## Exam #1

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## Exam #1

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Problems/Exams

- Pages:
- 2
- School:
- Iowa State University
- Course:
- Econ 600 - Quant Mthd Analy Ii

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Econ 600 Schroeter Fall 2011 Exam 1 Do all three problems Weights 1 30 2 35 3 35 Closed book closed notes Be sure your answers are presented in a neat and wellorganized manner 1 Answer True or False for each of the following statements If the statement is false indicate how it could be changed to a true statement with a small change in wording a Let U be a convex subset of n If F U is differentiable then F is concave if and only if F u F v F v u v for all u v U x b Let U be a convex subset of n If F U is differentiable then F is strictly concave if and only if 2F v is negative definite for all v U x 2 c Let F n be differentiable If x n is such that is positive semi definite then x is a local minimum of F F 2F x 0 and x x x 2 d Let F 2 and g 2 be differentiable and let b Let x 2 be a g point at which x 0 and a local solution to x max F x such that g x b w r t x Then there exists such that F g x x x x e F x is differentiable For each value of the scalar parameter a min F x a has a regular solution at x a Define F a F x a a Then for w r t x each value of the parameter a0 2 d2 F x a a F a 0 0 0 a 2 da 2 2 2 First recall the following definition of a quasi convex function Let U be a convex subset of n F U is quasi convex if for all u v U and for all h 0 1 F hu 1 h v max F u F v Now for U a convex subset of n let F m x U be such that F x a is quasiconvex in a for every x m Consider the problem max F x a and assume that it w r t x has a global solution x a for each a U Define the value function in the usual way F a F x a a Note that F a is well defined even if x a is not unique Prove that F U is quasi convex 3 Consider the problem min a1 x1 a 2 x 2 such that g x1 x 2 c w r t x1 x2 where c is a constant and a1 and a 2 are positive constants The constraint function g 2 is differentiable with first and second order partial derivatives denoted using subscripts that satisfy the following sign restrictions throughout 2 g1 g 2 0 g 11 g 22 0 g12 0 and g11 g 22 g122 0 a Write down

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