Find the Equilibrium Quantity
In a small town only two candy shops operate and they compete with each other in quantity. Consumers do not differentiate between candies sold by the two stores. The market demand for candies is given by D(p) = 126 - p. The cost functions of the two candy stores are C_{1}(q1) = 0.5q_{2}^{1} and C_{2}(q2) = 1.5q^{2}_{2}, respectively.
(a) Find the equilibrium quantity of candies sold by each shop, their profits and also the market price.
(b) Now imagine the game is repeated infinitely, so that collusion between the two shops may be possible. You may assume that, under collusion, the two candy stores divide the market unequally: the first store supplies twice as much as the second shop (q_{c}^{1} = 2q_{c}^{2}). Assume also that un- der collusion, the total amount of candies supplied to the market maximizes the joint profits of the two shops. Explain the trigger strategy that helps maintain collusion among the two shops in the market in the absence of discounting. Discuss and calculate the payoffs from cooperation, deviation and the non-cooperative outcome for each firm.
(c) Calculate the critical discount factors for both candy-shops.
(d) Assume that the frequency of interactions between the two shops changes. In particular, the candy stores only choose their quantities every two periods: the two shops thus compete in period 1, period 3, period 5, etc. Does this change affect the critical discount factor derived above? If yes, derive the new critical discount factor and comment briefly. If not, explain in detail why.