Deriving the Relationship Between Marginal Revenue and Price Elasticity of Demand
The algebraic relationship between marginal revenue () and the price elasticity of demand () is established through a specific derivation. Starting with the marginal revenue formula derived via the product rule, , and using the inverse demand function , we can factor out the price (). Substituting the formula for the price elasticity of demand, defined as , yields the final relationship:
For further details, refer to Section 6.4 of Mathematics for Economists: An Introductory Textbook by Malcolm Pemberton and Nicholas Rau.
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Deriving the Relationship Between Marginal Revenue and Price Elasticity of Demand
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A firm's total revenue (R) is the product of the quantity sold (Q) and the price (P). The price is determined by the inverse demand function, P = f(Q). To find the marginal revenue (MR), which is the rate of change of total revenue with respect to quantity, one must differentiate the total revenue function. Arrange the following conceptual steps in the correct logical order to derive the general formula for marginal revenue.
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A student is deriving the relationship between marginal revenue (MR) and price elasticity of demand (ε), starting from the total revenue function R = P × Q. Analyze the student's derivation below and identify the step containing a mathematical or conceptual error.
Step 1: Find marginal revenue by taking the derivative of total revenue with respect to quantity (Q) using the product rule. MR = dR/dQ = P + Q × (dP/dQ)
Step 2: Factor out Price (P) from the right side of the equation. MR = P × [1 + (Q/P) × (dP/dQ)]
Step 3: Identify the relationship between the term in the parentheses and price elasticity of demand (ε). Recognize that by definition, ε = (P/Q) × (dQ/dP), therefore its reciprocal is (Q/P) × (dP/dQ) = 1/ε.
Step 4: Substitute the elasticity term into the equation from Step 2. MR = P × [1 + 1/ε]
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