Second%20order%20condition.RRex

# Second%20order%20condition.RRex - Consider an indi/erence...

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Second order condition Consider the following optimization problem: maximize u ( x;y ) subject to p x x + p y y = m; x ± 0 ;y ± 0 To look for an optimal solution ( x ;y ) with x > 0 ;y > 0 , we can use the budget equation to get y = [ m ² p x x ] =p y (1) and, upon substitution get the problem maximize H ( x ) = u ( x; [ m ² p x x ] =p y ) on 0 < x < m=p x This is a maximization problem with a single variable x . If an optimal x exists ; condition must hold at x : [using the "relevant rules from calculus"] dH dx x = x = u x + u y ( ² p x =p y ) = 0 (2) or, u x =u y = p x =p y (3) The second order condition for a maximum : it su¢ ces to have d 2 H dx 2 x = x < 0 which, from (2), means u xx + u xy ( ² p x =p y ) + u yx ( ² p x =p y ) + u yy ( p x =p y ) 2 < 0 (4) But, according to a general calculus theorem, u xy = u yx : So, from (4) we get u xx + 2 u xy [ ² ( p x =p y )] + u yy ( p x =p y ) 2 < 0 Using (3) we now get : u xx + 2 u xy [ ² ( u x = u y )] + u yy ( u x =u y ) 2 < 0 Multiplying by ( u y ) 2 u xx ( u y ) 2 ² 2 u xy u x u y + u yy ( u x ) 2 < 0 (5) Convexity of the indi/erence curve:

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Unformatted text preview: Consider an indi/erence curve through ( x &amp; ;y &amp; ) : f ( x;y ) : u ( x;y ) = u ( x &amp; ;y &amp; ) g dy=dx = [ u x = u y ] Convexity is equivalent to d 2 y=dx 2 &gt; : Di/erentiating with respect to x and using the &quot;chain rule&quot; we get 1 d 2 y=dx 2 = &amp; [ f u xx + u xy ( dy=dx ) g u y &amp; u x f u yx + u yy ( dy=dx ) g ] = ( u y ) 2 = &amp; [ f u xx + u xy ( &amp; [ u x =u y ] g u y &amp; u x f u yx + u yy ( &amp; [ u x =u y ] g ] = ( u y ) 2 = &amp; [ u xx u y &amp; u xy u x &amp; u x u yx + u x u yy ([ u x =u y ] g ] = ( u y ) 2 = &amp; [(1 = ( u y ) 3 ][( u xx ( u y ) 2 &amp; 2 u xy u x u y + u yy ( u x ) 2 ] (6) Now, from (6) if d 2 y=dx 2 &gt; ; we also have (5). 2...
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## This note was uploaded on 12/23/2009 for the course ECON 3130 at Cornell University (Engineering School).

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Second%20order%20condition.RRex - Consider an indi/erence...

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