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GREG HUTKO: Welcome
back to the 14.01

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problem-solving videos.

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Today we're going to work on
Fall 2010 Problem Set 4,

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Problem Number 3.

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And this problem is really
going to take

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us through two scenarios.

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We're dealing with producer
decisions.

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So now instead of dealing with
utilities, we're going to be

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working with cost functions.

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And we're going to
first go through

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the short-run scenario.

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And then we're going to talk
about the long-run scenario

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and the implications of
both of these cases.

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Problem Number 3 says, suppose
the process of producing corn

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on a farm is described by the
function q equals 8k to the

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1/3 times quantity L minus 40
raised to the 2/3, where q is

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the number of units of corn
produced, k is the number of

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machine hours used, and
L is the number of

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person-hours of labor.

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In addition to capital and
labor, the farmer needs to pay

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a $15 transportation fee to
deliver corn to downtown.

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So the cost can be written as
total cost equals 15 times the

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quantity produced plus the
rental cost of capital plus

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the wage rate times the
quantity of labor.

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Part A says, suppose in the
short-run the machine hours

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rented are fixed at k equals 8,
and its rental rate equals

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64, and the wage
rate equals 16.

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Derive the short-run total cost
and the average costs as

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a function of output level q.

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So to start off this problem,
we're going to start by

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working with the short-run
scenario.

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And typically, the only
difference between the

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short-run scenario and the
long-run scenario in economics

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problems is that in the
short-run, the amount of

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capital that a firm can use
is going to be fixed.

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It means that because machines
are a fixed cost in the

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short-run, you can't actually
change how many machines or

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how often you use
the machines.

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So we're going to set that equal
to 8 for this scenario.

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And we also know that each hour
that we use this machine

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is going to cost us 64.

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And we know that for each hour
that we're using labor, it's

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going to cost us 16.

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We also know that in addition to
the cost of the capital and

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the cost of the labor, which
is represented in our total

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cost function here, for each
unit q that we produce, we

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have to transport
it to market.

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So we also have this 15 times
q added into our total cost

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function, which is something
that we might not always see

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in all of our cost functions.

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So let's start off by solving
for the total cost function.

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And to do this, the first thing
that we're going to do

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is we're going to plug in to our
production function here

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what we know the capital
is fixed at.

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And we're going to solve for
labor, or L, in terms of q.

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So plugging in for k
we're going to be

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left with this equation.

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And from here, we can solve
for L in terms of q.

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And this is going to be useful
for us because what we're

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going to do is we're going to
take this L and we're going to

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plug it into the total cost
function, so that our cost

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function is no longer in terms
of k and L. But it's only

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going to be in terms of q.

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So isolating L in this equation,
we're going to have

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that L equals 40 plus q/16
raised to the 3/2.

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So now let's go to our
total cost function.

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We're going to plug
in for k and r.

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So we know that r is
64 and k is 8.

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We're going to plug
in for w 16.

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And now for L we're going to
plug in what we solved for

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using our production function.

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So from this equation when we do
the algebraic manipulation,

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we're going to get the total
cost function in

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terms of only q.

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And solving out for this, we
find that the total cost--

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and I'm going to denote that
this is in the short-run with

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the capital fixed by a little
sr as a subscript.

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The total cost is going to be
equal to 1,152 plus 15q plus

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16 times quantity q/16
raised to the 3/2.

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Now to find the average cost,
all the average cost is it's

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the total cost divided by q.

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So it's per unit, how much on
average does the producer have

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to spend to actually
produce one unit?

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So to find the average cost,
we're going to divide this

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whole thing through by q.

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And we're going to find the
average cost in the short-run

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is going to be equal to 1,152
divided by q plus 15 divided

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by q/16 raised to the 1/2.

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In part B, what we're going to
do is we're going to take the

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total cost function, we're
going to plug in a fixed

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quantity, and we're going to
find, what is the actual

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amount of money that a producer
would have to pay to

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produce a fixed quantity?

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Part B says, suppose
the farm wants to

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produce 64 units of corn.

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Based on the answer to
part A, what is the

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total short-run cost?

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So using our solution from part
A, all you have to do now

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for part B is you're going to
plug in for q the number 64.

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So for part B, plugging in for
q, you're going to find that

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the total cost in the
short-run is going

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to be equal to 2,240.

