Derive the input requirement set from production set

Question

This question relates to the book Varian Microeconomic Analysis 3rd edition exercise 1.1. Much like this question 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.?

Answer 1

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.