Note2 - 1. Every point on the map sits on an indi/erence...

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Unformatted text preview: 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 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 ( x )& x: Examples: f ( x ) = x; f ( x ) = x 2 ; f ( x;y ) = xy; f ( x ) = x 2 y 2 & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ¡...
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Note2 - 1. Every point on the map sits on an indi/erence...

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