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midterm07 - ECON 831 Mathematical Economics MIDTERM Fall...

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MIDTERM Fall 2007 B. Antoine Time: 1h50 1. [5] Consider the set A = S { 1 /k,k N } = { 1 , 1 / 2 , 1 / 3 ,... } . (a) Show that A is not open in R . (b) Show that A is not closed in R . (c) Intuitively, what is the closure of A and why? No formal proof required here. (d) Find the closure of A . Provide a formal proof here. 2. [4] Consider the correspondence Γ : [0 , 10] R defined by Γ( x ) = x + 1 0 x < 3 { 6 } ∪ [4 , 5] x = 3 [ - 5 / 8 x + 39 / 8 , 5 / 8 x + 33 / 8] 3 < x < 7 [1 / 2 , 9] \ { 5 } x = 7 9 7 < x 10 (a) At x=3: is Γ upper hemicontinuous? lower hemicontinuous? (b) At x=7: is Γ upper hemicontinuous? lower hemicontinuous? 3. [4] We want to show that Hicksian demand curves are downward-sloping. The consumer expenditure problem is min x R n p.x s.t. u ( x ) u where p = ( p 1 , ··· ,p n ) is the price of each good and x 1 , ··· ,x n the quantities consumed. Your goal is to show that the optimal demand function is non-increasing in
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midterm07 - ECON 831 Mathematical Economics MIDTERM Fall...

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