Let's break down the problem and verify the result step-by-step: **Statement to Verify:** The marginal revenue (MR) of the (q+1)st unit equals the price of that unit minus the loss in revenue on previous units resulting from a price reduction. --- ### **Step 1: Define Total Revenue** Let the price function be \( p(q) \), where \( q \) is the quantity sold. Total revenue (TR) when selling \( q \) units: \[ TR = p(q) \cdot q \] --- ### **Step 2: Marginal Revenue Definition** Marginal Revenue (MR) is the change in total revenue when one more unit is sold: \[ MR = \frac{d(TR)}{dq} \] \[ MR = \frac{d}{dq}[p(q) \cdot q] \] --- ### **Step 3: Differentiate Using Product Rule** Apply the product rule: \[ MR = p(q) \cdot \frac{d}{dq}(q) + q \cdot \frac{d}{dq}[p(q)] \] \[ MR = p(q) \cdot 1 + q \cdot p'(q) \] \[ MR = p(q) + q \cdot p'(q) \] --- ### **Step 4: Interpret the Terms** - **\( p(q) \):** The price at which the (q+1)st unit is sold. - **\( q \cdot p'(q) \):** The change in price (\( p'(q) \)) multiplied by the number of units previously sold (\( q \)). This represents the total loss in revenue on the previous \( q \) units due to the price reduction required to sell one more unit. --- ### **Step 5: Final Explanation** So, \[ MR = \text{Price of (q+1)st unit} - \text{Loss in revenue on previous units due to price reduction} \] - The **price** of the (q+1)st unit is \( p(q) \). - The **loss on previous units** due to price reduction is \( -q \cdot |p'(q)| \) (since \( p'(q) < 0 \) for a downward-sloping demand curve). --- ### **Step 6: Final Statement** \[ \boxed{ MR = p(q) + q \cdot p'(q) } \] - **MR of the (q+1)st unit = Price of that unit – Loss in revenue on previous units from price reduction.** **This verifies the result as required.** --- **Summary Table:** | Term | Expression | Meaning | |-----------------------|-------------------|--------------------------------------------------------| | Total Revenue (TR) | \( p(q) \cdot q \) | Total revenue from selling \( q \) units | | Marginal Revenue (MR) | \( p(q) + q p'(q) \) | Additional revenue from selling one more unit | | Price of (q+1)st unit | \( p(q) \) | Price at which (q+1)st unit is sold | | Loss in revenue | \( q \cdot p'(q) \) | Loss on previous \( q \) units due to price reduction | --- **If you need the same with a numerical example, let me know!**