A variable introduced to solve a problem involving constrained optimization.
The Lagrange multiplier, λ, measures the increasde in the ojective function (ƒ(x,y)) that is obtained through a marginal relaxation in the constraint (an increase in k). For this reason, the Lagrange multiplier is often termed a shadow price. For example, if ƒ(x,y) is a utility function, which is maximized subject to the constraint that total spending on x and y is less than or equal to income, k, then X measures the marginal utility of income — the additional utility provided by one more unit of income.
Reference: Oxford Press Dictonary of Economics, 5th edt.