Part IIA_ Paper 2 Consumer and Producer Theory201045161030

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Part IIA_ Paper 2 Consumer and Producer Theory201045161030 Powered By Docstoc
					      Part IIA, Paper 1
Consumer and Producer Theory

                Lecture 2
   Direct and Indirect Utility Functions
             Flavio Toxvaerd
              Today’s Outline
   Indifference curves
   Marginal rates of substitution
   Marshallian demand functions
   Types of goods
   Indirect utility function
   Consumer surplus and welfare
   Some mathematical results (Envelope Thm.)
   Roy’s Identity
                Utility Function
 Recall from last lecture – given Axioms of Choice,
  continuity and local non-satiation - a consumer’s
  preference ordering can be represented by a utility
  function
 Simplifying Assumption: The domain consists of only two
  types of commodities, types 1 and 2
 A specific consumption bundle, x, will be represented by
  a vector x = (x1,x2) and we can write the utility function
  as u(x1,x2)
              Indifference Curves
 Indifference curves show combinations of
  commodities for which utility is constant
  Mathematic ally, we require that du ( x1, x2 )  0
  u ( x1, x2 )       u ( x1, x2 )
                dx1                dx2  0
      x1                 x2
           u ( x1, x2 )
    dx2        x1
       
    dx1    u ( x1, x2 )
               x1
                                Slope of Indifference Curve =
                                Marginal Rate of Substitution
            Utility Maximisation
                    max u ( x1 , x 2 )
                    x1 , x2

            subject to p1 x1  p 2 x 2  m
         L  u ( x1 , x2 )  [m  p1 x1  p 2 x2 ]

                              u ( x1 , x 2 )
First order conditions:                        p1
                                  x1
                              u ( x1 , x 2 )
                                               p 2
                                  x 2
                              p1 x1  p 2 x 2  m
                Utility Maximisation
Eliminating the Lagrange Multiplier () gives
u ( x1, x2 )                MRS = Ratio of Prices
    x2              p2
                            Slope of Indifference Curve
u ( x1, x2 )        p1
                                   = Slope of Budget Line
    x1

 and      p1x1  p2 x2  m   Budget Line

Note, with more than two commodities have FOC
 u
 xi MUxi
              MU of income
  pi     pi
               Utility Maximisation
     To solve: move along budget line, until point of
     tangency with the indifference curve

x2                               x2

           A

                                           D
                  C
           B



                            x1                          x1
            Demand Function
When the indifference curves are strictly convex
the solution is unique, say x*, where

     x1*= x1(p1,p2,m) , x2*= x2(p1,p2,m)

giving demand as a function of prices and income

       Marshallian Demand Functions
' Ordinary' Good :
                            Goods
    dx1 ( p1 , p2 , m)
                       0        x2
           dp1
Giffen Good :               m                       Price expansion path
                                p2
  dx1 ( p1 , p2 , m)
                     0
         dp1

Complements :
  dx2 ( p1 , p2 , m)
                     0                               m
                                                          p1
                                                                    m
                                                                        p '1
                                                                               x1
         dp1
Substitutes :                   p1
                                                          Demand Curve
  dx2 ( p1 , p2 , m)
                     0         p’1
         dp1
                                      x1(p1,p2,m)    x1(p’1,p2,m)              x1
                      Goods
                   0 : Inferior Good  < 0
dx1 ( p1, p2 , m)
       dm
dx1 ( p1, p2 , m)
                   0 : Normal Good  > 0
       dm
dx1 ( p1, p2 , m) x1
                      : Superior Good  > 1
       dm           m
IncomeElasticity of Demand :
                                       Graphically:
             dx1 m
                .                    Engel Curve
             dm x1
                     Practice
 Problem 1:
    Show that the Marshallian Demand Function is
     homogenous degree zero. (So consumers never
     suffer from Money Illusion)
 Problem 2:
    Show that consumers’ purchase decisions are
     unaffected by any monotonic transformation of the
     utility function.
    (Hint: A monotonic transformation can be
     represented by a strictly increasing function f(.).
     Use the chain rule to show that the MRS remains
     unaffected)
     Convex Indifference Curves
 Convex indifference curves means that the
  (absolute value of the) slope of the indifference
  curve is decreasing as x1 increases
 That is: Diminishing MRS
 Problem 3:
    Show that diminishing marginal utilities is
     neither a necessary nor a sufficient condition
     for convex indifference curves
   Example: Cobb-Douglas Utility
                                  1
              u ( x1, x2 )    x1 x2
                        1
 Lagrangian : L    x 1 x2       ( p1x1  p2 x2  m)

