What does the numeric value of a Lagrange multiplier tell you about the optimum?

## The multiplier is a price, not a leftover When you solve a constrained problem with a Lagrangian, the multiplier drops out of the algebra alongside the optimal point. Candidates often discard it. It is in fact the most decision-relevant number in the solution: the **shadow price** of the constraint, meaning the marginal value of relaxing it. ## The precise statement Write the problem as `maximize f(x) subject to g(x) = c`, where `c` is the constraint level - a budget, a capacity, a total that must be hit. Build `L = f(x) - lambda*(g(x) - c)`. Define the **value function** `f*(c)` as the optimal objective value achievable at constraint level `c`. Then, where things are smooth, `d f*(c) / dc = lambda` This is the envelope theorem: when `c` nudges up, the optimal point moves too, but because that point is already stationary the movement contributes nothing to first order. All that survives is the direct effect through the constraint, weighted by `lambda`. ## Check it on a concrete case Take `maximize x*y subject to x + y = c`. The stationarity equations give `x = y = c/2` and `lambda = c/2`, so the optimal value is `f*(c) = c^2/4`. Differentiate the value function directly: `d f*/dc = c/2`. At `c = 10` both routes agree on 5. Push the total from 10 to 11 and the true optimum rises from 25 to `121/4 = 30.25`, an increase of 5.25 - close to the multiplier's prediction of 5, and off only by the second-order curvature you would expect from a derivative. ## Units, and why they matter The multiplier's units are *objective units per constraint unit*. If the objective is revenue in dollars and the constraint is labour hours, the multiplier is dollars per hour: directly comparable to what an extra hour costs, and therefore directly actionable. If two constraints bind and one has a multiplier three times the other's, an extra unit of the first is worth three times an extra unit of the second - that is how a shadow price turns an optimization output into a capacity-planning decision. A multiplier is not unitless and not comparable across problems with different scalings. ## The zero multiplier A multiplier of zero says the constraint is not doing any work at the optimum: loosening it changes nothing, because the solution is not pressed against it. For inequality constraints this is the content of complementary slackness, and it is the standard way to tell which requirements actually bind and which are decorative. When you solve a problem with a dozen constraints and only three come back with nonzero multipliers, you have learned that nine of them are, at this operating point, irrelevant. ## Where the reading breaks down **It is local.** `f*(c)` is generally a nonlinear function, so multiplying the shadow price by a large change overstates or understates the true gain. In the example above, going from `c = 10` to `c = 20` raises the optimum from 25 to 100 - a gain of 75, not the 50 that naive linear extrapolation from `lambda = 5` predicts. Use it for the next unit, not the next order of magnitude. **The active set can change.** With inequality constraints, relaxing one constraint far enough makes a different constraint start to bind. At that point the value function kinks and the shadow price jumps; the derivative on one side of the kink says nothing about the other side. **Degeneracy.** If several constraints are redundant at the optimum, or the problem has multiple optima, the multipliers may not be unique, and the value function may have no derivative at all - only one-sided directional derivatives. A unique, stable shadow price is something you should confirm rather than assume. **Signs and conventions.** Whether `lambda` comes out positive or negative depends on whether you wrote `f - lambda*(g - c)` or `f + lambda*(g - c)`, and on whether you are maximizing or minimizing. Always sanity-check the direction against a small numerical perturbation before quoting the number to anyone, rather than trusting a remembered sign rule. ## How to use it In practice the workflow is: solve, read off the multipliers, rank the binding constraints by multiplier size, and take that ranking to whoever controls the constraint levels. It converts a mathematical solution into a sentence a decision-maker can act on - one more unit of this resource is worth roughly this much - which is usually more valuable than the optimal point itself, because the point changes with the data while the pricing insight tends to persist.