Working with production functions can be tricky. This post goes over the economics of average
and marginal products of labor (or any input for that matter) working with a
production function. To do this we are
going to use the following example:

Let Q = 1200L^2 –L^4 where L is an input (for example,
labor) and Q is the associated output.

First, we want to find the level of the input that will maximize the average
product of labor or APL.

Second, we want to find out how much output is produced at this level of labor.

Third, we want to actually calculate the average product of labor (APL).

And finally, we want to derive the marginal product of labor
(MPL) for this amount of labor.

In order to find the maximized average product of labor we
need to change the production function into an average product of labor
function by dividing both sides of the equation by labor or L.

This leaves us with:

APL = Q/L = 1200L – L^3

We want to know where this is maximized so we take the
derivative of APL with respect to L and set it equal to zero (the first order
condition for maximization). This gives
us:

1200 – 3L^2 = 0 now
we subtract 3L^2 from both sides, divide both sides by 3 and take the square
root of both sides to get:

L = 20 which is the average product of labor maximizing
amount of labor.

We can now calculate how much output we will produce with
this much labor by plugging it into our production function:

Q = 1200(20) – (20)^3
or

Q = 480,000 – 160,000 = 320,000

We can use the Q and L values we solved for to figure out
APL:

APL = Q/L = 320,000/20 = 16,000

The marginal product of labor is the output that the next
unit of labor produces. We can find this
by either solving for Q when we have 19 and 20 units of labor (this is not
exact) and finding the difference or by taking the derivative and plugging in
our optimal labor amount.

Remember our production function is: 1200L^2 –L^4

The derivative of the above production function with respect
to L will be:

MPL = 2400L – 4L^3

We then plug in our labor value of 20 to get:

48,000 – 32,000 = 16,000

Which is good that it equals our APL because they should
cross at APL’s max (the MPL curve crosses the APL curve from above).