<p>The input requirement set is defined as the set of all inputs required to produce at least the quantity <span class="math-container">$y$</span>, or in other words, with a production function given by <span class="math-container">$y=f(x),$</span>
<span class="math-container">$$ V(y)= \{x: f(x) \geq y \}.$$</span>
Note that <span class="math-container">$x$</span> can be a vector in this definition.<br />
If <span class="math-container">$f:\mathbb{R} \rightarrow \mathbb{R}$</span> and is increasing, the input requirement is equivalently written as
<span class="math-container">$$ V(y)= \{x: x \geq f^{-1}(y) \}.$$</span>
If <span class="math-container">$f:\mathbb{R}^2 \rightarrow \mathbb{R}$</span> and is increasing, the input requirement is equivalently written as
<span class="math-container">$$ V(y)= \{ ( x_1,x_2 ) : x_2 \geq f^{-1}(x_1,y) \}.$$</span></p>
<p>The minimum requirement on <span class="math-container">$f$</span> to obtain a convex set <span class="math-container">$V$</span>, is that <span class="math-container">$f$</span> is quasi-convave in <span class="math-container">$x$</span>. See for instance the Simon and Blume (1994, chapter 21.3) on this topic.</p>

CC BY-SA 4.0
2023-12-29T16:45:45.663Z
2023-12-29T16:45:45.663Z
Bertrand
<p>This question relates to the book Varian Microeconomic Analysis 3rd edition exercise 1.1. Much like <a href="https://economics.stackexchange.com/questions/54004/convexity-of-production-sets-and-input-requirement-sets">this question</a> but my emphasize is different.</p>
<p>Q: True or False? If V(y) is a convex set, then the associated production set must be convex.</p>
<p>Answer: as in the linked question above.</p>
<p>But I wonder how did Varian derive the input requirement set V(y) from his assumed production function and the associated production set? I see here there is obviously some relationship between the production function being quadratic, and the production set being the area below this graph and bounded by the x-axis, and V(y) being constrained by the inverse, the square root function, but I still do not get it. And also how to <em>generally</em> do it, say if we have a production function with two inputs etc.?</p>

<microeconomics><production-function>
Derive the input requirement set from production set
yomath
# Derive the input requirement set from production set

(Asked by: username_0 on 2023-12-26)

This question relates to the book Varian Microeconomic Analysis 3rd edition exercise 1.1. Much like [this question](https://economics.stackexchange.com/questions/54004/convexity-of-production-sets-and-input-requirement-sets) but my emphasize is different.

Q: True or False? If V(y) is a convex set, then the associated production set must be convex.

Answer: as in the linked question above.

But I wonder how did Varian derive the input requirement set V(y) from his assumed production function and the associated production set? I see here there is obviously some relationship between the production function being quadratic, and the production set being the area below this graph and bounded by the x-axis, and V(y) being constrained by the inverse, the square root function, but I still do not get it. And also how to *generally* do it, say if we have a production function with two inputs etc.?

**Tags:** microeconomics, production-function

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## 1 answers

### Answer by username_1 on 2023-12-29. Score: 2

The input requirement set is defined as the set of all inputs required to produce at least the quantity $y$, or in other words, with a production function given by $y=f(x),$
$$ V(y)= \{x: f(x) \geq y \}.$$
Note that $x$ can be a vector in this definition.  
If $f:\mathbb{R} \rightarrow \mathbb{R}$ and is increasing, the input requirement is equivalently written as
$$ V(y)= \{x: x \geq f^{-1}(y) \}.$$
If $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ and is increasing, the input requirement is equivalently written as
$$ V(y)= \{ ( x\_1,x\_2 ) : x\_2 \geq f^{-1}(x\_1,y) \}.$$

The minimum requirement on $f$ to obtain a convex set $V$, is that $f$ is quasi-convave in $x$. See for instance the Simon and Blume (1994, chapter 21.3) on this topic.


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