Course:MATH110/Archive/2010-2011/003/Notes/Elasticity

Theoretical background

If $D(p)$ is a demand function which describes the demand $D$ for a price $p$; The elasticity of demand is defined as the ratio of the relative change (or percent change if you prefer) in demand to the relative change in price. More precisely, we have that:

the relative change in demand is given by  $\frac{\mathrm{\Delta }D}{D}$

the relative change in price is given by  $\frac{\mathrm{\Delta }p}{p}$

Hence, we'll define (for now) the elasticity of demand as the ratio:

$E=\frac{\mathrm{\Delta }D/D}{\mathrm{\Delta }p/p}$

Example

The demand for processed pork in Canada is modelled by the demand function $D(p)=286-20p$ Where the demand is in millions of kilograms per year and the price in dollars per kilogram. At the price of $4 per kilogram (so $p=4$), the demand is of 206 millions kilograms per year, since $D(4)=286-80=206$. We want to know how an increase of price will influence the demand. We look for the demand at a price of $5 per kilogram, $4.5 per kilogram and $4.1 per kilogram to see what is happening if we compare it back to our initial price of $4 per kilogram: At $p=5$, the demand is $D(5)=286-100=186$. The change of price is $\mathrm{\Delta }p=5-4=1$ and the change of demand is $\mathrm{\Delta }D=186-206=-20$. Hence the relative change of price and the relative change of demand are: $\frac{\mathrm{\Delta }D}{D}=\frac{-20}{206}\approx -9.71\mathrm{\%}$   and   $\frac{\mathrm{\Delta }p}{p}=\frac{1}{4}=25\mathrm{\%}$ And so the elasticity of the demand for these changes is $E=\frac{\mathrm{\Delta }D/D}{\mathrm{\Delta }p/p}\approx \frac{-9.71\mathrm{\%}}{25\mathrm{\%}}\approx -0.39$ This means that for each 1% increase of the price, the demand will decrease by around 0.39%. The fact that the change of demand is of a much lower magnitude compared to the change of price is described by saying that the demand is inelastic at this price.

In the above example, we see that this ratio is interesting to compute, but seems to depend on the choice of price change. In the above example, we decided that $\mathrm{\Delta }p=1$, what if we took a smaller change? We would like to measure how a very small change to the price will influence the demand, so we're actually going to let the price change be almost zero by taking a limit. We'll define more formally the elasticity of demand at a price $p$ as:

$E(p)=\underset{\mathrm{\Delta }p\to 0}{\lim }\frac{\mathrm{\Delta }D/D}{\mathrm{\Delta }p/p}$

We can play with this definition and actually compute the limit which turns out to contain (in disguise) the derivative of the demand.

$E(p)=\underset{\mathrm{\Delta }p\to 0}{\lim }\frac{\mathrm{\Delta }D/D}{\mathrm{\Delta }p/p}=\underset{\mathrm{\Delta }p\to 0}{\lim }\frac{\mathrm{\Delta }D\cdot p}{\mathrm{\Delta }p\cdot D}=\frac{p}{D}\left(\underset{\mathrm{\Delta }p\to 0}{\lim }\frac{\mathrm{\Delta }D}{\mathrm{\Delta }p}\right)=\frac{p}{D}{D}^{\prime }$

Rewriting this using better notation gives us a formula for the elasticity that we can work with:

$E(p)=\frac{p{D}^{\prime }(p)}{D(p)}$