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Econ 11  Additional Problems: Solutions
1. Toyota’s technology for producing cars is given by the production function
F
(
K,L
)=
min
{
K, L
}
,whe
re
K
denotes the amount of capital inputs, and
L
denotes the amount of
labor inputs necessary for producing cars.
(
i
)
For a given rental rate of capital
v
and a wage rate
w
,
f
nd Toyota’s input demands
of
K
and
L
necessary to produce a quantity
Q
of cars. Also,
f
nd Toyota’s cost function.
(
ii
)
Suppose Toyota takes the price of its cars as given. If
w
=10
and
v
=10
,
f
nd
Toyota’s longrun supply function of cars (as a function of its price
P
) in the longrun, when
Toyota can adjust both
K
and
L
.
(
iii
)
The demand for Toyota cars is given by
Q
=200
−
5
P
. A
twhatpr
icewou
ldthe
market clear in the long run, if Toyota took the market price as given? In general, how
would the marketclearing price depend on Toyota’s input prices?
(
iv
)
E
f
ectively, Toyota is the only supplier of Toyota cars, and therefore will price its
cars as a monopolist, taking into account the e
f
ecto
fthepr
iceondemand
.Whatquant
ity
will Toyota choose to produce in the long run?
(
v
)
In the short run,
K
is
f
xed at
40
, and input prices are
f
xat
w
=10
and
v
=10
.A
sa
function of
P
, what is the shortrun optimal supply of cars, as a well as the marketclearing
price for Toyotas?
(
i
)
In order to minimize costs for a given quantity of output, Toyota wants to use inputs in
the
f
xed proportions. To produce
Q
units of output, it needs
Q
units of capital and
Q
units
of labor. Hence, the inputs necessary for producing
Q
cars are
K
(
Q
)=
Q
and
L
(
Q
)=
Q
,
which are independent of the market prices (perfect complements  no substitution between
inputs). The Cost function is
C
(
Q
)=
vK
(
Q
)+
wL
(
Q
)=(
v
+
w
)
Q
(
ii
)
With
w
=
v
=10
, the cost function becomes
C
(
Q
)=20
Q
. The marginal cost is
MC
(
Q
)=20
(CRS implies constant marginal cost). Whenever
P<
20
, the longrun supply
is
0
the
f
rm
f
nds it optimal not to produce. Whenever
P>
20
, Toyota would want to
produce an in
f
nite amount. Whenever
P
=20
,Toyotaisind
i
f
erent between all quantities
(pro
f
ts are necessarily equal to zero). The longrun supply curve is
F
at at a price level of
P
=20
.
(
iii
)
If Toyota takes the market price as given, the market has to clear at
P
=20
at
that price, Toyota would be willing to produce any quantity
Q
. The corresponding
Q
that
is demanded in the market is
Q
=200
−
5
P
=100
. With CRS (and perfect complements),
the marginal cost is simply
MC
(
Q
)=
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This note was uploaded on 04/05/2009 for the course ECON 11 taught by Professor Cunningham during the Spring '08 term at UCLA.
 Spring '08
 cunningham

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