Econ 2ZZ3: Midterm - Summer 2021

4. Your pages must be numbered. When you finish your exam, at the top of the first page indicate the total number of pages.

Rules and Grading:

1. For each of the above instructions that you fail to meet, a 10% penalty will apply. For example, if you submit multiple files, the pages are not numbered, and you omit including your name in the first page, an immediate 30% penalty will apply. In such a scenario, even if your answers are all correct, your grade will be at most 70%.

2. You must complete the exam on your own and without the help of your classmates, friends, family, strangers on the internet, etc. Failure to do so may result in a grade reduction and/or academic dishonesty charges.

3. For full marks, you must explain your approach, have the correct calculations, and provide the correct economic interpretation. Your graphs must be fully and clearly labelled.

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Question 1 (8 marks)

Two firms are engaging in quantity competition. Firm 1 has the cost function c1(q1) = 8q1 forallq1 >0andFirm2hasthecostfunctionc(q2)=2q2 forallq2 >0. Firm1andFirm2 produce q1 and q2 respectively (units of output). Market demand is given by Q = 1, 200 − 2p where Q = q1 + q2.

a) Suppose that the two firms are playing a Stackelberg game and firm 1 moves first. Find the Stackelberg equilibrium. (2 marks)

Start with the follower’s problem. The marginal revenue function is MR2 = 600 − q2 − q1/2 0.5 mark. Set equal to the marginal costs of the follower and rearrange to obtain the follower’s best response function q2∗ = (1196 − q1)/2. 0.5 mark

Plug the follower’s BR function to the leader’s total revenue function and find the leader’s marginal revenues M R1 = 600 − q1 /2 − 1196/4. 0.5 mark Set equal to the leader’s marginal costs and solve to obtain q1∗ = 586. Plug q1∗ into the follower’s BR function to obtain q2∗ = 305. Thus, the Stackelberg equilibrium is (q1∗, q2∗) = (586, 305) 0.5 mark

b) Suppose that the two firms are playing a Stackelberg game and firm 2 moves first. Find the Stackelberg equilibrium. (2 marks)

Work as in part a but now firm 1 is the follower. The follower’s best response function is q1∗ = (1184 − q2)/2 1 mark

Plug the BR function into the leader’s total revenue function, find marginal revenues and set equal to marginal costs, rearrange to obtain q2∗ = 604. Thus, q1∗ = 290 and this pair of quantities is the Stackelberg equilibrium.

c) Explain the first mover advantage in the context of your answers in parts a) and b) above (2marks)

In part a, the equilibrium price is p = 600 − (586 + 305)/2 = 154.5 and in part b p = 153. 2

(0.5 mark)

Profits for firm 1 are (154.5 − 8) ∗ 586 = 85, 849 and for firm 2 (154.5 − 3) ∗ 305 = 46, 207 in part a (0.5 mark) and in part b, π1 = 42,050 and π2 = 90,600. 0.5 mark

We see that firm 1 makes more profits when it moves first (part a) relatively to its profits when it moves second (part b). Similarly, firm 2 makes more profits in part b relative to its profits in part a. This difference in profits illustrates the first mover’s advantage 0.5 mark.

d) Suppose now that the firms play a Cournot game. Find the Cournot equilibrium. (2 marks)

From part a we know the best response function of firm 2 q2 = (1196 − q1)/2 and from part b the best response of firm 1 q1 = (1184 − q2)/2. Solve these as a 2x2 system to find the Cournot equilibrium: (q1∗, q2∗) = (390.66, 402.66) 2 marks

Question 2 (4 marks total)

Consider a game of simultaneous moves between two players. Each player can choose between the same set of two actions: Action 1 and Action 2. The payoffs are (1, 0) if both players take the same action, and (0, 1) if they do not take the same action.

a. Write down the payoff matrix of the game and find the best responses of each player to every possible action of the other player. (1 mark) Is there an equilibrium in Dominant Strategies? (1 marks)

For player A: If player B plays A1, best response is A1. If player B plays A2, best response is A2 0.5 mark

For player B: If player A plays A1, best response is A2. If player A plays A2, best response is A1. 0.5 mark

Thus, both players do not have a dominant strategy and there is no DSNE. 1 mark 3

b. Find all Nash Equilibria in pure strategies, if they exist. (1 mark) Is there a Nash equilibrium in mixed strategies? (1 mark)

There is no NE in pure strategies, since the BR do not intersect in part a. 1 mark We know that there is a mixed strategy NE in every game, so there is an equilibrium in mixed strategies 0.5 mark, in which both players play A1 with 50% probability and A2 with 50% probability.

