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Physical Capital and Diminishing Returns - YouTube
Channel: Marginal Revolution University
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- [Alex] In our last video,
we introduced the variables
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in our Super Simple Solow Model.
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We have physical capital,
represented by "K,"
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human capital, represented by "e"
times "L," and ideas,
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represented by "A."
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In this video, we're going to hold
human capital and ideas constant.
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That will let us focus in on K
so we can show what happens
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to output when the amount
of physical capital changes.
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Since capital is the only input,
output is a function
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just of the quantity of capital.
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Let's write output
with the letter "Y."
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Then we can say that Y is
a function of K.
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Output is a function
of the quantity of capital.
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What properties should
our production function have?
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First, it makes sense
that more K increases output.
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Recall from our earlier
video, our farmer.
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A farmer with a tractor can
produce a lot more output
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than a farmer with just a shovel.
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Similarly, a farmer with two
tractors can produce more output
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than a farmer with just one tractor.
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If we graph capital
on the horizontal axis
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and output on the vertical axis,
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we're going to see
a positive relationship.
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As capital goes up,
output goes up.
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That seems pretty straightforward.
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The second property
our production function
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should have is
that while more capital
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produces more output,
it should do so
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at a diminishing rate.
What do I mean by that?
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Let's go back to our farmer.
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The first tractor he gets is
the most productive.
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It helps him grow
a lot more wheat.
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The second tractor he might use if
the first tractor -- it breaks down.
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So the second tractor is
less productive than the first.
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The third tractor is maybe just
a spare in case both break down.
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So the third tractor will boost
his output even less
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than did the second.
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Said another way, the farmer will
allocate his tractors
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so that the first tractor,
he's going to allocate
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to the most important,
the most productive task.
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Meaning that subsequent tractors --
the farmer will allocate them
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to less and less productive tasks.
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We call this the Iron Logic
of Diminishing Returns.
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To represent both
of these properties,
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we can use a simple
production function,
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one which we're already familiar
with: the square root function.
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Output equals the square root
of the capital inputs.
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So if we input 1 unit of capital,
output is 1.
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If we input 4 units of capital,
output is 2.
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If we input 9 units of capital,
output is… 3.
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The marginal product
of capital describes how much
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additional output is produced
with each additional unit
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of capital.
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Notice that the marginal product
of the first unit of capital
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is really high.
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But as the capital stock grows,
the marginal product of capital
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is less and less and less.
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Already, we can explain
one of our puzzles.
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Recall that growth was fast
in Germany and Japan
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after World War II.
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That makes sense,
because after the war,
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those countries -- they
didn't have a lot of capital.
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So that meant that the first units
of capital had a very high
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marginal product.
The first road between two cities
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or the first tractor on a farm,
or the first new steel factory --
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that gets you a lot
of additional output.
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Capital's very productive
when you don't have a lot of it.
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But don't forget that Germany
and Japan were growing
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from a low base.
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You can grow fast
when you don't have a lot,
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but all else being the same,
you'd rather have more
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and grow slower.
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So, capital can drive growth,
but because of the iron logic
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of diminishing returns,
the same additions
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to the capital stock may get
you less and less output.
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Unfortunately for K,
in the next video we'll show
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that capital has another
problem to deal with.
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- [Announcer] If you want to test
yourself, click “Practice Questions.”
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Or, if you're ready to move on, you
can click “Go to the Next Video.”
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to see our entire library
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of videos and resources.
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