Economic Interpretation

Calculus 3 · Axiom Academy

LESSON Lagrange Multipliers: Economic Interpretation Understanding as the marginal value of relaxing a constraint — the shadow price that reveals how much your objective improves per unit change in the constraint. Consider a typical constrained optimization problem: maximize an objective function subject to a constraint. Maximize an objective f(x, y) subject to a budget constraint g(x, y) = b . At the optimum the objective's level curve is tangent to the constraint, so the gradients line up. 2. The Shadow Price Interpretation 3. Concrete Example: Production Optimization A company produces two products with profit function f(x, y) = 100x + 150y (dollars), subject to a labor constraint 2x + 3y = 1200 (hours available). Solving this problem yields . This means each additional hour of labor increases maximum profit by about 50 — the "shadow price" of labor, or what the company should be willing to pay for one more hour. 1200 hours → max profit = 60,000 1201 hours → max profit ≈ 60,050 The Lagrange multiplier drives sensitivity analysis : understanding how the optimum responds when constraint parameters vary. Line the shadow prices up side by side and the biggest one wins. Marginal value: how much does relaxing the constraint improve the objective? Resource allocation: which constraints are most limiting? (Higher = more valuable to relax.) Economic decisions: should we invest in expanding a constraint? (Compare to the cost.)

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