# Why in the input requirement set the x is written with a minus sign?

My question comes from the textbook "Microeconomic analysis" by Varian, 1992. The textbook defines the set of all technologically feasible production plans as $$Y\in \mathbb{R}^n$$. It is said that $$Y$$ is supposed to describe all the production plans that are technologically feasible. After that, I’m seeing the initial definition of an input requirement set: $$V(y)=\{x\in \mathbb{R}^n_+:(y,-x) \in Y\}$$ And my question is about the graphical interpretation of this idea in 2D. I don't really get the reasoning behind the minus sign before the $$x$$.

• Look at how $Y$ is defined. Commented Jul 3, 2022 at 0:35
• The IS cannot be $\{2,-1\}$. Look at its definition again. Commented Jul 3, 2022 at 16:13

Intuitively, the definition of our production set and plans within it allows for a simple way of writing profit function and more generally making the restriction that all prices must be positive.

for example consider the case where: $$P=\left[\matrix{p \\w_1\\w_2}\right]$$ $$Y=\left[\matrix{q \\-x_1\\-x_2}\right]$$

we cam simply define the profit function as:

$$\pi=P^T\cdot Y$$

which is just: $$\pi=pq-w_1 x_1-w_2x_2$$

Hope this helps.

In order to understand the reason of the minus sign before x in the input requirement set, you have to look at the definition of production plan and production possibilities set, given before by Varian.

A production plan is represented by a vector in $$R^n$$, where a good is represented by a positive number if it is a net output, and negative if it is a net input. So, a production plan has both positive and negative components. The positive or negative sign for output and input is conventional. The production possibilities set is the set of all technological feasible production plans. That is, not any plan can be realized through the available technology, so the impossible plans are discarded from the production possibilities set.

The input requirement set is then defined in a particular case, when a firm produces only one output, say $$y$$. In this case, it is convenient to describe a production plan (Varian says 'the net output bundle') in a slight different way, that is considering an input $$x$$ as a positive number and placing the minus sign before it. That is, in this case x is in $$R^n_+$$.

This way, in this representation, you can immediately see, formally, which goods are inputs (minus sign before) and which is the output.

Then you have the definition of input requirement set:

𝑉(𝑦)={𝑥∈ℝ𝑛+:(𝑦,−𝑥)∈𝑌}.

That is, all the combinations of inputs (taken with positive sign) that can produce at least a quantity $$y$$ of the output. It can be seen, as $$y$$ varies, as a slight different way to describe a production possibilities set.

Then you wrote:

And my question is about the graphical interpretation of this idea in 2D.

Actually, you cannot have a representation in two dimensions, for the simple reason that x is in $$R^n_+$$, not in $$R^2$$. A (imaginary) graphic representation of the x vector should be in $$R^n_+$$, that is in the positive quadrant of $$R^n$$.

But, perhaps, you ask if there can be a graphic representation in $$R^2$$ in the case there are only two inputs. I think you can have an idea of such a representation when you will look at isoquants in two dimensions, for two inputs only.