<p>First, to give a little more background to 123's answer, a production set $Y$ is the set of all feasible production values. With $y \in Y$, $y$ is a vector in $\mathbb{R}^L$, where positive elements indicate outputs, negative values indicate inputs. The inputs you put in that get you outputs can be expressed with either a transformation or production function.</p>

<p>123 is right when stating the formal microeconomic axiom:</p>

<blockquote>
  <p>If that $y \in Y$ and $y \geq 0$, then $y = 0$.</p>
</blockquote>

<p>However, no free lunch axiom needs more than merely "passing through the origin."<a href="https://i.sstatic.net/PlzPG.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PlzPG.png" alt="beautiful"></a></p>

<p>The first image shows a production set (two dimensions) that satisfies no free lunch. The second image shows a production set that does not, even though it passes through the origin.</p>

<p>(yes this image is a bad rendition of the one from Mas-Colell)</p>

<hr>

<p>Interestingly though, the author has not tagged this with a microeconomics tag, but with a macro and financial econ tag. The closest thing I can thing of to a no free lunch axiom here is some sort of transversality condition. For example, in simple money market models with borrowing with households, you might have a condition that you have to pay off all debt eventually (or just pay the interest for the rest of your life). So that would look something like,</p>

<p>$$\lim_{t \rightarrow \infty} \frac{b_t}{(1+r)^t} = 0$$</p>

<p>where $b$ is the amount of borrowing in a period. I do not hear conditions like these called "no free lunch" however. Something more like "no ponzi scheme condition" or "borrowing constraints" (snoree) are more apt.</p>

CC BY-SA 3.0
2016-07-02T04:01:34.820Z
Just to clarify - I've not at all implied that passing through the origin is the sufficient condition for satisfying this axiom. Rather, I stated any function satisfying this axiom will pass through the origin.
123
CC BY-SA 3.0
2016-07-02T05:26:57.117Z
Fair enough. and I'm just clarifying the condition :P
Kitsune Cavalry
CC BY-SA 3.0
2016-07-01T21:33:25.577Z
2016-07-01T21:33:25.577Z
Kitsune Cavalry
<p>No Free Lunch : Let Y be the production set and $y \in Y$ denote an element of the production set Y. If $y\geq 0$ then $y=0$. </p>

<p>Recall that for all elements $y \in Y$ we have that y comprises inputs and outputs. The inputs should be negative values anytime there is a positive output. That is, you can never produce something from nothing. </p>

<p>To say y is greater than or equal to zero means y has no non-negative elements. To rephrase - y has nothing acting as an input. So, it must be true that nothing in y can be strictly positive. To rephrase - we can't have produced a positive amount of something if we don't have any inputs with which to produce outputs. </p>

<p>Production functions satisfying this property should pass through the origin. </p>

CC BY-SA 3.0
2016-07-01T20:47:21.613Z
2016-07-01T20:47:21.613Z
123
<p>I have tried to know what exactly is the no free lunch axiom.</p>

<p>I got multiple links but none of them explain (a bit rigorously, with examples and mathematical formulation) what is this axiom and how I can formulate it in a problem.</p>

<p>Any explanation/formulation is very much welcome. Thanks.</p>

<macroeconomics><mathematical-economics><financial-economics>
What is the No Free Lunch Axiom?
ferrer
# What is the No Free Lunch Axiom?

(Asked by: username_0 on 2016-07-01)

I have tried to know what exactly is the no free lunch axiom.

I got multiple links but none of them explain (a bit rigorously, with examples and mathematical formulation) what is this axiom and how I can formulate it in a problem.

Any explanation/formulation is very much welcome. Thanks.

**Tags:** macroeconomics, mathematical-economics, financial-economics

---

## 2 answers

### Answer by username_1 on 2016-07-01. Score: 3

First, to give a little more background to 123's answer, a production set $Y$ is the set of all feasible production values. With $y \in Y$, $y$ is a vector in $\mathbb{R}^L$, where positive elements indicate outputs, negative values indicate inputs. The inputs you put in that get you outputs can be expressed with either a transformation or production function.

123 is right when stating the formal microeconomic axiom:

> If that $y \in Y$ and $y \geq 0$, then $y = 0$.

However, no free lunch axiom needs more than merely "passing through the origin."[![beautiful](https://i.sstatic.net/PlzPG.png)](https://i.sstatic.net/PlzPG.png)

The first image shows a production set (two dimensions) that satisfies no free lunch. The second image shows a production set that does not, even though it passes through the origin.

(yes this image is a bad rendition of the one from Mas-Colell)

---

Interestingly though, the author has not tagged this with a microeconomics tag, but with a macro and financial econ tag. The closest thing I can thing of to a no free lunch axiom here is some sort of transversality condition. For example, in simple money market models with borrowing with households, you might have a condition that you have to pay off all debt eventually (or just pay the interest for the rest of your life). So that would look something like,

$$\lim\_{t \rightarrow \infty} \frac{b\_t}{(1+r)^t} = 0$$

where $b$ is the amount of borrowing in a period. I do not hear conditions like these called "no free lunch" however. Something more like "no ponzi scheme condition" or "borrowing constraints" (snoree) are more apt.

> **Comments:**
> **username_2**: Just to clarify - I've not at all implied that passing through the origin is the sufficient condition for satisfying this axiom. Rather, I stated any function satisfying this axiom will pass through the origin.  
> **username_1**: Fair enough. and I'm just clarifying the condition :P  

---

### Answer by username_2 on 2016-07-01. Score: 2

No Free Lunch : Let Y be the production set and $y \in Y$ denote an element of the production set Y. If $y\geq 0$ then $y=0$.

Recall that for all elements $y \in Y$ we have that y comprises inputs and outputs. The inputs should be negative values anytime there is a positive output. That is, you can never produce something from nothing.

To say y is greater than or equal to zero means y has no non-negative elements. To rephrase - y has nothing acting as an input. So, it must be true that nothing in y can be strictly positive. To rephrase - we can't have produced a positive amount of something if we don't have any inputs with which to produce outputs.

Production functions satisfying this property should pass through the origin.


---

