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 total revenue.
\[ R=pq \]

Step 2: Differentiate revenue with respect to price.
\[ \frac{\partial R}{\partial p} = q+p\frac{\partial q}{\partial p} \]

Step 3: Use elasticity formula.
\[ e=\frac{\partial q}{\partial p}\cdot\frac{p}{q} \]
So,
\[ p\frac{\partial q}{\partial p}=eq \]

Step 4: Substitute in revenue derivative.
\[ \frac{\partial R}{\partial p}=q+eq \] \[ \frac{\partial R}{\partial p}=q(1+e) \]

Step 5: Apply given condition.
Given
\[ \frac{\partial R}{\partial p}<0 \] Since \(q>0\),
\[ 1+e<0 \]

Step 6: Solve for elasticity.
\[ e<-1 \] Therefore,
\[ |e|>1 \]

Step 7: Final conclusion.
\[ \boxed{|e|>1} \]
Hence, the correct answer is option (A).