Question 3 (3 marks total)

Player A and Player B are playing a game of sequential moves.

Player A moves first and has two actions: {Aggressive, Safe}, and if he chooses Safe, the game ends and the payoffs are (1, 2) i.e. Player A receives 1 dollar and player B receives 2 dollars. If he plays Aggressive, then it’s Player B’s turn.

Player B has two actions: {Left, Right}. If he plays Left, the game ends and the payoffs are (2, 1) for (A, B). If he plays Right, the game ends and the payoffs are (0, 0).

a. Write down the game in its extensive form (i.e. “game tree”) and find all Subgame- Perfect Nash Equilibria. (1 mark)

The game tree is (0.5 marks)

The unique SPNE is {aggressive,safe} (0.5 marks)
b. Suppose that before the game is played, Player B announces to Player A that he will

always play Right. What will Player A do if he believes this statement is true? (1

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marks) Should Player A believe Player B? Why or why not? (1 mark)

If Player A believes this announcement, he would play {safe} (0.5 marks), in order to get a payoff of 1 instead of 0 from Player B playing Right after Player A has choosen {aggressive}. (0.5 marks for comparing the two payoffs and justifying the answer)

Player A should not believe Player B because he knows that Right is not a best response to Aggressive, B would get a payoff of zero so he’d play Left to get a payoff of 1 (1 mark).

Question 4 (3.5 marks total)

Two firms are engaging in price competition. Firm 1 has the cost function c1(q1) = 8q1 for all q1 >0andFirm2hasthecostfunctionc(q2)=2q2 forallq2 >0. Firm1andFirm2charge a price of p1 and p2 respectively (per unit). Market demand is given by Q = 1, 200 − 2p where p = min{p1, p2}. If p1 = p2 both firms supply half of the total output Q, and if p1 = p2 the firm with the lowest price supplies all of the output.

a) Provide the definition of a Bertrand equilibrium. (0.5 mark)

A Bertrand equilibrium is a pair of prices (p∗1, p∗2) such that no firm can increase its own profits by changing its own price given the price of the other firm (0.5 mark - the definition must be precise for full marks)

b) Consider the pair of prices (p1, p2) = (9, 9). Calculate profits for both firms under these prices. (0.5 mark). Use your definition from part a to prove that (p1, p2) = (9, 9) does not form a Bertrand equilibrium (1 mark)

At these prices, total market demand is Q(9) = 1,182 units and each firm supplies 591 units. Thus,profitsforfirm1areπ1(9,9)=(9−8)∗591=$591andforfirm2π2(9,9)=

(9 − 2)591 = $4, 137. (0.5 mark)
To prove that this is not a Bertrand equilibrium, we need to show that at least one firm can

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increase their profits by changing their price, given that the price of the other firm remains constant.

Suppose that firm 1 decreases their price by a small amount, e.g. to p1 = 8.5. Given that p2 remains fixed at 9, firm 1 captures all of the demand Q(8.5) = 1, 183. Profits for firm 1 are now π1(8.5, 9) = (8.5 − 8) ∗ 1183 = $591.5 > 591. We have showed that firm 1 can do better by changing its own price given the price of the other firm, therefore (9, 9) does not form a Bertrand equilibrium. (1 mark - Must use formal arguments for full marks. Alternatively, we can show that firm 2 can do better by decreasing its price by a small amount, given the price of firm 1)

c) Find a Bertrand equilibrium for this game. (0.5 mark). Use your definition from part a to prove that your answer is indeed a Bertrand equilibrium. (1 mark)

The unique Bertrand equilibrium is for firm 2 to price just below the marginal cost of firm 1, and firm 2 prices at its marginal cost: (p1, p2) = (8, 7.99). (1 mark)

Any pair of prices in which firm 1 prices above 8 cannot be supported as a Bertrand equilibrium: firm 2 would price just below firm 1 and firm 1 would keep reducing its price to capture some of the demand. For the same reason, firm 2 will not price above 8 in any Bertrand equilibrium.

Firm 1 will never price below 8 as it would make negative profits. Firm 2 would then never price below 7.99 as it would make lower profits 1 mark

End of Exam

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