Question

Demand function x, for a certain commodity is given as x = 200 – 4p where p is the unit price.
Find:
a. elasticity of demand as a function of p.
b. elasticity of demand when p = 10, interpret your result.

Answer 1

Hint: Use the formula for elasticity of demand \[=\dfrac{dp}{dx}.\dfrac{x}{p}\]. Differentiate the given equation with respect to p.

Complete step-by-step answer:
Here, we are given that x = 200 – 4p where x is demand function and p is the unit price.
a. Here we have to find the elasticity of demand as a function of p.
We know that elasticity of demand,
\[E=\dfrac{dx}{dp}.\dfrac{p}{x}....\left( i \right)\]
Now, we will consider the given equation that is
\[x=200-4p\]
We know that \[\dfrac{d}{dx}\left( x \right)=1\]
And \[\dfrac{d}{dx}\left( \text{constant} \right)=0\]
Therefore, by differentiating the above equation with respect to p, we get
\[\dfrac{dx}{dp}=0-4\]
Or, \[\dfrac{dx}{dp}=-4\]
By putting the value of \[\dfrac{dx}{dp}\] and x in equation (i), we get,
Elasticity of demand as a function of p, \[E=\left( -4 \right).\dfrac{p}{200-4p}\]
Therefore, we get \[E=\dfrac{-4p}{4\left( 50-p \right)}\]
Or, \[E=\dfrac{-p}{50-p}\]
b. Here we have to find the elasticity of demand when p = 10
We have already found the elasticity of demand, that is
\[E=\dfrac{-p}{50-p}\]
Now, we will put p = 10 to find the elasticity of demand when p = 10.
Therefore, we get
\[E=\dfrac{-10}{50-10}\]
\[=\dfrac{-10}{40}\]
Therefore, we get \[E=\dfrac{-1}{4}\].
The elasticity of demand is change in demand due to change in demand function (x) and price per unit (p). Here, we have elasticity of demand \[E=\dfrac{-1}{4}\] which is less than 1. When E < 1, it means that demand is inelastic. Inelastic demand is when people buy about the same amount, whether the price drops or rises. This situation happens with things that people must have, like gasoline and food. Drivers must purchase the same amount even when the price increases. Likewise, they don't buy much more even if the price drops

Note: Students must know the meaning of elasticity of demand. Also, they should not blindly follow the formula \[E=\dfrac{dx}{dp}.\dfrac{p}{x}\] but also take care that here x = demand function and p = price per unit. The variable could change in different questions.