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Now the more interesting part
of this problem is what

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actually happens when instead
of having to fix the capital

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at 8, what happens when the
producer can change the amount

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of capital that they're
producing?

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Part D says, in the long-run,
the farm can change its

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capital level by minimizing
the cost subject to the

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production function, derive the
cost-minimizing demands

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for k and L as a function of
output q, the wage rates w,

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and the rental rates
of machine r.

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So now we're going to go back
to a similar problem like we

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saw with consumer theory where
we saw the marginal rate of

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substitution had to be equal
to the price ratio.

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In this case, we're going to
use something called the

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marginal rate of technical
substitution.

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Simply put, the marginal rate of
technical substitution asks

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us, how many machines or how
many people would we be

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willing to basically lay off
for one additional machine?

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And we call that the marginal
product of capital, or how

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much we're actually getting
from each unit of capital,

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divided by the marginal
product of labor.

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We're going to set that equal
to the price of the capital

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and the price of the labor.

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To find these marginal products
of capital and the

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margin marginal product of
labor, what we're going to do

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is we're going to take the
derivative of our production

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function with respect to q, or
with respect to k, and with

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respect to L. And when we do
that, we find that the

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marginal product of capital
is going to be equal to.

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And we can also solve for the
marginal product of labor.

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And we're simply going to divide
here and we can find

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that L minus 40 over 2k
is going to equal r/w.

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And finally, the last thing
we're going to do here is

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we're going to solve for the
amount of labor and the amount

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of capital rented in terms
of the other variables.

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We're going to plug this back
into our production function

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that we have over here in
the middle of the board.

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And then we can solve for how
much capital and how much

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labor is demanded in terms
of w and in terms of r.

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So I'll go through this process
first for solving for

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the demand function
for capital.

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To start off, we're going to get
this equation in terms of

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L. So we have that L is equal
to 2rk over w plus 40.

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And then we're going to take
this L and we're going to plug

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it back into our production
function.

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And when we do this, we're going
to have that 8k to the

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1/3 times quantity 2rk over
w raised to the 2/3.

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And now we're just going to
solve through for rk.

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And when we solve through for k,
we're going to find that k

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equals q/8 w over 2r
raised to the 2/3.

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This is our conditional
demand for capital.

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And we call it the conditional
demand because it's dependent

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on the price that we're going
to pay for labor, the price

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that we're going to pay for
labor, the price we're going

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to pay for capital, and
the quantity that

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we're going to produce.

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Now to do this same process
only solving for the

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conditional demand for labor,
you're going to go back to

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this equation right here.

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And instead of solving through
for L, you're going to solve

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through for k.

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And when you solve through for
k, you're going to find that k

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equals quantity L minus
40 times w over 2r.

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You're going to take this k,
just like we did for labor

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you're going to plug it back
into the production function.

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And you can solve
through for the

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conditional demand for labor.

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And when you do that, you're
going to find that labor is

00:13:44.050 --> 00:14:06.140
just going to equal q/8 times
quantity 2rw raised to

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the 1/3 plus 40.

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So again, to summarize this
problem, we started off with

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the short-run scenario.

00:14:14.700 --> 00:14:17.240
And what we found with the
short-run scenario, we were

00:14:17.240 --> 00:14:20.870
able to plug-in for the capital
that was fixed at 8.

00:14:20.870 --> 00:14:23.250
We were able to find the
total cost and the

00:14:23.250 --> 00:14:25.590
average cost functions.

00:14:25.590 --> 00:14:28.410
Given a fixed quantity that
they wanted to produce, we

00:14:28.410 --> 00:14:31.510
solved for the total cost
in the short-run.

00:14:31.510 --> 00:14:33.900
And then finally we said, let's
let the producer change

00:14:33.900 --> 00:14:36.840
how much capital they
are actually using.

00:14:36.840 --> 00:14:41.500
And let's figure out based on
letting the wage rate and the

00:14:41.500 --> 00:14:44.660
rental rate of capital change,
let's get a conditional demand

00:14:44.660 --> 00:14:47.400
that lets those variables change
that would let us then

00:14:47.400 --> 00:14:49.700
solve for the amount of capital
and the amount of

00:14:49.700 --> 00:14:51.690
labor that would be needed
to produce a

00:14:51.690 --> 00:14:52.940
certain amount of quantity.