We're given a Cobb-Douglas Production
function P of L comma K.
We want to find in the
marginal productivity of labor
and the marginal productivity
of capital functions
which would be the partial
derivative with respect to L
and the partial derivative
with respect to K.
Before we do this though, let's talk more
about the Cobb-Douglas
Production function.
In economics, the Cobb-Douglas
Production function
is a form of the production
function often used
to represent the relationship
between physical capital and labor
and the amount of output
that can be produced by those inputs.
So this is the form of the
Cobb-Douglas Production function
where P is the total production,
L is the labor input,
K is the capital input
A is the total factor productivity.
And notice alpha and beta, the exponents,
are the output and elasticity of labor
and the output and elasticity of capital.
So going back to our example,
we'll now find the marginal
productivity of labor function
by determining the partial
derivative with respect to L.
So to do this, we'll
differentiate with respect to L
treating K as a constant.
So we'd have 12 times, if we differentiate
with respect to L, we
would multiply by 0.7,
and then we'd have L raised
to the power of 0.7 minus one,
that's negative 0.3 and we're
treating K as a constant.
Now let's go ahead and simplify this
and write this only
using positive exponents.
So 12 times 0.7 is 8.4.
So we'd have 8.4.
K has a positive exponent,
so that would be in the numerator.
So K to the 0.3.
And because L has a negative exponent,
we'll move it down to the denominator
which would change the
sign of the exponent.
So we'd have L to the positive
0.3 in the denominator.
So this is the marginal
productivity of labor function.
And then we'll find the partial
derivative with respect to K
which would be the marginal productivity
of capital function.
So we differentiate with respect to K,
now treating L as a constant.
So we'd have 12 and we
multiply by the exponent on K
which would be 0.3.
L will stay this same, L to the 0.7.
And then for the exponent
on K, we would subtract one,
0.3 minus one is negative 0.7.
Let's go ahead and simplify.
12 times 0.3 is 3.6.
L has a positive exponent.
The denominator would be K
raised to the power of 0.7.
Now that we have our two functions,
we're asked to evaluate them
when L equals 400 and K equals 1,000.
After we do this, we'll
explain what this means.
So the partial with respect to L,
notice how the coordinates are L comma K.
So it'd be 400 comma 1,000.
Again, where L equals
400 and K equals 1,000.
So for the partial with respect to L
we would have 8.4
times K or 1,000
raised to the power of
0.3 divided by L or 400
to the 0.3.
We'll evaluate this in just a moment.
Let's go ahead and set up
the partial with respect to K
at the same point.
So we'll have 3.6 times L or 400
raised to the power of 0.7
divided by K or 1,000
raised to the power of 0.7.
Now we'll evaluate
these on the calculator.
So first we have in the numerator 8.4
and then times 1,000.
This is raised to the power of .3,
right arrow, closed parenthesis,
that's the numerator,
divided by 400
raised to the power of .3,
Enter.
If we round to three decimal places,
this would be approximately 11.058.
This is the change in production
with respect to labor at this point.
And next we have, and now for the partial
with respect to K, we
have in the numerator 3.6
times 400
raised to the power of .7,
right arrow, closed parenthesis,
there's the numerator,
divided by 1,000
raised to the power of .7,
Enter.
To three decimal places we'd
have approximately, 1.896.
Now let's emphasize what
these two values represent.
The value from the
partial with respect to L
which we found here is additional output
that results from employing
an additional unit of labor
when L equals 400 and K equals 1,000.
So we can say at this point
production is increasing
at approximately 11 units per worker.
And then for the partial
with respect to K.
This represents the additional output
resulting from the use of an
additional unit of capital,
again when L equals
400 and K equals 1,000.
So the units on this value would be
units of production per let's say dollar.
I hope you found this helpful.
