6.2 Optimization with equality constraints: n variables, m constraints
The Lagrangean for this problem is
j=1λ_{j}(g_{j}(x) − c_{j}).
As in the case of a problem with two variables and one constraint, the firstorder condition is that x* be a stationary point of the Lagrangean. The “nondegeneracy” condition in the two variable case—namely that at least one of g'_{1}(x_{1}, x_{2}) and g'_{2}(x_{1}, x_{2}) is nonzero—is less straightforward to generalize. The appropriate generalization involves the Jacobian matrix of the constraint functions (g_{1}, ..., g_{m}), named in honor of Carl Gustav Jacob Jacobi (1804–1851) and defined as follows.
 Definition

For j = 1, ..., m let g_{j} be a differentiable function of n variables. The Jacobian matrix of (g_{1}, ..., g_{m}) at the point x is
(∂g_{1}/∂x_{1})(x) ... (∂g_{1}/∂x_{n})(x) ... ... ... (∂g_{m}/∂x_{1})(x) ... (∂g_{m}/∂x_{n})(x) .
 Proposition (Necessary conditions for an extremum)

Let f and g_{j} for j = 1, ..., m be continuously differentiable functions of n variables defined on the set S, with m ≤ n, let c_{j} for j = 1, ..., m be numbers, and
suppose that x* is an interior point of S that solves the problem
max_{x} f(x) subject to g_{j}(x) = c_{j} for j = 1, ..., m.or the problemmin_{x} f(x) subject to g_{j}(x) = c_{j} for j = 1, ..., m.or is a local maximizer or minimizer of f(x) subject to g_{j}(x) = c_{j} for j = 1, ..., m. Suppose also that the rank of the Jacobian matrix of (g_{1}, ..., g_{m}) at the point x* is m.
Then there exist unique numbers λ_{1}, ..., λ_{m} such that x* is a stationary point of the Lagrangean function L defined by
L(x) = f(x) − ∑mThat is, x* satisfies the firstorder conditions
j=1λ_{j}(g_{j}(x) − c_{j}).L'_{i}(x*) = f_{i}'(x*) − ∑m 0 for i = 1, ..., n.
j=1λ_{j}(∂g_{j}/∂x_{i})(x*) =
 Source
 For proofs, see Sydsæter (1981), Theorem 5.20 (p. 275) and Simon and Blume (1994), pp. 478–480. (Only Sydsæter argues explicitly that the Lagrange multipliers are unique.)
 Proposition (Conditions under which necessary conditions for an extremum are sufficient)

Let f and g_{j} for j = 1, ..., m be continuously differentiable functions of n variables defined on the open convex set S. Let x* be an interior point
of S that is a stationary point of the Lagrangean
L(x) = f(x) − ∑mSuppose further that g_{j}(x*) = c_{j} for j = 1, ..., m. Then
j=1λ*_{j} (g_{j}(x) − c_{j}).
 Example

Consider the problem
min_{x,y,z} x^{2} + y^{2} + z^{2} subject to x + 2y + z = 1 and 2x − y − 3z = 4.The Lagrangean is
L(x, y, z) = x^{2} + y^{2} + z^{2} − λ_{1}(x + 2y + z − 1) − λ_{2}(2x − y − 3z − 4). The firstorder conditions are
2x − λ_{1} − 2 λ_{2} = 0 2y − 2 λ_{1} + λ_{2} = 0 2z − λ_{1} + 3 λ_{2} = 0 x + 2 y + z = 1 2 x − y − 3 z = 4 λ_{1} = (2/5)x + (4/5)y λ_{2} = (4/5)x − (2/5)y. x = 16/15, y = 1/3, z = −11/15,with λ_{1} = 52/75 and λ_{2} = 54/75.We conclude that (x, y, z) = (16/15, 1/3, −11/15) is the unique solution of the problem.
Economic interpretation of Lagrange multipliers
In the case of a problem with two variables and one constraint we saw that the Lagrange multiplier has an interesting economic interpretation. This interpretation generalizes to the case of a problem with n variables and m constraints.Consider the problem
That is:
the value of the Lagrange multiplier on the jth constraint at the solution of the problem is equal to the rate of change in the maximal value of the objective function as the jth constraint is relaxed.If the jth constraint arises because of a limit on the amount of some resource, then we refer to λ_{j}(c) as the shadow price of the jth resource.