Word Count : 1238

---

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 x = 0, and μy y = 0. These conditions ensure that a multiplier is positive only when its corresponding constraint is tight, and it is zero when the constraint is slack. Consider first an interior candidate, x > 0 and y > 0. Then μx = μy = 0 by complementary slackness. The two stationarity equations simplify to 0.7 x^−0.3 y^0.3 ____ ___ _________ __________ _____ _________ ________ ______.
_______ ________ _______ _____ ____ ___ ______ __________ _____.
________ _____ ______ ________ ____ _________ _________.
__________ ______ ________ ____ _________ _______.
__________ __________ ____ _____ ____ ____ ____ _______.
________ ______ ________ ________ __________ ______ _________.
_______ _____ __________ ______ ________ _____.
________ __________ _______ ________ __________ _____ __________ _____ _________ ____ _______ _______.
_______ ________ _________ ___ ____ _________ ____.
________ __________ _____ _____ ________ _______ __________ _________ __________ __________.
_____ ______ ___ ________ ___ __________ ____ _____ __________ ___.
____ __________ _______ _______ __________ _____ __________ ____ ________ ________.
____ _________ _________ ________ _________ ___ ____.
____ __________ ____ ____ ______ _____ ________ ________.
_________ ___ ______ __________ _______ _______ _____ _____ _____ __________ _______ _________.
______ ______ ________ ________ _________ ____.
_________ ________ _________ ______ _______ _____ ________.
__________ _____ ____ __________ _____ _____ ______.
__________ _________ __________ ____ ______ ____ __________.
______ __________ __________ __________ __________ ________ _______ _______.
_____ _____ ___ ____ ______.
_____ ______ ____ ____ ____ _______ ___ ______ _________.
__________ _____ _____ _________ __________ __________ ______.
____ __________ _______ ____ ______ ________.
_______ ___ ________ ___ ______.
________ __________ __________ ______ ___.
_________ _____ ____ ___ ___ ___ _________ ____.
_____ _____ ______ _________ _______ __________ _________ ____.
_________ ___ _________ _________ _______ ____.
_________ __________ _________ ___ _________ ________ _________ _________ _____.
_________ _________ _____ ________ ____.
________ ___ ______ _______ __________.
________ ___ ________ ___ _____ _________ _________ _______.
________ _____ __________ __________ ______ ____ _________ ________ ___.
____ __________ ____ ___ _________ __________ ______ ____ ____ _____ ____.
_______ ______ ____ ________ ____ __________ _______ ________ __________ ______.
___ _________ _____ ____ ____ ______ ____.
____ ________ ______ __________ _____ ________ _______ ________ _________.
______ _____ _________ _______ _______ __________ ______ ____ ___ ____ _____ ________.
________ _____ ___ ____ _________ ______ ___ ___ ___ _______ _________.
_______ _____ _____ ____ __________ __________ ___.
______ _____ ___ _____ _____.
_______ ________ ________ _____ ______ _______ ___ _______ _____ _________ _______.
__________ _________ ______ ____ __________ ____ _____ _____.
________ _______ ________ ____ ______ _______ ______.
_____ ___ ___ ________ __________ ______ ______ ____ __________ _____.
____ __________ _________ ______ ______ __________ ___ ________ _________ ___ __________ ______.
_____ ______ ________ ______ ___ ________ _________ _____.
__________ _________ _____ _____ _______ _________ __________ ____ _______.
______ ___ _______ ______ _____ ___ __________ ______ _____ ____.
___ _________ _______ ______ _______ _________ ___ ______ ___ _________ ____ ____.
___ _____ ________ ________ ____ ______ ___ _______ _________.
___ _________ ___ ___ ______ ______ _______ ________ _______ _______.
_________ ________ _____ _____ _________ ________.
______ ______ _____ _______ _________ ______ ________ _______ _____ _______ _________.
__________ ____ _________ _____ _____.
_______ ________ ____ ________ _____ _________.
__________ _____ _________ _____ __________ ___ _________ _________ _________.
_______ ___ ______ ___ __________ ___ ____ ______.
________ ________ _____ _______ ____ ____ ________ _______ _____ _______ _________.
______ ___ _______ __________ _____ _______ ____ _____ ______.
_____ ____ ____ ____ __________ _____.
____ ___ __________ ________ ______ ______.
______ _____ ___ ________ ______ _____ _____.
_____ ________ _________ ________ ___ _______ _________ _______.
__________ _____ __________ __________ _____ ______.
_____ __________ ___ __________ _________ _________.
____ _____ ________ ____ ________ ________ _______ ___ ______ _______ ______.
__________ ________ _________ _____ ________ __________ _________ _____ __________.
_____ ________ _______ ___ __________ _________.
________ ________ _______ __________ ______ _____ ____ ______ ______ ________.
___ _____ ______ ____ ____.
______ ___ ______ ______ ________.
_____ _____ _________ ___ ____ ________ _____ __________ _______.
_________ ____ ____ _______ ____ _________ ______ _________ ____ _____ ______ ______.
___ ___ __________ ______ _________ _______.
________ ___ _____ ________ ___ _________.
_________ ________ __________ _______ ______ ________ ________.
___ ______ ____ ________ _________ ___ __________.
______ ____ ________ _________ _____ ___ ________ _______.
__________ ________ ____ ____ ____ ____ ___ ____ _______.
___ _________ _____ _____ _________ ___ _______ ______ ______ ______ _____ _____.
____ _____ ________ ________ ________.
_________ ___ ________ __________ ________ ________ _________ ____ __________ ___ _________.
_________ ___ ____ __________ _________.
_______ _______ ___ ______ __________ _______ ________ _____ ___ _____ ______.
____ _____ ___ ______ ______ ________ _______.
____ _________ ___ _____ _______ ______ _____ __________ _________ ____ _____ ___.
___ ______ _______ _________ _____.
_______ ________ _____ ___ __________ ____ ___.
______ ______ _________ ______ ________ ______ __________ ____ ______ ___ ___ _____.
_____ _________ _______ ____ _________.
_________ ____ ___ _____ ______ ______ __________ ________ ________.
_______ _____ ______ ____ ____ _____.
______ __________ ___ ____ __________ ______.
________ _____ ____ __________ ______ _____ _______ _______ _______.
________ ____ ______ __________ _________ __________ ________ ________ ____ _______ ______.
__________ ____ _____ ___ ___ __________ ___.
_____ _____ __________ ____ _________ _____ ______ ___ ___ _______ ________.
_________ ______ _________ _______ ______.
___ _____ _______ ___ ____ __________ ___ __________.
___ _____ ___ ______ _____.
_________ _____ ________ ___ ___ _________.
________ _____ ____ _____ __________ ____ _______.
________ _________ _______ ___ ____ _______ __________.
__________ ______ __________ _________ _________ ____ ___.
____ _______ ________ ________ __________ _________.
_______ __________ __________ _______ ____ __________ ________.
______ _____ _________ ___ _______ ___ ___ ________.
________ ____ ______ _________ _________ ______ ________ ________.
_____ ___ ___ _______ ______ _____ __________ _____ _________ ______ _____.
___ ______ ____ __________ __________ ________.
_________ _________ ______ ___ ______ _____ __________ _________ _______ _____.
_____ _____ _________ _____ ____ _________.
________ ______ _____ ____ ______ _____ ___ ______ _____ _______ ______.
_____ ______ ____ ________ _______ ____ _______ ________ __________ _____.
___.
Get Full Answer on WhatsApp