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Lagrange multiplier

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A variable introduced to solve a problem involving constrained optimization. Suppose that the function f(x, y) has to be maximized by choice of x and y subject to the constraint that g(x, y) ≤ k. The solution can be found by constructing the Lagrangean function

L(x, y, λ) = f(x, y) + λ[kg(x, y)]

where λ is the Lagrange multiplier. Let fx denote

∂f(x, y)/∂x,

etc. The optimal values of x, y, and λ solve the necessary conditions

Lxfxλgx = 0, Lyfyλgy = 0, λ[kg(x, y)] = 0, and λ ≥ 0.

If the constraint is not binding, λ = 0 and the maximum occurs where fx = fy = 0. If the constraint is binding, λ > 0 and the optimum is found by solving the three equations Lx = 0, Ly = 0, and g(x, y) − k = 0. The Lagrange multiplier, λ, measures the increase in the objective function (f(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 f(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 λ measures the marginal utility of income—the additional utility provided by one more unit of income.

Subjects: Science and Mathematics — Economics.

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