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= 21 1 2 12 1 2 (,) . dx u x x dx u x x −= 1 11 2 () 2 2 . u u du du d dx dx dMRS dx dx uu u == Now, 1 2 2 22 1 2 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 dx du dx But, along the indifference curve, 2 2 0 . dx u dx u du u dx u dx = ⇒= Therefore, 11 12 11 12 21 22 21 22 dx u dx u dx u dx u du u u u u dx u du u u u u dx u 1 2 1 2 1 1 2 2 2 1 Thus, 1 2 2 1 2 11 2 12 1 21 2 22 1 2 2 11 2 12 1 21 1 22 2 2 1 1 212 2 3 2 2 . u u d dMRS dx dx u u uu uu u uu uu uu u u uuu uu u ⎛⎞ −− ⎜⎟ ⎝⎠ −−+ = −+ =
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Note that, by assumption, both goods have positive marginal utilities. Hence, and the sign of 3 2 0 u > 1 dMRS dx depends on the sign of the numerator. Focus now on the numerator () 2 12 12 12 11 11 11 12 11 22 11 2 12 1 2 22 1 2 11 2 1 2 1 22 1 1 2 2 2 11 22 12 11 2 1 1 11 2 2 . uu u u u u uuu uu u u u uu u u u u u −+ =− + + =−+ 2 2 In the sum in this last expression, the first term is negative so long as
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This note was uploaded on 10/19/2010 for the course ECON 1202 taught by Professor Matel during the Fall '08 term at UConn.

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

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