The short run is considered the period of time where fixed costs are still fixed, which basically means that, if you have a factory, you have to make do with it because you can neither sell it, nor make it bigger, nor rent half of it: you are stuck with it for the time being. Capital is also considered fixed, meaning that, in the short run, all you can play around with are your variable costs, being labour the most commonly used variable cost.
If we look at the adjacent graphs, we can see how marginal productivity (cyan, second graph) drops with each added unit of labour, under the law of diminishing returns. The two diagrams allow us to see three clear phases, characterised by the point of inflection that marks the law of diminishing returns. In the first diagram, we see how production increases exponentially with each added unit of labour, up to a point where added units begin to offer less and less return. In the second diagram, we introduce average productivity too; to define even further at what level we should produce.
In phase I, average and marginal productivity increase with each added unit, corresponding with the first diagram up to the point of inflection. Phase I ends in the extensive margin (or technical optimum), being this the line that splits phases I and II. Phase II is where we should ideally be. Average productivity drops with each added unit, but marginal productivity is still positive. The line that separates phases II and III is called the intensive margin (or technical maximum), from which there will be too much variable input for each level of fixed input. When marginal productivity becomes negative, we enter phase III, and we should consider reducing labour. We can also see that, at optimum level, marginal and average productivity coincide.
This all relates to the productive elasticity of our product, which is the difference in quantity that occurs as the result of a change in the level of productive inputs. Obviously, in phase I, a change of one unit in labour input will produce a much larger return in production levels than itself: εL>1. In phase two, 1> εL>0, and in phase III, 0> εL.
Algebraically, the elasticity of production can be defined as:
It must be noted that the elasticity of production for a given input (here labour, L) is the first half of the elasticity of scale, which in turn gives us the returns to scale.