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JONATHAN GRUBER: All right.
Why don't we get started?
Today, we're going to continue
our discussion of producer
theory.
Once again, to remember
to put this in context,
the first few lectures were
working consumer theory
to help us derive
a demand curve.
Now we're working
on producer theory
to help us come up
with a supply curve.
We started last time
by talking about how
producers profit maximizes,
and the profit maximization
implies cost minimization.
Therefore, to maximize
profits, you're
going to want to produce
as efficiently as possible.
And basically, to
do that, we need
to understand how your
costs vary with your output.
If you're going to produce
at an efficient level,
you need to understand how what
your costs are going to vary
with your level of production.
So essentially, our
goal of this lecture
is to develop a cost curve--
develop a curve which tells you
how the cost of your production
varies with how
much you produce.
And that's what we're
after in this lecture.
OK?
So we're going to start
with the short run
and then turn to the long run.
So start with developing
a short-run cost curve
and then turn to a long-run.
OK?
And to make this lecture sort
of mathematically coherent,
throughout the
whole lecture, we'll
work with our favorite
functional form--
a production form
of the function q
equals the square root
of L times K. Remember,
firms produce goods--
little q-- using two
inputs, labor and capital.
Labor is a variable input.
That means you can change it in
the short run and the long run.
Capital is a fixed input,
which means that you can only
change it in the long run.
OK?
So in the short run, we're going
to have two kinds of costs.
We're going to have
fixed costs, which
are going to come from a
fixed level of capital,
and we're going to
variable costs, which are
going to come from our labor.
So we're going to have two
kinds of costs: fixed costs--
they're fixed, because
in the short run,
you can't change the
level of capital--
and variable costs, which
are the costs of our labor.
And then we're going
to have total costs.
They're simply going to be
fixed costs plus variable costs.
OK?
So that's how we
think about costs.
The costs of a firm's
production in the short run
is the sum of
their fixed costs--
i.e., their capital costs--
and the variable costs--
i.e., their labor costs.
Now, we're going to show you
how you can turn a production
function into a cost function.
And to do so, you simply
need to recognize that cost--
the costs of firms' production--
are simply the amount
of capital that
uses k bar times the price
of that capital, which we'll
call r, plus the amount
of labor it uses,
times the price of that labor,
which we'll call W, the wage.
OK?
This is the easy part.
Think of the amount
of hours of work
you use times the wage per
hour, or the amount of workers
times the salary per year.
In any case, this is
easier to understand.
Every additional
unit of labor comes
with a cost that is the
wage of that unit of labor.
Think about an hourly model.
Every hour you work at
your convenience store,
they have to pay you the
minimum wage for that hour.
OK?
That's the cost of
that hour of labor.
Capital is harder.
We call r the rental rate.
And the reason is because we
don't think of buying capital.
Don't think of buying a machine.
Think of renting a machine.
The reason we do that is to
make the periodicity work.
OK?
You don't buy a
worker, thank god.
You rent that worker.
And you rent that
worker at a price, w.
So when the firm uses
your time, they're
renting your time an hour
at a time at a price, w.
When we get machines,
think of a firm
as renting machines for a price,
r per machine per time period.
OK?
So I understand firms
usually don't do this.
They usually buy machines.
And we'll come
back to even if you
buy a machine how it
effectively is like renting it,
but for ease of
thinking about this,
you want to think about
flows, not stocks.
Think about the firm's
decision as renting a worker
at the price, w, or renting
a machine at the price, r.
Yeah.
AUDIENCE: Kind of
like also we consider
the gas and electricity
a machine can
use in productivity?
JONATHAN GRUBER: All
that would be in there,
and we'll come back to that.
That's right.
This is sort of the per period
cost of using a machine.
This is the per period cost of
using a worker, is the wage.
This is the per
period cost of using
a machine, which will include
all the costs of running
the machine as well as the costs
of renting the machine itself.
OK?
So later, we'll talk about
how to own the machine.
And we'll come back to the
fact that you can actually
use r as a representation.
It's not bad.
But for now, just think
of renting a machine.
Or if it's a building,
think of this as the rent
you pay on that building, OK?
Not the cost to
build the building.
Now, armed with our
production function--
and let's also say,
to make life easy--
I did this wrong on my notes.
So I hope the other
teachers figured it out.
To make life easy, let's
say that the rental rate, r,
we're going to say is
$10, and the wage rate, w,
is going to be $5.
Now, armed with a
production function,
if you simply have the
production function
and these two prices, you
can derive the short-run cost
function.
How do you do that?
Well, just look at the math.
We know that q equals
square root of L times
k bar in the short-run.
L times k bar.
OK?
So inverting that simply
means that L equals q
squared over k bar.
L equals q squared over k bar.
OK?
And that means that cost
can be written as 10 k bar--
the price the rental
rate is 10, we
have k bar amount of capital--
plus 5q squared over k bar.
That's our cost function.
I just plugged in for L and
multiplied by the wage rate, w,
which is 5.
