University of California, Davis - Department of Economics SPRING 2012

ECN / ARE 200C: MICROECONOMIC THEORY

Professor Giacomo Bonanno

====================================================================== TEST # 1 Total 50 points

1.

[22 points] In a second price auction (where the winner is the player who submits the highest bid,

but the price she pays is the second highest) for each player bidding her true value is a dominant strategy. Consider the following modification of this auction. For simplicity we consider only two players. The player who submits the highest bid wins (in case of ties, player 1 wins). The winner pays not her bid but the average of her bid and the other player’s bid. The true value of the object to player i is vi > 0. A player’s payoff is the difference between her true value and what she has to pay, if she wins the object, and zero, if she does not win the object. Bids can be any non-negative numbers. (a) [4 points] Write down the payoff function of Player 1 and that of Player 2. (b) [8 points] Assume now that v1 = $80. Is bidding $80 a dominant strategy for player 1? If your answer is “Yes” you have to prove it and if your answer is “No” then you still have to prove it! (c) [10 points] Now consider the case where v1 and v2 are any two positive numbers with 0 < v2 < v1. Find a Nash equilibrium.

2. [28 points] A homogeneous-product industry consists of two firms. Inverse demand is given by P = 90 − 3Q (where P is price and Q is industry output). Both firms have the same cost function, given by C (q ) = 9q .

(a) [8 points] Calculate the Cournot-Nash equilibrium. (b) The owner of firm 1 decides to retire and appoints a manager to run the firm. The manager’s compensation is a percentage of the revenue of the firm. Thus the manager’s objective is to maximize the revenue of the firm. This is common knowledge between the manager of firm 1 and the owner of firm 2 (who still manages firm 2 and whose objective is to maximize the profits of firm 2).

(b.1) [8 points] Calculate the Nash equilibrium of the new game that results after the change in the management of firm 1.

(b.2) [4 points] Are consumers hurt by or do they benefit from the change? (b.3) [4 points] Is the owner of firm 2 hurt by or does she benefit from the change? (b.4) [4 points] Does the owner of firm 1 earn more or less after retirement if he pays the manager 1% of the revenue of the firm?

University of California, Davis -- Department of Economics

SPRING 2012

ECON. 200C: MICROECONOMIC THEORY

Giacomo Bonanno

Test # 1 ANSWERS

1. (a) The payoff functions are as follows (v1 is the true value of the object to player 1 and v2 the true value to player 2; b1 denotes player 1’s bid and b2 denotes player 2’s bid):

R|v − FG b + b IJ H 2 K Payoff of player 1: S |T0 1

2

1

R|0 Payoff of player 2: S b +b |Tv − FGH 2 IJK 1

2

2

if b1 ≥ b2 if b1 < b2 if b1 ≥ b2 if b1 < b2

(b) No. For example, suppose player 2 bids b2 = 10. Then with b1 = 80 the average bid is 45 and player 1’s payoff is 80 − 45 = 35. If, instead, he were to bid b1 = 20 then the average bid would be 15 and his payoff would be 80 − 15 = 65 > 35. (c) For every b ∈ [v2,v1], b1 = b2 = b is a Nash equilibrium. Given the tie-breaking rule, the object goes to player 1 for a payment of b. Hence his payoff is v1 − b ≥ 0. If player 1 were to reduce his bid he would not win the object and his payoff would be 0. If he bid more than b he would still win the object but would have to pay more. As for player 2, her payoff is 0. If she reduced her bid, she would still have a payoff of 0. If she increased her bid, she would win the object but would have to pay more than v2 hence her payoff would be negative.

2.

(a) The profit functions are π 1 = q1[90 − 3(q1 + q2 )] − 9q1 and π 2 = q2 [90 − 3(q1 + q2 )] − 9q2 . The ∂π 1 ∂π 2 Cournot-Nash equilibrium is given by the solution to = 0 and = 0 . The solution is ∂q1 ∂q2 q1 = q2 = 9 with corresponding profits π 1 = π 2 = 243 .

(b) (b.1) Now firm 1 chooses q1 to maximize R1 = q1[90 − 3(q1 + q2 )] while firm 2 still chooses q2 to maximize π 2 = q2 [90 − 3(q1 + q2 )] − 9q2 . The Nash equilibrium is given by the solution to ∂R1 ∂π 2 = 0 and = 0 . The solution is q1 = 11 and q2 = 8 with corresponding profits ∂q1 ∂q2 π 1 = 264 and π 2 = 192 . (b.2) Since total output has increased (from 18 to 19) price is lower and thus consumers are better off. (b.3) The owner of firm 2 is hurt by the change: her profits have decreased from 243 to 192. (b.4) The owner of firm 1 earns more. The firm’s revenue is 363, thus the manager’s remuneration is 3.63. Hence the income of the owner of firm 1 increases from 243 to 264− 3.63=260.37

Page 1 of 1

University of California, Davis - Department of Economics SPRING 2012

ECN / ARE 200C: MICROECONOMIC THEORY

Professor Giacomo Bonanno

====================================================================== TEST # 2 Answer all questions. Explain your answers. Total 50 points

1. [20 points] Consider the following game, where the payoffs are given in the following order (from top to bottom): player 1, player 2, player 3.

