# The Elasticity Conjecture

If you know economists, you probably have noticed that when they discuss microeconomics, they almost certainly end up talking about elasticity. In fact, I have summed up this simple observation in my very own conjecture:

The longer a discussion about microeconomics between two economists is, the probability that elasticity is mentionned exponentially approaches 1.

Of course, if you are not acquainted with this notion, it may seem far-fetched to talk about the elasticty of a curve and discard the comments about the demand for insulin being perfectly inelastic as yet another surrealistic comment about the strange nature of markets. Well, not really.

But what exactly is elasticty ? Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$. We define an operator $e_{x_i}$ which we call $f$-elasticty of parameter $x_i$ as:

$e_{x_i} f(x_1, ..., x_n) =\left |\dfrac{\partial f}{\partial x_i} \dfrac{x_i}{f} \right |$

Simply put, it is the product of the partial derivative in regard to parameter $x_i$ and the ratio of $x_i$ and $f$, which, you probably have noticed, means elasticity has no unit and is exactly the reason why its use is so widespread in economics. Elasticity is the measure of the relative effect the change in a variable has on another variable, regardless of the units employed. The elasticity of a parameter is classified in the following categories:

• $e = 0$ : perfectly inelastic: a change in the parameter has no effect on the other
• $0 < |e| < 1$: inelastic: a change in the parameter has a small effect on the other
• $e = 1$: unit elastic: a change in the parameter has a proportional effect on the other
• $e \geq 1$: elastic: a change in the parameter has a more than proportional effect on the other
• $e = \infty$: perfectly elastic: a change in the parameter nullifies the other

To better illustrate the notion, let us take a very straightforward, thus bogus but instructive exemple. Let us imagine the demand for wheat is described by the following demand curve:

$Q = -2P + 15$

We can calculate the price elasticity of demand:

$e_P Q = \dfrac{-2P}{-2P+15}$

Thus, at point $P = 1$, elasticity is:

$e_P Q(1) \approx 0.13$

Which means that the demand is quite inelastic at this point, i.e. that a change in price will only have a small effect on the demanded quantity. Insulin is a very good example of an almost inelastic good: whatever the price, someone with diabetes will pay this much to get his dose, as it is of vital importance to him. In contrario, an example of a very elastic price demand is the demand for leisure goods, such as DVDs and books.

Microeconomists study many kinds of demand elasticities (e.g. income elasticty of demand, cross-price elasticity of demand, etc.) and it is of peculiar interest to observe the empirical measures of such elasticities. In a future blog post, I will introduce you to a selected collection of elasticity data taken from litterature.