Assignment:
Problem 1
Consider the demand system of tutorial problem 2.5 and assume g = 1/2.
(a) Suppose the two goods are produced by one firm. What are the optimal prices for the two goods? What are the profits?
(b) Now suppose again that the products are produced by two different firms. Furthermore, the firms play the price game with an infinite horizon and a discount rate of.
Construct a subgame perfect equilibrium with trigger strategies in which both firms charge the price you found in a) and punish deviation by reverting forever to the equilibrium in 2.5c. Under which condition does this equilibrium hold?
Problem 2
Consider an industry with three active firms competing in quantities. The inverse market demand is given by
p = 60 - Q/2
where Q is the sum of the quantities of all firms. Firms have a marginal cost of $12 and compete with an infinite horizon and a discount rate of δ.
(a) Construct a subgame perfect equilibrium with trigger strategies in which firms collude on the industry-profit maximizing quantities and punish deviation by reverting forever to the static Nash equilibrium. Under which condition does this equilibrium hold?
(b) How would your result in a) change if firms could observe each others' quantities only with a one period lag (that is, period t's quantities are observed only before period t + 2's quantities are set)?
Problem 3
Two electricity retail companies, Flower Power (F) and Mighty Shiver (M), operate in a local market. Demand for electricity is inelastic and uncertain. Households value electricity supply at 10 dollars per KwH at any time. However, consumption is subject to weather-dependent shocks. Suppose with probability 1/4 households demand 120 KwH, and with probability 3/4 they demand 200 KwH. The retail companies compete in prices and despite all the marketing efforts, consumers consider the electricity supplied by the two companies as perfect substitutes. The marginal cost of the electricity retailers is zero and both companies maximize their profits with an infinite horizon and a discount factor of , with 0 < δ≤ 1.
(a) Construct a subgame perfect equilibrium of the infinitely repeated price game in which the companies charge the monopoly price in each of the two states of demand and use trigger strategies with eternal reversion to the Nash equilibrium. Show under which conditions this equilibrium holds. Explain briefly.
(b) Suppose the two companies are also supplying gas. Households value gas supply at 6 dollars per KwH. Consumption is constant over time at 100 KwH. Gas is a homogenous good and the companies compete in prices. The marginal cost of gas is zero. Construct a subgame perfect equilibrium of the infinitely repeated price game in which the companies charge the monopoly price in the electricity AND the gas market and use trigger strategies with eternal reversion to the Nash equilibrium. Show under which conditions collusion in both markets is sustainable. Compare your result with (a) and explain the difference.
Problem 4
Consider a market for a homogenous product with three active companies. Consumers value the product at 110 dollars and buy at most one unit. Total demand in the market is fluctuating. In any given period, demand is high with probability 1/2 and there are 60 potential customers. With probability 1/2, demand is low and there are 30 potential customers. Firms know the current demand level but not the level of demand in future periods. They compete in prices with an infinite horizon. Suppose all firms face the same constant marginal cost of 10 dollars and a discount factor δ with 0 < δ < 1.
(a) Construct a subgame perfect equilibrium of the infinitely repeated price game in which the companies charge the monopoly price in each of the two states of demand and use trigger strategies with eternal reversion to the static Nash equilibrium. Show under which conditions this equilibrium holds. Explain your calculations.
(b) Suppose the discount factor of all firms is δ= 2/3. For this value collusion would not be sustainable with the strategies you specified in (a). Construct a subgame perfect equilibrium of the infinitely repeated price game in which the firms charge the monopoly price in low demand states and a price p in high demand states and use trigger strategies with eternal reversion to the static Nash equilibrium. Give the highest price p, for which this equilibrium is sustainable. Explain your calculations.
(c) Suppose the discount factor of all firms is δ= 5/6. Consider the following cyclical demand structure. Firms know that a low demand period is always followed by a high demand period and a high demand period is always followed by a low demand period. Construct trigger strategies of the infinitely repeated price game in which the companies charge the monopoly price in each of the two demand states and punish deviation by eternal reversion to the static Nash equilibrium. Check whether these strategies are sustainable in a subgame perfect equilibrium. Explain your calculations.
(d) Answer each of the following three questions in less than 100 words. No calculations are required.
(i) Suppose the probability of a high demand period in this market increases. How would this affect the sustainability of collusion in subquestion (a)?
(ii) Suppose the firms cannot observe the current level of demand before setting their prices. How would this affect the sustainability of collusion in sub-question.