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ans_midterm - ECON 831 Mathematical Economics ANSWERS...

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ECON 831 - Mathematical Economics ANSWERS MIDTERM 1. (Budget correspondence) (see your notes) 2. (Metric) The function d is a non-negative real valued function and we show that it is a metric. We have to check the 3 properties: (i) We show that x,y R n , d ( x,y ) = 0 x = y . d ( x,y ) = 0 n X i =1 | x i - y i | = 0 ⇔ | x i - y i | = 0 i = 1 , ··· ,n since | x i - y i | ≥ 0 x i - y i = 0 i = 1 , ··· ,n x i = y i i = 1 , ··· ,n x = y Note: you can also prove the equivalence by proving both " " and " ". (ii) We show that x,y R n , d ( x,y ) = d ( y,x ) . d ( y,x ) = n X i =1 | y i - x i | = n X i =1 | x i - y i | b/c for any real number a , | a | = | - a | = d ( x,y ) (iii) We show that x,y,z R n , d ( x,y ) d ( x,z ) + d ( z,y ) . d ( x,y ) d ( x,z ) + d ( z,y ) n X i =1 | x i - y i | ≤ n X i =1 | x i - z i | + n X i =1 | z i - y i | 1
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n X i =1 | x i - y i | = n X i =1 | ( x i - z i ) + ( z i - y i ) | n X i =1 ( | x i - z i | + | z i - y i | ) b/c for any 2 real numbers a,b, | a + b | ≤ | a | + | b | = n X i =1 | x i - z
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