Competitive markets
Specification of the economy
⑧
We have L commodities (products) Lex apples )
⑧
We have consumers i I
for which the consumption set (what they consume) is specified as
=
1
,
000
,
Xi
(ex how
many apples consumers consume
⑧
The total amount of each commodity initially available is the endowment Jex how many apples where there a
·
5 for which the production set (what they produce) is specified as (ex how firms
Firms j many apples
=
1 ,
00
,
produce) transforms inputs into outputs
-
Economic allocation is a specification of X1 , 000
,
XI ,
Y2 ,
000
, ys)
· the allocation is feasible if
:
i = 1Xli -we +
yg ↳ er kan nict meer
geconsumeerd worden dan
geproduced
wat geproduceerd wordt
↳ wat er al was
wat geeonsumeerd wordt
A feasible allocation (x1 .....
K ,
yr ,
00
-, ys) is Pareto optimal if there is no alternative
way
to organize the production and distribution of goods that makes
some consumer better off without making some other consumer worse off
Competitive market
economy
↳ firms and endowments owned
assume are by consumers
↳
Je
Gli =
We
> Fij =
1 (consumers all own shares (
In competitive market also have Competitive equilibrium
a
,
we an
equilibrium :
" .
profit maximazation : max
p
*.
y; N
2 .
utility maximization : max vi(xi) s .
A
p
*
xi
p
*. Wi +j *.
y
market
clearing
.
.
3 :
= we y when these 3 hold ,
we have
equillibrium
, 2 characteristics
1 .
Prices are relative p (pY pr) is an equilibrium vector, then so is ap Capi, ap)
·
*
=
, ....
*
=
....
↳
Prices can be normalized
2
.
If allocation satisfies market clearing conditions for all goods l k and if consumer's budget equation is
=
satisfied p p
.
+jp yj then the market for good k also clears
Xi = .
wi .
When searching for competitive equilibrium it is sufficient to find prices that L-p markets clear
Partial equilibrium competitive analyse
↳ In competitive equilibrium typically all markets are related to each other. However in this case
it is convenient to look at one market in isolation.
↳ This is reasonable because the market is small because of that substitution and wealth
effects are limited.
~
Two - good quasilinear model
In this model we look at good l in the small market in isolation compared to all goods in other markets
⑧
Other markets: we call the goods the numeraire
↳
Any good whose price is fixed as so all other prizes are in
terms with the numeraire good, we set price numeraire as 1
O h
Small isolated market: good , we say price is p i =1
XI
i = 2
i= 3 i =I
Consumer D
1 - 0
**
T
↓
Model # 3 Xi
xi = qj
& ①
x
2
Numeraire g2 Good h j=
~ 21 23T -
qj
E
D ↓ 22 -
93
j = 1
j = 2 j = 3
000
* j = J
Firms
ni (mi ,
xi) =
mi + Di (xi) gi zj
Graphical representation economy
7
Consumption good small market
7 Function with di(0) = Di'(xi)
0 ,
> 0 0:"(xi) < o
2 Consumption numeraire good
> Utility consumer
T
l
Firms produce good from good M
Price good l
·
p
:
Units produced 9ood l
· :
q
Units used to produce good h
· :
2
O
e(q) Costs to produce good h hence amount of numeraire good to produce
:
good h -
, 7 How to find equilibrium &, ,
000
*
,
X ,
gi , ..., ge)
-
Solve profit maximization, find qj
I
ma * p - e;
(g)
f
foe p ej(qj)
*
-
= 0
to
soe-ej(qj) -o
2 .
Solve utility maximization, find (mi *, xi
*
)
max Pi(xi) -
p
*
xi + m max mi + bi(xi)
> =
+ j (p** ej(qj))
Ami A pxi -
mi-mitj[g ej(q)]
Or s .
+ -
V
Note it does not matter what the value of is as it disappear with differentiating M
foe di(xi) -
p
o
=
p
=
bi(xi)
↓
soe bi (xi) o
.
3
Calculate market equilibrium Note
*
The market equilibrium condition is independent of
the distribution of endowments and ownership
shares, therefore the equilibrium price and
Aggregate demand = Aggregate supply allocation is also in dependent, thus the market can
be studied in isolation.
