Problem Set 4Microeconomic Theory: A Mathematical Approach (Monopoly, Game Theory1)

1. Price Discrimination. There is one product provided by a firm. There are two consumers each ofwhom demands at most two units of the product. Consumers are characterised by their marginalvaluations vi1 and vi2, where vij is consumer i’s marginal valuation for jth unit of the product.That is, if consumer i consumes one unit of the good, her valuation is vi1 and if she consumes twounits, her valuation is vi1 +vi2. Consumers make the choice that maximizes the difference betweentheir valuation and the price they pay for it. Assume that firm can costlessly provide any amountof the product. Let v11 = 5, v12 = 3, v21 = 6 and v22 = 1. Solve the firm’s profit maximizationproblem in each of the following cases:

(a) Firm is a first-degree price discriminating monopolist.

(b) Firm is a second-degree price discriminating monopolist.

(c) Firm is a third-degree price discrminating monopolist.

(d) Firm is a non-discriminating monopolist.

(e) Firm is competitive.

2. Multi-plant Monopolist. Consider a multi-plant monopolist with two production units.

(a) The cost function of the i-th plant is 1 + qi, if qi > 0 and zero otherwise. Let the demandfunction facing the monopolist be q = max{10 − p, 0}. Solve for the optimal monopolyoutcome.

(b) How would your answer change if the cost function of the i-th plant is 1+qi, qi ≥ 0? Explain.

(c) How would your answer change if the cost function of the i-th plant is 1 + q2i , if qi > 0 andzero otherwise? Explain.

3. Monopolist problem: A Variant. There are 20 buyers with low demand price function p1(q1) =20 − 2q1 and 10 with high demand price function p2(q2) = 40 − 2q2. The monopoly seller has q̄units available.

(a) Write down the monopolist’s optimization problem.

(b) If the seller has q̄ = 240 units available, are either of the following output vectors optimal?(q1, q2) = (3, 18), (q1, q2) = (4, 16)

(c) For what values of q̄ will the monopolist sell only to the 10 high demanders?

4. Vertically-related Firms. Consider two vertically related firms, A and B. Each is a monopolist. Aproduces good F which is used by B as input in his own production of good X. B’s productionfunction is given by

X = f(F ) = F .

The demand function for X is given by

p(X) = a− bX.

The marginal cost of monopolist A is constant and given by v. Monopolist B incurs a cost of cper unit produced in addition to the cost of the input from A. Show that the final output levelswhen the firms are integrated are twice the output levels when they are not integrated.

1Ref: An Introduction to Game Theory by Osborne, Contact: econschool@gmail.com

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5. Third-degree price discrimination. In market 1 we have Qd(p) = 10−p/2 while in market 2 we haveQd(p) = 32−2p. The monopolist’s total cost function is TC(y) = y2. What outputs does the third-degree price discriminating monopolist sell in each market? What if monopolist is not allowed todiscriminate and has to sell the commodity at uniform price in both markets? What if monopolistis a first-degree price discriminating monopolist? What if monopolist acts as a competitive firm.Compare the quantity sold and profits in the four cases. Also check for the welfare (Welfare =Consumer surplus + Producer surplus). Which of the outcomes is pareto efficient?

6. Second-Degree Price Discrimination. A monopolist faces two consumers, one with demand functionDH(P ) = 12 − bHP and the other with demand function DL(P ) = 12 − bLP , where 0 < bH <bL < 2bH . The monopolist cannot tell which consumer has the higher demand and which has thelower demand, although he knows the two demand functions. The monopolist decides to sell thegood in bundles or packages. Denote a package as a pair (Q,V ) where Q is the number of units ofthe product and V is the price of the entire package (not the price per unit). He considers threeoptions.Option 1: sell only one package, targeted to the consumer with high demand.Option 2: sell two identical packages, designed in such a way that each consumer will buy onepackage.Option 3: sell two different packages, one targeted to the high-demand consumer and the otherto the low-demand consumer.The monopolist has the following cost function: C(Q) = Q

(a) Determine the profit maximizing package for option 1 and calculate the corresponding profits.

(b) Determine the profit maximizing package for option 2 and calculate the corresponding profits.

(c) For option 3 write the constraints that must be satisfied in order for each consumer to endup buying the package which is designed for her.

(d) Determine the profit maximizing package for option 3.

(e) Now assume that bH = 2 and bL = 3. Rank the three options based on the profits they yield.Calculate total surplus with the best (in terms of profit-maximization) of the three options.Calculate also the effective price(s) per unit. What would the monopolist’s profits be if hewere able to use the first-degree price discrimination?

