Let's solve this step-by-step: --- ## 1. **Inverse Demand Function** Given: \[ P = 60 - Q \] --- ## 2. **Total Revenue (TR) Function** Total Revenue = \( P \times Q \) \[ TR = (60 - Q) Q = 60Q - Q^2 \] --- ## 3. **Marginal Revenue (MR) Function** Marginal Revenue is the derivative of TR with respect to Q: \[ MR = \frac{d(TR)}{dQ} = \frac{d}{dQ}(60Q - Q^2) = 60 - 2Q \] **Equation of the MR function:** \[ \boxed{MR = 60 - 2Q} \] --- ## 4. **Calculate MR for the 31st Unit** We want the MR when output increases from 30 to 31 units. Plug \( Q = 31 \) into the MR function: \[ MR_{31} = 60 - 2 \times 31 = 60 - 62 = -2 \] --- ## 5. **Graphically Identifying Components** When output increases by 1 unit (from 30 to 31): ### Total Revenue Change: \[ \Delta TR = TR(31) - TR(30) \] \[ TR(31) = (60 - 31) \times 31 = 29 \times 31 = 899 \] \[ TR(30) = (60 - 30) \times 30 = 30 \times 30 = 900 \] \[ \Delta TR = 899 - 900 = -1 \] ### Decomposition into Components: - **Positive component:** Revenue from selling the 31st unit at the new price = \( P(31) = 29 \) - **Negative component:** Revenue lost due to lowering the price on all previous 30 units. - Price drop per unit: \( P(31) - P(30) = 29 - 30 = -1 \) - Revenue lost: \( 30 \times -1 = -30 \) #### So: \[ \Delta TR = \text{Positive component} + \text{Negative component} = 29 + (-30) = -1 \] --- ## 6. **Which Component is Larger?** - The negative component (\(-30\)) is larger in magnitude than the positive component (\(29\)). --- ## 7. **Summary Table** | Component | Value | |-------------------------------|-------| | Positive (from 31st unit) | 29 | | Negative (lost on first 30) | -30 | | **Total (MR\(_{31}\))** | -1 | --- ## 8. **Final Answers** **a) MR Equation:** \[ \boxed{MR = 60 - 2Q} \] **b) Numerical value of MR\(_{31}\):** \[ \boxed{-2} \] **c) Positive component:** 29 **d) Negative component:** -30 **e) Which component is larger?** The **negative component** is larger in magnitude. --- ## 9. **Graphical Representation (Described)** - **Positive component:** Height of the demand curve at \( Q = 31 \) (i.e., 29). - **Negative component:** Area representing the revenue lost on the first 30 units due to the price drop. - **MR:** Net change in total revenue, which is negative. --- ### **Summary** - **Marginal Revenue function:** \( MR = 60 - 2Q \) - **MR at 31st unit:** \( -2 \) - **Positive component:** \( 29 \) - **Negative component:** \( -30 \) - **Negative component is larger in magnitude.**

## Graphical Representation ### Graph Description 1. **X-axis:** Quantity (Q) 2. **Y-axis:** Price (P) and Marginal Revenue (MR) 3. **Demand Curve:** Downward sloping line representing the inverse demand function \( P = 60 - Q \). 4. **Marginal Revenue Curve:** A downward sloping line representing \( MR = 60 - 2Q \). ### Components on the Graph - **Demand Curve at Q=31:** Show price \( P = 29 \). - **Area for Positive Component (Revenue from 31st Unit):** A rectangle with height 29 and width 1 unit. - **Area for Negative Component (Revenue Lost):** A rectangle with height -30 (the price drop impact on 30 units) and width 30 units. **Note:** An actual graph would visually depict these areas and curves. --- ## Economic Context This question pertains to **Microeconomics** as it deals with the behavior of a monopolistic firm, its pricing strategy, output decisions, and revenue implications. ### Key Microeconomic Concepts: - **Monopoly Market Structure:** Analysis of a single seller dominating the market. - **Demand and Marginal Revenue:** Understanding how demand affects pricing and revenue. - **Profit Maximization:** The relationship between output levels, price changes, and total revenue. --- ### Conclusion - The analysis focuses on the monopolist's behavior in response to changes in output and demonstrates fundamental microeconomic principles related to pricing and revenue.