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GREG HUTKO: Welcome
back to the 14.01
problem-solving videos.
Today we're going to work on
Fall 2010 Problem Set 4,
Problem Number 3.
And this problem is really
going to take
us through two scenarios.
We're dealing with producer
decisions.
So now instead of dealing with
utilities, we're going to be
working with cost functions.
And we're going to
first go through
the short-run scenario.
And then we're going to talk
about the long-run scenario
and the implications of
both of these cases.
Problem Number 3 says, suppose
the process of producing corn
on a farm is described by the
function q equals 8k to the
1/3 times quantity L minus 40
raised to the 2/3, where q is
the number of units of corn
produced, k is the number of
machine hours used, and
L is the number of
person-hours of labor.
In addition to capital and
labor, the farmer needs to pay
a $15 transportation fee to
deliver corn to downtown.
So the cost can be written as
total cost equals 15 times the
quantity produced plus the
rental cost of capital plus
the wage rate times the
quantity of labor.
Part A says, suppose in the
short-run the machine hours
rented are fixed at k equals 8,
and its rental rate equals
64, and the wage
rate equals 16.
Derive the short-run total cost
and the average costs as
a function of output level q.
So to start off this problem,
we're going to start by
working with the short-run
scenario.
And typically, the only
difference between the
short-run scenario and the
long-run scenario in economics
problems is that in the
short-run, the amount of
capital that a firm can use
is going to be fixed.
It means that because machines
are a fixed cost in the
short-run, you can't actually
change how many machines or
how often you use
the machines.
So we're going to set that equal
to 8 for this scenario.
And we also know that each hour
that we use this machine
is going to cost us 64.
And we know that for each hour
that we're using labor, it's
going to cost us 16.
We also know that in addition to
the cost of the capital and
the cost of the labor, which
is represented in our total
cost function here, for each
unit q that we produce, we
have to transport
it to market.
So we also have this 15 times
q added into our total cost
function, which is something
that we might not always see
in all of our cost functions.
So let's start off by solving
for the total cost function.
And to do this, the first thing
that we're going to do
is we're going to plug in to our
production function here
what we know the capital
is fixed at.
And we're going to solve for
labor, or L, in terms of q.
So plugging in for k
we're going to be
left with this equation.
And from here, we can solve
for L in terms of q.
And this is going to be useful
for us because what we're
going to do is we're going to
take this L and we're going to
plug it into the total cost
function, so that our cost
function is no longer in terms
of k and L. But it's only
going to be in terms of q.
So isolating L in this equation,
we're going to have
that L equals 40 plus q/16
raised to the 3/2.
So now let's go to our
total cost function.
We're going to plug
in for k and r.
So we know that r is
64 and k is 8.
We're going to plug
in for w 16.
And now for L we're going to
plug in what we solved for
using our production function.
So from this equation when we do
the algebraic manipulation,
we're going to get the total
cost function in
terms of only q.
And solving out for this, we
find that the total cost--
and I'm going to denote that
this is in the short-run with
the capital fixed by a little
sr as a subscript.
The total cost is going to be
equal to 1,152 plus 15q plus
16 times quantity q/16
raised to the 3/2.
Now to find the average cost,
all the average cost is it's
the total cost divided by q.
So it's per unit, how much on
average does the producer have
to spend to actually
produce one unit?
So to find the average cost,
we're going to divide this
whole thing through by q.
And we're going to find the
average cost in the short-run
is going to be equal to 1,152
divided by q plus 15 divided
by q/16 raised to the 1/2.
In part B, what we're going to
do is we're going to take the
total cost function, we're
going to plug in a fixed
quantity, and we're going to
find, what is the actual
amount of money that a producer
would have to pay to
produce a fixed quantity?
Part B says, suppose
the farm wants to
produce 64 units of corn.
Based on the answer to
part A, what is the
total short-run cost?
So using our solution from part
A, all you have to do now
for part B is you're going to
plug in for q the number 64.
So for part B, plugging in for
q, you're going to find that
the total cost in the
short-run is going
to be equal to 2,240.
Now the more interesting part
of this problem is what
actually happens when instead
of having to fix the capital
at 8, what happens when the
producer can change the amount
of capital that they're
producing?
Part D says, in the long-run,
the farm can change its
capital level by minimizing
the cost subject to the
production function, derive the
cost-minimizing demands
for k and L as a function of
output q, the wage rates w,
and the rental rates
of machine r.
So now we're going to go back
to a similar problem like we
saw with consumer theory where
we saw the marginal rate of
substitution had to be equal
to the price ratio.
In this case, we're going to
use something called the
marginal rate of technical
substitution.
Simply put, the marginal rate of
technical substitution asks
us, how many machines or how
many people would we be
willing to basically lay off
for one additional machine?
And we call that the marginal
product of capital, or how
much we're actually getting
from each unit of capital,
divided by the marginal
product of labor.
We're going to set that equal
to the price of the capital
and the price of the labor.
To find these marginal products
of capital and the
margin marginal product of
labor, what we're going to do
is we're going to take the
derivative of our production
function with respect to q, or
with respect to k, and with
respect to L. And when we do
that, we find that the
marginal product of capital
is going to be equal to.
And we can also solve for the
marginal product of labor.
And we're simply going to divide
here and we can find
that L minus 40 over 2k
is going to equal r/w.
And finally, the last thing
we're going to do here is
we're going to solve for the
amount of labor and the amount
of capital rented in terms
of the other variables.
We're going to plug this back
into our production function
that we have over here in
the middle of the board.
And then we can solve for how
much capital and how much
labor is demanded in terms
of w and in terms of r.
So I'll go through this process
first for solving for
the demand function
for capital.
To start off, we're going to get
this equation in terms of
L. So we have that L is equal
to 2rk over w plus 40.
And then we're going to take
this L and we're going to plug
it back into our production
function.
And when we do this, we're going
to have that 8k to the
1/3 times quantity 2rk over
w raised to the 2/3.
And now we're just going to
solve through for rk.
And when we solve through for k,
we're going to find that k
equals q/8 w over 2r
raised to the 2/3.
This is our conditional
demand for capital.
And we call it the conditional
demand because it's dependent
on the price that we're going
to pay for labor, the price
that we're going to pay for
labor, the price we're going
to pay for capital, and
the quantity that
we're going to produce.
Now to do this same process
only solving for the
conditional demand for labor,
you're going to go back to
this equation right here.
And instead of solving through
for L, you're going to solve
through for k.
And when you solve through for
k, you're going to find that k
equals quantity L minus
40 times w over 2r.
You're going to take this k,
just like we did for labor
you're going to plug it back
into the production function.
And you can solve
through for the
conditional demand for labor.
And when you do that, you're
going to find that labor is
just going to equal q/8 times
quantity 2rw raised to
the 1/3 plus 40.
So again, to summarize this
problem, we started off with
the short-run scenario.
And what we found with the
short-run scenario, we were
able to plug-in for the capital
that was fixed at 8.
We were able to find the
total cost and the
average cost functions.
Given a fixed quantity that
they wanted to produce, we
solved for the total cost
in the short-run.
And then finally we said, let's
let the producer change
how much capital they
are actually using.
And let's figure out based on
letting the wage rate and the
rental rate of capital change,
let's get a conditional demand
that lets those variables change
that would let us then
solve for the amount of capital
and the amount of
labor that would be needed
to produce a
certain amount of quantity.
