Lecture10:Competitive Market
local equilibrium theory I
Content
Competitive equilibrium
Local analysis
Complete compete market
Monopoly
Competitive equilibrium
An allocation A= (x1,…xI;y1……yJ) is a
combine of consumption vector x and
production vector y,A is feasible if
11
,f o r a n y 1,
IJ
l i l l j
ij
x y l L?
??
? ? ??? L
IJL??
Competitive equilibrium
Pareto Optimal ( Pareto efficient ),
An allocation is Pareto
efficient ( optimal ) if there isn’t any the
other feasible allocation,
made for any i and
for some i,See the fig,
11(,;,)IJx x y y? ? ? ?LL
11(,;,)IJx x y yLL
( ) ( )iiuu? ?xx
( ) ( )iiuu? ?xx
Competitive equilibrium
Competitive equilibrium,
? An allocation and price
are a competitive (Walrasian) equilibrium,
if,
?Profit maximization
?Utility maximization
?Market clearing
11(,;,)IJx x y y? ? ? ?LL
L? ??p
m a x
jjjyY
y p y j???? ? ?
1
m a x ( ),,
i
J
i i i i i i j jxX
j
x u x i s t p x p p y??? ? ? ? ?
? ?
? ? ? ? ? ??
11
IJ
l i l l j
ij
xy???
??
????
Local analysis
Hicksian Separability
? Divide the consumption bundle into two
sub-bundles,and price
? The prices of z are change homogenously
? Choice,
? Let then
? so
(,)x?xz (,)p? zpp
0t?zpp
,(,) m a x (,),, xx u x s t p x t w
?? ? ? ?0
zz z p z
zw?0pz m a x (,),, z
xx u x t s t p x tw w
? ? ? ?
(,) (,) ( ) ( ) ( ) (,)zu x u x t w t w x m x u x m??? ? ? ? ? ? ?z
Local analysis
For every i=1,… I,they have the quasi-
linear utility function,
and
(Inada condition.)
Standardization the price of m as 1,and
pricing commodity l as p,
(,) ( )i i i i i iu x m m x???
( ) 0 ; ( ) 0 ; ( 0 ) 0i i i i ixx? ? ?? ??? ? ?
Local analysis
For the firm j
Profit maximization
First order condition,
{ (,), 0 a n d ( ) }j j j j j j jY z q q z c q? ? ? ?
0m a x ( )j j j jq p q c q? ??
( ),f o r 0jjp c q q?????
Local analysis
For the consumer i
Utility maximization
First order condition,
{ (,), 0 a n d ( ) }j j j j j j jY z q q z c q? ? ? ?
,
1
m a x ( )
., ( ( ) )
ii
i i i
mx
J
i i m i i j j j j
j
mx
s t m p x p q c q
?
??
?
?
? ? ? ? ? ??
( ) for 0i i ix p x? ??? ??
Local analysis
Market clearing
i’s Demand function
11
IJ
ij
ij
xq??
??
???
1 ( ) (0 )
()
0 (0 )
i i i
i
i
x i f p
xp
i f p
??
?
?? ?? ??
? ?
????
1( ) < 0 (0 )
( ) iiii
x p i f p
x
?
?
????
??
Local analysis
Aggregation demand function for l,
It’s continuous and non-increasing for
See the fig,
1
( ) ( )
I
i
i
x p x p
?
? ?
0 m a x ( 0)iip ? ???
Local analysis
j’s supply function
Aggregation supply function
It’s continuous and non-increasing for
See the fig,
1 ( ) ( 0 )
()
0 ( 0 )
j j j
j
j
c q if p c
qp
if p c
?? ?? ??
? ?
????
1( ) < 0 ( 0 )
( ) jjjj
q p if p c
cq
????
??
1
( ) ( )
J
j
j
q p q p
?
? ?
m in ( 0 )jjpc ??
Local analysis
Welfare
So we can metric the welfare by
Marshallian aggregate surplus,
1
1 1 1
m a x (,)
., ( ) ( )
I
I I J
i m i i j j
i i j
W u u
s t u x c q??
