# Free Disposal: the production set less the vector space of positive reals

In Microeconomic Theory (3rd Edition), Mas-Collel, Whinston and Green state that the property of free disposal implies the following:

$$Y - \mathbb{R}^L_+ \subset Y$$

Y: Production set

y: Production vector

L: Amount of commodities considered

Why does this formula means that the extra amount of inputs or outputs can be disposed of or eliminated at no cost?

So the way that $$Y$$ is defined is such that negative values are inputs and positive values are outputs. For example, if I used 3 units of labor, and 2 of capital to produce 10 pencils, that will be reflected as a single vector in $$\mathbb{R}^3$$ equal to $$[-3;-2;10]\in Y$$.

Notice that subtracting 1 unit to either the first or the second element of that vector will mean that I used more inputs to produce the same output, for example, $$[-3;-3;-10]$$. Similarly, if I subtract one unit to the third element, it would mean that I produced less output with the same inputs. For example $$[-3;-2;9]$$.

Both of these operations, respect the idea of free disposal (so long as you are subtracting units, not adding to them). Moreover, any notion of free disposal can be represented by one of these operations: i.e producing less with the same inputs, or using more inputs to produce the same.

Therefore you can conclude that a production set, $$Y$$, satisfies free disposal if and only if $$Y-\mathbb{R}^L_+\subset Y$$.