Econ 521 - Week 5:
Exercises 1- Bertrand Model
1. Bertrand's oligopoly game- Suppose that there are n firms, with n ≥ 3, each of whose cost functions has constant unit cost c. Assume that the demand function is D(p) = α - p for p ≤ α and 0 for all p > α, and that c < a.
(a) Show that the set of Nash equilibria is the set of profiles (p1, ..., pn) of prices for which pi ≥ c for all i and at least two prices are equal to c.
(b) Show that every other profile different from profiles described in (a) is not a Nash equilibrium.
2. Bertrand's duopoly game with c1 ≠ c2- Consider Bertrand's duopoly game under a variant of the assumptions. Now, firms' unit costs are different, equal to c1 and c2, where c1 < c2. Denote by p1m the price that maximizes (p-c1)(α-p), and assume that c2 < p1m and that the function (p-c1)(α-p) is increasing in p up to p1m. Based on this answer the following,
(a) Suppose that the rule for splitting up consumers when the prices are equal assigns all consumers to firm 1 when both firms charge the price c2. Show that (p1, p2) = (c2, c2) is a Nash equilibrium and that no other pair of prices is a Nash equilibrium.
(b) Show that no Nash equilibrium exists if the rule for splitting up consumers when the prices are equal assigns some consumers to firm 2 when both firms charge c2.
3. Bertrand's duopoly game with fixed costs- Consider Bertrand's game under a variant of the usual assumptions in which the cost function of each firm i is given by Ci(qi) = f + cqi for qi > 0, and Ci(0) = 0, where f is positive and less than the maximum of (p - c)(α - p) with respect to p. Denote by p- the price p that satisfies (p - c)(α - p) = f and is less than the maximizer of (p - c)(α - p).
(a) Show that if firm 1 gets all the demand when both firms charge the same price then (p¯, p¯) is a Nash equilibrium.
(b) Show also that no other pair of prices is a Nash equilibrium. (Hint: First consider cases in which the firms charge the same price, then cases in which they charge different prices.)
Exercises 2 - Electoral Competition
1. Electoral competition with asymmetric preferences- Consider a variant of Hotelling's model in which voters' preferences are asymmetric. Specifically, suppose that each voter cares twice as much about policy differences to the left of her favorite position than about policy differences to the right of her favorite position. How does this affect the Nash equilibrium?
2. U.S. presidential election- Consider a variant of Hotelling's model that captures features of a US presidential election. Voters are divided between two states. State 1 has more Electoral College votes than does state 2. The winner is the candidate who obtains the most Electoral College votes. Denote by mi the median favorite position among the citizens of state i, for i = 1, 2; assume that m2 < m1.
Each of two candidates chooses a single position. Each citizen votes (non-strategically) for the candidate whose position is closest to her favorite position. The candidate who wins a majority of the votes in a state obtains all the Electoral College votes of that state; if for some state the candidates obtain the same number of votes, they each obtain half of the Electoral College votes of that state. Find the Nash equilibrium (equilibria?) of the strategic game that models this situation.
3. Electoral Competition between candidates who care only about winning position- Consider a variant of Hotelling's model in which the candidates (like citizens) care about the winner's position, and not at all about winning per se. There are two candidates. Each candidate has a favorite position; her dislike for other positions increases with their distance from her favorite position. Assume that the favorite position of one candidate is less than m and the favorite position of the other candidate is greater than m. Assume also that if the candidate tie when they take the positions x1 and x2, then the outcome is the policy ½(x1 + x2). Find the set of Nash equilibria of the game that model this situation.