How to Calculate the Law of Diminishing Returns

One example of diminishing returns is a factory with too many workers.

The Law of Diminishing Returns states that at some point the amount of output you get for your inputs is going to reduce, and eventually your inputs actually make your output go down. For example, you have a factory, and you find that adding workers (inputs) makes your factory's productivity (output) go up. Add too many workers, though, and the factory will become crowded. At some point you'll add another new employee only to find that the productivity actually goes down, not up.

Instructions

1. Determine your return function. Either get it from the math problem you're trying to solve, or calculate a function based on historical data of inputs and outputs. For example, you may have a parabolic return function of f(x) = -x^2 + 20x + 10.

2. Find the derivative of your function from Step 1. In the example, f'(x) = -2x + 20.

3. To find when additional inputs actually make overall output go down, maximize your function by solving the derivative in step 2 for 0.
   0 = -2x + 20
   x = 10
   After 10 units of input, the output begins to go down instead of up.

Tips & Warnings

• If you want to find the point at which output begins to go up less per unit of input, indtead of when it begins to go down, solve the derivative for <1.

References

• The Law of Diminishing Returns