Consider the following duopoly with differentiated goods where x_{1} and x_{2} denote the amounts of the goods 1 and 2 respectively, with prices p_{1} and p_{2}. The demand functions are:
x_{1} = 1-2p_{1} + p_{2 }
x_{2} = 1-2p_{2} + p_{1}
And the corresponding unitary production costs are c_{1}= 0 and c_{2}= 0.5
i) Determine the solution under perfect competition and if there is collusive behaviour between the two firms (i.e., prices, quantities and profits).
ii) The two firms decide now simultaneously. If the firms could determine if they compete in prices or quantities which variable would they choose? Determine the corresponding equilibria (prices, quantities and the profits) and provide an interpretation of the results.
iii) Do we get the Bertrand paradox here? Comment on the result.
iv) Check that the Stakelberg-Bertrand solution is p_{1} = 3/7 y p_{2} = 17/28. Compare the profits of the leader (firm 1) and the follower. Any surprise? Comment on it.