1 A C x 2 2

B 2

2 D

F

G

3 L

H

6 0 0

E

0 4 0

0 2 y

2 6 2

0 4 4

Note: to answer the following questions you don’t need to write the normal form (you will save a lot of time if you reason on the extensive form without constructing the normal form). (a) [2 points] Are there values of y for which Player 3 has a strictly dominant strategy? If your answer is Yes, say what values and what strategy, if your answer is No explain why not. (b) [2 points] Does Player 2 have weakly dominated strategies? (If your answer is Yes, name the strategies and the strategies that dominate them; if your answer is No prove your claim.) (c) [3 points] For what values of y does Player 3 have a weakly dominated strategy? Name the strategy. (d) [8 points] Find all the backward-induction solutions when x = 1 and y = 2? (e) [2 points] Assume that x = 1 and y = 1. Explain why (B,D,L) is not a backward-induction solution. (f) [3 points] Assume that x = 1 and y = 1. Is there a Nash equilibrium where Player 1 plays A? If Yes, then say what the equilibrium is, if No then explain why not.

Page 1 of 2

2. [15 points] Consider the following game. Janet is a contestant in a popular game show and her task is to guess behind which door Liz, another contestant, is standing. With Janet out of the room, Liz chooses a door behind which to stand - either door A or door B. The host, Monty, observes this choice. Janet, not having observed Liz's choice, enters the room. Monty says to Janet either “Red cabbage” or “Green beans” (which sounds silly, of course, but it is as good as anything else you see on TV these days!). After hearing Monty’s statement, Janet picks a door (says either “A” or “B”). If she picks the correct door, then she wins $100. If she picks the wrong door then she wins nothing. Liz wins $100 if Janet picks the wrong door and nothing if Janet picks the correct door. Thus, Liz would like to hide from Janet, while Janet would like to find Liz. What about Monty? Well, he likes the letter A. If Janet selects door A, then Monty gets $10, while if Janet selects door B then he gets nothing. (a) [6 points] Represent the game in extensive form. (b) [4 points] Write the corresponding normal (or strategic) form. Let Liz be the row player, Monty the column player and Janet the remaining player. (c) [3 points] Does this game have any pure-strategy Nash equilibria? If your answer is Yes, then name them. (d) [2 points] Does this game have any subgame-perfect equilibria? If your answer is Yes, then name them. If your answer is No, then explain why not.

3. [15 points] Kidding Inc. has discovered a new substance, called IWISH, which eliminates baldness. There are 100 consumers who are interested in this product. The fraction q (with 0 < q < 1) of these consumers are (each) willing to pay up to $15 for it, while the remaining consumers are (each) willing to pay up to $10. Kidding is going to sell IWISH in packets to a retailer at a price of $w per packet (where w is an integer with 0 ≤ w ≤ 15). The retailer takes the price w as given and chooses the price p (per packet) to consumers. Every consumer is going to buy at most one packet. Both the manufacturer and the retailer know what has been said so far and nothing more (in particular, the retailer cannot tell if a particular consumer has a reservation price of 15 or 10). Production and retailing costs are zero (of course, w is a cost for the retailer). Both Kidding’s and the retailer’s objective is to maximize their own profits. Assume that both w and p can only be integers. (a) [3 points] Represent this as a two-player extensive game with perfect information, where the two players are Kidding and the retailer. Do not draw the entire game, just give a brief (partial) sketch. (b) [9 points] Find the backward-induction solution. [Hint: it will depend on the value of q.] 1 (c) [3 points] What is the backward induction solution when q = 10 ?

Page 2 of 2

University of California, Davis -- Department of Economics

SPRING 2012

ECON. 200C: MICROECONOMIC THEORY

Giacomo Bonanno

Test # 2 ANSWERS

1. (a) No, because if player 1 plays A the payoff of player 3 is the same (namely 2) no matter what strategy player 3 chooses. (b) Yes (F,D) is weakly dominated by (G,D) and (F,E) is weakly dominated by (G,E). (c) For y ≠ 2 (and only those values of y): if y > 2 then H weakly dominates L and if y < 2 then L weakly dominates H. (d) One solution is (A, (G,E), H) and the other is (B, (G,D), L). (e) Because it is not a strategy profile. (f) Yes: (A,(G,E),H) and also (A,(G,E),L) and (A,(G,D),H).

2. (a) The extensive form is shown on the next page. (b) The normal form is as follows (for Monty we go left to right, thus RG means ‘Red if A and Green if B’; for Janet we go top to bottom, thus BA means ‘B if Green and A if Red’).

Liz

A

RR 0,10,100

Monty RG GR 0,10,100 0,10,100

GG 0,10,100

B

100,10,0

100,10,0

100,10,0

100,10,0

Liz

A

RR 0,10,100

Monty RG 0,10,100

GR 100,0,0

GG 100,0,0

B

100,10,0

0,0,100

100,10,0

0,0,100

Janet: AA

Liz

A B

RR 100,0,0 0,0,100

Monty RG GR 100,0,0 0,10,100 100,10,0 0,0,100

Janet: BA

GG 0,10,100 100,10,0

Liz

A B

RR 100,0,0 0,0,100

Monty RG 100,0,0 0,0,100

Janet: AB

GR 100,0,0 0,0,100

GG 100,0,0 0,0,100

Janet: BB

(c) There are two pure-strategy Nash equilibria: (A,RG,BA) and (A,GR,AB). (d) Since there are no proper subgames, the pure-strategy Nash equilibria coincide with the purestrategy subgame-perfect equilibria. Hence there are two subgame-perfect equilibria: (A,RG,BA) and (A,GR,AB).