OK?
So for example, for a fixed
level of capital, this is cost.
So for example, let's imagine
our short-run level of capital
is 1.
Let's imagine there's 1 unit
of capital in the short run,
just to make the math easy.
Then that simply says that
cost equals 10 plus 5q squared.
And that's our cost function.
10 plus 5q squared.
We've just derived the
short-run cost function.
10 is the fixed cost component.
That doesn't vary with
the amount you produce.
So there's no q part of this.
5q squared is the
variable component.
That varies with how
much you produce.
So the cost function
is a fixed part,
which comes from that one
fixed unit of capital,
and the varying part,
which comes from the fact
that the amount of q drives
the amount of labor we need.
OK?
Questions about that math?
All right.
Now, armed with
this cost function,
let's write that
down again here.
So C equals 10 plus 5q squared.
That's going to be our
short-run cost function
we're going to work with.
And remember, that short-run
cost function came directly
from that production function.
To derive this
equation, all I needed
was that production function
and those two prices,
and I derived it.
OK.
Armed with that,
we can define some
of the key concepts
that will drive
our entire analysis of firms.
And the single most
important concept
is marginal cost, which
is what it sounds like--
the derivative costs
with respect to quantity.
OK?
So in this, the marginal
cost is delta c delta q.
OK?
That's marginal cost.
OK.
We'll also care about average
cost, which is just c over q.
It's very important
in this class
to keep our marginals
separate from our averages.
The average is simply over the
entire range of production.
What is the average cost
to produce each unit?
The marginal is what's the cost
of producing the next unit?
And since production
functions are nonlinear,
those will not be
the same, generally.
OK.
Average will not equal
marginal in general,
because a nonlinear function
delta c delta q is not
the same as c/q.
OK?
So we can actually graph
these in figure 6-1.
Figure 6-1 shows the cost
curves for this cost function
I just wrote down, which comes
from that production function.
OK?
So you can see that the
marginal cost, as I said,
is delta c delta q.
Well, that's 10q.
So the marginal cost, the cost
of producing the next unit,
rises with the number of units.
Which makes sense.
The cost has a q
squared term in it,
So obviously, the
marginal cost is
going to have a q term in it.
So basically, the
more you produce,
the higher your marginal cost.
The more unit you need to
produce, the more the little q,
the higher your marginal cost.
Average cost is this sort
of funky shaped thing,
where it's--
which is 10 over q plus 5q--
I just divided this by q--
where it's first declining
and then increasing.
Why is that?
Why is average cost first
declining and then increasing?
We've seen what--
just intuitively?
Why is that?
Why in general in the short
run would we expect that?
Average cost first to
fall and then increase.
Anyone have ideas?
Yeah.
AUDIENCE: [INAUDIBLE]
start up [INAUDIBLE]..
JONATHAN GRUBER: Well--
no, but it's falling first.
So why is it falling first?
It's about that-- yeah.
AUDIENCE: They have a
really high average--
or first fixed cost is not
[INAUDIBLE] the more you make,
the lower that is.
JONATHAN GRUBER: Right.
The first units are paying
off your fixed costs
if you think about it.
Well, the first unit
you sell, basically
you start with this
huge fixed cost.
So actually, by
selling two units, yes,
you get the variable
cost, second unit,
but you get to pay off the
fixed cost, the first unit.
So look at here on this graph.
We show average fixed costs
and average variable costs
Average fixed costs are 10/q.
If you only produce one unit,
your average fixed cost is $10.
To produce two units it's $5.
With every unit you produce,
your average fixed cost
is falling.
You're paying off
that fixed cost.
Average variable cost rises.
Every unit you
produce, you're getting
more and more variable cost.
You put those together,
and you get a function that
first declines and then rises.
You first pay off
your fixed costs,
so your average
costs are falling,
then your marginal cost--
then you start to rise,
because you've got marginal
costs that increase
with quantity produced.
And critically,
the marginal cost
intersects the
average cost curve
at the minimum of average cost.
And that's just mathematical.
If you have any function,
and you take the average,
then the minimum is going to be
a derivative that basically--
before you get to here, 1.5
units, before 1.5 units,
average cost is
above marginal cost,
because you're paying
off your fixed costs.
Once you get beyond
1.5 units, average cost
is below marginal cost.
So average cost
hits marginal cost
at the minimum of average cost.
Yeah.
AUDIENCE: In a
relatively large company,
eventually doesn't really
worry about their fixed cost,
because if they're having
a lot more workers,
their entire cost is going
to be considered basically--
JONATHAN GRUBER:
In the short run.
It depends on this function.
You said large, but large
can be defined in two ways.
But absolute true.
Certainly, if we take
this very function,
for large enough q's average
fixed cost asymptotes to 0.
10 over infinity is 0.
So for large enough q's, the
average fixed cost goes away--
in the short run.
Remember, we're in the short
run with these fixed costs.
OK.
Now, so other questions.
Good question.
Other questions about that?
