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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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