Optimal Pricing Strategy

Business Calculus · Axiom Academy

EXAMPLE Optimal Pricing Example Finding the price that maximizes profit using calculus A coffee shop is trying to determine the optimal price for their specialty latte. Market research reveals: Current situation: At 5 per latte, they sell 200 lattes per day. Price sensitivity: For every 0.50 price increase, sales drop by 20 lattes. Cost: Each latte costs 1.50 to make (ingredients + labor). Find: What price maximizes daily profit? Let p = price per latte. We need to express quantity as a function of price. Slope: -20 lattes per 0.50 = -40 lattes per dollar Demand Function: quantity as a function of price Cost: 1.50 per latte × Quantity Take the derivative with respect to price: Verify maximum: P''(p) = -80 ✓ (concave down = maximum) By raising the price from 5.00 to 5.75: Sales decrease by 30 lattes (15% drop) Revenue decreases by 22.50 (2.25% drop) Profit increases by 22.50 (3.2% gain) The 0.75 price increase more than compensates for the lost sales because each remaining sale is more profitable. You've learned how to optimize pricing using calculus-based analysis. Based on the example above, which approach is correct?

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