# Calculating Price Elasticity of Demand

Hi I was given the following price vs quantity values.

Price   Quantity Demanded
4   221
5   210
6   185
7   162
8   144
9   122
10  102
11  81
12  61
13  46
14  25


The graph was plotted as shown below. The equation was, $$Y = -0.0496 X + 15.133$$. What I need to know is, I was asked to find the PED when price is $$\7.5$$.

Then what I did was, I found the quantity at price $$7.5$$ substituting to the price quantity equation. And then found the PED using the equation The quantity derived for price $$7.5$$ was $$153.89$$. Then I calculated the PED as below. Is it correct?

$$\frac{(153.89-144)/144}{(7.5-8)/8} = 1.099$$

May I know whether this calculation is correct?

• Is Y = -0.0496 X + 15.133 your formula for quantity demanded? – user20105 Apr 22 '19 at 16:51

Arc elasticity (or midpoint elasticity) uses the formula $$\begin{equation} \epsilon^\text{arc}=\frac{Q_1-Q_0}{P_1-P_0}\cdot\frac{\frac12(Q_0+Q_1)}{\frac12(P_0+P_1)}, \end{equation}$$ where $$\frac12(Q_0+Q_1)$$ is the midpoint between $$Q_0$$ and $$Q_1$$. Note that in your case, $$7.5$$ is the midpoint between $$7$$ and $$8$$.
Point elasticity uses the formula $$\begin{equation} \epsilon^\text{point}=\frac{\mathrm dQ}{\mathrm dP}\cdot\frac{P}{Q}. \end{equation}$$ Here, $$\frac{\mathrm dQ}{\mathrm dP}$$ is the derivative of the demand function (evaluated at the point you want to calculate the elasticity, but it will be constant if demand is linear), and in your case, $$P=7.5$$ and $$Q$$ is the quantity demanded corresponding to that price.
• @Hiru: If you assume linear demand, then there is a nice property you can use. Suppose $Q^*$ is the quantity where $\epsilon=1$, then we must have $\epsilon >1$ for all $Q<Q^*$ and $\epsilon <1$ for all $Q>Q^*$. – Herr K. Apr 23 '19 at 3:42
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