MEPARTMENT
ICROECONOMICS
D
OF E CONOMICS
PROFESSOR VASILIKI SKRETA
SAMPLE MIDTERM
M ICROECONOMICS , S PRING 2010
!"#$ % &'!""""""""""""""""""""""""""""""""""""""""""
P ROBLEM S ET 10: G AME T HEORY A ND B ETRAND C OMPETITION
()* !"#$+"""""""""""""""""""""""""""""""""""""""""""
DUE in Class on April 14 (Skreta Sections) or April 15 (Bowmaker and Collard-Wexler Sections)
&,*-./0-12,*+ (3$ /*$ 24 05"** ,2-$*6 -$7-8229* 1* ,2- :$.#1--$;< =5$"*$
>.1-$ 5$?185@A @2/ >155 .$0$1B$ 0.$;1- 42. -3$ :".-* 24 @2/. >2.9 -3"- ".$
5$?185$< C$"; -3$ D/$*-12,* 0".$4/55@
",; :.2B1;$ 02,01*$ 8/- E/*-141$; ",F
1. S HORT A NSWER P ROBLEMS
*>$.*< GHIJ )KK LIMC JICN< O22; K/09P
1. Let us assume the following game where player 1 and 2 think about playing UP or DOWN. Are there any
dominant strategies in this game and what is a dominant strategy? What is the Nash equilibrium in this
==".- &+G32.- ),*>$. Q/$*-12,*
game?
#$ ! "#!$ %&' %" (
1\2
Up
Down
Up
10,5 7,6
! "# )* +,- *.&/- &%""""""""""""""""""""""""""""
Down
4,9
2,5
2. True or false. Entry deterrence can only occur in a game where one player moves before the other.
0$ "& #!' %&' (" #
3. Take the game described in the following matrix. And let player 1 now move before player 2. What will be
"& # )* +,- *.&/- &%"""""""""""""""""""""""""""""
the credible Nash equilibrium? Use the extensive form of the game to solve this.
1$ ") & 2$ %&' %" #
1\2
Up
Down
Up
10,5
7,6
Down 4,9
2,5
") & )* +,- *.&/- &%"""""""""""""""""""""""""""""
3$ 4'5- &' %6.*-7 8&+)96+-$ :,-; ;&8);6. <6=- );>'-6*-7 <&'?-'* 6'- 6.<6@* A-++-' &%$
2. L ONG -A NSWER P ROBLEMS
B$ 4'5- &' %6.*-7 8&+)96+-$ 4,- 86'=);6. '6+- &% *5A*+)+5+)&; )* C ); =-;-'6. C ;&+ >&;*+6;+
6.&;= +,- );D)E-'-;>- >5'9-$
QUESTION 1
1. Rock, Paper, Scissors!
#
Background: jan-ken-pon (http://en.wikipedia.org/wiki/Rock_paper_scissors)
1
4. Rock, Paper, Scissors!
Background: jan-ken-pon ( http://en.wikipedia.org/wiki/Rock_paper_scissors )
Pokemon (aka, the ‘Underdog’)
Rock
Paper
Rock
(0,0)
(-1,1)
Paper
(1,-1)
(0,0)
Scissors
(1,-1)
(-1,1)
(0,0)
Pokemon
(aka,
the ‘Underdog’)
Pokemon
(aka,
the ‘Underdog’)
Pikachu (aka, ‘Princess’)
Pikachu
(aka, ‘Princess’)
Rock
Rock
Paper
(0,0)
(1,-1)
Scissors
(-1,1)
(1,-1)
Scissors
(-1,1)
Rock
Paper
a)In the payoff schedule, it isPaper
stipulated
that
the
winner
receives
$1
from
the loser,
(0,0)
(1,-1)
(-1,1)Rock
(0,0)
(1,-1)
under the hierarchy (akin to the ‘intransitivity’
of
what
is
stronger,
rock
bends
(-1,1)
(0,0)
Scissors (1,-1)Paper
(-1,1)
(0,0)
scissors, while scissors shears through paper, and paper covers rock), can you
produce a flow diagram of the one dollar passed on for each scenario? Hint: can you
(a) What is the Nash Equilibrium in Pure Strategies?
see how this matches the payoffs in the matrix?
Scissors
(1,-1)
(-1,1)
(b) What are the mixed strategies in this game?
Scissors
(-1,1)
(1,-1)
(0,0)
b)What can you state about the zero profiles along the diagonal and what payments
Pikachu
areStrategies
made
if they(aka,
adopt‘Princess’)
the strategy? Can you illustrate the flow diagram for these
2. Credible
kinds of ties?
player A\player B L
M
R
a)In the payoff schedule, it is stipulated that the winner receives $1 from the loser,
L
0,10
under
the hierarchy
to not
the-10,-10
‘intransitivity’
of10,20
what is stronger, rock bends
c)List all the
strategic
profiles (akin
that are
stable.
M scissors shears 10,0
-20,-20and-5,15
scissors, while
through paper,
paper covers rock), can you
a flow
diagram of the 20,10
one dollar
passed -30,-30
on for each scenario? Hint: can you
d)What areproduce
the mixed
Rstrategies?
15,-5
see how this matches the payoffs in the matrix?
(a) Are there any dominant strategies in this game?
canEquilibrium
you state about
profiles
c) What isb)What
the Nash
and the
whyzero
or why
not?along the diagonal and what payments
are made if they adopt the strategy? Can you illustrate the flow diagram for these
kinds
of ties?
7)Bon
Jovi
,
Ozzy
Osbourne,
Shania Twain?: B Or S (B.attle O.f the S.exes!):
(c) Find the pure
strategy
Nash equilibria.
(b) Are there any dominated strategies?
(d) Find the credible
of theprofiles
game if player
A move
first.
c)Listequilibrium
all the strategic
that are
not stable.
(e) Find the credible equilibrium if player B moves first.
d)What are the mixed strategies?
QUESTION 2: B ERTRAND - T HE H OMOGENOUS P RODUCTS P RICE G AME OF O LIGOPOLY T HEORY
c) What
is the
Equilibrium
and firms
why or
whysimultaneous
not?
Following the Bertrand
model
of Nash
oligopoly,
assume that
make
price decisions with
constant marginal costs Suppose that there are two producers of Cement for the U.S. market. The demand
7)Bon Jovi , Ozzy Osbourne, Shania Twain?: B Or S (B.attle O.f the S.exes!):
for cement is:
Q d = 100 −
p
4
There are two producers, Holcim which produces in the United States and Cemex which produces in
Mexico. Holcim has costs T C = 20Q, while Cemex which has to import Cement from abroad has costs
T C = 40Q.
(a) What are the marginal costs of each firm?
(b) What is Cemex’s best response given the price that Holcim is charging? Remember that there are 3
cases we need to consider...
2
(c) What is Holcim’s best response given the price that Cemex is charging? Note how Cemex and Holcim’s
best responses are different.
(d) Plot Holcim and Cemex’s Best-Response Curves. What will be the Nash Equilibrium in this Game? At
the Nash Equilibrium what is the quantity of cement imported from Mexico.
(e) What would happen if Cemex was unable to ship cement to the U.S. so Holcim was a monopolist.
Compare consumer surplus when Holcim is a Monopolist versus when it competes against Cemex.
(f) What is the Nash Equilibrium if Cemex’s Cost Function is C (Q) = 240Q?
3