What point am I calculating the PED (Price Elasticity of demand)

Question

I'm at a very basic level of economics (highschool), so what I write may not be very coherent, i'm just a bit confused and I can't get any book or source to awnser the question I have.

See, the problems (excercises) I have ask me for the PED and the data they give me to obtain it is two points. For example:

In october the price for cigarretes was 5 USD and the Quantity demanded was 1000 packs, in december the price for cigarretes increased to 6 USD and the quantity demanded decreased to 500 packs. Calculate the price elasticity of demand:

%ΔP= 1/5 %ΔD=-500/100=

PED=(-500/100)/(1/5)= -2.5

I'm using PED = %ΔD/%ΔP

Using the midpoint method the awnser would be

PED= -3.6667.

Using this first method (not midpoint method) I dont understand for which point im getting the PED, since PED is different at each point of the curve. Is it perhaps the average PED between those points? Otherwise, what is it?

Also, what is the difference between the first method and the midpoint method?.

And finally, if they tell that a product has an elasticity of for example PED= -0.5, why is it that I can say that if I rase the price 1 % the the demand decreases -0.5? Because by raising the price 1% the elasticity has already changed (since elasticity is different at each point of the demand curve) and hence, I cant use that previous elasticity of -0.5 to find out the the demand decrease .

Thanks in advance, and sorry if the question doesn't make any sense.

Answer 1

Formally, the price elasticity of demand $D$ with respect to the price $p$ is defined as: $$ \frac{\partial D}{\partial p} \frac{p}{D} $$ In reality, it is impossible to compute derivatives, so in practice one estimates them using finite differences. $$ \frac{\partial D}{\partial p} \approx \frac{\Delta D}{\Delta p}. $$ Here $\Delta D$ is the difference of demand in two time periods, say $D_1 - D_0$ and $\Delta p$ is the difference of the price in two time periods, say $p_1 - p_0$. In your setting $D_0 = 1000, D_1 = 500, p_0 = 5$ and $p_1 = 6$.

This gives: $$ \frac{\partial D}{\partial p} \frac{p}{D}\approx \frac{\Delta D}{\Delta p} \frac{p}{D} = \frac{-500}{1} \frac{p}{D} $$ Now, we still need to get the values of $p$ and $D$. For this, there is no real best option. One idea could be to take $p = p_0$ and $D = D_0$ which gives $$ \frac{-500}{1} \frac{5}{1000}= -2.5 $$ Another option could be tot take $p = p_1$ and $D = D_1$ which gives: $$ \frac{-500}{1} \frac{6}{500} = -6 $$ A final option could be tot take the midpoint $p = \frac{p_1 + p_0}{2}$ and $D = \frac{D_1 + D_0}{2}$, which gives: $$ \frac{-500}{1} \frac{5.5}{750} = -3.66\ldots $$ Normall, the closer $p_0$ to $p_1$ (and hence $D_1$ to $D_0$), the closer the number will be to the true elasiticity (in terms of derivatives).

For your last question, notice that we can rewrite: $$ \frac{\Delta D}{\Delta p} \frac{p}{D} = \frac{\frac{\Delta D}{D}}{\frac{\Delta p}{p}}. $$ The numerator $\frac{\Delta D}{D}$ is the % change in $D$ while $\frac{\Delta p}{p}$ is the percentage change in $p$. As such, the elasticity can be seen as the number of percentage-point changes in $D$ per 1%-point change in $p$.

In summary: ideally, we would like to compute elasiticites for a certain price $p$ and associated demand $D$. This would require us to compute derivatives. In reality, we cannot do this, so we need to approximate the derivative with discrete changes. This requires us to settle on a base period. Depending on this choice, the resulting computation may change.