requires a strictly positive level of capital and labor.
Inputs are chosen to maximize profits. Each firm can be viewed
as maximizing profits in two steps. First, it chooses the input
bundle (
j
, k
j
) that minimizes the cost of producing
y
j
units of
output. The corresponding cost function is given by
C
j
(
w, r
)
y
j
≡
min
j
,k
j
w
j
+
rk
j
:
y
j
≤
f
j
(
j
, k
j
)
, j
= 1
,
2
and satisfies conditions
C
1 –
C
6. In the second step, given the
cost function
C
j
(
·
)
y
j
, the firm solves the optimization problem
Π
j
(
p
j
, w, r
)
≡
max
y
j
p
j
y
j
−
C
j
(
·
)
y
j
The optimal choice of
y
j
must satisfy the following complemen-
tary slackness condition
y
j
≥
0;
p
j
−
C
j
(
·
)
≤
0; and
p
j
−
C
j
(
·
)
y
j
= 0
Hence,
in
a
competitive
equilibrium
only
zero
profits
are
possible.

2.
The Preliminaries
21
2.2.3
Characterization of equilibrium
Restricting our analysis to the case where both sectors are open,
i.e.
Y
1
, Y
2
>
0
,
equilibrium is defined by a set of factor prices
and output levels (
w, r, Y
1
, Y
2
)
∈
R
4
++
satisfying the following
four conditions:
Firms earn zero profits in each output market,
C
1
(
w, r
)
−
p
1
=
0
(2.8)
C
2
(
w, r
)
−
p
2
=
0
(2.9)
Labor and capital markets clear,
2
j
=1
C
j
w
(
w, r
)
Y
j
=
L
(2.10)
2
j
=1
C
j
r
(
w, r
)
Y
j
=
K
(2.11)
Expressions (2.8) and (2.9) require that the marginal cost of
production in sector
j
be equal to the per-unit output price for
the sector. Expression (2.10) ensures the aggregate demand for
labor from the two sectors is equal to the endowment of labor
L.
Likewise, expression (2.11) ensures the capital market clears.
In principle, since (2.8) and (2.9) consists of two equations in
the unknowns
w
and
r,
the solution may be written as
w
=
W
(
p
1
, p
2
)
(2.12)
r
=
R
(
p
1
, p
2
)
(2.13)
Notice that endowments do not appear as arguments in these
equations. This result obtains because the number of traded
goods equals the number of endowed factors of production.
Substituting (2.12) and (2.13) into the factor market clear-
ing conditions (2.10) and (2.11) yields two linear equations with
input-output coeﬃcients
C
j
i
(
W
(
p
1
, p
2
)
, R
(
p
1
, p
2
)),
i, j
= 1
,
2
,
and unknowns
Y
1
and
Y
2
.
The system is linear because the prices
p
1
, p
2
are exogenous in which case
C
j
i
(
·
) is a scalar value. Assum-
ing both sectors produce at positive levels, denote the solution

22
2.
The Preliminaries
to the resulting system as
Y
j
=
Y
j
(
p
1
, p
2
, L, K
)
, j
= 1
,
2
(2.14)
When both sectors produce at positive levels, and the elas-
ticity of factor substitution between
L
j
and
K
j
is the same for
both technologies, then the solution
w
∗
, r
∗
satisfying the zero
profit conditions is unique. Furthermore, it follows from (2.5)
that
W
(
·
) and
R
(
·
) are homogeneous of degree one in
p
1
and
p
2
,
3
while the supply functions (2.14) are homogeneous of degree
zero in prices, and of degree one in endowments
L
and
K.
The factor rental rate equations (2.12), (2.13) and the sup-
ply functions (2.14) can each be used to determine the gross
domestic product function
G
(
p
1
, p
2
, L, K
)
=
W
(
p
1
, p
2
)
L
+
R
(
p
1
, p
2
)
K
(2.15)
=
p
1
Y
1
(
p
1
, p
2
, L, K
) +
p
2
Y
2
(
p
1
, p
2
, L, K
)
Equation (2.15) indicates that in this model, GDP measured
via the cost of production or via the value of output, yields the
same result. As noted in the previous section, we can also derive

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