Page 1 of 2

LIZ door A

door B

green

green

MONTY

MONTY

JANET red

A

B

0 10 100

100 0 0

A

100 10 0

red

B

0 0 100

JANET A

B

0 10 100

100 0 0

A

100 10 0

B

0 0 100

p=1 Retailer w=1 p=15

3. (a)

Manufacturer w=2

Retailer p=2 p=15

(b) Given w, the retailer will only choose between p = 15 [with profits of q100(15 − w)] = 1500q − 100qw and p = 10 [with profits of 100(10 − w)] = 1000 − 100w. Thus the retailer

R| p = 10 if | chooses S || p = 15 if T

10 − 15q 1− q * (and will be indifferent between p = 10 and p = 15 otherwise). Let w 10 − 15q w> 1− q 10 − 15q 10 − 15q be the largest integer less than or equal to (i.e. the integer part of ). Then the manufacturer 1− q 1− q w
1500q if 100w* < 1500q

(and will be indifferent between the two otherwise). Hence the backward*

*

induction solution is as follows: (1) if 100w > 1500q then the manufacturer sets w = w , the retailer sets p = * 10 and quantity sold is 100; (2) if 100w < 1500q then the manufacturer sets w = 15, the retailer sets p = 15 and quantity sold is 100q. * * 1 10 −15q 85 (c) When q = 10 , = 9 = 9.44 hence w = 9 and 100w = 900 > 1500q = 150. Thus at the 1−q backward induction solution the manufacturer sets w = 9, the retailer sets p = 10, the manufacturer’s profits are 900 and the retailer’s profits are (10 − 9)100 = 100. [If the manufacturer were to set w = 15, the retailer would set p = 15 and the manufacturer’s profits would be only 150.]

Page 2 of 2

University of California, Davis - Department of Economics ECN / ARE 200C: MICROECONOMIC THEORY

SPRING 2012

Professor Giacomo Bonanno

====================================================================== TEST # 3 Answer all questions. Explain your answers. Total 50 points

1. [17 points] Consider the following multi-stage game. In the first stage an incumbent monopolist decides whether to be passive or committed. Commitment costs $C and is irreversible. In stage two Nature (i.e. a random mechanism) selects the opportunity cost of entry k ∈ K (that is, the profit that the potential entrant could make in the best alternative investment) according to the cumulative distribution function F [thus, for every number x, F(x) is the probability that the opportunity cost of entry k is less than or equal to x]. In stage three the potential entrant observes the opportunity cost of entry which Nature selected and decides whether to enter or not. If she doesn't enter, the incumbent remains the only firm in the market. Monopoly profits are given by $M (not affected by the commitment that the incumbent is considering). If entry occurs, a duopoly game between the two firms follows. Let DI and DE be the incumbent's and entrant's profits, respectively, at the Nash equilibrium of the duopoly game following entry with a passive incumbent, and HI and HE be their respective profits at the Nash equilibrium of the duopoly game following entry with a committed incumbent (HI includes the commitment cost C). Assume that if indifferent between entering and not entering, the potential entrant will enter. (a) [6 points] Assume that K = [a, b] (the closed interval between a and b, 0 < a < b) and a < HE < DE < b. Under what conditions is there commitment at every subgame-perfect equilibrium? Under what conditions are the subgame-perfect equilibria characterized by the fact that the incumbent is passive? (b) [3 points] Draw the extensive form of this multistage game for the case where K = {k1, k2} (replace each duopoly game with the corresponding equilibrium payoffs). (c) [4 points] Suppose that K = {1, 4, 6, 12}, all the values in K are equally likely, M = 12, C = 1, DI = DE = 5, HI = HE = 2. Would a rational incumbent choose commitment? (Identify rationality with subgame-perfect equilibrium behavior.) (d) [4 points] Suppose that K = {1, 2, 4, 7}, Prob{1} = Prob{2} = Prob{4} = 1/5, Prob{7} = 2/5, M = 8, C = 2, DI = DE = 7/2, HI = HE = 3/2. Would a rational incumbent choose commitment?

2. [16 points] Consider the following game. 1 L

R

1 1 A

2 D

C

D

0 1 2

0 2 2

2 0 0

3

w

2 3 1

1 2 2

G

E

y

L 1 0 0

H

B

x C

2

F

z

M

0 2 0

L

0 3 3

M

2 1 2

(a) [6 points] Find three subgame-perfect equilibria. [Use pure strategies wherever possible.] (b) [6 points] For each of the three equilibria you found in part (a), explain if it can be written as a weak sequential equilibrium. (c) [4 points] Find a sequential equilibrium. [Use pure strategies wherever possible.]

3.

