Deriving the Relationship Between Marginal Revenue and Price Elasticity of Demand
A direct relationship between marginal revenue (MR) and the price elasticity of demand (ε) is established through a specific derivation. This process involves taking the expression for marginal revenue, derived using the product rule, and rewriting it by substituting the formula for price elasticity, expressed as . The derivation also relies on using the inverse demand function, , to represent the price. For further reading on the connection between marginal revenue and elasticity, refer to Section 6.4 of 'Mathematics for Economists: An Introductory Textbook' by Malcolm Pemberton and Nicholas Rau.
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Ch.7 The firm and its customers - The Economy 2.0 Microeconomics @ CORE Econ
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Deriving the Relationship Between Marginal Revenue and Price Elasticity of Demand
A firm faces an inverse demand function given by P = 150 - 3Q, where P is the price per unit and Q is the quantity of units sold. The firm's total revenue (R) is calculated as the product of price and quantity (P × Q). Using the principles of differentiation to find the rate of change of total revenue with respect to quantity, what is the firm's marginal revenue (MR) function?
Critique of a Marginal Revenue Calculation
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.
A firm's total revenue (R) is determined by multiplying the quantity of goods sold (Q) by the price (P), where the price is a function of quantity given by the inverse demand function P = f(Q). A student claims that the marginal revenue (MR), which is the derivative of total revenue with respect to quantity, is simply equal to the derivative of the price function, f'(Q). Is this claim correct?
A firm's total revenue is given by R(Q) = P × Q, where P = f(Q) is the inverse demand function. The marginal revenue (MR) is the derivative of the total revenue function with respect to quantity (Q). Using the product rule, the MR function is derived as: MR = f(Q) + Q × f'(Q). Match each mathematical component of this derived formula to its correct economic interpretation.
Derivation of the Marginal Revenue Formula
Evaluating a Consultant's Revenue Analysis
A company's total revenue (R) is determined by the product of the quantity sold (Q) and the price (P). The price is determined by the inverse demand function P = 100 - Q². The marginal revenue is the derivative of the total revenue function with respect to quantity. The resulting marginal revenue function is MR = ____.
Applicability of the Marginal Revenue Derivation
A firm's marginal revenue (MR) function is given by MR = 200 - 8Q. Assuming the firm faces a linear inverse demand function of the form P = a - bQ, where P is price and Q is quantity, which of the following represents the correct inverse demand function for this firm?
Learn After
The Sign of Marginal Revenue based on Price Elasticity of Demand
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]
A key relationship in microeconomics connects a firm's marginal revenue (MR) to the price elasticity of demand (ε). Arrange the following steps to show the correct logical sequence for deriving this relationship, starting from the definition of total revenue (R).
Evaluating Competing Marginal Revenue Formulas
Justifying the Key Step in the MR-Elasticity Derivation
A firm is analyzing its pricing strategy and wants to understand the relationship between its marginal revenue (MR) and the price elasticity of demand (ε). Starting with the marginal revenue expression derived from the product rule,
MR = P + Q(dP/dQ), which of the following algebraic manipulations is the crucial next step to reformulate this expression into the standard formula that links marginal revenue to price elasticity?In the process of deriving the formula that links marginal revenue to price elasticity of demand, the expression
(Q/P) * (dP/dQ), which arises after applying the product rule and factoring out price, is mathematically equivalent to the price elasticity of demand (ε).A microeconomist is deriving the formula that connects marginal revenue (MR) to the price elasticity of demand (ε). Match each mathematical step in the derivation with its corresponding conceptual or procedural justification.
Deriving the Marginal Revenue and Elasticity Formula
In the derivation of the relationship between marginal revenue and price, we begin with the expression
MR = P + Q(dP/dQ). This equation can be algebraically restructured into a factored form that isolates the price variable (P). Complete the following identity by providing the correct mathematical expression for the blank space:MR = P * [1 + ________]Adapting the Revenue Derivation for Marketing Expenditure