Definitions and Some Theorems

Then aif 0 bif 12 2 quasiconcave and superior

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Unformatted text preview: is represented by a function real valued u: → , then ■ u( is strictly increasing if and only if ≿ is strictly monotonic. ≡ ⋮ ∈ , , where ∈ ⊂ ■ u( is quasiconcave if and only if ≿ is convex. ≡ ⋮ ∈, , where ∈ ⊂ ■ u( is stictly quasiconcave if and only if ≿ is strictly convex. Summary Conditions for Optima 5. Some useful tools to study multivariate concavity and convexity. be Theorem: Let ∶ → be twice continuously differentiable, and let the ith‐order principal matrix of the Hessian . Then A. If 0, B. If 1,2, … , 2. Quasiconcave and superior sets. A real value function is quasiconcave iff its superior sets are convex. is positive definite. alternates in sign, starting with 0, is negative definite. Theorem (for a simple choice problem of 2 variables with 1 with equality constraints). Consider the bordered Hessian: ∗ 0 ∗ , ∗ , ∗ ∗ ∗ If ∗ , ∗ , ∗ , , ∗ ∗ ∗ ∗ , , ∗ , ∗ ∗ ∗ ∗ ∗ , , ∗ ∗ , ∗ , , ∗ , , ∗ ∗ 0, the objective function attains a local (constrained) maximum at the critical point ∗ , ∗ , (constrained) minimum a that point. ∗ ∗ . If , ∗ , ∗ 0, it attains a local A variation of this theorem holds for problems with 2 variables and more than 1 constraint (provided the number of cons...
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