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Intermediate Economics 100B

Winter 2023

PROBLEM SET 1

Due on January 26^th, 2023

(1) Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by  and , where  is the output of Firm 1 and  is the output of Firm 2. Price is determined by the following demand curve:

P = 300 – Q

Where

a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.

b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit.

c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm 1’s profit differ from that found in part (b) above?

d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm’s profit?

 (2) Two competing firms are each planning to introduce a new product. Each will decide whether to produce Product A, Product B, or Product C. They will make their choices at the same time. The resulting payoffs are shown below.

	Firm 2
Firm 1		A	B	C
	A	-10, -10	0, 10	10, 20
	B	10, 0	-20, -20	-5, 15
	C	20, 10	15, -5	-30, -30

a. Are there any Nash equilibria in pure strategies? If so, what are they?

b. If both firms use maxmin strategies, what outcome will result?

c. If Firm 1 uses a maxmin strategy and Firm 2 knows this, what will Firm 2 do?

(3) Joe and Sarah’s Investment Dilemma

Suppose Joe and Sarah each have a patent on their respective product: no other supplier can provide their particular product. However, Joe and Sarah’s products are imperfect substitutes for each other. Consequently, they face the following respective consumer demand

Qjoe = 300 − 15 Pjoe + 10 Psarah

Qsarah = 300 − 15 Psarah + 10 Pjoe

They face the following costs characterized by constant marginal cost and no fixed costs

C(Qjoe) = 8 Qjoe

C(Qsarah) = 8 Qsarah

(3a) Assuming that each supplier charges marginal cost Pjoe = Psarah = $8, calculate the own-price and cross-price elasticities for Joe. (Sarah’s are the same due to symmetry)

(3b) Solve for Joe’s and Sarah’s respective reaction curves, assuming a Bertrand game.

(3c) Solve for the Bertrand-Nash Equilibrium.

(3d) Under the Betrand-Nash Equilibrium, how much does each supplier earn (payoff)?

(3e) Suppose Joe has the opportunity to invest and lower his costs as follows:

C*(Qjoe) = 4 Qjoe

If Joe invests in this new technology and Sarah is stuck with her current costs (constant marginal cost of $8), what would the new Bertrand-Nash Equilibrium be?

(3f) How much does each supplier earn under the Bertrand-Nash Equilibrium in (2e) given that the investment cost for Joe is $500 ? Assuming that Sarah is stuck with her current costs, what is the most that Joe would have been willing to spend for the new technology?

(3g) Suppose Sarah has the same opportunity to invest in the lower cost technology (at $500). If both Joe and Sarah make the investment and lower their marginal costs to $4, what is the new Bertrand-Nash Equilibrium?

(3h) How much does each supplier earn under the Bertrand-Nash Equilibrium in (2g) (accounting for the investment cost)? Are payoffs lower or higher in this equilibrium than in the other two Bertrand-Nash equilibria (considered above)?

Consider the following two-stage game.

1. First, each supplier simultaneously and independently decides whether to invest in the new technology at a cost of $500.

2. Second, with each supplier having made his/her investment decision (revealed to each other), each supplier simultaneously and independently decides on the price to charge for his/her product.

So the second stage of this game is the Bertrand game, conditional on each supplier’s first stage investment decision.

This two-stage game can be solved using backward induction ... first solve the second stage game for each possible outcome of the first stage and then solve the first stage. You’ve actually done the first step of the induction in the earlier parts of this question ...

(3i) Fill-in the following payoff matrix for the first-stage “investment game” using your answers from (2d), (2f), (2h)

Sarah
ddddd…dd              Joe		Not Invest	Invest
	Not Invest	? , ?	? , ?
	Invest	? , ?	? , ?

HINT: The payoffs from the Nash-Bertrand Equilibrium where Sarah invests but Joe does not are symmetric to the case where Joe invests but Sarah does not.

(3j) Solve for the pure strategy Nash equilibrium for this “investment game.”

(3k) Explain why this “investment game” is a “Prisoner’s Dilemma” game. Who benefits the most from the availability of this new low cost technology: Joe, Sarah, or consumers?

(4) Solving for Nash Equilibria

Consider the following two-player game represented in normal form.

