Econ303 A Simple Model of GDP Determination
1
We make the following assumptions
Assumption 1
All countries use the same technology,
Y
=
F
(
K,L
)
.
Assumption 2
The supplies of labor and capital are exogenously given.
Let
L
s
denote the supply of labor and
K
s
the supply of capital. Then
L
s
=
¯
L
and
K
s
=
¯
K
.
Both
¯
L
and
¯
K
are some numbers.
Firms in a competitive market maximize proﬁts. Their optimal demand for capital and labor
are
K
d
and
L
d
, which is determined by
marginal conditions
:
MP
K
= ˆ
r
MP
L
=
w
At the equilibrium on the labor and capital markets, demands equal supplies. These are
called
market clearing conditions
L
d
=
L
s
K
d
=
K
s
Deﬁnition 1
A
competitive equilibrium
consists of two sets of numbers: quantities
(
Y,K
d
,L
d
,K
s
,L
s
)
and prices
(ˆ
r,w
)
. They satisﬁes the following requirements:
1. Given prices
(ˆ
r,w
)
, quantities
(
Y,K
d
,L
d
)
maximize ﬁrms proﬁts;
2.
(ˆ
r,w
)
are prices that clear markets.
In another word, the equilibrium is determined by both marginal conditions and market
clearing conditions. Mathematically, (
Y,L
d
,K
d
,L
s
,K
s
,w,
ˆ
r
) satisﬁes the following sets of
equations
MP
K
(
K
=
K
d
,L
=
L
d
) = ˆ
r
MP
L
(
K
=
K
d
,L
=
L
d
) =
w
L
d
=
L
s
=
¯
L
K
d
=
K
s
=
¯
K
Y
=
F
(
¯
K,
¯
L
)
Below is how we ﬁnd a competitive equilibrium for an economy:
1. The equilibrium quantities are given by the exogenous supplies of factors. (We use
market clearing condition here.) At equilibrium
K
d
=
K
s
=
¯
K
,
L
d
=
L
s
=
¯
L
and
Y
=
F
(
¯
K,
¯
L
).