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The consumer faces the problem of maximizing U(x, y) = x^0.7 y^0.3 subject to the linear budget 3x + 2y ≤ 200 with non-negativity x ≥ 0, y ≥ 0. Because the exponents are positive, the utility function is strictly increasing in each good over the non-negative orthant. Intuitively, if the budget were slack, the consumer could purchase a little more of at least one good and strictly increase utility, so an optimum cannot occur with unspent income. The Kuhn–Tucker method formalizes this intuition and handles, in one framework, the possibilities that the budget binds or not and that non-negativity constraints might bind or not. Form the Lagrangian with multipliers λ ≥ 0 for the budget, μx ≥ 0 for x ≥ 0, and μy ≥ 0 for y ≥ 0: L(x, y, λ, μx, μy) = x^0.7 y^0.3 + λ(200 − 3x − 2y) + μx x + μy y. The Karush–Kuhn–Tucker conditions consist of feasibility, stationarity, and complementary slackness. Feasibility requires x ≥ 0, y ≥ 0, and 3x + 2y ≤ 200. Stationarity gives the first-order derivatives equal to zero: ∂L/∂x = 0.7 x^−0.3 y^0.3 − 3λ + μx = 0, ∂L/∂y = 0.3 x^0.7 y^−0.7 − 2λ + μy = 0, ∂L/∂λ = 200 − 3x − 2y ≥ 0, with λ ≥ 0, ∂L/∂μx = x ≥ 0, with μx ≥ 0, ∂L/∂μy = y ≥ 0, with μy ≥ 0. Complementary slackness requires λ(200 − 3x − 2y) = 0, μx ________ _________ ________ ______ __________ ____ ________ ___ ___.
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