? ? ?
? ? ?? ? ?
L
1
11
( ) ( ) (,)
IJ
i i j j I
ij
x c q W u u?
??
? ? ??? L
11
( ) ( ) ( )
IJ
i i j j
ij
S x c q?
??
????x,q
Local analysis
if
Because of
Then
11
( ) (,) ( )
IJ
ij
ij
S S x q S x
??
????x,q
( ) ( )iiC q c q?? ?
0
( ) [ ( ) ( ) ]xS x p s C s d s????
Complete compete market
Free entrance (same cost function）
If,new firms enter,
otherwise,some firms will exit,See the fig,
equilibrium,
( ) /p c q q c??
(,,)p q J? ? ?
0m a x ( )qq p q c q
??
?? ? ?
( ) 0p q c q? ? ?? ? ?
()x p J q? ? ??
Monopoly
Market power (to control the price)
First order condition
Welfare lost
,
m a x ( ) ( )
., ( )
pq
p x p c q
s t x p q
??
?
( ) ( ) ( )p q q p q c q? ? ? ???? ? ?( ) ( ) ( ) 0p q c q f o r p q? ? ?????
[ ( ) ( ) ] 0q
q
p s c s ds? ????
o
Monopoly
Consider the first order condition,we
can get,
? If,no distorts occurs,
? if,we have,
(as is a
constant )
1( ) [ 1 ] ( )
()p q c qq? ???
()q? ??
()c q c q??
1
() 12
()
dp
pqdc q
pq
?
?
????
??
? ()q?
Monopoly
Price discrimination(Pigou,1920)
? First degree price discrimination
The firm set a,take-it-or-leave-it” contract
with q and it’s total price r,
,
m ax ( )
., ( )
rq
r c q
s t u q r
?
?
()( ) ( ) ; ( ) ; uqu q c q r u q p
q
?
? ? ? ?
?? ? ?? ? ?
Monopoly
Price discrimination
? second degree price discrimination
? For group 1,set (q1,r1) and (q2,r2) group2,
1 2 1
1 1 1
2 2 2
1 1 1 1 2 2
2 2 2 2 1 1
m a x ( )
()..
P.C, ( p a r tic ip a tio n c o n st r a in ts )
()
( ) ( )
I, C, ( in c e n tiv e c o n st r a in ts )
( ) ( )
r r c q q
u q rst
u q r
u q r u q r
u q r u q r
? ? ?
? ?
?
? ?
? ? ? ?
?
? ? ? ?
Monopoly
Price discrimination
? second degree price discrimination
? If c ( q ) = c q,u2 > u1,and we have,
See the fig,
21uu???
1 1 2 1 1 1
22
( ) ( ) ( )
()
u x c u x u x
u x c
? ? ?? ? ?
? ?
Monopoly
Price discrimination
? Third degree price discrimination
1 1 1 2 2 1 2
1 1 1 1 1 2 2 2 2 2
1 1 2 2
12
1 1 2 2 1 2
m a x ( ) ( ) ( )
( ) ( ) ( ) ( )
11
( ) [ 1 ] ( ) [ 1 ]
( ) ( )
p q q p q c q q
p q p q q p q p q q c
p q p q
p q p q if
??
??
? ? ?
??? ? ? ? ?
? ? ? ?
? ? ?
Assignment
Textbook,ex.13.3;ex.14.19
Pareto Optimal u
2
u1
Feasible utility sets
The utility combine of Pareto optimal
demand function
12( ) ( ) ( )x p x p x p??
1()xp
1(0)??
2()xp
2(0)??
supply function
12( ) ( ) ( )q p q p q p??
1()qp
1(0)c?
2()qp
2(0)c?
equilibrium
p
p
,xq,xQ
First degree price
discrimination
()cq?
()uq?
qq?
second degree price
discrimination
q?
p?
q
second degree price
discrimination
cc
Third degree price discrimination