So that's our basic
intuition, the short run--
is that at first our
costs are super high,
because you've got to pay--
you had to build the plant.
But then over time,
that plant cost falls,
and then your only
costs is basically
the fact you've got to
hire more workers if you
want to produce more.
Now, what we want to notice
is that in the short run
there is a really
close relationship--
one was the key relationship
between marginal cost
and the marginal
product of labor,
which we defined last time.
Remember, the marginal
product of labor was dq dL.
How much-- remember
digging a hole
and diminishing
marginal product?
That each additional worker,
for a fixed level of capital,
is less and less
productive, right?
We talked about that last time.
Well, the marginal cost of
production is, as I said,
equal to delta c over delta q.
OK?
Well, we know from last
time that delta q over delta
L we defined as the
marginal product of labor.
Plugging those and-- so we know
marginal cost is delta c delta
q.
And we can write this-- if you
take the derivative of the cost
function, so our general
cost function up there--
see the cost function
at the top there?
Take the derivative of that with
respect to the amount of labor.
Well, the first term drops
out, because it's fixed.
So you can rewrite delta c delta
q as w times delta L delta q--
w times delta L delta q.
Right.
I just rewrote delta
c-- q delta w-- god.
Brutal.
Sorry, guys.
w times delta L over delta q.
OK.
A little bit better.
OK.
I read that, because I just
took the derivative of the cost
function.
First term drops out,
we take the derivative.
Second term, I just took
the derivative here,
or the discrete derivative, and
I just said, delta c delta q
is w times delta L delta q.
Well, we know the
marginal product
is delta q over
delta L. So we can
rewrite marginal cost as w over
the marginal product of labor.
The marginal cost
is equal to w over
the marginal product of labor.
And that makes sense.
The marginal cost
of the next unit
will be higher the
higher the wage
and lower the more
productive the worker.
Making another unit with a super
productive worker is cheap.
Making another unit with
a very unproductive worker
is expensive.
So essentially, the more you
pay them for each hour of work,
the higher your marginal cost.
But the less they get
done each hour of work,
the higher your marginal cost.
So that's why, roughly
speaking, firms
might want to pay a lot to
people who are high skilled.
So you might say, gee, why are
they paying my friend twice
what I'm paid?
Well, maybe your friend's
twice as productive as you.
That would make-- or two and
a half times as productive
as you.
And that would make sense.
So we can't just say
that it's a mistake
to pay someone higher wages.
We have to consider their wages
relative to how productive they
are.
And that's a key relationship
we'll come back to.
OK?
Question about that?
All right.
That's the short run.
Now let's go to
the long run, which
gets a little more interesting.
Actually, let's
write in that side.
I'll switch to this side today.
Switch things up.
This way, this side, I
block you less, probably.
So I should do this side.
OK.
Long run cost curves.
OK.
Long run cost.
Now, here what gets interesting,
is now K is no longer fixed.
Now we get to choose
our input mix.
And now our goal is going to
be, how do we choose our input
mix to minimize costs?
That's our goal here.
How we going to choose the
mix of workers and machines
to produce a given
quantity most efficiently?
Now, that optimal mix may
change with the quantity.
So we're going to start.
We're going to do
this in two steps.
First, we're going to say, for
a given quantity picked out
of a hat, what's the right
mix of labor and capital
that minimizes the cost of
producing that quantity, given
our production function?
Then we're going to
say, as the quantity
varies how does that change
the optimal mix of L and K?
And does it?
So two steps.
First say, for a
given quantity, what's
the right L and K
to minimize costs
of producing that quantity?
Then ask, well, as we vary
the quantity how does L--
how do L and K vary optimally?
OK.
So basically, we want to find
the economically efficient
combination of L and K
which is a combination that
produces goods at minimum cost.
And to do this, we are
going to write down--
to derive this we're
going to write down
what we call isocost curves.
Isocost curves.
Remember last time
we did isoquants,
Which felt a lot like
the difference curves?
Isocosts are going to feel a
lot like budget constraints.
Isocost curves are essentially
the firm's budget constraint.
They're essentially
mappings of the function
c equals wL plus rK--
essentially, mappings for
different amounts of K and L
of the function c
equals rK plus wL.
So if you look at figure 6-2,
here we see our isocost lines.
OK.
So let's talk about
this for a second.
So for example, take the
middle one, the $100 isocost.
This is saying, what
combinations of labor
and capital cost you $100?
Well, with a rental rate
of $10 and a wage of $5,
that means you can have
10 machines and no workers
or 20 workers and no machines,
or some combination in between.
This is just a
budget constraints.
It's just saying, given
the amount of money
you want to spend,
given your cost,
how many machines and
workers can you have?
The difference is,
we don't start--
I didn't start this
example by saying,
your parents give you x dollars.
That's why firm theory is
harder than consumer theory.
I pin down the consumer theory
problem much more easily
by saying, your parents
give you x dollars, which
told you which line to derive--
graph.
I don't have that here.
