exam 2 2012

Dont forget to check the constraint qualification

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. Don’t forget to check the constraint qualification conditions. Answer: The derivative of the constraints is dg = 2 1 1 1 1 3 - 1 0 0 - 1 . A quick look at the constraints shows that at most two can bind simultaneously. Since any two rows of dg are linearly independent, NDCQ is statisfied. Alternatively, you can use the fact that all the constraints are linear. The Lagrangian is L = x +2 y - λ 1 (2 x + y - 6) - λ 2 ( x + y - 4) - λ 3 ( x +3 y - 9)+ μ x x + μ y y . The first order conditions are 0 = 1 - 2 λ 1 - λ 2 - λ 3 + μ x 0 = 2 - λ 1 - λ 2 - 3 λ 3 + μ y If x = 0, the first two constraints are slack ( y 6 and y 4), so λ 1 = λ 2 = 0. Then λ 3 > 0, so y = 3. This implies λ 3 = 2 / 3 and μ x = - 1 / 3 < 0, so it is not a solution. Now we know x > 0 and μ x = 0. Suppose y = 0. Then constraints (2) and (3) are slack ( x 4 and x 9) since constraint (1) tells us x 3. Now λ 2 = λ 3 = 0. Here λ 1 = 1 / 2 and μ x = - 3 / 2 < 0, so this is not a solution either.
MATHEMATICAL ECONOMICS EXAM #2, NOVEMBER 9, 2012 Page 3 We must have both x > 0 and y > 0, so μ x = μ y = 0. If constraint (1) binds, constraint (2) implies x 2 while constraint 3 say x 9 / 5. Constraint (3) cannot bind, so λ 3 = 0. This implies 1 = 2 λ 1 + λ 2 and 2 = λ 1 + λ 2 , so λ 1 = - 1, which is impossible. This shows that constraint (1) cannot bind.

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