[17 points] Each of two players is given five tokens, numbered 1 to 5. At stage t ∈ {1,...,5} the two players simultaneously select a token and put it on the table. If the two tokens have the same number printed on them, each player gets 1 point; if one token shows a higher number than the other then the player who put down the token with the higher number wins 2 points and the other player gets zero points. In either case, the tokens are removed and the game proceeds to stage t + 1 . The player who collects the largest number of points wins. Thus the possible outcomes for each player are: win, draw and lose. Each player prefers winning to drawing and drawing to losing. We shall think of this game in different ways. (A) Imagine first that each player, before the game begins, has to choose a sequence in which to arrange the tokens and then at stage t ∈ {1,...,5} the tokens in position t in the sequences are compared. For example, if player 1 chooses the sequence 3,1, 2,5, 4 and player 2 chooses the

round 1 2 3 4 5 sequence 4,1, 2,3,5 then the scores are as follows: player 1's score 0 1 1 2 0 and the player 2's score 2 1 1 0 2 outcome is that player 2 wins (6 points to 4). (a.1) [2 points] How many strategies does each player have? (a.2) [1 points] If you were to draw the strategic form as a matrix, how many cells would you have to fill in? (a.3) [6 points] Prove that there is no pure-strategy Nash equilibrium. [Hint: think in terms of best replies.] (a.4) [2 points] Does this game have a mixed-strategy Nash equilibrium? [You don’t have to find it, just give a Yes or No answer and explain your answer.] (B) Now we think of the game in a way which is closer to the verbal description above. The game is played sequentially and at each stage t ∈ {1,...,5} the two players simultaneously choose a token among the remaining ones and scores are determined as explained above. (b.1) [2 points] Sketch the extensive form. [The extensive form is huge, so simply sketch the initial part in a way that it is clear to the reader how the rest would be drawn.] (b.2) [2 points] Which of the following numbers is closer to the number of strategies that Player 4 12 36 1 has in this game? (1) 10 (2) 10 (3) 10 Give some explanation for your answer. (b.3) [2 point] Does the strategic form of part A correspond to the strategic form of this extensive-form game?

University of California, Davis -- Department of Economics

SPRING 2012

ECON. 200C: MICROECONOMIC THEORY

Giacomo Bonanno

Test # 3 ANSWERS

1. (a)

Suppose the Incumbent is passive. Then the entrant will enter iff DE ≥ k. Thus ex ante the

probability of entry, if the Incumbent is passive, is Prob{k ≤ DE} = F(DE). Thus the incumbent’s expected profits if he chooses to be passive are: F(DE) DI + [1 − F(DE)] M

(1).

Similarly, if the incumbent is committed, entry occurs with probability F(HE) and the Incumbent’s expected profits are: F(HE) HI + [1 − F(HE)] (M − C)

(2).

Thus the incumbent will choose to be passive if (1) > (2), will choose to commit if (2) > (1) and be indifferent if (1) = (2) (b) Let I = incumbent, N = Nature, E = entrant, p1 = Prob{k1}, p2 = Prob{k2}. I committed

passive

N k1

p

1

N p

D D

I E

k1 p

2

E in

k2 E

E out

in

M

D

k

D

1

I E

out

in

M

H

k2

H

I

E

k2 1

p E

2

out

in

out

M-C

HI

M-C

k

H

k

1

E

2

(c) In this case we have that F(1) = 1/4, F(4) = 2/4, F(6) = 3/4 and F(12) = 1. Thus F(DE) = F(5) = F(4) = 2/4 and F(HE) = F(2) = F(1) = 1/4. Thus (1) above becomes: (2/4) (5) + [1 − 2/4] 12 = 17/2 = 8.5 while (2) above becomes: (1/4) (2) + [1 − 1/4] (12 − 1) = 35/4 = 8.75 Hence the Incumbent will choose commitment. (d) In this case we have that F(1)=1/5, F(2)=2/5, F(4)=3/5 and F(7)=1. Thus F(DE) = F(7/2) = F(2) = 2/5 and F(HE) = F(3/2) = F(1) = 1/5. Thus (1) above becomes: (2/5) (7/2) + [1 − 2/5] 8 = 17/2 = 31/5 = 6.2 while (2) above becomes: (1/5) (3/2) + [1 − 1/5] (8 − 2) = 51/10 = 5.1 Hence the Incumbent will choose to be passive.

Page 1 of 3

2. (a) To find a subgame-perfect equilibrium first we solve the subgame on the left: Player 2 C D Player 1

A

1,0

0,1

B

0,2

2,0

There is no pure-strategy Nash equilibrium. Let p be the probability of A and q the probability of C. Then at a Nash equilibrium it must be that q = 2(1− q) and 2(1− p) = p. Thus there is a unique Nash A B C D  equilibrium given by  2 1 2 1  with an expected payoff for Player 1 of 23 . 3 3 3 3  Next consider the subgame on the right. In this subgame the following are pure-strategy Nash equilibria: (F,H,M) (where Player 1 gets 1), (E,L,H) (where Player 1 gets 2), and (E,L,G) (where Player 1 gets 2). Thus the following are subgame-perfect equilibria:

