Econ 400B Lecture Note.pdf - Chapter 3 Preferences and Utility There are n commodities(also include time and location A consumption bundle is denoted as

Econ 400B Lecture Note.pdf - Chapter 3 Preferences and...

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Chapter 3: Preferences and Utility ° There are n commodities (also include time and location). A consumption bundle is denoted as X = ( x 1 ; x 2 ; : : : ; x n ) 2 R n + : ° A consumer °should be able±to make simple choice between two consumption bundles, known as preference relation . ° If the consumer likes bundle A better than bundle B , then ° A is (strictly) preferred to B ±, denote as A ± B . ° If the consumer thinks that A and B are equally attractive, then ° A is indi/erent from B ±, denote as A ² B . ° If either A ± B or A ² B , then ° A is weakly preferred to B ±, or A is at least as good as B , denote as A % B: ° Axioms of Rational Choice ° Complete: For all A and B , either A ± B , or B ± A , or A ² B . ° Transitive: If A ± B and B ± C , then A ± C . ° Continuous: If A ± B , and C is °very close to± A , then C ± B: Lexicographic preference is not continuous! ° U ( ³ ) : R n ! R is a utility function if U ( A ) > U ( B ) if and only A ± B , and U ( A ) = U ( B ) if and only A ² B . ° If there is a utility function, then preference must be complete and transitive. ° If preference is complete, transitive, and continuous , then there is a utility function. ° Any monotonic transformation of a utility function is also a utility function. ° With n commodities, denote a bundle as A = ( x 1 ; x 2 ; : : : ; x n ) , and utility as U ( A ) = U ( x 1 ; x 2 ; : : : ; x n ) : With two commodities ( n = 2 ), simply write a utility function as U ( x; y ) . ° An indi/erence curve is a set of all bundles that are indi/erent from each other. ° All bundles on the same indi/erence curve have the same utility. ° Given utility function U ( x; y ) , an indi/erence curve is f ( x; y ) : U ( x; y ) = ° g , where ° is the °common utility±of the bundles on the indi/erence curve. ° How to ²nd an indi/erence curve? Set U ( x; y ) = ° , then solve y as a function of x (and ° ). ° Marginal Rate of Substitution ( MRS ) ° The slope of an indi/erence curve (at a point/bundle) MRS = ´ dy dx ° ° U = ° : 1
° What is the meaning of MRS ? The consumer is willing to exchange one unit of good X with MRS units of good Y . ° Indi/erence curve map ° Strictly and weakly convex preference ° A convex combination of two bundles ( x 1 ; y 1 ) and ( x 2 ; y 2 ) is, for ± 2 (0 ; 1) , ± ( x 1 ; y 1 ) + (1 ´ ± )( x 2 ; y 2 ) = ( ±x 1 + (1 ´ ± ) x 2 ; ±y 1 + (1 ´ ± ) y 2 ) ° Preference µ is strictly convex if for ( x 1 ; y 1 ) ² ( x 2 ; y 2 ) and ± 2 (0 ; 1) , ± ( x 1 ; y 1 ) + (1 ´ ± )( x 2 ; y 2 ) ± ( x 1 ; y 1 ) ² ( x 2 ; y 2 ) : Preference µ is weakly convex if for ( x 1 ; y 1 ) ² ( x 2 ; y 2 ) and ± 2 (0 ; 1) , ± ( x 1 ; y 1 ) + (1 ´ ± )( x 2 ; y 2 ) µ ( x 1 ; y 1 ) ² ( x 2 ; y 2 ) : - 6 x y 0 r U 1 @ @ @ @ @ @ r r r ( x 2 ; y 2 ) ( x C ; y C ) ( x 1 ; y 1 ) convex - 6 x y 0 r U 1 @ @ @ @ @ @ r r r ( x 2 ; y 2 ) ( x C ; y C ) ( x 1 ; y 1 ) nonconvex ° Example 3.1 Utility function U ( x; y ) = p xy ° To ²nd the indi/erence curve with utility 10 ; p xy = 10 ) y = 100 x ° To ²nd marginal rate of substitution: MRS = ´ dy dx = 100 x 2 ° To demonstrate convexity: Note that (20 ; 5) ² (5 ; 20) as U (20 ; 5) = U (5 ; 20) = 10 : U ± 1 2 (20 ; 5) + 1 2 (5 ; 20) ² = U (12 : 5 ; 12 ; 5) = p 12 : 5 12 : 5 = 12 : 5 > 10 . ° Marginal utilities: Additional utility from consuming more but small amount of one commodity MU x = @U ( x; y ) @x and MU y = @U ( x; y ) @y : ° How to ²nd marginal utility? Two ways: derivative of an indi/erence curve, or ° Marginal rate of substitution and marginal utility: MRS = ´ dy dx ° ° U ( x;y )= ° = MU x MU y Note that U ( x; y ) = ° , then @U ( x;y ) @x dx + @U ( x;y ) @y dy = 0 ) MRS = ´ dy dx = MU x MU y : 2
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