Utility by huanghengdong

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									Utility
     UTILITY FUNCTIONS
 A preference relation that is complete,
  reflexive, transitive and continuous can be
  represented by a continuous utility
  function (as an alternative, or as a
  complement, to the indifference “map” of
  the previous lecture).
 Continuity means that small changes to a
  consumption bundle cause only small
  changes to the preference (utility) level.
     UTILITY FUNCTIONS
A utility function U(x) represents a
 preference relation if and only if:
           p
      x0       x1     U(x0) > U(x1)

      x0 p x1         U(x0) < U(x1)

      x0 ~ x1         U(x0) = U(x1)
     UTILITY FUNCTIONS
        is an ordinal (i.e. ordering or
 Utility
  ranking) concept.
 For example, if U(x) = 6 and U(y) = 2
  then bundle x is strictly preferred to
  bundle y. However, x is not
  necessarily “three times better” than
  y.
           UTILITY FUNCTIONS
       and INDIFFERENCE CURVES

 Consider   the bundles (4,1), (2,3) and
  (2,2).
 Suppose (2,3) (4,1) ~ (2,2).
                 p
 Assign to these bundles any
  numbers that preserve the
  preference ordering;
  e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.
 Call these numbers utility levels.
           UTILITY FUNCTIONS
       and INDIFFERENCE CURVES

 Anindifference curve contains
 equally preferred bundles.
 Equal   preference  same utility level.
 Therefore, all bundles on an
  indifference curve have the same
  utility level.
            UTILITY FUNCTIONS
        and INDIFFERENCE CURVES

 So the bundles (4,1) and (2,2) are on
  the indifference curve with utility
  level U  
 But the bundle (2,3) is on the
  indifference curve with utility level U
  6
     UTILITY FUNCTIONS
 and INDIFFERENCE CURVES


 x2          (2,3) > (2,2) = (4,1)
x2




                        U6
                        U4

                          x1
                         xx11
          UTILITY FUNCTIONS
      and INDIFFERENCE CURVES

 Comparing   more bundles will create
 a larger collection of all indifference
 curves and a better description of the
 consumer’s preferences.
     UTILITY FUNCTIONS
 and INDIFFERENCE CURVES


x2x2




                    U6
                    U4
                    U2
                    xx1
                    1
          UTILITY FUNCTIONS
      and INDIFFERENCE CURVES

 Comparing   all possible consumption
  bundles gives the complete collection
  of the consumer’s indifference curves,
  each with its assigned utility level.
 This complete collection of
  indifference curves completely
  represents the consumer’s
  preferences.
            UTILITY FUNCTIONS
        and INDIFFERENCE CURVES


 The  collection of all indifference
  curves for a given preference relation
  is an indifference map.
 An indifference map is equivalent to
  a utility function; each is the other.
       UTILITY FUNCTIONS
   If
    (i) U is a utility function that represents a
    preference relation; and (ii) f is a strictly
    increasing function,
    then
    V = f(U) is also a utility function
    representing the original preference
    function.
    Example? V = 2.U
  GOODS, BADS and NEUTRALS
A  good is a commodity unit which
  increases utility (gives a more
  preferred bundle).
 A bad is a commodity unit which
  decreases utility (gives a less
  preferred bundle).
 A neutral is a commodity unit which
  does not change utility (gives an
  equally preferred bundle).
   GOODS, BADS and NEUTRALS

    Utility
                                     Utility
                                     function
              Units of      Units of
              water are     water are
              goods         bads


                       x’            Water
Around x’ units, a little extra water is a neutral.
     UTILITY FUNCTIONS
Cobb-Douglas Utility Function

U x1 , x2   x x (a  0, b  0)
                    a b
                    1 2
Perfect Substitutes Utility Function

 U  x1 , x2   ax1  bx2      Note: MRS = (-)a/b

Perfect Complements Utility Function
 U  x1 , x2   minx1, x2      Note: MRS = ?
                   UTILITY
     Preferences can be represented by a
     utility function if the functional form has
     certain “nice” properties

     Example: Consider U(x1,x2)= x1.x2

1.   u/x1>0      and     u/x2>0
2.   Along a particular indifference curve
             x1.x2 = constant  x2=c/x1
     As      x1          x 2
     i.e. downward sloping indifference curve
             UTILITY
Example U(x1,x2)= x1.x2 =16
    X1       X2             MRS
    1        16
    2        8              (-) 8
    3        5.3            (-) 2.7
    4        4              (-) 1.3
    5        3.2            (-) 0.8
 3. As X1  MRS  (in absolute terms),
    i.e convex preferences
COBB DOUGLAS UTILITY FUNCTION

  Any   utility function of the form

           U(x1,x2) = x1a x2b

   with a > 0 and b > 0 is called a Cobb-
   Douglas utility function.
  Examples
   U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
   V(x1,x2) = x1 x23      (a = 1, b = 3)
COBB DOUBLAS INDIFFERENCE CURVES

   x2
            All curves are “hyperbolic”,
            asymptoting to, but never
            touching any axis.




