to 2.
Firm 2: Collude
Firm 1: Collude Profit 1: 3
2: 3
Firm 1: Compete Profit 1: 6
Profit 2: 0
Firm 2: Compete
1: 0 Profit 2: 6
Profit 1: 2
Profit 2: 2
a) If the firms play a repeated game, how high would the discount factor of each player need to be in order for the companies to find it rational to cooperate/collude with each other? (0.5 points)
When deciding whether to compete or cooperate, firms compare the discounted profit flows from each action.
The payoff from cooperating is:
b) Suppose that the pay offs change, with Firm 2 being the smaller company and hence making lower sales when prices are matched. How does the lower bound of the discount rate change for each player, if their payoffs are now specified by the matrix below? (0.5 points)
c) Example theory question (DO NOT ANSWER IN SUBMITTED HOMEWORK): Which company would be less likely to join the collusion, based on the minimum discount rates that you calculated? Why do you think this is the case?