Price Optimization Under Constraints

From a demand curve to a chosen price — with fairness, competitive, inventory, and legal constraints turning it into a constrained optimization

Once you have a demand curve and an objective, "find the best price" looks like one line of calculus: argmax over P of (P − c)·Q(P). In production it almost never is, because the unconstrained optimum is usually a price you are not allowed to ship. The real problem is a *constrained* optimization, and the constraints are where the engineering lives.

Start with the clean unconstrained problem. Given estimated demand Q(P) and cost c, maximize expected profit π(P) = (P − c)·Q(P). Set dπ/dP = 0 → this recovers the standard markup rule P* = c·|ε| / (|ε| − 1) when elasticity is constant. That P* is the textbook answer — and it is frequently outside the feasible set: too high to be fair or legal, too low to protect margin, or inconsistent with sibling products.

Then layer the constraints that make it shippable — this is the actual job. Real optimizers add: price floors/ceilings (regulatory caps, MAP agreements, brand-minimum prices); a min-margin guarantee (P ≥ c·(1 + m), never sell below a floor markup); catalog consistency (a 500ml pack can't cost more per-ml than the 1L; variants must order sensibly); capacity/inventory limits (if you can only fulfill K units, don't set a price whose Q(P) far exceeds K); and no-surge-above-X legal caps in regulated markets. Each turns the free argmax into a bounded/constrained one.

Mechanically, you handle it with bounded search or Lagrangian methods. If the constraint is a simple box (P ∈ [P_min, P_max]), clip: the constrained optimum is the unconstrained P* if it's inside the box, else the nearer boundary (profit is concave, so the best feasible price is the closest allowed one to P*). For coupled constraints (a total-inventory or catalog-consistency limit spanning many prices), use a Lagrangian: add λ·(constraint) to the objective, and λ is the *shadow price* — how much profit one more unit of slack (one more unit of inventory, one more dollar of allowed ceiling) would buy. The headline lesson: the unconstrained optimum is a starting point, not the answer, and the binding constraints — not the calculus — usually determine the shipped price.

Key points

• The unconstrained problem is argmax over P of (P − c)·Q(P). With constant elasticity this gives the markup rule P* = c·|ε|/(|ε| − 1). That P* is the textbook answer and is frequently outside the feasible set — too high, too low, or inconsistent with sibling SKUs.
• Constraints are the real job: floors/ceilings, min-margin, catalog consistency, capacity, legal caps. A min-margin guarantee (P ≥ c·(1+m)), a per-unit consistency rule across pack sizes, an inventory limit K, and regulatory no-surge caps each shrink the feasible region — and the binding one usually sets the shipped price.
• For box constraints, clip to the boundary. Profit is concave in P, so if the unconstrained P* violates [P_min, P_max], the best feasible price is the nearer boundary — not an interior compromise. Moving toward P* is always improving until you hit the wall.
• For coupled constraints, use a Lagrangian and read λ as a shadow price. Adding λ·(constraint) to the objective, λ tells you the marginal profit of relaxing the constraint by one unit — the value of one more unit of inventory or one more dollar of allowed ceiling. It converts "what's the price" into "what's the constraint worth."

Recap

• Unconstrained problem: argmax over P of (P − c)·Q(P). Constant elasticity → markup rule P* = c·|ε|/(|ε| − 1). This P* is a starting point, not the shipped price.
• Constraints are the real job: price floors/ceilings, min-margin (P ≥ c·(1+m)), catalog consistency across pack sizes, capacity/inventory K, and legal no-surge caps. The binding one usually sets the price.
• Box constraints → clip to the boundary: profit is concave in P, so a violated optimum resolves to the nearer allowed bound, never an interior compromise.
• Coupled constraints → Lagrangian: add λ·(constraint); λ is the shadow price — the marginal profit of relaxing the constraint by one unit (one more unit of inventory, one more dollar of ceiling).
• Headline: the unconstrained optimum is where you start; the binding constraint — not the calculus — is usually what determines the price you actually ship.

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