Question

Let \(e\), \(R\), and \(p\) represent own price elasticity of demand, total revenue, and price, respectively for a good. If \[ \frac{\partial R}{\partial p}<0, \] then

• A. \(|e|>1\)
• B. \(|e|=1\)
• C. \(|e|<1\)
• D. \(|e|=0\)

Hint:

When demand is elastic, \(|e|>1\), an increase in price reduces total revenue.

Answer 1

The Correct Option is A

Step 1: Write revenue and its change.
Revenue is price times quantity, $R=pq$. We want how $R$ moves when price moves.

Step 2: Use the product rule.
\[ \frac{\partial R}{\partial p}=q+p\frac{\partial q}{\partial p} \]

Step 3: Bring in elasticity.
Since $e=\dfrac{\partial q}{\partial p}\cdot\dfrac{p}{q}$, we have $p\dfrac{\partial q}{\partial p}=eq$. Putting this in,
\[ \frac{\partial R}{\partial p}=q(1+e) \]

Step 4: Apply the given sign.
We are told this is negative. As $q>0$, we need
\[ 1+e<0 \]

Step 5: Read off the size.
This gives $e<-1$, so the size of elasticity is bigger than one. In plain words, when demand is elastic, a price rise pulls revenue down.
\[ \boxed{|e|>1} \]