Revenue Maximization

Business Calculus · Axiom Academy

Finding the price and quantity that generate the most revenue 1 The Revenue Maximization Problem Revenue is the total money coming in from sales. To maximize revenue, we need to find the sweet spot between price and quantity. Higher prices mean fewer units sold, but more money per unit. Lower prices mean more units sold, but less money per unit. Maximum revenue occurs where these effects balance perfectly. Usually we're given a demand function that relates price to quantity: To find revenue as a function of quantity alone: Revenue as a function of quantity When the demand function is linear, R(x) becomes a quadratic that opens downward. This guarantees a single maximum point. Write the revenue function: R(x) = p(x) x Take the derivative: Find R'(x) Set derivative to zero: Solve R'(x) = 0 Verify it's a maximum: Check R''(x) Find the optimal price: Plug x back into demand function Optimal quantity: 100 units at 50 each = 5,000 revenue Maximum revenue occurs exactly where elasticity = 1 (unit elastic). When E = -1, MR = 0, which is our condition for maximum revenue! 6 Revenue vs. Profit Maximization Maximum revenue ≠ Maximum profit Revenue maximization ignores costs. In most cases, the profit-maximizing quantity is different from the revenue-maximizing quantity. When Revenue Maximization Makes Sense Subscription businesses maximizing subscribers Startups prioritizing market share over profit Nonprofits maximizing donations/funding

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