                                      
                                    x1 1x1  p1
                                           2
                                          
First Order Conditions          (1   ) x1 x2  p2
                                 p1 x1  p2 x2  m
   Example: Cobb-Douglas Utility
Eliminating  gives:
      (1   ) x1 p2
MRS             
         x2       p1
Substitution into the budget constraint gives the solution
                             m
x *1  x1 ( p1 , p2 , m) 
                             p1
                           (1   ) m
x *2  x2 ( p1 , p2 , m) 
                               p2
With Cobb-Douglas Utility the consumer spends a
fixed proportion of income on each commodity
            Indirect Utility Function
It is often useful to consider the utility obtained by
a consumer indirectly, as a function of prices and
income rather than the quantities actually
consumed

Let v( p1 , p2 , m)  u ( x *1 , x *2 )  u ( x1 ( p1 , p2 , m), x2 ( p1 , p2 , m))
                        max u ( x1 , x2 ) such that p1 x1  p2 x2  m


      v( p1 , p2 , m) is called the Indirect U tility Function
  Properties of Indirect Utility Fn
 Property 1: v(p,m) is non-increasing in prices
  (p), and non-decreasing in income (m).
   Proof: Diagramatically, it is clear that any increase in
    prices or decrease in income contracts the ‘affordable’
    set of commodities – as nothing new is available to
    the consumer utility cannot increase
 Property 2: v(p,m) is homogeneous degree
  zero.
    Proof: No change in the affordable set, or in
     preferences
Properties of Indirect Utility Fn

These two are General Properties and
NOT reliant on additional restrictions
  such as convexity of indifference
      curve, more is better etc.
       Direct and Indirect Utility
We will see that direct and indirect utility functions are
closely related - and that any preference ordering that can
be represented by a utility function can also be represented
by an indirect utility function. This means we are free to use
whichever specification we please

For Example:
If the price of commodity 1 changes from, say, p1=a to
p1=b, we may want to use the indirect utility function to
measure the change in consumer welfare:

        (utility)  v(b, p2 , m)  v(a, p2 , m)
           Mathematical Digression
 The Envelope Theorem:
  If M (a )  max x , x g ( x1 , x2 , a ) s.t. h( x1 , x2 , a )  0
                          1   2




                 g ( x *1 , x *2 , a )
    for which the Lagrangian can be written
       L  g ( x1 , x2 , a ) -  h( x1 , x2 , a )
    then
      M L                                           Evaluated at the
          ( x *1 , x *2 , a )                          maximising
       a a                                              values


   Proof: See Varian Microeconomic Analysis, p. 502.
                  Application 1
 Marginal utility of Income
  As v( p1, p2 , m)  max x , x u ( x1, x2 ) s.t. p1x1  p2 x2  m  0
                              1   2




  and the Lagrangian
     L  u ( x1, x2 ) -  ( p1x1  p2 x2  m)
  then, by the Envelope Theorem,
     v L
          ( x *1, x *2 , p1, p2 , m)  
     m m

          The marginal utility of income is
          given by the Lagrange multiplier
                   Application 2
 Roy’s Identity
  Similarly, by the Envelope Theorem,
    v( p1, p2 , m) L( x1 , x1 )
                         * *
                                   x1( p1, p2 , m)
         p1           p1
  and so,
                        v( p1, p2 , m)
                             p1                  Roy’s
    x1( p1, p2 , m)  
                        v( p1, p2 , m)           Identity
                             m
  Consumer Surplus and Welfare
We saw earlier that, following a price change,
      (utility)  v(b, p2 , m)  v(a, p2 , m)
                      v( p1, p2 , m)
                       a
                                   dp1
                    b      p1
                   a
                    v( p1, p2 , m)
                                  .x1 ( p1, p2 , m) dp1
                  b      m
                       a
                    x1 ( p1, p2 , m) dp1 ,
                       b
                              v( p1, p2 , m)
                           if                 constant
                                   m
Consumer Surplus and Welfare
  a

   x ( p , p , m)dp
  b
      1    1   2   1    (Consumer Surplus)

  So, ( Utility)  (Consumer Surplus)
      p1


      a


      b                    Marshallian Demand


                                x1
                    Summary
   Indifference curves
   Marginal rates of substitution
   Marshallian demand functions
   Types of goods
   Indirect utility function
   Consumer surplus and welfare
   Roy’s Identity
                Readings
 Texts:
   Varian, Intermediate Economics (7th ed.)
    chapters 4, 5, 6, 14.
   Varian, Microeconomic Analysis, chapter 7

				
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