I haven't told you that here.
So you have to graph a
series of isocost curves,
because you don't know what the
optimal cost is going to be.
That's to be pinned down later.
That's what makes
supply theory harder.
You have to draw the series.
So you draw these series
of isocost curves--
different combinations
that represent different
amounts, different
totals of cost.
And of course, the slope
of that isocost curve
is delta K delta L,
or minus w over r.
That's the slope.
Or in this case, minus 0.5.
Now, those of you
thinking ahead--
I know you guys are very
insightful as a class,
I'm sure many of you
are thinking ahead--
might think, gee,
that slope might
change as the number of
workers and machines change.
Could you imagine the relative
price of capital labor changes
in different costs-- and it
might, and we'll put that aside
for now.
For now, assuming for every
relevant quantity these prices
$5 and $10 are fixed--
let's ignore where
those prices come from.
They're just given now.
We'll come back to that later.
Like I said, this course is
sort of like peeling an onion.
We raise things, then we come
back go to the next layer.
Where'd that come from?
Right now-- we'll tell you
where w and r come from.
Right now we're just
going to take them fixed.
And we'll assume they're
always $5 and $10,
regardless of the
amount produced.
OK.
Now, here's the question.
You're a firm that
wants to release
a certain amount of units.
You have a production
function and a cost function.
How do you graphically figure
out the right combination
of capital labor
to use to produce
a certain amount of units?
Yeah.
AUDIENCE: Could it be the
tension between the isoquant
and the isocost curves?
JONATHAN GRUBER:
That's exactly right.
Just as I asked you, what is
the right combination of pizza
and cookies, and
you told me that it
was the tangency
of the indifference
curve and the budget constraint,
it's the exact same logic here.
The optimal mix of
capital and labor
comes from the tangency of
the isoquant with the isocost,
as we see in figure 6-3.
And ignore the, like--
somehow that curve is sort
of connected at the top.
It's sort of a
glitch a PowerPoint.
Just ignore that.
It's not actually like
a square or a trapezoid.
It's just a curve.
What would you call that--
the curve and the two sides.
Is that a-- that's
not a trapezoid.
That's not-- it's not a polygon.
It's just a line.
All right.
So basically-- is
there a name for that?
A curve and two lines?
I don't think so.
It's just a polygon, right?
OK.
So it's not a polygon,
it's just a curve.
So the curve is the isoquant
for the square root of 12.5.
What do I mean by that?
I mean that is the combination
of capital and labor
that delivers square root
of 12.5 units of production.
So what that curve is
all possible combinations
of capital and
labor that deliver
square root of 12.5 units.
Just like the indifference curve
is all possible combinations
of pizza and cookies that
leaves you equally happy,
this is all possible
combinations
of capital and
labor that leads you
to a given production level.
And as we said, the
further out the isoquant,
the more you can produce.
So you want to produce as much
as you can given the prices you
face in the market.
Well, those prices
you face in the market
are delivered by
the isocost curve.
So the tangency is the best--
is the cost minimizing point.
That's when you're producing
the most you can given the costs
you face in the market--
the most you can, given the
costs you face in the market.
And that tangency
condition-- once again,
considering our parallels
to consumer theory,
the tangency condition is going
to deliver that the marginal
product--
is going to deliver that the
marginal product of labor
over the marginal product of
capital, which we remember
called last time
the marginal rate
of technical substitution,
is going to be equal to w/r--
actually, the negative
of these is going
to be equal to each other.
But we'll just cross
out the negatives.
So the negative of
MPL over MPK, which
we called the marginal rate
of technical substitution,
is equal to the negative of w/r.
The slope, the optimal
point, is where
the marginal rate of
technical subsection
equals the slope, which is
the wage to rental rate ratio.
Alternatively-- I don't know
if anyone besides me likes this
intuition-- we can rewrite
this as MPL/w equals MPK/r,
my bang-for-the-buck
formulation that I like--
that the next dollar
of wages, if you ask,
should I spend next dollar
on wages or machines,
you should do it until
the next dollar of wages
delivers you the same return
as the next dollar of machines.
This is what you get for the
next dollar of wages, MPL/w.
This it what you get for the
next dollar of machines, MPK/r.
You want to continue to trade
off machines and workers
until that condition is true.
So let's actually now solve
for this for our example.
Let's actually solve for that.
If we solve for this, we know
that the marginal product
of labor, which is dq dL, is 0.5
times K over square root of K
times L. And the marginal
product of capital,
which is dq dK, equals 0.5
times L over the square root--
0.5 times L over the square
root of K times L. Just taking
the derivative of
the-- all I did
was take the derivative of
the production function.
So therefore, putting
these together,
we're going to-- we
know that marginal rate
of technical substitution
in this example
is equal to minus K over L.
That's not a general formula.
That's just this example.
The marginal rate of
technical substitution
is equal to the negative of
the ratio of capital to labor.
We also know that
the wage rate--
we also know we want to set
this equal to the negative
of the wage rate--
I'm sorry, the wage
rental rate ratio.
