By Vinogradov V.
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Extra resources for A cookbook of mathematics
In order to interpret the Kuhn-Tucker conditions, we first have to note the meanings of the following variables: • fj = ∂f ∂xj is the marginal profit (revenue) of product j; • λi is the shadow price of resource i; • gji = ∂g i ∂xj is the amount of resource i used in producing a marginal unit of product j; • λi gji is the imputed cost of resource i incurred in the production of a marginal unit of product j. ∂L i The KT condition ∂x ≤ 0 can be written as fj ≤ m i=1 λi gj and it says that the j marginal profit of the jth product cannot exceed the aggregate marginal imputed cost of the jth product.
F (x0 ) = 0. 26 ! Proposition 19 (The First-Derivative Test for a) changes its sign from If at a stationary positive to negative, point x0 the first b) changes its sign from derivative of a negative to positive, function f c) does not change its sign, Local Extremum) then the value of the function at x0 , f (x0 ), will be a) a local maximum; b) a local minimum; c) neither a local maximum nor a local minimum. Proposition 20 (Second-Derivative Test for Local Extremum) A stationary point x0 of f (x) will be a local maximum if f (x0 ) < 0 and a local minimum if f (x0 ) > 0.
He will choose the bundle from his budget set that maximizes his utility. Since U (x, y) is an increasing function in both x and y (over the domain x ≥ 0 and y ≥ 0), the budget constraint should be satisfied with equality. The consumer’s optimization problem can be stated as follows: max U (x, y) subject to x + 2y = 30. The Lagrangian function is L = U (x, y) − λ(x + 2y − 30) = xy + x + y + 1 − λ(x + 2y − 30). The first-order necessary conditions are ∂L ∂x ∂L ∂y ∂L ∂z = 0 = 0 = 0 y+1−λ = 0 or x + 1 − 2λ = 0 x + 2y − 30 = 0 From the first equation, we get λ = y + 1; substituting y + 1 for λ in the second equation, , we obtain x − 2y − 1 = 0.
A cookbook of mathematics by Vinogradov V.