Profit Max with Two Products

Business Calculus · Axiom Academy

EXAMPLE Profit Maximization: Two Products A complete worked example of multivariable optimization A company manufactures two products: wireless earbuds (Product A) and portable chargers (Product B). The marketing team has determined the following: Interaction cost with B: 0.10 per unit of each Additionally, there are diminishing returns: producing more units increases per-unit costs due to overtime, rush orders, etc. The profit function is: (where x = units of Product A, y = units of Product B) Compute the partial derivatives Marginal profit with respect to Product A: Marginal profit with respect to Product B: Compute second partial derivatives Apply the Second Derivative Test Since D = 0.38 > 0 and P xx = -0.4 , this is a local maximum! Step 3: Calculate Maximum Profit Substitute optimal values into profit function Negative second partials (-0.4 and -1.0): Confirm diminishing returns—producing more of either product eventually hurts profitability Negative cross partial (-0.1): Products compete for resources; making more of one slightly decreases marginal profit of the other Higher output of A than B: Earbuds have higher profit margin ( 50 vs 30), so optimal mix favors Product A Sensitivity Analysis Questions What if earbud revenue increases to 90? (Higher x* expected) What if fixed costs rise to 3,000? (Same optimal x*, y*, but lower profit) What if production interaction increases? (Products become more substitutable)

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