Econ 404 A: Practice Questions
Yuya
Takahashi
7/2/
201
9
Solve
the
following
three
big
questions.
1. (45 points in total)
Answer the following questions.
(a) (5 points) Consider the market for tomatoes, which are considered to be homogenous products.
There are only two producers in the market. The cost functions of producer 1 and producer 2 are
given by
TC
1
(
q
1
) =
q
1
and
TC
2
(
q
2
) = 4
q
2
;
respectively. Let the demand be
p
= 10
°
Q:
Find the
pro°t level of each producer in a Cournot-Nash equilibrium.
(b) (5 points) Consider a Cournot game with homogenous products. There are
N
°rms in the market,
and
N
is °xed. The demand function is given by
p
(
Q
) = 100
°
Q
and the cost function is given
by
TC
(
q
) = 2
q:
First, °nd a competitive equilibrium outcome (the market price, quantity of each
°rm, and pro°t). Then, calculate the price, quantity, and pro°t in the Cournot-Nash equilibrium.
What is the relationship between the competitive equilibrium outcome and the outcome in the
Cournot-Nash equilibrium when
N
is large?
(c) (5 points) Consider two °rms producing a homogeneous product. Let
q
1
and
q
2
be the quantities
produced by °rm 1 and °rm 2, respectively. Demand is given by
Q
=
5
2
°
p
, where
Q
=
q
1
+
q
2
:
The
total cost functions of °rm 1 and °rm 2 are given by
C
(
q
1
) =
q
1
and
C
(
q
2
) = 2
q
2
;
respectively.
Suppose that competition is Bertrand and that each °rm has to choose its price level from a discrete
set with equal intervals,
P
=
f
0
;
0
:
1
;
0
:
2
; :::;
9
:
8
;
9
:
9
;
10
g
; that is, for instance, choosing
p
1
= 4
:
3
is
allowed but choosing
p
1
= 0
:
17
is not allowed.
Find a Bertrand Nash equilibrium.
Justify your
answer.
(d) (5 points) Explain why a Bertrand competition in di/erentiated product markets with identical
°rms (identical cost functions) does not achieve the
p
=
MC
condition.
(e) (5 points) Consider a two-°rm industry producing two di/erentiated products indexed by
i
= 1
;
2
:
The inverse demand functions are given by
p
1
=
3
°
2
q
1
°
q
2
p
2
=
3
°
q
1
°
2
q
2
:
Assuming zero production cost, calculate Cournot-Nash equilibrium prices (
p
C
1
; p
C
2
).
1

(f) (5 points) Consider a model of vertical product di/erentiation.
The conditional indirect utility
function is given by
u
=
8
<
:
°
i
z
j
°
p
j
if
i
buys product
j
0
if
i
does not by anything
where
z
j
is the quality of product
j; p
j
is the price of product
j;
and
°
i
measures how much consumer
i
cares about quality. Assume that there are only two products and that
z
1
= 2
; z
2
= 4
; p
1
= 0
:
5
;
and
p
2
= 2
:
Assume also that
°
i
is uniformly distributed between 0 and 1. Calculate the market
share of each good and the outside option.

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- Economics, Game Theory, Bertrand Nash