2
As before, we need to satisfy two conditions
As before, we need to satisfy two
of producer equilibrium.
conditions of producer equilibrium.
MRTS = _r_
MRTS = _r_
w
w
rK + wL =
!
f
(K , L) = Q
"
Solving these two equations, we get the
Solving these two equations, we get
producer demands for K & L as functions
producer demands for K & L as
of factor prices r , w, and cost level,
!
.
functions of r,w, and output level, Q
"
.
K = K(r , w ,
!
)
K = K(r , w , Q
"
)
L = L(r , w ,
!
)
L = L(r , w , Q
"
)
If we substitute these demands into
If we substitute these demands into
the objective function, we get the
the objective function, we get the
maximum output.
minimum cost.
Q =
f
(K(r , w ,
!
) , L(r , w ,
!
))
C = r K(r , w , Q
"
) + w L(r , w , Q
"
)
Q = Q(r , w ,
!
)
C = C(r , w , Q
"
)
Q = Q(
!
)
C = C(Q
"
)
Quantity can now be expressed as a function
Cost can be expressed as a function
of the given cost level.
of the given output level.
PRODUCER’S DUALITY EXAMPLE
Duality theory provides us with a procedure to construct a cost function from only two
pieces of information, namely, a production function and factor prices.
For example,
suppose we have unit factor prices
r = w = 1
and the following Cobb-Douglas production function
Q = K
4/5
L
1/5
Q = K
0.8
L
0.2