Marginal Revenue

Business Calculus · Axiom Academy

Understanding the revenue from selling one more unit Before we find marginal revenue, let's understand how revenue works: Revenue = Price per unit × Quantity sold But here's the key insight: in most markets, price depends on quantity ! Price decreases as quantity increases The demand function p(x) captures a fundamental economic law: to sell more units, you typically need to lower your price. This creates an interesting optimization problem! Marginal Revenue is the additional revenue gained from selling one more unit. It is found by taking the derivative of the revenue function. MR = Marginal Revenue = Derivative of Total Revenue Why isn't MR just equal to price? You might think: "If I sell one more item for 50, I get 50 more revenue, right?" Not quite! To sell that extra unit, you often need to lower the price on all units. So the marginal revenue accounts for: The revenue from the additional unit sold (+) The revenue lost from lowering prices on previous units (-) Marginal Revenue is typically less than Price because selling more requires lowering prices. A company sells tablets with demand function: where x is thousands of units and p is price in dollars. Find the marginal revenue function: For a linear demand function p = a - bx , the marginal revenue is always MR = a - 2bx . Notice the coefficient doubles! Let's see how demand, revenue, and marginal revenue relate: MR starts at the same point as demand (when x = 0) MR falls twice as fast as the demand curve

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