And we know that's negative 1/2.
We know that's a 1/2, because
that's 5 and that's 10.
So we want to set the marginal
rate of technical substitution
to the wage rental
rate ratio, which
means we set minus K over
L equal to minus 1/2.
Or at the optimum, that means
that your labor, your capital,
at the optimum-- that means
the amount of capital--
should be half as much
as the amount of labor.
In this example, we just solved
for the efficient combination
of inputs, the
efficient combination
is you should use capital
to labor in a ratio of 1/2.
You should use half as
much capital as use labor.
So let me pause there.
And let's talk about
where this is coming from.
Yeah.
AUDIENCE: So does that mean
that at any given price of cost
[INAUDIBLE] line, that is
the optimal point where
it will be tangent [INAUDIBLE]
JONATHAN GRUBER: Exactly.
Exactly.
That's the graphic intuition.
Let's come to the
economics intuition.
The economics intuition
is the following.
The production function
delivers this relationship--
that the marginal rate
of technical substitution
was minus K over
L. In other words,
when you're producing goods,
given this production function
you're indifferent
between the next machine
and the next worker.
That's just the way this
production function worked
out--
that one more machine
delivers you the same amount
as one more worker.
Now, I've just told
you that one machine
costs half of one more worker.
So which you want more of?
No?
One machine delivers the
same return as one worker.
You want more workers.
Workers cost half machines,
they're equally productive,
so you want more workers.
So the optimal
amount of machines
is going to be half as many
as the number of workers.
You want more workers,
because you're indifferent--
look at that
production function.
You're indifferent.
You don't give a shit
about L versus K.
They're the same to you.
You're a hard capitalist, man.
Machine or worker,
you don't care.
But the market's telling you
you can get a worker for half
the price of a machine.
So you, as a good cost
minimizing capitalist,
take twice as many
workers as machines.
And that's the outcome
that you get here.
Questions about that?
OK.
So that's basically what
we do to derive this.
Now, what I want to do is
take this and then derive
our ultimate goal, which is,
what is the long run cost
function?
That's sort of what we--
why we started this lecture.
What is the long
run cost function?
Let's do the math.
We'll do the math in five steps.
Step one, q equals square
root of K times L. Step two,
we know from up there
that K/L equals w/r.
We derived that, leading us
to the conclusion that K--
lead us to the
conclusion that K equals
1/2 L. We just derived that.
Therefore, we can rewrite q as
the square root of 1/2 times L
squared, just substituting
it, because K equals 1/2 L.
Therefore, we can
solve for L is going
to be square root of 2 over q.
And K is going to be square
root of 2 over 2 over q.
L is square root of 2 over q,
K is square root of 2 over 2
over q.
I'm sorry-- no, not over q.
I'm sorry.
That's my bed.
Error.
Error.
Go back.
Should always look at my notes.
It's square root of 2 times q.
My bad.
L is square root of 2 times q.
And K is square root
of 2 over 2 times q.
OK.
Therefore, armed
with this L and K,
we can rewrite
our cost function.
So step five is that the cost
function-- given this stuff,
the cost function equals r
times square root of 2 times q--
I'm sorry, r times square
root of 2 over 2 times q--
plus w times square
root of 2 times q.
I just plugged in the optimal
L and K into my cost function.
Now I can plug in
the 10 and the 5
to get C equals 10 times
square root of 2 times q.
And I'm done.
I just derived
the cost function.
That's what we came here for.
This is what you got up this
morning and wanted to see.
You got up this
morning, you said,
I want to know how
does the cost of a firm
vary with the
quantity it produces?
And I've just told you.
This tells you how
the costs that you pay
vary the quantity you produce.
And I did that by deriving
the optimal mix of L and K
you want to use, and
then simply imposing
the prices of those two,
and I get a cost function.
Yeah.
AUDIENCE: Wouldn't you
be adding the two terms?
JONATHAN GRUBER: I'm sorry?
AUDIENCE: Wouldn't you
be adding the two terms?
JONATHAN GRUBER:
Which two terms?
Oh, I see.
Yeah, plus.
I'm sorry.
You're right.
My bad.
That's a plus.
Thanks.
This is the most math I'll
do in a lecture all year,
you'll be pleased to know.
It's why it's my least
favorite lecture.
Yeah?
AUDIENCE: So is
five generally true?
JONATHAN GRUBER: You mean this
particular functional form?
AUDIENCE: Yeah.
JONATHAN GRUBER: No.
This is all dependent
on that production
function I wrote down.
What's generally true is this--
or actually, K wouldn't be--
what's generally true
is just C equals--
in the long run,
what's generally true
is C equals wL plus rK.
That's what's generally true.
I just made--
But what I've showed you
is, given three things--
a production-- all I gave you
was a production function,
a wage rate, and a rental rate.
Given those three
things, you can then
derive the cost function.
Given those three things, you
can derive the cost function.