                         x1
  PERFECT SUBSITITUTES
 Instead   of U(x1,x2) = x1x2 consider

            V(x1,x2) = x1 + x2.
 PERFECT SUBSITITUTES
x2
                 x1 + x2 = 5
13
                    x1 + x2 = 9
 9
                         x1 + x2 = 13
 5
                        V(x1,x2) = x1 + x2

        5    9     13    x1
     All are linear and parallel.
 PERFECT COMPLEMENTS
 Instead of U(x1,x2) = x1x2 or
 V(x1,x2) = x1 + x2, consider

            W(x1,x2) = min{x1,x2}.
 PERFECT COMPLEMENTS
 x2
                     45o
                       W(x1,x2) = min{x1,x2}

  8                        min{x1,x2} = 8
  5                   min{x1,x2} = 5
  3                 min{x1,x2} = 3

        3 5 8              x1
All are right-angled with vertices/corners
on a ray from the origin.
     MARGINAL UTILITY

 Marginal  means “incremental”.
 The marginal utility of product i is the
  rate-of-change of total utility as the
  quantity of product i consumed
  changes by one unit; i.e.
                     U
              MU i 
                      xi
     MARGINAL UTILITY
U=(x1,x2)
MU1=U/x1  U=MU1.x1
MU2=U/x2  U=MU2.x2

Along a particular indifference curve

     U = 0 = MU1(x1) + MU2(x2)

    x2/x1 {= MRS} = (-)MU1/MU2
     MARGINAL UTILITY

 E.g.   if U(x1,x2) = x11/2 x22 then


                 U 1 1/ 2 2
         MU 1       x1 x2
                 x1 2
     MARGINAL UTILITY
 E.g.   if U(x1,x2) = x11/2 x22 then


                 U 1 1/ 2 2
          MU1       x1 x2
                 x1 2
     MARGINAL UTILITY
 E.g.   if U(x1,x2) = x11/2 x22 then



                U        1/ 2
         MU 2        2 x1 x2
                 x2
     MARGINAL UTILITY
 E.g.   if U(x1,x2) = x11/2 x22   then



                U        1/ 2
         MU 2        2 x1 x2
                 x2
    MARGINAL UTILITY
 So,   if U(x1,x2) = x11/2 x22 then

                   U 1 1 / 2 2
         MU 1         x1 x 2
                   x1 2
                   U
         MU 2          2 x1 / 2 x 2
                            1

                   x2
   MARGINAL UTLITIES AND MARGINAL
        RATE OF SUBISITUTION

 The  general equation for an
  indifference curve is
      U(x1,x2)  k, a constant
 Totally differentiating this identity gives



        U         U
             dx1       dx2  0
         x1        x2
MARGINAL UTLITIES AND MARGINAL
     RATE OF SUBISITUTION

    U         U
         dx1       dx2  0
     x1        x2
rearranging
     U           U
          dx2        dx1
      x2          x1
MARGINAL UTLITIES AND MARGINAL
     RATE OF SUBISITUTION


      U           U
           dx2        dx1
       x2          x1
rearranging
      d x2     U /  x1
           
      d x1     U /  x2
This is the MRS.
 MU’s and MRS: An example
Suppose U(x1,x2) = x1x2. Then

        U
              (1)( x2 )    x2
         x1
        U
              ( x1 )(1)    x1
         x2
           d x2     U /  x1    x2
so   MRS                   
           d x1     U /  x2    x1
 MU’s and MRS: An example

                                      x2
x2           U(x1,x2) = x1x2; MRS  
                                      x1
     8               MRS(1,8) = - 8/1 = -8
     6               MRS(6,6) = - 6/6 = -1.


                           U = 36
                           U=8
         1      6           x1
MONOTONIC TRANSFORMATIONS AND MRS

 Applying    a monotonic transformation
  to a utility function representing a
  preference relation simply creates
  another utility function representing
  the same preference relation.
 What happens to marginal rates-of-
  substitution when a monotonic
  transformation is applied?
  (Hopefully, nothing)
MONOTONIC TRANSFORMATIONS AND MRS

   For U(x1,x2) = x1x2 the MRS = (-) x2/x1

   Create V = U2; i.e. V(x1,x2) = x12x22 What is
    the MRS for V?


         V /  x1          2
                      2 x1 x2    x2
MRS                       
         V /  x2       2
                      2 x1 x2    x1

which is the same as the MRS for U.
MONOTONIC TRANSFORMATIONS AND MRS

 More   generally, if V = f(U) where f is a
  strictly increasing function, then

         V /  x1     f  (U )   U / x1
MRS               
         V /  x2     f '(U )   U / x2
         U /  x1
                 .
         U /  x2
  So MRS is unchanged by a positive
  monotonic transformation.

								
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