Consider a duopoly formed by firms 1 and 2, producing a homogeneous good. The demand in this market is represented by the following function:

P = 30 - Q

And the cost of both firms is given by

C(q_i) = q_i , i=1,2.

(i) Assume that firms compete in quantity. Calculate the Cournot equilibrium: quantity produced and profit of each firm and the market price.

In the following items, firms compete on price.

(ii) Calculate the Bertrand equilibrium: quantity produced and profit of each firm and the market price. (iii) Now the firms decide to form a cartel. Find the produced quantity and profit of each firm and the market price of the cartel. (iv) Now consider the infinite replay of these firms, and that they adopt the trigger strategy. Find the discount rate that makes the cartel price sustainable on a repeat game equilibrium.

Now firms have invested in differentiating their products so that they are no longer homogeneous. Market demand can be represented by the following system of equations,

q₁ = 30 - p₁ + 0.5 p2 q₂ = 30 - p₂ + 0.5 p₁

and the costs have not changed.

(v) Calculate the Bertrand equilibrium with differentiation: prices, quantities and profits of both firms.