<p>So the way that <span class="math-container">$Y$</span> 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 <span class="math-container">$\mathbb{R}^3$</span> equal to <span class="math-container">$[-3;-2;10]\in Y$</span>. </p>

<p>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, <span class="math-container">$[-3;-3;-10]$</span>. Similarly, if I subtract one unit to the third element, it would mean that I produced less output with the same inputs. For example <span class="math-container">$[-3;-2;9]$</span>. </p>

<p>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. </p>

<p>Therefore you can conclude that a production set, <span class="math-container">$Y$</span>, satisfies free disposal if and only if <span class="math-container">$Y-\mathbb{R}^L_+\subset Y$</span>.</p>

CC BY-SA 4.0
2019-06-24T19:24:49.023Z
2019-06-24T19:24:49.023Z
Regio
<p>In Microeconomic Theory (3rd Edition), Mas-Collel, Whinston and Green state that the property of free disposal implies the following:</p>

<p><span class="math-container">$Y - \mathbb{R}^L_+ \subset Y $</span></p>

<p><a href="https://i.sstatic.net/g3qmb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/g3qmb.png" alt="enter image description here"></a></p>

<p>Y: Production set</p>

<p>y: Production vector</p>

<p>L: Amount of commodities considered</p>

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

<microeconomics>
Free Disposal: the production set less the vector space of positive reals
Incognito
# Free Disposal: the production set less the vector space of positive reals

(Asked by: username_0 on 2019-06-21)

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 $

[![enter image description here](https://i.sstatic.net/g3qmb.png)](https://i.sstatic.net/g3qmb.png)

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?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2019-06-24. Score: 3 | Accepted answer

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 $Y-\mathbb{R}^L\_+\subset Y$.


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