# Note2 - 1 Every point on the map sits on an indierence...

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1. Every point on the map sits on an indi/erence curve 2. Utility increases in the northeast direction 3. Indi/erence curves do not cross 4. Indi/erence curves are not bands 5. Indi/erence curves are continuous 6. Indi/erence curves are downward sloping 7. Indi/erence curves are convex 8. The slope of an indi/erence curve is invariant to monotonic transformation 6

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Proof of property 6: 1. Intuition Diagrammatic proof by contradiction 2. Mathematical Proof The total di/erential of U ( x; y ) is given by dU ( x; y ) = @U ( x; y ) @x dx + @U ( x; y ) @y dy: (1) ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ± Digression: The total di/erential of a function f ( x; y ) is given by df ( x; y ) = @f ( x; y ) @x dx + @f ( x; y ) @y dy: (2) Heuristic proof: Let ° denote change, then ° f ( x; y ) = f ( x + ° x; y + ° y ) ° f ( x; y ) = f ( x + ° x; y ) ° f ( x; y ) + f ( x + ° x; y + ° y ) ° f ( x + ° x; y ) = f ( x + ° x; y ) ° f ( x; y ) ° x ° x + f ( x + ° x; y + ° y ) ° f ( x + ° x; y ) ° y ° y = ° f ( x; y ) ° x ° ° ° ° only x varies ° x + ° f ( x; y ) ° y ° ° ° ° only y varies ° y (3) For small changes ° x and ° y , df ( x; y ) = @f ( x; y ) @x dx + @f ( x; y ) @y dy: In the case of a function with only one argument, ° f ( x ) = f ( x + ° x ) ° f ( x ) = f ( x x ) ° f ( x ) ° x ° x + f 0 ( x x: Examples: f ( x ) = x; f ( x ) = x 2 ; f ( x; y ) = xy; f ( x ) = x 2 y 2 ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ± Since U ( x; y ) = U 0
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