Cost Minimization

Business Calculus · Axiom Academy

Finding the most efficient way to achieve a business goal 1 The Cost Minimization Problem While profit maximization asks "how much should we produce?", cost minimization asks "how can we produce a given amount most efficiently?" Two Types of Cost Minimization 1. Minimize Average Cost: Find the production level where cost per unit is lowest. 2. Minimize Total Cost for Fixed Output: Find the best input combination to produce a target quantity. Finding the optimal batch size to minimize production cost per unit. Minimizing shipping costs while meeting delivery requirements. Balancing ordering costs against holding costs. Minimizing material costs for a container of given volume. The average cost function divides total cost by quantity: Average Cost = Total Cost / Quantity To find the minimum, we take the derivative and set it to zero: Setting '(x) = 0 and simplifying gives us: At minimum average cost: MC = AC Condition for Minimum Average Cost The graph shows that AC is minimized exactly where MC intersects AC from below. 3 Finding Minimum Average Cost Write the average cost function: (x) = x Take the derivative: Use the quotient rule Set derivative to zero: Solve '(x) = 0 Verify it's a minimum: Check ''(x) > 0 OR use the shortcut: Solve MC = AC Example: Given C(x) = x^3 - 6x^2 + 15x + 10 Minimum average cost occurs at x = 5 units 4 Cost Minimization vs. Profit Maximization Minimizing average cost does NOT maximize profit! These are different optimization problems.

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