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 didn't have a lot of
capital. So that meant that the first units

of capital had 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 go 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.

- [Narrator] If you want to test yourself,
click “Practice Questions.” Or, if you're

ready to move on you can click
“Go to the Next Video.”

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