In fact, given two things you
can derive the cost function.
You could actually derive the
cost function given one thing--
given just the
production function,
derive the cost function as a
function of these input prices.
OK.
So that's a lot of
results from one function,
from one production function.
Now, other question about this?
Other math I got wrong?
Sorry about that.
Yeah.
AUDIENCE: Sorry, could you
just repeat the three inputs
that you were using?
JONATHAN GRUBER:
[INAUDIBLE] all I used.
Somebody tell me.
What do you need to get
the magical cost function?
What three things do you need?
Yeah.
AUDIENCE: w and r and then q.
JONATHAN GRUBER: No, w, r, and--
what about q?
You need q, but what
about-- what do you need?
w, r, and the
production function.
So armed with the
production function,
that mathematical equation, this
mathematical equation, w and r,
I'm done.
Everything I've
done in this lecture
comes from those three things.
The math is hard and annoying.
We will have you practice it.
You will not like it.
OK.
It's just kind of
what you've got to do.
All right?
I don't like it, you're
not going to like it.
It's just what we've
got to do to get
to the more interesting stuff.
OK?
Yeah.
AUDIENCE: Is the bar on the K?
JONATHAN GRUBER: It's
fixed in the short run.
The bar on the--
shouldn't see a bar
on a K over here.
That's all over here when
I was doing the short run.
Yeah?
AUDIENCE: Can you use r and w in
order to get to the fifth step?
JONATHAN GRUBER: Yeah,
we needed r and w.
AUDIENCE: Well, in the
sense that, to simplify,
K equals 1/2 L?
JONATHAN GRUBER: To K--
oh, you're right.
That's a good point.
I needed r and w back here.
You're right.
That's a good point.
Good point.
But I still could have done this
whole thing as a function of r
and w if I wanted to--
if I wanted to really
screw up my math.
All right?
OK.
So now, armed with
this, let's talk
about what happens when
input prices change.
We talked about with
consumer theory,
what happens when the price
of pizza and cookies change.
What happens when the price
of labor and capital changes?
What does that do?
So let's talk about
changes in input prices.
OK.
Let's go to figure 6-4.
And let's look at, with the
same production function,
square root of L times K-- we're
not changing our production
function-- we're going to
change the wage rental ratio.
So line-- we have our initial
line, our initial wage rental
ratio, which is that basically
you have a wage rate,
but the budget constraint,
essentially, that's flatter
is our original
budget constraint.
The flatter budget constraint is
our original budget constraint.
That's the budget constraint
with the price of capital
of $10 and a wage of $5.
And that intersects our
isoquant at point x.
So we chose five units of labor.
Now we have a new--
we chose five units of labor
and two and a half machines.
That was our original.
This is sort of a
messed up graph.
But our original
intersection was at point x.
The cost minimizing
combination, the square root
of 12.5 production, was to have
five workers and two and a half
machines.
Now let's say the price
of workers rises to $10.
The wage rate rises
to $10 an hour.
So now, workers and
machines cost the same.
What is now the optimal mix
of workers and machines?
Well, graphically
we know we still
want to produce the
square root of 12.5.
So we want to stay tangent
to the same isoquant.
So based on-- what
we're saying is,
this is as if we said to
consumers, keep your utility
changed, change the price.
What do we call that?
Remember what we called that?
Keeping utility constant,
changing the price?
Anyone remember
what we call that?
Who said that?
All right.
Raise your hand next.
Be proud.
Substitution effect.
That's the substation effect.
It's the same idea here.
We want to know, for a
given level of production,
what happens as the price
of the inputs change?
And so we shift along
the isoquant for point x
to point y.
And you'll see we
choose a mix where
we use fewer workers
and more machines.
And just as the substitution
effect is always nonpositive,
this shift as the wage
rate rises, the price
of the good, the x-axis rises.
You will unambiguously use
no more, and almost certainly
less, workers.
OK.
And you can see
that graphically--
think about graphically,
you're looking
for the tangency between
this curve and the line.
The slope of the line
just got steeper.
Therefore, you must move
to the left on the curve.
It's the same proof as we
used the substitution effect,
where substitution effect
was always nonpositive.
OK.
It's the same intuition
here we use for y.
A rise in the wage
rate will lead
you to hire fewer workers
and more machines.
Guess what?
You just entered the
debate on the minimum wage.
And if you follow the
debate in minimum wage, what
do people say?
Well, if you raise the wage
you have to pay workers,
they'll be replaced by machines.
That's this.
This is the math-- this is
the mathematical and graphical
intuition behind the debate
on minimum wage, which we'll
get into later in the semester.
But the basic idea
of that debate
is, gee, if you force firms
to pay more to workers,
they're going to substitute
towards machines.
That's exactly right, in theory.
And practice, there's
a lot of complications.
But this gives you the theory of
why people make that argument.
Yeah.
AUDIENCE: So in this example,
only the wage for workers
[INAUDIBLE] not the machines.
JONATHAN GRUBER:
Not the machine.
