4
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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

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.

| improve this answer | |
$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – 123 Jul 2 '16 at 4:01
  • $\begingroup$ Fair enough. and I'm just clarifying the condition :P $\endgroup$ – Kitsune Cavalry Jul 2 '16 at 5:26
2
$\begingroup$

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.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.