Winter 2014 - Economics 2P30 Foundations of Economic...

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Economics 2P30 Foundations of Economic Analysis Lester M.K. Kwong Department of Economics Brock University Winter 2014 Final Examination - Suggested Solutions Time: 3 Hours This examination will not be deposited in the library reserve Section A: Definitions * * * * * * * Define 6 of the following 8 terms in two sentences or less. * * * * * * * * * * * * * * This Section of the exam is worth 10%. * * * * ** 1. Maximum of a function f 2. Continuous Function at x 3. Differentiable Function at x 4. Weakly Convex Function 5. Interior of a Set 6. Convex Set 7. Contradiction 8. Contrapositive of P Q Solution: 1. If x * is the maximum of a function f over domain D then f ( x * ) f ( x ) for all x D . 2. A function is continuous at x if for every > 0, there exists a δ > 0 so that y B δ ( x ) implies f ( y ) B ( f ( x )) . 3. A function is differentiable at x if: lim 0 f ( x + ) - f ( x ) exists. 1
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4. A function f X R is said to be weakly convex if for all x,y X with x y : f ( αx + ( 1 - α ) y ) αf ( x ) + ( 1 - α ) f ( y ) for all α ( 0 , 1 ) . 5. The interior of a set X is given by int ( X ) = { x X ∶ ∃ > 0 ,B ( x ) X } . 6. A set X is said to be convex if for all x,y X , αx + ( 1 - α ) y X for all α ( 0 , 1 ) . 7. A contradiction is a propositional form that is always false. 8. The contrapositive of P Q is the proposition Q P Section B: Proofs * * * * * * * Choose 6 of the following 8 questions. * * * * * * * * * * * ** This section of the exam is worth 40%. * * * * ** For each of the following, answer true or false. If true, prove. If false, derive a counterexample 1. Suppose f C 2 is strictly concave and x * so that f ( x * ) = 0. If g C 2 and g > 0 and g ′′ < 0 for all x , then x * is the maximum of g ( f ( x )) . Solution: True. Note that the function g ( f ( x )) is strictly concave since if h ( x ) = g ( f ( x )) then: h ( x ) = g ( f ( x )) f ( x ) and: h ′′ ( x ) = g ′′ ( f ( x )) f ( x ) 2 + g ( f ( x )) f ′′ ( x ) By assumption g ′′ < 0, g > 0, and f ′′ < 0 and note that f ( x ) > 0 for all x . Hence, h ′′ < 0 for all x and therefore strictly concave. It follows that h ( x * ) = g ( f ( x * )) f ( x * ) = 0 and therefore, x * is a maximum. 2. Suppose a global maximum exists for the function f ( x ) over the domain D R . Then for all D so that D D , a global maximum exists over D .
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  • Derivative, lim, Convex function

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