Basic MONOPOLY MODEL
1
2
For the long run, equating MR = LMC gives:
Let the market demand
curve be:
" v!a %
Q=$
# 2s + b '
&
"
P = $v !
#
P = v ! sQ
or:
&v'P#
Q=$
!
% s "
#
(2)
s (v ! a) %
(3)
2s + b '
&
use a = a and b = b for ( 0 < Q < ! )
!
!
" use a =
Demand Shifts in the BERTRAND MODEL
Nash equilibrium An outcome is said to be a Nash equilibrium if no firm would find it
beneficial to make a change provided that all other firms do not
make a change from their situations at this current outcome.
Short R
Innovation in the Spence Model
Deriving Our Rule for Fixed Cost
~
Q
Fig. 1
0Q
a
a0
a1
1Q
Fig. 3
D
MR
Qb
LAC
Q
Qa
LAC
LMC
LMC
Fig. 2
K cost
~
Q
as a i , SXCi
L cost
TVC
0
a0
a1
a2
D
MR
QM
Q
0
Q
Proof that Spence > Limit
Proof that LXC > SXC
Fig. 3
Fig. 1
Q
XC
a
Q
a
LMC
(or a)
K cost
L cost
MR
0
Qb Excess Q
0
capacity
(either model)
Q
Fig. 4
SAC
Q
a
Fig. 2
Q
d
a
a
(or a)
Q
Fixed cost
a
f
b
LMC
c
TVC
0
Qsr
0
Q
e
MR
QM
Qa
Qb
Fig. 5
Q
Q
a
a
s
d
f
b
Why qx = qy in Equilibrium in the COURNOT MODEL
1
Mathematics for the COURNOT MODEL
(with two firms, firm X and firm Y )
(with two firms, firm X and firm Y )
Period 1
Y sets price
p1
(short run and long run market conditions)
(
P = v s qy + qx
demand curv
3
4
The Difference between Firms Quantities
The Industry Demand Curve (continued)
If we again define the overall quantity demanded in the industry as Q = ( qy + qx )
but now add equations 1y and 1x assuming the firms charge a different price:
What is the
5
6
Product Differentiation and the Value of
Monopolistic Competition
Fig. 1Each firm wishes to raise price
P
P1
P0
1
1
i
0
MC
0
D
d0
mr
qi q1 q0
0
d2
Q
7
8
Monopolistic Competition
Monopolistic Competition
Fig. 2Each firm wishes to lower price
Fig. 3Neither firm wishes to change price
P
P0
P1
P
0
0
i
1
Pf
1
MC
f
f
D
MC
d
0
df
mr 0
mr f
d2
q0 q1 qi
D
Q
qf
d2
Q
The Long run COURNOT solution
7
8
Things to note
1 Using the equilibrium , one of the (n+1) pieces are accounted for by the
horizontal distance between df and mrf , the rest by the n firms.
# 1 &
2 Pcour = " + %
!
(v ) " )
$ n + 1(
'
" n %
Q as n increase
3
Monopolistic Competition Price Reaction Function
Using the Price Reaction Function
The Price Reaction Function Derived
!p $
To find the equation for Ys price reaction function, substitute qy = # y &
"
%
(from the previous page) into equation 1y) to get:
1
MONOPOLY Responses to changes in Demand
Two ways to increase demand:
Parallel shift (increase v)
2
Some Range III results
Rotation (decrease s)
In Range III, we have a = a and b = 0 since ( Q ! " ) so:
"v ! a %
Q=$
'
# 2s &
(2a)
"
s (v ! a ) %
"v + a%
P
BERTRAND MATHEMATICS
Long run solutions for two different demand curves (demand 0 and demand 1):
"v !a%
Q0 = $ 0
# s0 '
&
"v !a%
Q1 = $ 1
# s1 '
&
Short run solutions for demand 1 (after initial equilibrium at demand 0) :
Supply:
! a $
Psr = # & Qsr
" Q
VARIABLE RETURNS TO SCALE PRODUCTION AND COST FUNCTIONS
The simplest, widely used production function that has classical properties and constant
returns to scale (CRTS) is the Cobb-Douglas form:
Q = D LK
(1)
where Q , L and K represent output, labor and c
VRTS COST CURVE MODEL
Suppose that the firm has made long-run plans to produce Q = Qi . As long as it is minimizing long run cost it will have average
and marginal costs of LACi and LMCi . If not, it must use the SACi and SMCi curves associated with this
The role of technical progress in the Dominant Firm Model
5
6
Entry in the Dominant Firm Model
We can rewrite equation 3 as:
qdom
1 a + s
pdom
= .5
=
<0
a
2 sa
a
1 v
a
Pdom = + qdom =
( v s qdom ) = ( v s qdom )
s a s
a + s
qdom 1 a
3
Mathematics of the Dominant Firm Model
Let the market demand curve be:
P = v sQ
Define:
v P
Q=
s
or
4
More on the DOMINANT FIRM MODEL
3 = ( pdom a ) ( qdom )
1 = (a a ) Q
Equation for S2 :
To eliminate the other firm(s) the dominant firm must set p
3
4
from Firm 2 but cannot. We call this difference the residual demand . Thus Firm 1 has
maximum output of q1 = 30 and, at p = 180, profit less than = 1650 . It would be very
unhappy and have little choice but to match p = 170.
Now the two firms will be
COMPETITIVE DUOPOLY (the Bertrand Model)
1
Profit and the Price Taking Firm
Two conditions define a price taking firm:
Price-taking firm 1 The firm cannot (or will not) change price from current P.
2 The firm can sell more than it wishes to sell at curren
Mathematics of the Limit Price and Spence Models
1
2
Limit Price and Spence Models PRACTICE QUESTION
Consider a situation where : v = 40, s = .1, = 80, a = 22 and a = 24 . What will the
Limit Price and Spence Models predict? Find price and profit in each
THE COURNOT MODEL OF OLIGOPOLY
In order to avoid competitive price cutting, one firm sets
the market price and the other firm accepts that price. The
price setting firm leaves a fixed quantity of output (sales)
for the other firm and then adjusts market p
Technique for finding Short run COURNOT reactions
4
3
Nash equilibrium An outcome is said to be a Nash equilibrium if no firm would
find it beneficial to make a change provided that all other firms
do not make a change from their situations at this curren
5
Finding a Long run COURNOT solution
2
6
qy
Beginning at an initial short run equilibrium as described above, there are two
different ways that long run Cournot adjustments might occur. We can use the
diagram below for either.
SMC
p
qy
sr
D
df
LAC
mrf
q
RETURNS TO SCALE AND COSTS
Production function:
To find short run costs, first solve for L with K = K : L = g (Q, K )
Then write short run total cost, average cost and marginal cost as:
Q = f ( L, K )
STC = rK + wL = rK + wg (Q, K )
Q
STC
rK wg (Q, K )
Simple Model for Two DIFFERENTIATED PRODUCTS
1
Suppose that two firms in an industry each produce and sell a somewhat
different product. The products are, however, similar enough that it makes sense to
add their quantities (two brands of cars, two types o