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Numerical Problems

Numerical Problems

Consider the following Oligopoly market with two firms which produce a differentiated product and have the following demand functions.

q​1=​ 12 – P​1​ + (P​2/​ 2)

q​2​ =12 – P​2 +​ (P​1/​ 2)

Total Fixed Costs to either firm is 10 dollars and there are no variable costs.
(a) (5 points) Derive the equilibrium prices and quantities that will prevail if the firms do not collude (P*,q*). What amount of profit will each firm gain.

(b) (5 points) If the two firms decide to collude what will be the equilibrium price and total quantity produced in the market (P**,2q**). Find the profits for each firm.

(c) (10 points) Set up a normal form game (box game) which allows each firm (firm 1 and firm 2) to decide between setting the price derived in part (a) or the price derived in part (b). The payoffs to each firm should be the profit they would receive in each scenario.

(d) (5 points) Show clearly the Cournot-Nash equilibrium of the game you created from part