L R 0 1 

A B C 2 3

1 3

L R and  0 1

2 3

D E F 1 0 1 3

A B C 2 3

1 3

2 3

G H 0 1

L M 0 1  ,

L R 0 1 

A B C 2 3

1 3

2 3

D E F 1 1 0 3

G H L M 0 1 1 0 

D E F G H L M 1 1 0 1 0 1 0  3

L R A B C D E F G H L M  (b)  0 1 2 1 2 1 0 1 0 1 0 1  cannot be part of a perfect Bayesian 3 3 3 3   equilibrium, because choice M is strictly dominated by L and thus there are no beliefs at player 3’s information set that justify choosing M. L R A B C D E F G H L M   0 1 2 1 2 1 1 0 0 1 1 0  cannot be part of a perfect Bayesian equilibrium, 3 3 3 3   because – given that Player 3 chooses L, H is not a sequentially rational choice for Player 2 at his singleton node. L R 0 1 

D E F G H L M is a perfect Bayesian equilibrium with the 1 1 0 1 0 1 0  3 x y w z following system of beliefs:  2 1   3 3 1 0 A B C 2 3

1 3

2 3

(b) The perfect Bayesian equilibrium given above is also a sequential equilibrium. Proof …

Page 2 of 3

3. (a.1) There are 5! = 120 possible ways of arranging the 5 tokens in a sequence. Thus each player has 120 possible strategies. (a.2) 120 × 120 = 14,400. (a.3) Take a strategy of the opponent. The best reply to it is to match his 5 with your 1, his 4 with your 5, his 3 with your 4, his 2 with your 3 and his 1 with your 2, thus totaling 8 points in your favor (against 2 for the opponent). For example, the best reply to 4,1, 2,3,5 of the opponent is

5, 2,3, 4,1 . If there were a Nash equilibrium then each strategy would be a best reply to the other, implying that each player would win (by scoring 8 points to 2), but this is impossible. (a.4) Yes, because by Nash’s theorem every finite game has an equilibrium in mixed strategies. (b.1) The extensive form is as follows (at every stage it is common knowledge what tokens were chosen at the previous stage):

1 5

1 2

4

3

2

1

1

5

1

5

1

1 1 2

3

4 5

2 1

2 3

4

(b.2) Consider player 1. At the initial node he has to choose among 5 possibilities. After a similar choice of player 2 there are 25 nodes where player 1 has to choose among 4 possibilities. After a similar choice of player 2 there are 16 × 25 = 400 nodes where player 1 has to choose among 3 possibilities. After a similar choice of player 2 there are 9 × 16 × 25 = 3,600 nodes where player 1 has to choose among 2 possibilities. After a similar choice of player 2 there are 4 × 9 × 16 × 25 = 14,400 nodes where player 1 has only one token left and thus only one choice. Thus he has the following 25 400 3,600 307 number of strategies: 5 × 4 × 3 × 2 = which is a number greater than 10 (this is what Mathcad tells me after refusing to compute this number!). (b.3) Obviously not, since the number of strategies is vastly different!

Page 3 of 3

University of California, Davis - Department of Economics ECN / ARE 200C: MICROECONOMIC THEORY

SPRING 2012

Professor Giacomo Bonanno

====================================================================== TEST # 4 Answer all questions. Explain your answers. Total 50 points 1. [17 points] Ann, Bob and Carla are taken to a room where there is a tray with two white balls and two red balls. Thus it becomes common knowledge among them that there are four balls, two of which are white and two are red. Then they leave the room. A referee then leads Ann to the room and asks her to pick one ball and hide it in her pocket. Then it becomes Bob’s turn to enter the room and pick one of the remaining balls and put it in his pocket. Finally, Carla enters the room, picks one of the remaining balls and puts it in her pocket. It is commonly known among them that this is what happened, that is, that first Ann, then Bob, then Carla went to the room and picked a ball. Nobody gets to see the balls picked by the other two. However, they do see what balls are left on the tray when they make their own choice. (a) [6 points] Using a set of states and information partitions, represent the state of knowledge of Ann, Bob and Carla (concerning who has what balls). (b) [2 points] Draw the partition representing what is common knowledge between Ann and Bob. (c) [2 points] Draw the partition representing what is common knowledge between Bob and Carla. (d) [3 points] Find a state a and an event E such that at a all the following are true: (1) Bob knows E, (2) Ann knows that Bob knows E, (3) Bob knows that Carla does not know E. (e) [4 points] What strategy should Bob use in picking his ball if he wants to prevent Carla from knowing what color ball Ann has? And what if he wants to prevent Carla from knowing what color ball he himself has?

2. [15 points] Consider the following game (where the payoffs are von Neumann-Morgenstern payoffs) and the following model of it Player 2

Player 1

E

F

G

H

A

3 , 1

4 , 0

1 , 1

1 , 2

B

2 , 5

6 , 1

0 , 2

1 , 1

C

2 , 2

8 , 0

4 , 1

1 , 1

D

2 , 4

2 , 2

1 , 3

2 , 3

1/2

1/2

Player 1 a

b 1/2

Player 2

c 1/2

d 1/2

1/2

1/2 e

1/2

1's strategy:

A

C

C

D

D

2's strategy:

E

E

F

F

E

Page 1 of 2

(a) [12 points] Let RATi be the event that player i is rational (i =1,2) and RAT = RAT1∩RAT2 be the event that both players are rational. Find the events: (1) RAT1 , (2) RAT2 , (3) K1 RAT , (4) K2 RAT , (5) K1K2 RAT , (6) K2K1 RAT . (b) [3 points] Suppose that you found a model (of the game given above) and a state w in that model where there was common knowledge of rationality. What strategy profile could be associated with that state?

