Final Exam
Economics 501b
Microeconomic Theory
Dirk Bergemann
May 2006
Suggested Solutions
for
Final Exam Microeconomic Theory 501b
2005/06
1. Consider the following model of Cournot competition with di¤erentiated products and I
…rms. The price for the product of …rm i is given by
X
qj
pi (q) = a qi
j6=i
with 0 <
< 1. The marginal cost of production is equal to zero.
1. De…ne the notion of iterative elimination of strictly dominated strategies for this game.
The process of iterated deletion of strictly dominated strategies proceeds as follows: Set
Si0 = Si . De…ne Sin recursively by
Sin = si 2 Sin
1
@s0i 2 Sin 1 , s.th. ui (s0i ; s i ) > ui (si ; s i ) , 8s
set
Si1
=
1
\
i
2 Sni 1 ;
Sin :
n=0
Si1
The set
is then the set of pure strategies that survive iterated deletion of strictly
dominated strategies.
2. Find a Nash equilibrium of the Cournot game. We can look for the symmetricequilibrium
by solving the …rst order conditions
qi (a
qi
q i)
and hence
(a
2qi
q i) = 0
or
(a
2q
(I
or
1) q) = 0
a
.
(I 1) + 2
q=
3. Is the Nash equilibrium unique? Yes, it is unique, the best response is strict and hence
there do not mixed strategy equilibria, and asymmetric equilibria do not exist either as
the matrix of best responses is of full rank, and hence permitting only a unique solution.
4. The process of iterated deletion of strictly dominated strategies either starts and ends
at the unique outcome and it does not lead to any prediction over the set of feasible
strategies. We recall that this is a game of strategic substitutes, so that the elimation
starts with:
h ai
.
S1 = q1 ; q1 = 0;
2
The remaining question is then when is
BRi
or
a
a
; ::::;
>0
2
2
1) a2
(I
a
qi =
2
2
or
1>
(I
1)
2
>0
;
and hence
(I
1) < 2
is a necessary and su¢ cient condition for the process of iterated deletion of dominated
strategies to converge to the unique Nash equilibrium. Otherwise the process stops at
S1 .
2
2. (45) Consider the following card game between two players, Ann (A) and Bob (B). Each
player has four cards: one, two, three, and a joker. Ann wins the game if there is either a
match of jokers (two jokers played) or a mismatch of numbered cards (one, two for example).
In all other circumstances, Bob wins the game. Ann and Bob have to simultaneously choose
a card, and the winner receives $1 from the loser according to the above rules of winner
determination.
1. De…ne and carefully write down the normal form of this game:
Ann = Bob
1
2
3
J
1
1; 1 1; 1 1; 1
1; 1
2
1; 1
1; 1 1; 1
1; 1
3
1; 1 1; 1
1; 1
1; 1
J
1; 1
1; 1
1; 1 1; 1
2. Completely describe the set of pure strategy equilibria. It is immediate to verify that a
pure strategy equilibrium does not exist in this game.
3. Completely describe the set of all strictly mixed strategy equilibria. Does the game
favor either Ann or Bob? There is a unique mixed strategy equilibria, and it is almost
symmetric. It puts equal probabibility on the numbers and a di¤erent probability on
the joker. The mixed equilibrium is unique and identical across players 51 ; 51 ; 15 ; 25 . As
you might have guessed by jsut looking at the number of winning …elds, Ann is at a
disadvantage and will win 40% of all the times.
3. (45) Consider a second price auction with two bidders and an entrance fee f . The valuation
for each bidder is uniformly distributed in [0; 1]. The timing is as follows. First the seller o¤ers
a second price auction with an entrance fee f > 0 which is a fee that the bidders have to pay
if they participate in the auction, independently of the outcome of the auction. Second, given
the auction format, the bidders have to simultaneously decide whether or not to participate,
and if so, how much to bid. If they don’t bid at all (or equivalently bid zero) then they don’t
have to pay the fee but will also never get the object.
1. Describe the payo¤ function of each bidder as a function of an
8
< vi bj f if bi > bj > 0
vi f if bi > 0; bj = 0
ui (bi ; bj ) =
:
0 if bi = 0
arbitrary pair of bids.
9
=
;
2. De…ne the notion of a strategy and the notion of a Bayesian Nash equilibrium for the
auction game with the entry fee. Describe the expected payo¤ of the auctioneer as a
function of the bidding strategies and the reserve price.
BN E : (b1 (v1 ) ; b2 (v2 )) : Evj [ui (bi (vi ) ; bj (vj ))]
3
Evj [ui (b0i ; bj (vj ))] 8b0i ; 8vi
E [ (f )] = f (Pr (bi > 0) + Pr (bj > 0)) + Pr (min fbi ; bj g > 0) min fbi ; bj g
3. Does the imposition of an entrance fee change the bidding strategy of the agents?
No, because conditional on bidding (b positive) the entry fee is sunk and the game
looks like the standard second price auction. So if bi > 0 then bi (vi ) = vi is a
dominant strategy.