7 Unique equilibrium
A unique equalibrium exists if
max Di (0) <min ej(o) with D1(xi) To and
ej(qj) zo
P
max di'(d) g(p)
From the graph we see that
if maxdi(o) min ej'(o) , there is a
<
minej' (0) unique equilibrium
x(x)
X ,
G
Specification of the economy
⑧
We have L commodities (products) Lex apples )
⑧
We have consumers i I
for which the consumption set (what they consume) is specified as
=
1
,
000
,
Xi
(ex how
many apples consumers consume
⑧
The total amount of each commodity initially available is the endowment Jex how many apples where there a
·
5 for which the production set (what they produce) is specified as (ex how firms
Firms j many apples
=
1 ,
00
,
produce) transforms inputs into outputs
-
Economic allocation is a specification of X1 , 000
,
XI ,
Y2 ,
000
, ys)
· the allocation is feasible if
:
i = 1Xli -we +
yg ↳ er kan nict meer
geconsumeerd worden dan
geproduced
wat geproduceerd wordt
↳ wat er al was
wat geeonsumeerd wordt
A feasible allocation (x1 .....
K ,
yr ,
00
-, ys) is Pareto optimal if there is no alternative
way
to organize the production and distribution of goods that makes
some consumer better off without making some other consumer worse off
Competitive market
economy
↳ firms and endowments owned
assume are by consumers
↳
Je
Gli =
We
> Fij =
1 (consumers all own shares (
In competitive market also have Competitive equilibrium
a
,
we an
equilibrium :
" .
profit maximazation : max
p
*.
y; N
2 .
utility maximization : max vi(xi) s .
A
p
*
xi
p
*. Wi +j *.
y
market
clearing
.
.
3 :
= we y when these 3 hold ,
we have
equillibrium
, 2 characteristics
1 .
Prices are relative p (pY pr) is an equilibrium vector, then so is ap Capi, ap)
·
*
=
, ....
*
=
....
↳
Prices can be normalized
2
.
If allocation satisfies market clearing conditions for all goods l k and if consumer's budget equation is
=
satisfied p p
.
+jp yj then the market for good k also clears
Xi = .
wi .
When searching for competitive equilibrium it is sufficient to find prices that L-p markets clear
Partial equilibrium competitive analyse
↳ In competitive equilibrium typically all markets are related to each other. However in this case
it is convenient to look at one market in isolation.
↳ This is reasonable because the market is small because of that substitution and wealth
effects are limited.
~
Two - good quasilinear model
In this model we look at good l in the small market in isolation compared to all goods in other markets
⑧
Other markets: we call the goods the numeraire
↳
Any good whose price is fixed as so all other prizes are in
terms with the numeraire good, we set price numeraire as 1
O h
Small isolated market: good , we say price is p i =1
XI
i = 2
i= 3 i =I
Consumer D
1 - 0
**
T
↓
Model # 3 Xi
xi = qj
& ①
x
2
Numeraire g2 Good h j=
~ 21 23T -
qj
E
D ↓ 22 -
93
j = 1
j = 2 j = 3
000
* j = J
Firms
ni (mi ,
xi) =
mi + Di (xi) gi zj
Graphical representation economy
7
Consumption good small market
7 Function with di(0) = Di'(xi)
0 ,
> 0 0:"(xi) < o
2 Consumption numeraire good
> Utility consumer
T
l
Firms produce good from good M
Price good l
·
p
:
Units produced 9ood l
· :
q
Units used to produce good h
· :
2
O
e(q) Costs to produce good h hence amount of numeraire good to produce
:
good h -
, 7 How to find equilibrium &, ,
000
*
,
X ,
gi , ..., ge)
-
Solve profit maximization, find qj
I
ma * p - e;
(g)
f
foe p ej(qj)
*
-
= 0
to
soe-ej(qj) -o
2 .
Solve utility maximization, find (mi *, xi
*
)
max Pi(xi) -
p
*
xi + m max mi + bi(xi)
> =
+ j (p** ej(qj))
Ami A pxi -
mi-mitj[g ej(q)]
Or s .
+ -
V
Note it does not matter what the value of is as it disappear with differentiating M
foe di(xi) -
p
o
=
p
=
bi(xi)
↓
soe bi (xi) o
.
3
Calculate market equilibrium Note
*
The market equilibrium condition is independent of
the distribution of endowments and ownership
shares, therefore the equilibrium price and
Aggregate demand = Aggregate supply allocation is also in dependent, thus the market can
be studied in isolation.
7 Unique equilibrium
A unique equalibrium exists if
max Di (0) <min ej(o) with D1(xi) To and
ej(qj) zo
P
max di'(d) g(p)
From the graph we see that
if maxdi(o) min ej'(o) , there is a
<
minej' (0) unique equilibrium
x(x)
X ,
G