3.

[18 points] Consider the following situation of incomplete information (in case you are wondering, the underlying interpretation is one based on uncertainty about outcomes rather than uncertainty about preferences): 1

T

2

1

A

T

2 1 B

D

1 0

2

A 2 2

B

0 2

D

1 0

0 1

1: 2:

p

(1−p)

(a) [6 points] Apply the Harsanyi transformation and represent this situation as a game with imperfect information. (b) [6 points] Write the corresponding strategic form. (c) [6 points] Find all the pure-strategy Nash equilibria for every possible value of p (with 0 < p < 1).

Page 2 of 2

University of California, Davis -- Department of Economics

SPRING 2012

ECON. 200C: MICROECONOMIC THEORY

Giacomo Bonanno

Test # 4 ANSWERS

1. (a) Let wwr represent the state where Ann has a white ball, Bob has a white ball and Carla has a red ball, etc. Thus the set of possible states (given that there are only two white balls and two red balls) is {wwr, wrw, wrr, rww, rwr, rrw}. The information partitions are as follows:

Ann

wwr

wrw

wrr

rww

rwr

rrw

Bob

wwr

wrw

wrr

rww

rwr

rrw

Carla

wwr

wrw

wrr

rww

rwr

rrw

(b) The common knowledge partition between Ann and Bob coincides with Ann’s partition (thus whatever Ann knows is common knowledge between Ann and Bob). (c) What is common knowledge between Bob and Carla is shown by the following partition:

common knowledge partition for Bob and Carla

wwr

wrw

wrr

rww

rwr

rrw

(d) Let a = wrw and E = {wwr, wrw, wrr}. Then KBobE = KAnn KBobE = E. Furthermore, ¬KCarlaE = {wrw, wrr, rww, rwr, rrw} = KBob¬KCarlaE. Thus a ∈ KBobE ∩ KAnn KBobE ∩ KBob¬KCarlaE. (e) First of all, by looking at the balls that are left, Bob can always tell what color ball Ann has. His strategy is thus the following: if Ann picked white, then pick red and if Ann picked red, then pick white (that is, pick a different color from Ann). Since Carla knows what color Ann has if and only if she knows what color Bob has, the same strategy works for both cases.

2. (a) RAT1 = {a,b,c}, RAT2 = {a,b,e} so that RAT = {a,b}. Thus K1RAT = {a}, K2RAT = {a,b}, K1K2 RAT = {a} and K2K1RAT = ∅. (b) The only strategy profiles that can be associated with states where there is common knowledge of rationality are those that remain after the iterated deletion of strictly dominated strategies (allowing domination by mixed strategies). In the original game strategy F is strictly dominated by strategy G. Eliminating F leads to the game shown in part (a) in the following table. In that game B is strictly dominated by the mixed strategy ( 21 A, 21 D). Eliminating B leads to the game shown in part (b). In that game G is strictly dominated by the mixed strategy ( 21 E,

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1 2

H). Eliminating G

leads to the game shown in part (c). In that game C is strictly dominated by the mixed strategy ( 21 A, 21 D). Eliminating C leads to the game shown in part (d). No more eliminations are possible. Thus the only strategy profiles compatible with common knowledge of rationality are ((A,E), (A,H), (D,E) and (D,H).

A Player 1

E

Player 2 G

H

3 , 1

1 , 1

1 , 2

B

2 , 5

0 , 2

1 , 1

C

2 , 2

4 , 1

1 , 1

D

2 , 4

1 , 3

2 , 3

Player 1

E

G

H

A

3 , 1

1 , 1

1 , 2

C

2 , 2

4 , 1

1 , 1

D

2 , 4

1 , 3

2 , 3

(b)

(a)

Player 1

E

H

A

3 , 1

1 , 2

C

2 , 2

1 , 1

D

2 , 4

2 , 3

Player 1

A D

E 3 , 1 2 , 4

H 1 , 2 2 , 3

(d)

(c)

3. (a) The game is as follows: NATURE

1−p

p B 1 0

B T

1

1

1 0

T 2

D

A

D

A

0 2

2 1

0 1

2 2

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(b) The strategic form is as follows:

Player 2 D

A

BB

1

0

1

0

BT

p

1-p

2-p

2(1-p)

TB

1-p

2p

1+p

p

TT

0

1+p

2

2-p

Player 1

(c) (BB,D) is a Nash equilibrium for every possible value of p between 0 and 1. 1 (TT,A) is also a Nash equilibrium if and only if 2 − p ≥ 1 + p , that is, if p ≤ . 2 There are no other pure-strategy Nash equilibria.

