Econ 11 - Additional Problems: Solutions
1. Toyota’s technology for producing cars is given by the production function
F
(
K, L
) =
min
{
K, L
}
, where
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
,
fi
nd Toyota’s input demands
of
K
and
L
necessary to produce a quantity
Q
of cars. Also,
fi
nd Toyota’s cost function.
(
ii
)
Suppose Toyota takes the price of its cars as given.
If
w
= 10
and
v
= 10
,
fi
nd
Toyota’s long-run supply function of cars (as a function of its price
P
) in the long-run, when
Toyota can adjust both
K
and
L
.
(
iii
)
The demand for Toyota cars is given by
Q
= 200
−
5
P
. At what price would the
market clear in the long run, if Toyota took the market price as given?
In general, how
would the market-clearing price depend on Toyota’s input prices?
(
iv
)
E
ff
ectively, Toyota is the only supplier of Toyota cars, and therefore will price its
cars as a monopolist, taking into account the e
ff
ect of the price on demand. What quantity
will Toyota choose to produce in the long run?
(
v
)
In the short run,
K
is
fi
xed at
40
, and input prices are
fi
x at
w
= 10
and
v
= 10
. As a
function of
P
, what is the short-run optimal supply of cars, as a well as the market-clearing
price for Toyotas?
(
i
)
In order to minimize costs for a given quantity of output, Toyota wants to use inputs in
the
fi
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 long-run supply
is
0
- the
fi
rm
fi
nds it optimal not to produce. Whenever
P >
20
, Toyota would want to
produce an in
fi
nite amount. Whenever
P
= 20
, Toyota is indi
ff
erent between all quantities
(pro
fi
ts are necessarily equal to zero). The long-run supply curve is
fl
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
) =
v
+
w
, i.e. given by the input prices. An increase
in the input prices would lead to an increase in the market-clearing price (Toyota’s supply
function shifts to a higher level). Toyota’s total pro
fi
ts in equilibrium are zero.
(
iv
)
If Toyota acts as a monopolist, it will set its price and quantity taking into account
the market demand.
The inverse demand function for Toyota’s cars is
P
=
200
−
Q
5
, and
1