Final Exam Solns 212 Fall 2020 PDF

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Solutions to practice questions for the course...

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Economics 212

Steven A. Matthews

Fall 2020

University of Pennsylvania

Final Exam Solutions

Open book, notes, and the Canvas course page. Do not communicate with anyone or consult the internet except Canvas. Indicate your reasoning. 2 hours, 90 points.

1. (20 pts) Consider the following normal form game. L

C

T

0; 2

0; 2

M

0; 1

1; 1

B

1; 0

0; 1

R

1 1 ;

0; 3

0; 0

(a) (4.5 pts) Redraw the matrix, but without payo¤s. Instead, in each box write “PO” if the corresponding strategy pro…le is Pareto optimal, and leave the box blank if it is not Pareto optimal.

(One-half point for each correctly

labeled box.) Soln:

(M ; C ) and (M ; R ) are e¢cient: any move away from one of them

will make at least one player worse o¤. No other pro…le is e¢cient: for any other pro…le, a move to either (M ; C ) or (M ; R) will make at least one player better o¤ and no player worse o¤. (b) (7 pts) Find all rationalizable strategy pro…les. Soln: T

is strictly dominated by any mixture of

it. In the resulting game,

L

M

and

B;

so we can delete

is strictly dominated by any mixture of

so we can now delete it. The resulting 2

C

and

R;

 2 game has no strictly dominated

strategies. Hence, the rationalizable pro…les are

(M ; C ); (M ; R ); (B; C ); and (B; R ):

(c) (8.5 pts) Find all pure and mixed strategy Nash equilibria. Soln:

We know that a strategy pro…le is a NE i¤ it is a NE of the game

consisting of just the rationalizable strategies: C

R

M

1; 1

0; 3

B

0; 1

0; 0

The only pure strategy equilibrium is (M ; R): There is no equilibrium in which player 2 plays

C

For, if

M;

to

M

 2 (C ) >

is

R;

0; player 1’s best reply would be

contrary to

 2 (C ) >

0:

with positive probability. and player 2’s best reply

But there are mixed strategy equilibria of the form (p; R); where Any

p

2

[0; 1] is a best reply to

best reply to

i¤

p

Hence, (p; R) is a NE i¤

p

2

3p

R;

since

u1 (M ; R)

 ,  1

=

u1 (B; R ):

p

=

 1 (M ):

But

R

is a

1 3:

p

1

[ 3 ; 1]:

1 = 2 ; play the in…nitely repeated game based on the stage game shown below. Let hC = ((C; C ); (C; C ); : : :)

2. (20 pts) Two players, each with the discount factor



denote the outcome in which (C; C ) is played forever. Find all values of

which the grim trigger strategy pro…le that has outcome

C

h

equilibrium outcome.

Soln:

C

D

C

10; 10

0; 6

D

11; 0

x; x

In any subgame that follows the play of a

D;

for

x

is a subgame perfect

the grim trigger pro…le calls

for (D; D ) to be played all the time, i.e., at any date after any history of that subgame. Thus, in such a subgame a player should always play his myopic best reply to

since he cannot a¤ect the future play of the other player.

D;

Hence,

(D; D ) must be an equilibrium of the stage game. This gives us one restriction: x



0:

Now, player 1 will not want to make a one-shot deviation from playing

C

if and

only if

 

10 1



11 + 

 ,  ,  x

1

20



9

11 + x

x;

1 = 2 : Similarly, player 2 will not want to make a one-shot deviation from playing C if and only if

using the fact that player 1’s discount factor is

 

10 1

Conclusion:



6+

 ,  ,  x

1





20

6+x

14

x:

The grim trigger pro…le is a SPE if and only if

x

2

3. (25 pts) Consider an industry with the inverse demand function where

Q

is the sum of the outputs

q1

and

q2

[0; 9]: P (Q )

=

a



Q;

of the two …rms in the industry.

