ECON 2160 Lecture Notes - Lecture 3: Bertrand Competition, Substitute Good, Inverse Demand Function
2 May 2018
School
Department
Course
Professor
Cournot
.
Stackleberg.
8r
Bertrand
Firmsrn
olrg'opoly.
Few
firms.
interdependent.
Firms
in
olrgopo‘ly
have
three
general
strategies
they
can
follow
regarding
the
other
firms
or
its
rivals.
I
know
-
Doesn't
take
the
other
firm's
actions
into
consideration
at
all.
2
Collusion
—
Agrees
to
lulty
cooperate
with
the
other
firms,
forming
a
monopoly
to
supply
the
market
3.
Compete
-
The
firms
can
compete
either
through
quantity
or
price.
l'he
lorms
of
Competition
There
are
two
'classlc'
models
of
simultaneous
decisions
we
can
study.
In
the
first:
Cournot
olrg’opoly
(Core
no)
in
this
model,
firms
take
the
market
price
as
given
and
compete
on
the
quantity
supplied
to
the
market.
The
second:
Bertrand.
in
this
model,
firms
compete
on
price
and
let
the
market
determine
the
quantity.
The
model
where
we
have
sequential
decisions
or
a
"leader-follower”
scenario
is
called
the
Stackleberg
model.
Before
we
begin
to
investigate
these
three
models,
let’s
take
a
bit
of
time
to
review
the
firm's
problem.
How
a
firm
chooses
its
output
or
price
to
maximize
profit.
We
are
working
with
oligopoly
markets.
Unlike
perfect
completion
or
highly
competitive
markets,
the
firms
have
some
control
over
their
pricing.
However,
there
a
only
a
few
firms
in
the
market.
They
may
produce
products
that
are
perfect
substitutes
or
imperfect
substitutes.
There
must
be
some
substitutability
in
the
products.
If
not,
then
the
firms
would
not
be
in
competition
with
one
another
and
may
be
monopolies
instead.
The
few
firms
part
is
important,
because
if
there
were
many
firms
we
would
find
our
monopolistic
models
to
be
a
better
fit
to
describe
the
market.
The
firm’s
problem.
The
firm
will
maximize
its
profit
function
given
the
constraint
of
its
cost.
A
quick
example.
The
inverse
demand
the
firm
faces
is
P:
100-Q
and
the
firm
has
a
cost
that
equals
100
The
firms
profit
is
then
its
total
revenue
minus
its
total
cost
or
II
=
PQ
—
Cost
for
our
example,
in
terms
of
quanity,
profit
could
be
expressed
as
1r
=
(100
—
Q)Q
-
100
=
100Q
-
Q2
-
100
=
900
—
Q2
In
terms
of
price,
our
profit
function
would
look
like
PM
Hum
all
radium;
.n
or
gum};
but
hr
may
9N6
yw
son’bfl-
Mow
LJVTGll’
ycudcm
WQVXHJC’O
don‘t
do
rm,
mole
Ming
We
need
the
demand
equation
as
a
function
of
price,so
the
invese
of
the
inverse
demand
-
1qu
the
demand
function.
Paroo-o
noeroo-P
n
a
Price
-P)
—
10(100—
P)
It
=100P
—
P‘
-1000+10P
=110P
—
P‘
—1000
v:
.y'rlflmég;
V
v
out-W
We
then
mumue
whatever
function
we
are
working
with
by
taking
the
first
derivative
in
relation
to
the
variable
with
which
we
are
concerned,
setting
it
equal
to
zero,
and
solving
that
equation.
r
z-jv
.7'
fir
'
let's
first
do
this
question
in
terms
of
the
firm
maximiting
its
profit
in
by
choosing its
output
or
its
quantity.
d900
—Q‘
do
90:20
0:45
=9o—2Q=0
A
“gigs-mm
“M
Find
the
price
by
going
back
into
the
demand
equation
F:
100—4S=$55
u‘
S
a
find
the
firm’s
profit
ft
=
45(55)
-
10(45)
=
$2025
Doing
the
same
problem
but
using
profit
expressed
as
a
function
of
price
dIOOP—Pz—IOOO
dQ
P=$55
Q=100—P
Q=100——55=45
=110—2P=0
We
could
find
the
profit
with
the
same
equation
above,
but
let’“
5
use
or
profit
function
in
terms
of
price
just
for
kicks
II
=
110P
—
P2
—
1000
TI
=
110(55)
—
552
—
1000
=
$2025
Collusion
Relatively
easy
to
solve.
Assume
the
when
the
firms
agree
to
fully
cooperate,
they
share
technology
giving
them
all
the
same
cost
function.
P=100-—Q
whereQ=q1+q2
Costl
=
Costz
=
10q
If
I
‘
1
ll‘llrllll'lltr;
since
the
[mm
are
now
a
monopoly.
we
simply
solve
the
monopoly
problem.
(or
you
can
just
maximize
ll"
Dtolit
lunction
You‘ll
notice,
in
this
use.
it it
the
same
as
our
firm
problem
example
above.
The
only
change
mu
have
to
maiie
n
dm‘de
the
profit
by
the
number
of
firms.)
TR
=
P0
=
(lOO—Q)0
=1000-Q‘
MR—dTR-IOO
2
..
dQ_
Q
Profit
is
maximized
where
MR=MC
100
—20
=
10
Q
=
45
q:
=
q:
=
22-5
P‘
=
100-
45
=
555
n,
=
n,
=
22.5(55)-10(22.5)=
$1012.50
Competition
7
(Qournot
let‘s
start
with
Cournotwhere
the
firms
make
their
choices
at
the
same
time
and
compete
on
quantity.
The
question
we
are
trying
to
answer
is
how
much
each
firm
will
produce.
Example:
P
=100-Q
whereQ
=q1+q2
C
ostiz
10q1
C
ostz:
20q2
Notice
our
inverse
demand
function
is
going
to
include
information
from
both
firms.
it
becomes
P=
100—(q1+q2)=
100-q1-q2
Firm
1
takes
Firm
2’s
output
as
a
given
(it
has
no
control
over
what
the
other
firm
produces).
Firm
1's
profits:
"'1
=
qu
_
CI
=
(100
—
q1
—
‘12)‘11
‘
10‘11
=
90q1
-
q?
-
qlqz
Maximization
problem
for
firm
1
(notice
you
have
a
partial
derivative.
You
treat
the
quantity
produced
b
the
other
firm
as
a
constant
:
y
)
'
371
chnm
.flnwx
*erp,(,.m1mr
MW
3>
subsmma-
now“
4)
"fwd
w"
(Ml—90
2
—0
6%"
Q1
‘12-
2%:
90-42
—
90‘qz
‘11-
2
=
R1012)
