Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual
The Lagrangian function and the Kuhn-Tucker conditions are Z = x1 + λ1 (−x21 + x2 ) ∂Z/∂x1 = 1 − 2λ1 x1 ≥ 0
plus the nonnegativity and
∂Z/∂x2 = λ1 ≥ 0
complementary slackness conditions
∂Z/∂λ1 =
−x21
+ x2 ≤ 0
At (0,0), the first and the third marginal conditions are duly satisfied. As long as we choose any value of λ∗1 ≥ 0, all the Kuhn-Tucker conditions are satisfied despite the cusp. 4. (a) Z = x1 + λ1 [x2 + (1 − x1 )3 ] Complementary slackness require that ∂Z/∂x1 vanish, but we actually find that, at the optimal solution (1,0), ∂Z/∂x1 = 1 − 3λ1 (1 − x1 )2 = 1. (b) Z0 = λ0 x1 + λ1 [x2 + (1 − x1 )3 ] The Kuhn-Tucker conditions are ∂Z0 /∂x1 = λ0 − 3λ1 (1 − x1 )2 ≥ 0
plus the nonnegativity and
∂Z0 /∂x2 = λ1 ≥ 0
complementary slackness conditions
∂Z0 /∂λ1 = x2 + (1 − x1 )3 ≤ 0
By choosing λ∗0 = 0 and λ∗1 ≥ 0, we can satisfy all of these conditions at the optimal solution.
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