1. Assume that the market demand is given by P(Q) = 100 - Q, and that the cost functions of the two firms in the industry are: Ci(qi) = $20 qi, i = 1,2.
1.1 Calculate the output of each firm, the market price and the profits of each firm that correspond to the Nash-Cournot equilibrium.
1.2 Calculate the output of each firm, the market price and the profits of each firm that correspond to the Perfect Nash-Stackelberg equilibrium if the firm 1 decides output first, and firm 2 decides output after firm 1.
1.3 Consider the following strategy of firm 2: q2 = 20 for any output level of firm 1. Calculate the output level of firm 1 that maximizes its profits. Is the pair of outputs a Nash equilibrium? Is it a Perfect Nash equilibrium? Why?
1.4. Consider the following strategy of firm 2: q2 = 40 - q1. Could it be the strategy associated with the Perfect Nash equilibrium? Why?
1.5 Compare the market outcome of each model (price and total output); the market share and profit level of each firm. Is there any advantage of being the first mover as compared with the follower's profits in the sequential game? Is there any advantage of the first mover as compared with the static game?
2. In an oligopolistic industry the market demand has a linear form: P = 100 - 2 Q. In that market two firms decide the level of output in a sequential way: first the firm 2, and second the firm 2. They have the same cost structure: Ci(qi) = 50 + 3 qi^2 , i = 1,2.
2.1 Find the equation of the best response function of firm 2.
2.2 Find the equation of the profit function of firm 1, using the function found in 2.1 above.
2.3 Find the Perfect Nash Equilibrium and the output levels associated to it.
3. In an oligopolistic market there are two firms: a leader in price fixing and a follower. The leader fixes the price first, and then the follower takes that price as given and decides the output level. The market demand is Q = 100 - P. The total cost function of the leader (L) is CL (qL) = $10 qL , and the cost function of the follower (f) is Cf(qf) = (1/2) (qf)^2.
3.1 Provide the equation of the follower´s supply function
3.2 Find the equation of the residual demand faced by the leader firm
3.3 Find the Perfect Nash Equilibrium, and the price level and output level of each firm