Each player has four possible pure strategies

		PLAYER 2
		L	M	N	O
PLAYER 1	A	4,12	7,2	3,4	5,14
	B	6,7	6,3	4,6	7,5
	C	3,7	2,11	1,8	4,6
	D	3,7	4,5	9,9	12,7

(4a) For each of player 1’s pure strategies, find player 2’s best response pure strategies.

(4b) For each of player 2’s pure strategies, find player 1’s best response pure strategies.

(4c) Find any dominant pure strategies for Player 1 and Player 2.

(4d) Find any strictly dominated pure strategies for Player 1 and Player 2.

(4e) Solve for any pure strategy Nash equilibria.

(5) For each situation, solve for the Bertrand-Nash Equilibrium (differentiated Product).

5a) Suppose Sarah’s constant marginal cost is $5 but Joe’s is $8

Recall that in a Bertrand model with differentiated product, each supplier faces his/her own demand:

Qjoe = 100 − 10 Pjoe + 5 Psarah

Qsarah = 100 − 10 Psarah + 5 Pjoe

5b) Suppose Joe and Sarah have the same cost functions as earlier (constant MC of $5) but asymmetric demand functions

Qjoe = 100 − 10 Pjoe + 5 Psarah

Qsarah = 160 − 10 Psarah + 5 Pjoe

(6) For each situation, solve for the Cournot-Nash equilibrium (Capacity Constraints)

6a) Suppose Sarah’s constant marginal cost is $5 but Joe’s is $8.

Recall that in a Cournot model, products are homogeneous. So each supplier faces the same aggregate demand function:

P(Q) = 20 − 0.1 Q

where Q = Qjoe + Qsarah

6b) Suppose Joe and Sarah have the same cost functions as earlier (constant MC of $5) but market inverse demand is now P(Q) = 30 − 0.2Q

(7) For each situation, solve for the Stackelberg equilibrium

7a) Suppose Sarah’s constant MC is $5 but Joe’s is $8. (Using P(Q) = 20−0.1Q again and assuming Joe goes first)

7b) Suppose Joe and Sarah have the same marginal cost ($5) but market inverse demand is now

P(Q) = 30 − 0.2 Q

(8) For each situation, solve whether collusion is sustainable using “trigger strategy” (infinitely repeated)

NOTE: For each of these exercises, you need to calculate {} and see if the incentive compatibility inequality (condition for collusion to occur as a Nash equilibrium) is satisfied.

8a) Stage Game is Bertrand with Homogeneous Product; cost same as before but overall demand is

Q(P) = 375 − 15 P and given

8b) Same as 4a) but given

8c) Stage Game is Cournot; Joe and Sarah have a constant MC

P(Q) = 20 − 0.1 × Q where Q = Qjoe + Qsarah and given

8d) Same as 4c) above but given

8e) Try 4c) and 4d) assuming Sarah’s MC = $5 but Joe’s MC = $8. They both collude on Sarah’s Monopoly Output.

(9) Suppose the market for tennis shoes has one dominant firm and five fringe firms. The market demand is Q = 400 - 2P. The dominant firm has a constant marginal cost of 20. The fringe firms each have a marginal cost of MC = 20 + 5q.

a. Verify that the total supply curve for the five fringe firms is .

b. Find the dominant firm’s demand curve.

c. Find the profit-maximizing quantity produced and price charged by the dominant firm, and the quantity produced and price charged by each of the fringe firms.

d. Suppose there are ten fringe firms instead of five. How does this change your results?

e. Suppose there continue to be five fringe firms but that each manages to reduce its marginal cost to MC = 20 + 2q. How does this change your results?

(10) Two firms compete by choosing price. Their demand functions are

Q₁ = 20 - P₁ + P₂   and     Q₂ = 20 + P₁ - P₂

where P₁ and P₂ are the prices charged by each firm, respectively, and Q₁ and Q₂ are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero.

a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.)

b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be?

c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

(11)   Assume  two firms with the same constant average and marginal cost, AC = MC = 5, facing the market demand curve Q₁ + Q₂ = 53 - P. Use the Stackelberg model to analyze what will happen if one of the firms makes its output decision before the other.

a. Suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions before Firm 2). Find the reaction curves that tell each firm how much to produce in terms of the output of its competitor.

b. How much will each firm produce, and what will its profit be? Compare your results to the Cournot Equilibrium.

(12) Suppose all firms in a monopolistically competitive industry were merged into one large firm. Would that new firm produce as many different brands? Would it produce only a single brand? Explain.