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2nd order condition for a maximum

# 2nd order condition for a maximum - Convex Indifference...

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Convex Indifference Curves and 2 nd Order Conditions for Utility Maximization MRS= 2 1 1 2 1 2 1 2 ( , ) ( , ) . dx u x x dx u x x = 1 2 1 1 1 2 1 1 ( ) 2 1 2 2 . u u du du d dx dx dMRS dx dx u u u = = Now, 1 11 1 12 2 2 21 1 22 2 du u dx u dx du u dx u dx = + = + Hence, 2 1 2 1 1 11 12 1 2 21 22 1 dx dx dx dx du u u dx du u u dx = + = + But, along the indifference curve, 2 1 1 2 1 1 2 2 0 . dx u dx u du u dx u dx = + = = − Therefore, 2 1 1 2 2 1 1 2 1 1 11 12 11 12 1 2 2 2 21 22 21 22 1 2 ( ) ( ) dx u dx u dx u dx u du u u u u u u u u dx u du u u u u u u u u dx u 1 2 12 1 1 2 22 1 = + = + = = + = + = Thus, ( ) 1 2 1 1 2 1 2 11 2 12 1 21 2 22 1 2 1 ( ) 2 2 2 2 11 2 12 1 21 1 22 2 2 2 2 11 2 12 1 2 22 1 3 2 2 . u u d dMRS dx dx u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u = = + = + =

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