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University of California, Davis - Department of Economics SPRING 2012

ECN / ARE 200C: MICROECONOMIC THEORY

Professor Giacomo Bonanno

====================================================================== TEST # 5 Answer all questions. Explain your answers. Total 50 points

1. [16 points] Imagine a world where a person’s productivity is decided at birth and education has no effect on it. However, employers don’t know this and believe that education is the determinant of productivity. There are two types of individuals. One type is born with a productivity of $20,000 and has the following cost of acquiring education (y denotes the number of years of schooling): CL(y) = a (y − 6). The other type is born with a productivity of $27,000 and has the following cost of acquiring education CH(y) = b (y − 6). The possible choices of y are 6, 12, 16, 18 and 21. Employers offer the following wage schedule (erroneously believing that education affects productivity):

y

wage

6

$8,000

12

$20,000

16

$24,000

18

$27,000

21

$30,300

(a) [12 points] Find all the values of a and b that give rise to a signaling equilibrium. [It is up to you to decide what an individual does when indifferent; state your assumption explicitly.] (b) [4 points] Assume that a and b are such that a signaling equilibrium exists and, currently, the economy is at a signaling equilibrium. Suppose that employers are riskneutral. Suppose also that the population is composed as follows: 40% with productivity 20,000 and 60% with productivity 27,000. Would it be desirable (that is, would it lead to a Pareto improvement) to (1) make it compulsory for everybody to complete 12th grade and (2) shut down all the institutions of higher education (that is, any type of education beyond 12th grade)? [If you think that the answer is a function of the values of a and b then be explicit about it; if not state it clearly.]

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2. [18 points]

There are two groups of individuals. All the individuals in Group 1 have the same utility function which is as follows (where α > 1): x + q  x

if owns a car of quality q and $x if owns $x but no car

All the individuals in Group 2 have the same utility function which is as follows: x + α q  x

if owns a car of quality q and $x if owns $x but no car

All cars are owned by individuals in Group 1. Each car is of one of the qualities in the set Q = {1,000, 2,000, …, 1,000n}, where n ≥ 2. There is an equal number of cars of each quality (that is, as many cars of quality 1,000i as cars of quality 1,000j, for every i,j ∈ {1,2,…,n}). The quality of each car is known to the owner but cannot be determined by the buyer. Thus there can be only one price for second-hand cars. Call this common price P. All agents are risk neutral. (a) [8 points] For every integer s ∈ {1, 2, …, n}, give a necessary and sufficient condition on the value of α for there to be an equilibrium where all and only the cars of quality up to (and including) 1,000s are traded at price P (assume that the number of individuals in Group 2 is sufficiently large for there to be a potential buyer for every car and that each member of this group has a sufficiently large endowment of money). (b) [5 points] For what values of α is there an equilibrium at which the final allocation of cars (and money) in the population is Pareto efficient? (c) [5 points] Suppose that there are 9 quality levels (n = 9) and α = 1.76. Furthermore, suppose that there are 2,000 cars of each quality (thus a total of 18,000 cars). What is the largest number of cars that can be traded at an equilibrium?

3. [16 points]

Consider a competitive insurance market facing two types of potential customers, H and L. They all have the same initial wealth W = $15,000 and they all face the same potential loss x = $6,000. 1 1 The probability of loss for H type is pH = and the probability of loss for L type is pL = . Let qH be 4 10 the proportion of H types in the population. They all have the same utility-of-wealth function U(m) = m.

Define a separating competitive equilibrium as a pair of contracts (CH,CL) (where each contract is described as a pair (h,D) where h is the premium and D is the deductible) such that (1) type H individuals purchase contract CH, which they find at least as good as contract CL and at least as good as no insurance, (2) type L individuals purchase contract CL, which they find at least as good as contract CH and at least as good as no insurance, (3) each contract yields zero expected profits, and (4) no insurance company could make positive profits by introducing a new contract.

(a) [5 points] Draw a diagram showing the equilibrium (using indifference curves, etc.), if it exists. (b) [9 points] Calculate the pair of contracts that constitutes an equilibrium, if an equilibrium exists. (c) [2 points] Explain what additional condition needs to be satisfied for the pair of contracts of part (b) to be an equilibrium (no need to do any calculations, just explain in words).

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University of California, Davis -- Department of Economics

SPRING 2012

ECON. 200C: MICROECONOMIC THEORY

Giacomo Bonanno

Test # 5 ANSWERS

1. Below it is assumed that – when indifferent − an individual chooses the level of education that gives her a salary equal to her true productivity. Hence inequalities will be taken to be weak inequalities. (a) Decision problem for person of productivity 20,000 y gross wage cost net income 6 8,000 0 8,000 12 20,000 6a 20,000-6a 16 24,000 10a 24,000-10a 18 27,000 12a 27,000-12a 21 30,300 15a 30,300-15a At a signaling equilibrium the best choice must be y = 12 (so that they get paid their true productivity). Hence we need all of the following inequalities to be satisfied: 20, 000 − 6a ≥ 8, 000 that is a ≤ 2, 000     a ≥ 1, 000   20, 000 − 6a ≥ 24, 000 − 10a that is  20, 000 − 6a ≥ 27, 000 − 12a that is a ≥ 1,166.66     20, 000 − 6a ≥ 30,300 − 15a that is a ≥ 1,144.44  Thus we need

