midterm08

# midterm08 - ECON 831 Mathematical Economics MIDTERM B...

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MIDTERM B. Antoine Fall 2008 Time: 1h50 Oct. 20th, 2008 1. [5] We de ne the diameter of any bounded subset S of a metric space ( X,d ) as: diam ( S ) = sup { d ( x,y ) : x,y S } . (a) Consider the metric space ( R ,d 1 ) (recall: d 1 ( x,y ) = | x - y | , x,y R ). Give an example of a sequence of subsets S k of R such that the following conditions are satis ed: (i) S k 6 = for any k . (ii) S k +1 S k for any k . (iii) diam ( S k ) 0 (iv) T k =1 S k = . (b) Consider now the metric space ((0 , 1) ,d 1 ) . Give an example of a sequence of subsets S k of (0 , 1) such that the above conditions (i) to (iv) are satis ed and in addition: (v) S k is closed in (0,1) for any k . 2. [5] Consider the budget correspondence when two goods are available: x 0 and y 0 are the associated quantities; p > 0 and q > 0 the associated prices; I > 0 is the available budget. For given prices ( p and q ) and budget I , the budget correspondence is de ned as:

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