University of California, Davis
Department of Agricultural and Resource Economics
ARE 100B
Winter 2009
Dr. Larson
Problem Set 6
Answers
1. Consider a firm that uses a single variable input, x, to produce output q, with production function
3 ln(
1).
qx
=
⋅+
(a) Derive the firm's inverse demand for the input x.
The firm's inverse demand for the input x comes directly from its first order condition for choosing it. The
firm’s short-run profit is
1)
,
P
S pq wx p
x
wx
=⋅−⋅=⋅
⋅
+−⋅
and its FOC for optimal choice of
x
is,
using the chain rule,
1
/
3
1
0.
1
PS
x
p
w
x
∂
∂= ⋅⋅
⋅− =
+
Rearranging this to solve for
w
, the firm's inverse demand for
x
is
3
.
1
p
w
x
⋅
=
+
(b) Derive the firm's direct demand for x.
The firm's direct demand for
x
comes from solving the inverse demand for
x
, so it is
3 ( / ) 1.
x
pw
=
⋅−
(c) Derive the firm's short-run supply function.
The firm's short-run supply function comes from
inserting the input demand for x into the production function:
3 ln(3
/ ).
q
=
⋅⋅
2. For an industry that uses one variable input, illustrate the general equilibrium supply function, as
contrasted with the partial equilibrium supply function, when the output price changes.
See the graph below. Suppose the output price falls, to
1
p
from
0
,
p
because output demand
shifts in from
0
D
to
1
D
.
The partial equilibrium supply curve
0
(, )
pe
q pw
predicts the change in quantity
produced if the price of input
x
is fixed at
0
.
w
In looking at the input market, the demand for
x
shifts
in, from
0
pe
x wp
to
1
( ,
),
pe
because output price has fallen.
This has an effect on the input price
w
, which falls from
0
w
to
1
,
w
because the supply of the input
x
is not perfectly elastic.
The change in
input price to
1
w
in turn causes the product supply curve to shift, from
0
pe
to
1
( ,
).
pe
When
all of these adjustments in price between inter-related markets cease, the general equilibrium supply curve
( )
( , ( )),
ge
ge
qpqp
w
p
=
which accounts for changes in the input market price, describes the actual
change in quantity that will be supplied when the output price changes.