Part IIA_ Paper 2 Consumer and Producer Theory201045161030

Document Sample

```					      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
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

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