Elasticity of demand the idea behind it.

Question

I have been looking at elasticity of demand, but I am struggling to understand the concept. Now I have taken a simple example in hope of beeing able to understand what is going on.

Eliacticity of demand is defined as: the percentage change in quantity demanded in response to a given small percentage change in price. $$\frac{dQ}{dP}*\frac{P}{Q}$$

For simplicity lets start by a linear equation $$Q=200-2P$$ where

Q=Quantity demanded.

P=Price

$$\frac{d(200-2 p)}{dp}=-2$$

Now I have tried to reason for myself in several ways but I am not able to get it the intuition to the written definition. I hope you could provide me with feedback of where my "logic" actually is not logical at all.

Here we go:

So -2 is constant (b) and is the rate of change in quantity demanded given the change in price. Now P/Q provides us with a portion of p for each unit provided. Given that p is the price and Q is the total amount supplied.

Now $$-\frac{\text{bP}}{200-\text{bP}}=1-\frac{\text{bP}}{200}$$

Now given that what I have done here is correct I end up with 1- (the rate of change in quantity demanded given price times price)/Total quantity.

Now if price is 6 I end up with a 6 percent reduction in price from its maximum value.

But I still am not able to understand the definition completely.

Next we can look at

$\frac{\text{dQ}}{Q}$ divided by $\frac{\text{dP}}{P}$, also defined as elasticity of demand.

But here I get stuck straight away cause I do not understand what dQ or dP by itself means I only understand it in terms of derivative where its $$\frac{dQ}{dP}$$,

It would be nice if someone could explain what dQ by it self mean and also when it is divided by total output Q. And the same with dp by itself.

Answer 1

$\dfrac {\mathrm dQ}P$ is an abuse of notation, trying to get the following point across:

The ratio of a change in $Q$ relative to $Q$, to the corresponding change in $P$ relative to $P$, is:

$(\Delta Q:Q):(\Delta P:P)$

You can write the ratio as a fraction:

$\displaystyle \frac{\frac{\Delta Q}{Q}}{\frac {\Delta P}{P}}$

which is the same as:

$\displaystyle \frac{\Delta Q}{Q} \frac {P}{\Delta P}$

You have the approximation:

$\displaystyle \frac{\Delta Q}{Q} \frac {P}{\Delta P} \approx \frac{\mathrm dQ}{\mathrm dP}\frac P Q$

if $\Delta Q$ and $\Delta P$ are small enough.