7, 000 = 1,166.66 ≤ a ≤ 2, 000 6

Decision problem for person of productivity 27,000 y gross wage cost net income 6 8,000 0 8,000 12 20,000 6b 20,000-6b 16 24,000 10b 24,000-10b 18 27,000 12b 27,000-12b 21 30,300 15b 30,300-15b At a signaling equilibrium the best choice must be y = 18 (so that they get paid their true productivity). Hence we need all of the following inequalities to be satisfied: 27, 000 − 12b ≥ 8, 000 that is b ≤ 1,583.33      27, 000 − 12b ≥ 20, 000 − 6b that is b ≤ 1,166.66   27, 000 − 12b ≥ 24, 000 − 10b that is b ≤ 1,500    b ≥ 1,100   27, 000 − 12b ≥ 30,300 − 15b that is Thus we need 1,100 ≤ b ≤ 1,166.66 =

7, 000 6

(b) If schools beyond 12th grade were eliminated, the employers would no longer have education as a signal of productivity. Hiring an employee would then be the same as playing the lottery  productivity 20, 000 27, 000    which has an expected value of 24,200. Thus employers (being probability 0.4 0.6   Page 1 of 3

risk-neutral) would pay everybody $24,200. Employees of type L would be better off (their salaries would increase from 20,000 − 6a to 24,200 − 6a). Employees of type H would be better off if and 2,800 = 466.66 . Since a signaling only if 24, 200 − 6b > 27, 000 − 12b , that is, if and only if b ≥ 6 equilibrium requires b to be larger than this, the proposed policy would lead to a Pareto improvement (all employees better off, employers as well off) and this is true for all the values of a and b that are consistent with a signaling equilibrium.

2. (a) A necessary and sufficient condition for cars of quality 1,000i to be offered for sale at price P is P > 1,000i. Fix s ∈ {1,2,…,n}. Thus all and only cars of quality i ∈ {1,…,s} will be offered for sale if and only if 1,000s < P < 1,000(s+1). Let m be the number of cars of each quality level and M the total number of cars. Then M = n m and the fraction of cars of any quality is m 1 = (uniform distribution). A buyer who purchases a car at a price P with 1,000s < P < mn n m 1 = 1,000(s+1) faces a lottery where he gets a car of quality i ∈ {1,…,s} with probability sm s (uniform distribution). Thus her expected utility if she buys is s 1, 000α s 1, 000α  s ( s + 1)  1, 000α ( s + 1) 1  i =ω − P + ω − P + ∑  α (1, 000i )  = ω − P + ∑   =ω −P+ s s  2  2  i =1  s i =1 (where ω is her initial endowment of money), whereas her utility if she does not buy is ω. Thus 1, 000α ( s + 1) she will buy if and only if P < . Hence necessary and sufficient conditions for all 2 and only cars of quality i ∈ {1,…,s} to be traded are (1) 1,000s < P < 1,000(s+1) and 1, 000α ( s + 1) . There is a P that satisfies both inequalities if and only if (2) P < 2 1, 000 s
is satisfies if and only if s ≤ 7. Thus there is an equilibrium where all cars of quality s +1 up to 7,000 are traded, so that the maximum number of cars that can be traded in equilibrium is 2,000(7) = 14,000. Such an equilibrium requires P to be such that 1, 000(1.76)(8) 7, 000 < P < = 7, 040. 2

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3. (a) The two contracts are represented as points H and L in the following figure: W

2 wealth in good state (no loss) probability of good state: 1−p

no insurance 45

o

line

L

W

P H

fair odds line for type L with slope p − L 1 − pL

A

indiff curve of type L indifference curve of H type

W W−x fair odds line for − pH type H, with slope 1 − pH

1

wealth in bad state (loss of x) probability of bad state: p

(b) H is the full insurance contract on the zero-profit (or fair odds) line for type H. This line has 1 p 1 slope − H = 43 = and goes through the no insurance point (W1 = 9,000, W2 = 15,000). 1 − pH 4 3 o 1 Thus it is the line of equation W2 = 18,000 − W1 . The intersection with the 45 line is given by 3 1 the solution to W1 = 18,000 − W1 which is W1 = 13,500. Thus CH is the full-insurance 3 contract with premium h = $1,500 (the solution to 15,000 − h = 13,500). To find contract CL we need the equation of the indifference curve of the H type through CH. This equation is 1 3 1 W1 + W2 = 13, 500 or W2 = 24, 000 + W1 − 103.28 W1 . Contract CL is given by the 4 4 9 intersection of this indifference curve and the zero-profit (or fair odds) line for type L. This line 1 p 1 has slope − L = − 109 = − and goes through the no insurance point (W1 = 9,000, W2 = 1 − pL 9 10 1 15,000). Thus it is the line of equation W2 = 16,000 − W1 . The intersection of the two occurs at 9 W1 = 9, 646.17 and W2 = 14,928.2 . Thus CL is the contract with premium h = 15,000 − 14,928.2 = $71.8 and deductible D = 15,000 − 71.8 − 9,646.17 = $5282.03. CH = (h = 1,500 , D = 0 ) CL = (h = 71.8 , D = 5,282.03). (c) it must be the case that the average fair odds line be below (or at most tangent to) the indifference curve of the L type through contract L. This amounts to saying that the fraction of H type in the population is sufficiently high.

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