AUDIENCE: So why
does the isocost not
have the same y-intercept?
JONATHAN GRUBER:
Ah, great point.
Because here I'm drawing a new--
I am drawing the isocost that I
would use while still producing
square root of 12.5.
So that's it's just the
substitution effect.
I'm not drawing the full set
of isocosts at the new price.
I'm just saying, to produce
the same amount what's
my new-- if I want to
produce the same amount, what
combination do I
now have to use?
OK?
Yeah.
AUDIENCE: The total cost
of production [INAUDIBLE]..
JONATHAN GRUBER: The total
cost of production-- let's see.
Yeah, it has to be-- no.
Let's see.
No, total cost doesn't
have to be the same.
The total cost used to be
five workers at $5 an hour,
that's $25.
So it used to be $50.
Now what is it?
Now it's $70.
The total cost has gone up.
AUDIENCE: So it's not like
the budget constraint or the--
JONATHAN GRUBER: Exactly.
It's not like the
budget constraint
where your income is fixed.
That's what's hard
about producer theory.
Because basically,
the budget constraint
was sort of asking,
keeping your budget fixed.
This is like asking, keeping
your total production fixed.
And so now you have
to pay more to get
that level of production.
OK?
Good questions.
OK.
Now, ultimately what
does this lead to?
So that gives us our
change in input prices.
Other questions about that?
Now, remember I said at the
beginning of the lecture,
we are first going
to solve for what
is the cost
minimizing combination
of inputs for a given quantity?
We derived that up there.
It's half as much--
going back to our
old prices, it's
half as much capital as labor.
Now we want to ask,
how does your cost
change as the quantity changes?
And we call that the
long run expansion path.
The long run
expansion path, which
is, how do your costs
expand as you produce more?
And we see that in figure 6-5.
In figure 6-5, we show
the particular case
of a linear long
run expansion path.
That's what you get
in this example.
It's a particular case.
What this case says is, at
any given level of production,
the optimal mix of labor
and capital is the same.
In other words, you
always want to have--
essentially, given
the price of labor
is half the price of
capital, you always
want to have twice as
many workers as machines.
So if you want to produce
square root of 12.5,
you want five workers and
two and a half machines.
If you want to produce
square root of 50,
you want 10 workers
and 5 machines.
If you want to produce
square root of 112.5,
you want 15 workers
and 7 and 1/2 machines.
So the long run-- given
this production function,
the long run expansion
path is linear.
You always want the same
ratio of workers to machines.
Yeah.
AUDIENCE: [INAUDIBLE]
consumer and firms,
is the reason why
we don't necessarily
have a strict budget,
per se, and then isn't
the idea that if we really
want increase production,
we can take a loan out?
JONATHAN GRUBER: This
is what I tried to say.
It's sort of-- I always say
it over here, but it's hard,
and we have to come back to it.
The reason producer
theory is harder
is because we're not
given a fixed constant we
are with consumers.
Consumers, we're saying, look--
you've got a resource, you've
got to constrain maximization.
We haven't constrained
the maximization yet.
There's another
constraint we need.
They have an extra degree of
freedom relative to consumers.
Now, in fact, consumers
have degree of freedom, too.
When you grow up, your
parents don't give you money.
You decide how much to make.
So in reality, consumers--
you can do this--
will have the same
degrees of freedom.
But we started with the
easy consumer theory
case, where you constrict-- we
took away a degree of freedom.
Now we're writing it
back, which is, you
can choose how much to produce.
Like, you being able choose
your income as a consumer.
That leads to long
run expansion path.
Let me go on, because
I want to make
sure I get through this stuff.
OK?
Now, the long run expansion
path does not have to be linear.
So think about-- look
at figure 6-5b and 6-5c.
So 6-5b is a long run expansion
path for a production function
such that capital becomes
less productive the more you
produce.
I don't have the example
of production function.
But when you write down
production functions which
have the feature that the
more you produce the less
productive capital
becomes, the less
each additional unit
capital helps [INAUDIBLE]
additional unit of workers.
So you know, we could
think of this roughly as
sort of like a fast
food restaurant.
That kind of-- you
know, each addition--
that basically,
there's so much stuff
to do where workers can
efficiently share tasks
and things.
Each additional worker-- that
the marginal product of labor
essentially diminishes
less quickly
than the marginal
product of capital.
On the other hand,
in figure 6-5c
we can have a long run expansion
path where labor becomes less
productive relative to capital.
Think of it as like
heavy machinery,
where basically all workers
can do is run the machine.
So that second worker--
workers don't really do much
but sit there and flip a switch.
You need the worker
to flip the switch.
That's all they do.
So the second worker, you're
already flipping the switch.
So really, adding more machines
is a more productive way
to expand the output.
None of these is right or wrong.
We're just saying that the
shape of this expansion path
can basically vary
with how much--
with different
production functions.
But they're all the same idea.
Now, so basically,
that tells you--
but here's the bottom line
that we wanted to come to.
