INTERMEDIATE MICROECONOMICS

Exercise 1 - Capturing surplus using a block pricing

Consider a monopolist facing a linear demand p(q) = 20 - q, and marginal cost MC = 2.

a) If the monopolist charges the same price for all units purchased (standard monopoly pricing), which is its optimal output and price? Which are its profits?

b) Assume now that the monopolist uses block pricing, i.e., charges a price p₁ for the first q₁ units sold, and a price p₂ for the remaining (q₂ - q₁) units sold. Set up the monopolist profit-maximizing problem.

c) Differentiate with respect to q₁ and q₂, and find the optimal values of q₁ and q₂.

d) Substitute the values of q₁ and q₂ you found in part (c) into the demand function to obtain the optimal prices p₁ and p₂.

e) Find the profits that the monopolist makes under block pricing you identified in parts (c)-(d), and compare them against the profits you found in part (a) when the monopolist charges the same price for all units sold.

Exercise 2 - Battle of the sexes game

Consider the following game, describing the so-called "Battle of the sexes" game: A husband and wife must simultaneously and independently choose whether to go to the opera or the football game. (They talked about going to either place last night, but none of them can remember which place they agreed to attend. To make matters worse, there has been a major phone and internet breakdown and cannot communicate before heading to either event.) The husband would prefer that both attend the football game, the wife prefers that both attend to the opera, but both would be better off attending each other's preferred event rather than being alone at their most preferred event. For instance, the husband obtains a payoff of 1 when attending the opera with his wife, but zero if he is at the football game, or at the opera, alone.

a) Show that there are no strictly dominated strategies for the husband or the wife.

b) Show that there are two NE in pure strategies, and that your equilibrium results resembles those in coordination games.

c) Find the mixed strategy Nash equilibrium of the game, where the Husband goes to the football game with probability p and the Wife with probability q.

Exercise 3 - Choice under uncertainty

Consider an individual with utility function u(x) = 5√x, where x ≥ 0 denotes money. He is considering to invest in two options: Option A provides him with a certain payoff of $440 at the end of the month, while option B generates a payoff of $800 with probability 0.4 or 200 with probability 0.6.

a) Find the expected value of each option.

b) Find the variance and standard deviation of each option.

c) Find the expected utility of option A, and option B. Which generates the highest expected utility?

d) Find the risk premium of option B. That is, by how much can the certain payoff of option A be reduced, and still induce the individual to choose option A?

Exercise 4 - Price competition with product differentiation.

Consider the following Bertrand game involving two firms (Coke and Pepsi) producing differentiated products. Firms have no costs of production. The demand for firm 1's product (Coke) is

q₁ = 1 - p₁ + 0.5p₂

Firm 2 (Pepsi) faces a demand

q₂ = 1 - p₂ + 0.3p₁

a. Find the best response function of Coke, i.e., the optimal price ????1 that Coke charges for every price charged by Pepsi, p₂.

b. Find the best response function of Pepsi, i.e., the optimal price ????2 that Pepsi charges for every price charged by Coke, p₁.

c. Solve for the Nash equilibrium of the simultaneous price-choice game, i.e., the price pair for which both best response functions cross each other.

d. Compute the firms' outputs and profits at the equilibrium prices you found in part (c).

Exercise 5 - Externalities and regulation

Consider an industry facing inverse demand function p = 20 - Q, and marginal cost of every firm is MC = 7 + Q. The industry produces an externality per unit of output (e.g., CO2 emissions). The negative effect of such externality are measured by the marginal external cost MEC = 1 + 2Q.

a. Find the output in the unregulated equilibrium (also referred to as "competitive equilibrium"). Which is the equilibrium price that emerges in this context?

b. Find the social optimal output, and compare your answer to that of part (a). Which is the price that emerges in this context?

c. Find the optimal tax or subsidy that induces firms to voluntarily produce the social optimum you found in part (b).