# One solution is v 2 1 t to find an eigenvector

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One solution is v = (2 , - 1) T . To find an eigenvector corresponding to λ = 5, we solve 4 - 2 - 2 1 u = 0 0 . One solution is u = (1 , 2) T . c ) Although u · v = 0, which shows the eigenvectors are orthogonal, they do not have unit norm. We divide by their norms to get orthonormal eigenvectors: 2 5 - 1 5 , 1 5 2 5 .

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MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 12, 2002 Page 2 3. Robinson Crusoe has utility function u ( x, y ) = x 2 + y 2 . Crusoe has a production possi- bilities set given by { ( x, y ) : x 0 , y 0 , x + y 2 4 } . a ) Is the production possibility set a closed set? Is it a bounded set? b ) Show that Crusoe’s problem of maximizing utility over his production set has a solution. Answer: a ) The set is closed. There are many ways to see this, one is to realize that the functions f ( x, y ) = x , g ( x, y ) = y , and h ( x, y ) = x + y 2 are all continuous. Then f - 1 ([0 , )), g - 1 ([0 , )) and h - 1 (( -∞ , 4]) are all closed sets (as the inverse image of closed intervals). The production possibility set is closed because it is the intersection of 3 closed sets.
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