That long run expansion path
is a long run cost curve.
So ultimately, if
you want to ask,
how do my costs vary
with how much I produce,
this curve tells you.
Because what it does, it says,
for every level of production
I'll tell you the optimal
combination of L and K.
Given the price of L and K,
that will tell you the costs.
And so you trace out the costs
with every level of production.
This is your cost curve.
This long run expansion
path tells you
what the costs are for
every level of production.
And it tells you
that, because you've
made-- you're doing the
efficient level of production.
That's what the long run
expansion path is telling you.
OK.
This is hard.
I'm about to make it harder.
Which is, we're
now going to talk
about the relationship between
short run costs and long run
costs.
And the key insight is that
long run costs are everywhere
lower than short run costs.
Without looking at the figure--
because the figure doesn't help
with this--
why does that make sense?
Why our long run costs--
why, if you can optimize
over the long run,
will you always
have costs that are
no higher, and in general lower,
than optimizing the short run?
Yeah.
AUDIENCE: [INAUDIBLE] already
had the right capital.
JONATHAN GRUBER:
Because you have
an extra degree of freedom.
I think that's
LeChatelier's Principle.
Is that right, for
the chemists among us?
That basically,
like-- essentially,
an extra degree
of freedom means,
the more you can optimize
over, the better you
can do in optimizing.
In the short run
you're constrained
by the size of the building.
In the long run,
you could choose.
So let's-- to see that,
let's go to figure 6-6.
This is a confusing figure.
So bear with me as I
walk you through it.
OK?
Consider a firm with three
possible sizes of plants.
They're going to build a plant.
So the capital here
is the building.
And there's three
possible sizes--
small, medium, and large.
The small plant has
the curve SRAC1.
What does that curve mean?
That means that
the small plant--
I'm sorry, the small
plant is SRAC2,
the medium plant is SRAC2,
and the large plant has SRAC3.
Compare SRAC1 to SRAC3.
What this is saying is, for
small quantities of production,
SRAC1 lies below SRAC3.
For small quantities production,
if you extend SRAC3 out,
you see at levels of
production like q1 or even q2,
SRAC3 is way above SRAC1.
When you go to a level
of production like q3,
SRAC1, if you extend
that dashed line out,
is going to be much,
much higher than SRAC3.
So the right-- and
SRAC2 is in between.
So essentially, for different
levels of production
these give the different
optimal short run cost curves.
In the long run,
you get to choose.
So the long run
average cost curve
is the lower envelope of the
short run average cost curves.
Because in the long run
you say, well, here's
my production level.
I know in the long run--
so if I know I'm going
to build a lot of things,
I choose SRAC3.
I choose the biggest plant.
If I know my production
is going to be low,
I choose the smallest plant.
But I can optimize
in the long run
by choosing the right sized
plant for my production level.
This is hard.
And I'm almost out
of time, so let
me end with an example that
perfectly illustrates this.
Tesla.
Elon Musk.
Everybody's favorite
guy these days.
Tesla, when they came
out, had to decide
how big a plant to build--
how many batteries to make.
Batteries are the key
[INAUDIBLE] Teslas.
And they expected to make--
to have demand for 20,000 cars
by the year 2017.
So they built a
plant like SRAC1.
They built a plant that
was the efficient plant
to produce 20,000 cars.
The problem is, demand
was for 200,000 cars.
And as a result, there's
a three-year waiting list
to get Teslas.
It turned out that was not
the right size to produce.
They lost money--
relative to the optimum.
They made money.
Musk is incredibly rich.
But they didn't do what was most
efficient given the underlying
demand.
But now, Musk can re-optimize.
Now he's saying, wait a second.
People want way more cars.
Well, producing them
at the tiny plant
was exorbitantly expensive.
I had to run it
over and overtime--
pay workers overtime.
To produce 200k cars
in that tiny plant
just was exorbitantly expensive.
That's if you take that dashed
line and extend it way the hell
up.
The SRAC1 extended way the
hell up, incredibly expensive.
So what is Musk doing now?
Building the largest
battery plant in the world.
In Nevada, he is
building a battery plant
that can produce batteries
for 500,000 cars.
So he shifted from
SRAC1 to SRAC3.
He's now saying
in the long run, I
can more-- if I'm going
to produce 200,000 cars,
I can do that more efficiently
with a giant battery plant.
And that's what he's doing.
So he's re-optimizing.
Now, what if Musk is wrong?
What if it turns out Teslas
suck and people are like, I
don't want them anymore?
Someone else-- or, you know,
Chevy finally figures it out
and makes a good electric car.
Then what's going to happen
is he's going to have
made a mistake in the long run.
Then the third period, he'll go
back to a smaller plant again.
But he always can do what's
efficient in the long run,
given the underlying demand.
So Tesla is an example of this
sort of long run, short run
dichotomy.
Anyway, it's a lot of
stuff for one lecture.
We'll come back next time,
talk more about costs.
And then we'll start
getting into competition.
