Physical Capital and Diminishing Returns

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- [Alex] In our last video,
we introduced the variables

in our Super Simple Solow Model.

We have physical capital,
represented by "K,"

human capital, represented by "e"
times "L," and ideas,

represented by "A."

In this video, we're going to hold
human capital and ideas constant.

That will let us focus in on K
so we can show what happens

to output when the amount
of physical capital changes.

Since capital is the only input,
output is a function

just of the quantity of capital.

Let's write output
with the letter "Y."

Then we can say that Y is
a function of K.

Output is a function
of the quantity of capital.

What properties should
our production function have?

First, it makes sense
that more K increases output.

Recall from our earlier
video, our farmer.

A farmer with a tractor can
produce a lot more output

than a farmer with just a shovel.

Similarly, a farmer with two
tractors can produce more output

than a farmer with just one tractor.

If we graph capital
on the horizontal axis

and output on the vertical axis,

we're going to see
a positive relationship.

As capital goes up,
output goes up.

That seems pretty straightforward.

The second property
our production function

should have is
that while more capital

produces more output,
it should do so

at a diminishing rate.
What do I mean by that?

Let's go back to our farmer.

The first tractor he gets is
the most productive.

It helps him grow
a lot more wheat.

The second tractor he might use if
the first tractor -- it breaks down.

So the second tractor is
less productive than the first.

The third tractor is maybe just
a spare in case both break down.

So the third tractor will boost
his output even less

than did the second.

Said another way, the farmer will
allocate his tractors

so that the first tractor,
he's going to allocate

to the most important,
the most productive task.

Meaning that subsequent tractors --
the farmer will allocate them

to less and less productive tasks.

We call this the Iron Logic
of Diminishing Returns.

To represent both
of these properties,

we can use a simple
production function,

one which we're already familiar
with: the square root function.

Output equals the square root
of the capital inputs.

So if we input 1 unit of capital,
output is 1.

If we input 4 units of capital,
output is 2.

If we input 9 units of capital,
output is… 3.

The marginal product
of capital describes how much

additional output is produced
with each additional unit

of capital.

Notice that the marginal product
of the first unit of capital

is really high.

But as the capital stock grows,
the marginal product of capital

is less and less and less.

Already, we can explain
one of our puzzles.

Recall that growth was fast
in Germany and Japan

after World War II.

That makes sense,
because after the war,

those countries -- they
didn't have a lot of capital.

So that meant that the first units
of capital had a very high

marginal product.
The first road between two cities

or the first tractor on a farm,
or the first new steel factory --

that gets you a lot
of additional output.

Capital's very productive
when you don't have a lot of it.

But don't forget that Germany
and Japan were growing

from a low base.

You can grow fast
when you don't have a lot,

but all else being the same,
you'd rather have more

and grow slower.

So, capital can drive growth,
but because of the iron logic

of diminishing returns,
the same additions

to the capital stock may get
you less and less output.

Unfortunately for K,
in the next video we'll show

that capital has another
problem to deal with.

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