Each …rm can produce any nonnegative amount of the output at zero cost. In addition, before they make their output decisions, …rm 1 can advertise to increase demand: it chooses 1

= (a

  q1

q 2 )q 1

1 = 48 a3 : Thus, given (a; q1 ; q2 ) their payo¤s are C (a) and  2 = (a q1 q 2 )q 2 :

at cost



a

C (a)

 

(a) (5 pts) Suppose …rm 1 chooses that …rm 2 sees both

q1

and

a

q1

at the same time as it chooses

before it chooses

q2 :

a;

and

Describe the nature of a

strategy for each …rm in this game. Soln:

A strategy for …rm 1 is a pair of numbers, (a; q1 ): A strategy for …rm

2 is a function possible (a; q1 ):

s2

:

R2+ ! R that speci…es an output 2

q2

=

s2 (a; q1 )

for each

(b) (20 pts) Find the subgame perfect equilibrium, and its outcome, of the game in (a). Denote the SPE as (a ; q1 ; s2): Use backwards induction to …nd it. As

Soln:

shown in the lecture slides, the sequentially rational strategy of …rm 2 is s

 1 2 (a; q1 ) = 2 (a



q 1 ):

(1)

Firm 1’s best reply to s2 is found by solving the problem max(a a;q1

 



s2 (a; q1 ))q1

q1



C (a)

= max 1 aq1 2

a;q1

  1 2 2 q1

1 3 48 a :

At the solution, the derivatives of the objective function with respect to and

q1

are equal to zero:

 

a

3 2 48 a = 0;

1 2 q1 1 2a

= 0:

q1

The solution of these two equations is (a ; q1 ) = (4; 2): So the SPE is (a ; q1; s2 ) = (4; 2; s2 ); where s2 is the function de…ned in (1). The SPE outcome is 

(a







;q ;q ) 1 2

= (a







; q1 ; s2 (a ; q

 1 )) = (4; 2; 1):

4. (25 pts) Two bidders participate in an all-pay common-value auction for an oil tract. Bidder

i

privately observes a signal s ~i : The signals s ~1 and s ~2 are uniformly

and independently distributed on [0; 1]: The value v ~ of the oil tract to whoever wins it is the product of the signals: v ~

If bidder is v ~



b:

bids

i

b

and wins (i.e.,

b

If she loses her payo¤ is

=s ~1 s ~2 :

is larger than the other bidder’s bid), her payo¤



b:

There is a strictly increasing di¤erentiable bid function is an equilibrium for each bidder

i

to bid

b (s i )

b

: [0; 1]

!R

such that it

when her signal is s ~i =

Derive

si :

this bid function. Soln: b(s)

1

= 3 s3 :

^ and bidder 2 uses the bid function bidder 1 bids b; ^ b( ): Then bidder 1 wins if b(~ s2 ) < b; or rather, if s ~2 < b1 (^ b) (since b( ) is strictly



Derivation.

Suppose s ~1 =



s;

increasing). Thus, the probability that bidder 1 wins is

 Pr

s ~2 < b

1 ^ (b)



 =

F

b

1 ^ (b)

 =

b

1 ^ ( b ):

The expected value of the oil tract to bidder 1 conditional on s ~1 = winning, i.e., on s ~ < b1 (^b); is

s

and on

2

E

h s ~1 s ~2

j

s ~1

1 ^ (b)

=

s; s ~2 < b

=

s; s ~2 < b

i

b

1 ^ ( b)

=

s



b 1 ^ ^ b = b ( b )s

2

:

So bidder 1’s expected payo¤ is

 Pr



^ b(~ s2 ) < b

E

h s ~1 s ~2

j

s ~1

3

1 ^ (b)

i

1 ^ (b)

2



^

b:

Let s ^ be the signal bidder

1

^ were to be her equilibrium would have to observe if b

bid, i.e., let s ^ be de…ned by b(^ s) the expression above for bidder terms of s ^

: s ^s

For b(

) to

s ^

2

= ^ b:

1’s

Then s ^

2 1 ss 2 ^

be the equilibrium bid function, s ^

0=

d ds ^

1

2 ^ 2 ss

b

1 (^

b): Substituting these into

expected pro…t gives us her expected pro…t in

 b(^s) =

pro…t. The …rst-order condition s ^

=

=

 b(^s): =

s must maximize this expected

s must satisfy is

  b(^s) s^=s = [ss^  b(^s)]js^=s = s2  b0 (s):

Hence, b for all s

(s ) =

s

2

2 [0; 1]: Integrating both sides yields b (s )

If b(0) >

0;

 b(0) =

a bidder with signal s

in which case she’d prefer to bid b(0)

0

= 0:

0

Z

= 0

s t

2

0

dt

would get the negative pro…t

to get

0

pro…t.

Thus, the equilibrium bid function is b(s)

4

1

= 3 s3 : 0

 b(0),

This contradiction implies

1

= 3 s3 :...