# Technology
## Production Vector

In an economy with **n** commodities, a **production vector/production plan** is a vector $y=(y_1,...,y_n)\in \mathbb{R}^n$. The vector represents the ability of a firm to transform commodities.

## Production Set

A **production set** $Y$ is the collection of all feasible production plans for a specific firm. In other words, $y \in Y$ and $Y \subset \mathbb{R}^n$.

## Properties of Production Sets

1. $Y$ is nonempty.

2. $Y$ is closed.

To ensure the existence of production function.

3. No free lunch.

If $y \in Y$ and $y \ge 0$, then $y=0$, i.e., the firm could not use zero inputs to produce and positive output.
4. Possibility of inaction.

$0 \in Y$, i.e., the firm can complete shutdown. It may not hold with sunk cost, but it is plausible in the long run.

5. Free disposal.

If $y \in Y$ and $y' \le y$, then $y' \in Y$, i.e., the firm can always waste inputs and throw out outputs.

6. Irreversibility.

If $y \in Y$ and $y \ne 0$, then $-y \notin Y$.
7. Nonincreasing return to scale.

If $y \in Y$, then $\alpha y \in Y$ for $\alpha \in [0,1]$. It is the small part of CRTS.
8. Nondecreasing return to scale.

If $y \in Y$, then $\alpha y \in Y$ for $\alpha \ge 1$. It is the large part of CRTS.
9. Constant return to scale.

If $y \in Y$, then $\alpha y \in Y$ for $\alpha \ge 0$. It is CRTS itself.
10. Additivity/free entry.

If $y, y' \in Y$, then $y+y' \in Y$.
11. Convexity.

If $y, y' \in Y$, then $\alpha y+(1-\alpha)y' \in Y$ for $\alpha \in [0,1]$. It assumes that "unbalanced" input combinations are not more productive than "balanced" ones.

12. $Y$ is a convex cone.

If $y, y' \in Y$, then $\alpha y+ \beta y' \in Y$ for $\alpha, \beta \in [0,1]$.