A firm’s total revenue function is given by:
Question
- Prove that the price elasticity of demand is approximately equal to 1 when the total revenue function reaches its maximum value.
Solutions
(1) Finding the Quantity that Maximizes Total Revenue
The total revenue function is:
To find the maximum total revenue, we take the derivative and set it to zero:
Solving for using the quadratic formula:
Approximating:
Thus, the revenue-maximizing quantity is .
Step 2: Calculate Price Elasticity of Demand at
The price elasticity of demand formula using total revenue is:
Since at revenue maximization, the price elasticity of demand equals -1:
Thus, we have proven that when reaches its maximum, demand elasticity is approximately equal to -1. ✅