7. Cournot Game. Consider a market in which there are three firms, each producing the same good.Firm i’s cost of producing qi units of the good is Ci(qi) = 16 for qi > 0 and Ci(0) = 0 for eachi ∈ {1, 2, 3}; the price at which output is sold when the total output is Q is P d(Q) = max{20−Q, 0},where Q = q1 + q2 + q3. Each firm’s strategic variable is output and the firms make their decisionssimultaneously. Find the Nash equilibria of Cournot’s oligopoly game.

8. Monopolistic Competition. Consider a monopolistically competitive industry in long-run equilib-rium. Each firm in the industry has the total cost function C = 10q + 100 where q is the firm’soutput level. Each firm faces the linear inverse demand function p = 200 − q − 0.5q∗ where p isthe firm’s price and q∗ is the total output produced by all other firms in the industry. Solve forthe long-run equilibrium levels of output produced by each firm, the price charged by each firm,and the total number of firms in the industry assuming that firms play Cournot-Nash on output.Include a diagram showing the firms long-run equilibrium price, average cost, and output level.

9. Bertrand Game. A single good is produced by two firms; each firm can produce qi units of thegood at a cost of Ci(qi) = 2qi. Each firm chooses a price, and produces enough output to meet thedemand it faces, given the price chosen by the other firm. If the good is available at the price pthen the total amount demanded is D(p) = max{10− p, 0}. Assume that if the firms set differentprices then all consumers purchase the good from the firm with the lowest price, which producesenough output to meet this demand. If both the firms sets the same price then they share thedemand at that price equally. A firm whose price is not the lowest price receives no demand andproduces no output. (Note that a firm does not choose its output strategically; it simply produces

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enough to satisfy all the demand it faces, given the prices, even if its price is below its unit cost,in which case it makes a loss. Bertrand’s oligopoly game is the following strategic game.

• Players The firms.

• Actions Each firm’s set of actions is the set of possible prices (nonnegative numbers).

• Preferences Firm i’s preferences are represented by its profit, equal to piD(pi)/m−Ci(D(pi)/m)if firm i is one of m firms setting the lowest price (m = 1 if firm i’s price pi is lower thanevery other price), and equal to zero if some firm’s price is lower than pi.

(a) Find the best response correspondences of the two firms.

(b) Find a Nash equilibrium.

10. Bertrand Game with integer prices. Consider Bertrand’s duopoly game where there are n = 2 firms,the cost functions are Ci(qi) = cqi for i ∈ {1, 2}, and the demand function is D(p) = max{0, α−p}.Restrict each firm to choose an integral price (e.g., representing an integral number of rupees).Assume that c is an integer and also assume that α > c + 1 and c > 0 so that positive profit ispossible. Find the set of all Nash equilibria.

11. Third-price auction. Consider a third-price sealed-bid auction, which differs from a first- and asecond-price auction only in that the winner (the person who submits the highest bid) pays thethird highest price. Assume that there are three bidders. Assume that the bidders’ valuations ofthe object are all different and all positive; number the players 1 through 3 in such a way thatv1 > v2 > v3 > 0.

(a) Show that the action profile in which each player bids her valuation is not a Nash equilibrium.

(b) Find a Nash equilibrium.

12. Joint venture. Two high tech firms (1 and 2) are considering a joint venture. Each firm i caninvest in a novel technology, and can choose a level of investment xi from 0 to 5 at a cost of

ci(xi) =x2i4

(think of x as how many hours to train employees, or how much capital to buy for R&D labs).The revenue of each firm depends both on its investment, and of the other firm’s investment. Inparticular, if firm i and j choose xi and xj respectively, then the gross revenue to firm i is

R(xi, xj) =

0, if xi < 1

2, if xi ≥ 1 and xj < 2

xi · xj , if xi ≥ 1 and xj ≥ 2

(a) What is the best response function of firm i?

(b) Find a Nash equilibrium.

13. Public Goods. An economy has n consumers. Each consumer belongs to one of the two possibletypes, type 1 and type 2. Consumers have preferences over a private good x, and a non-excludablepublic good G. The utility of a representative agent of type i is

ui(x,G) = lnx+ βi lnG

with β1 = 1, β2 ∈ [0, 1). Type 1 and 2 consumers respectively have an endowment of 1 unit and0.5 units of the private good. Each consumer j contributes an amount gj from his endowment forthe production of the public good. One unit of the private good can be costlessly transformed intoone unit of the public good, and vice versa. Hence the amount of public good produced is given

by G =

n∑j=1

gj .

In a symmetric Nash equilibrium in which all consumers of the same type contribute the sameamount for the public good, what will be the total production of the public good?

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