# Some doubts about netput vectors

I have started studying producer theory on my own and there are some confusions.

We know that a production plan is $y=(y_{1},y_{2},y_{3}....y_{L})$ where $y_{i}$ is an output if its greater than $0$ and input if it's less than $0$. The production set $Y$ is such that any $y \in Y$. The vector $y$ is also called a netput vector.

1. Take free-disposal: if $y \in Y$ and $y^{1} \leq y$ , then $y^{1} \in Y$. How are we comparing the vectors $y^{1}$ and $y$ here? Are we taking the cartesian length(norm) of both vectors and comparing? (or does it mean that each component of $y^{1}$ is smaller than the corresponding components in $y$) ?

2.Non-increasing returns to scale is given by:if $\forall y\in Y \implies ay\in Y$ for $a\in[0,1]$, then $Y$ shows non-increasing returns to scale.This would mean that vectors smaller than $y$ are also in $Y$. The way I knew it, decreasing returns to scale means output changes less than proportionately to change in inputs. How does these two ideas relate?

3.What is the difference between the transformation frontier and an isoquant? I am having a hard time vizualizing them. What would the frontier and isoquant look like if we draw $Y\in R^{3}$ using level curves for outputs and the two axes for the two inputs?

I hope I explained my questions clearly.

Thanks

1. We say that $x \le y$ when $x_i \le y_i$ for each $i=1,\ldots,n$.
2. If the production set can be represented by a production function $F$, and $F$ is homogeneous of degree $r < 1$ ($F(\lambda x) = \lambda^r F(x)$ for $\lambda >0$), then the production set satisfies the non-increasing returns to scale property you state. Moreover, if $F(0)=0$, and $F$ is concave, the production set also satisfies non-increasing returns to scale. Hence, you can think of the definition as a generalisation of the intuitive understanding you have.