## Elasticities

This is something which you will see a lot. The elasticity gives you a measure of proportionate response from one variable to another, it is commonly used for price and quantity. The elasticity of demand tells you how much quantity demanded changes when you change the price. Inelastic demand means you can raise the price and not suffer too much of a drop in output. Elastic demand means if you raise the price, people stop buying it and you suffer a larger proportionate drop in output. Obviously firms prefer to have inelastic demand. Various things affect elasticity of demand, the availability of substitutes is a big one.

Elasticity of demand is the proportional change in quantity demanded divided by the proportional change in price: $\frac{\triangle Q}{Q}\div\frac{\triangle P}{P} = \frac{\triangle Q}{Q}\frac{P}{\triangle P}$ which is generally expressed as $\frac{P}{Q}\frac{\triangle Q}{\triangle P}$. This will generally be a negative value, because when you increase the price, you decrease the quantity, so $\triangle Q$ will be negative.

As these changes tend to zero (ie at the margin) we can express the elasticity as $\epsilon=\frac{P}{Q}\frac{dQ}{dP}$.
If $-1<\epsilon<0$ then we say the demand is inelastic. If $\epsilon=1$ then it is 'unit elastic'. If $-\infty<\epsilon<-1$ then it is elastic.

Another little manoeuvre with elasticity involves using the chain rule again. Remember that a demand function expresses Q as a function of P. If we take the natural log of Q then we can differentiate it with respect to P by the chain rule:$\frac{d}{dP} ln (Q(P)) = \frac{1}{Q}\frac{dQ}{dP}$. You differentiate the outer function first (which is the natural log of Q) and multiply it by the differential of the inner function (which is Q in terms of P). At A-level you tend to get chain rule questions when you have to actually solve equations with values in them, but quite often in economics you get them written in this function notation so you have to get the hang of doing the chain rule that way.

Anyway since we just found that $\frac{d}{dP} ln (Q(P)) = \frac{1}{Q}\frac{dQ}{dP}$ we can multiply both sides by P to get $P\frac{d}{dP} ln (Q(P)) = \frac{P}{Q}\frac{dQ}{dP}$ which is the elasticity of demand again.

So $\epsilon=P\frac{d}{dP} ln Q$. This can be a useful way of calculating elasticities if the demand function is a bit complicated.

One area where this comes up is in the case of constant elasticity. Usually the elasticity will vary along different points of the demand curve (even if $\frac{dQ}{dP}$ is constant, at every point there will be a different combination of P and Q so the $\frac{P}{Q}$ part will vary. But there is a specific type of function that will give a constant elasticity all the way along the curve, it will look like this: $Q=aP^{-b}$

Now if we log both sides we will get $ln Q=ln a - b lnP$ (this is using rules of logs).
So our formula above of $\epsilon=P\frac{d}{dP} ln Q$ says that we need to differentiate that expression with respect to P, and multiply the whole thing by P. $\frac{d}{dP}(ln a - b ln P)=\frac{-b}{P}$ so $\epsilon=P\frac{-b}{P}=-b$. So the elasticity will always be the constant -b. That's why constant elasticity functions are always of that form.