# What precisely is scarcity?

I'm aware that certain terms in economics have precise definitions that don't necessarily line up with our intuition about what they mean. I'm currently writing a paper that heavily relies on the concept of scarcity, and I'd like to make sure I am using the term properly. Currently I have this sentence written:

Suppose widgets are a scarce resource that are distributed among agents

Then I rewrite this mathematically as $$\mu = \sum\limits_{i=0}^n x_i$$ where $$x_i$$ is the number of widgets agent $$i$$ has. I don't see any problems with this myself, but I'd like to make sure that economists view this as orthodox. Are there any references that I could use to verify this? Is this accurate? What is the precise definition of scarcity?

Edit

A Dictionary of Economics states scarcity is: "The property of being in excess demand at a zero price. This means that in equilibrium the price of a scarce good or factor must be positive" This is the most authoritative source I've found so far, but this doesn't seem like a rigorous definition to me - it clearly isn't mathematically defined. Am I wrong about this?

• Can you consider this from a post-scarcity standpoint, and see what needs to be changed to get there? The difference, will point you towards what scarcity is. – Mast Nov 11 at 19:15
• I don't understand this comment. Do you mean viewing the model from a post scarcity standpoint? – K Pomykala Nov 13 at 1:18

Your formula $$\mu = \sum\limits_{i=0}^n x_i$$ just denotes the total amount of goods, so this just means that $$\mu$$ units were distributed among $$n$$ agents. Scarcity (in this context) means that no matter how you distribute these $$\mu$$ units, as long as everything else is unchanged (ceteris paribus) at least one agent would like to get more, i.e. $$\forall (x_i)_{i = 0}^n$$ where $$\mu = \sum\limits_{i=0}^n x_i$$ there is an agent $$i$$ and a positive $$\epsilon$$ such that $$x_i \prec_i x_i + \epsilon.$$ This is context dependent, there may be other definitions as well (e.g. ones that allow for production).
There are two key properties for a scarce good. First that demand can get arbitrarily high if the price is low enough. So if you plot demand on the $$x$$-axis and price on the $$y$$-axis this function is decreasing for all $$x$$. Second is that production cost is non zero, producing more items will always cost more money. This is always the case for physical goods but not for things like intellectual property goods.