4. Compute the symmetric Bayesian Nash equilibrium.
Look for an equilibrium where each player uses a cuto¤ value: that is, if vi > vi then
player i places a bid (of exactly vi ). So if player j is using cuto¤ vj ; the expected
payo¤ of type vi from participating in the auction is
=
f + vi Pr vj vj
Z vi
(vi
f + vi vj +
+ Pr vj
vj
vi
vi
Evj j vj 2 vj ; vi
z) dz
vj
=
1
1
f + vi2 +
v
2
2 j
2
Clearly, this expression must be equal to zero for vi = vi : Imposing symmetry
vi = vj = v
f + (v )2 = 0
p
v =
f
Note that the cuto¤ type values the object more than f:
5. Compute the seller’s expected revenue from the auction with an arbitrary entrance fee
f > 0.
The seller’s expected revenue, including the fee is
Z 1 Z v
p
f + 2 p p xdxdv
2f 1
f
=
2 3
f2
3
f + 2f 1
p
f
f = E [ (f )]
6. Which entrance fee maximizes the expected revenue of the auctioneer? Is the resulting
allocation ex-post e¢ cient (assume the seller’s valuation of the good is zero)?
arg max E [ (f )] =
f
1
4
Obviously ine¢ cient, p
because it may fail to assign the object to those who value it
more, in the event of f > max fvi ; vj g
4
4. (45) Consider a …rm that can invest an amount I in a project generating a high observable
cash ‡ow C > 0 with probability and 0 otherwise. This probability depends on the …rm’s
e¢ ciency (type) 0 < l < h < 1: Let Pr ( = l ) = . The …rm needs to raise I from external
investors who do not observe the value of . Assume that l C I > 0: The external investors
are perfectly competitive.
1. Suppose that …rms can only promise to repay an amount R chosen by the …rm (with
0 R C) when cash ‡ow is C and 0 otherwise.
1. For this game carefully de…ne the notion of Perfect Bayesian Equilibrium.
Let (R) = ( h jR) : Then
P BE = fR ( ) 2 R+ ; (R) 2 [0; 1] ; I (R) 2 f0; Igg
s:t: I (R) = I 8R : E (R) R I
s:t: 8 ; R ( ) 2 arg max fE [C I (R)]g
R
s:t: if 9 : R ( ) = R )
p( )
( jR) = P
f 0 :R( )=Rg
p ( 0)
2. Can you …nd a separating PBE? In other words, can a good …rm signal its type?
Suppose in a PBE the two types were o¤ering di¤erent repayment amounts. By
the competitive markets assumption, these would have to make the investors
break even (in expectations). So they would have to be
Rh =
I
h
Rl =
I
l
The beliefs would be zero-one following each of these repayment o¤ers. But
then the ine¢ cient l type would want to o¤er Rh and only repay it with lower
probability - a pro…table deviation.
2. Suppose now that the …rm also has the possibility of pledging some assets as a collateral
for the loan. Should a “default” occur (the …rm being unable to repay R), an asset of
value K to the …rm is transferred to the creditor whose valuation is xK with 0 < x < 1.
The size of the collateral K is a choice variable of the entrepreneur.
1. For this extended game, de…ne the notion of a Perfect Bayesian Equilibrium.
P BE = f(R ( ) ; K ( )) ; (R; K) ; I (R; K)g
s:t: I (R; K) = I 8R; K : (R; K) R + (1
(R; K)) xK I
s:t: 8 ; (R; K) ( ) 2 arg max f [C I (R; K)] K (1
)g
R;K
s:t: if 9 : (R; K) ( ) = (R; K) )
5
( jR; K) = P
p( )
f 0 :(R;K)( )=(R;K)g
p ( 0)
2. Give a necessary and su¢ cient condition for the least cost separating equilibrium to
exist. How does it depend on and x?
The least cost separating equilibrium is one in which the bad …rm is identi…ed
to be such. Thus it puts down no collateral and o¤ers repayment Rl : There’s no
need to have the ine¢ cient …rm incur a cost to be recognized. The e¢ cient …rm
will have to put down as collateral the lowest K that won’t induce the l type
to mimic it. When the e¢ cient …rm puts down collateral K; the investors break
even at a lower repayment rate. This is given by
R
:
hR
I
R =
+ (1
(1
h ) xK
=I
h ) xK
h
note that the …rst condition implies that …rm h has to repay more than I in
equilibrium.
The relevant inequalities for the separating equilibrium are then:
l
C
I
C
l
I
(1
l
h
C
h ) xK
K (1
l)
K (1
h)
h
I
h
I
C
(1
l
h ) xK
h
note that the RHS of both inequalities is decreasing in K; but the RHS of the …rst
one is decreasing faster. So when the IC constraint for the l types is binding,
the IC of the h types is slack. Thus, the critical value for K will be
h
K =I
h
x
l
(1
l
x)
h l
This value depends positively on x (the more the investors value the collateral,
the less the e¢ cient …rm has to pay, the more attractive a deviation for the
ine¢ cient one). Now check that at this collateral level, the high-e¢ ciency …rm
would like to separate itself. At a least-cost (K = 0) pooling equilibrium, the
…rm obtains
I
h C
) h
l + (1
therefore the condition is
h
C
I
(1
h ) xK
K (1
h)
h
C
h
I h
l + (1
6
)
I + K (1
h
I
l + (1
)
h ) (1
x)
h
the higher ; the more pro…table it is for the high-cost …rm to separate itself.
Same thing for higher values of x (easier to put up collateral).
7