LESSON FOUR: THE CONSUMER
final User
Max U x X2
4.0. Introduction
So t M
t
In this lesson, we now turn out attention to another economic agent, who creates demand
Pix
PzXz
economic
for goods and services produced by the firm. The firm is studied
in the last environment
three lessons.
4.1. Objectives
At the end of the lesson, the learner should:
Pi
•
Distinguish between the different consumer demand functions.
•
Be able to derive the different demand functions.
•
Understand and demonstrate duality in consumption.
Pa and m
4.2. The Consumer Theory/Behavior.
The consumer behavior is modeled as a utility maximizing behavior. The
consumer is assumed to be rational i.e. given some income and prices of goods in the
market; the consumer is able to allocate the income between the goods so as to achieve
the highest possible satisfaction. All information about consumer level of satisfaction is
contained within a utility function. On the other hand, all the information pertaining to
the consumer’s ability to purchase the goods and services is contained in the budget line.
The consumer’s optimal choice is the point of tangency between the budget line
and the highest possible indifference curve. The convexity condition ensures a unique
solution for the consumer where the optimal choice involves a combination of goods.
Such a solution is referred to as an interior solution. There are however special cases,
where an interior solution is not possible.
1
Good X 2
C
A
B
Good X 1
(i)
The concave utility functions.
At point A, the first order condition is met i.e. point of tangency. However, it’s not
the optimal choice for the consumer. Points B & C are the best affordable choices.
However the consumer can only consume either at point B or at point C. The
consumption of such goods is said to be mutually exclusive i.e. when you consume one,
the other one cannot be consumed. Such a solution (B or C) is called a back boundary
solution.
(ii)
Kuhn Tucker Condition
This assumes that the consumer may not necessarily exhaust his/her income and
therefore the budget line is an inequality.
p1 x1 + p2 x2 £ M
The optimal choice occurs at a corner between the budget line and one of the
indifference curves and which is not a point of tangency. This is illustrated in the
diagram below.
2
Good X 2
Max U Xi
Boundary solution
S t P X tPzXz M
at
4.2.1. Consumer demand functions
There are two types of demand functions:
a
and
P M
I
Good X 1
X2
I I m
Marshallean demand
(i)
Ordinary demand functions / uncompensated / Marshallian demand functions
(ii)
Compensated / Hicksian demand functions
I
functions
4.2.1.1. The ordinary demand functions
A consumer’s ordinary demand function expresses the quantity demanded of a
product as a function of prices and income. xi ( pi m ) . It is derived from utility maximizing
find the marshalled
problem.
demand functions
y
e.g. Suppose the utility function is u ( x1 , x 2 ) = x1 x 2 where x1 and x 2 are two
goods. Let p1 and
p 2 represent the prices of the two goods respectively. Let M
represent the consumer’s income.
u 34,762
I
7C 262
The consumer seeks to maximize utility subject to the budget constraint.
max u = x1 x 2
st
p1 x1 + p 2 x 2 = M
3
L = x1 x 2 - l ( p1 x1 + p 2 x 2 - M )
¶L
= x 2 - lp1 = 0........(i )
¶x1
¶L
= x1 - lp 2 = 0............(ii )
¶x 2
¶L
= p1 x1 + p 2 x 2 - M = 0..........(iii )
¶l
Dividing the first two equations
x2
p
= 1
x1
p2
x1 =
p2 x2
p1
x2 =
p1 x1
p2
Substituting this into equation (iii)
æp x ö
p1 çç 2 2 ÷÷ + p 2 x 2 = m
è p1 ø
2 p2 x2 = m
x 2* =
m
2 p2
æpx ö
p1 x1 + p 2 çç 1 1 ÷÷ = m
è p2 ø
2 p1 x1 = m
x1* =
m
2 p1
x1* andx 2* are the ordinary or Marshalian demand functions and we could verify that by
showing that their slopes are negative ;
i.e.
¶x 2
-m
=
¶p 2 2 p 22
¶x1 - m
=
¶p1 2 p12
E
4
if you Income ism
There are two basic properties of the ordinary demand functions;
P andPa
Toxin
(i)
They are single valued functions of prices and income.
(ii)
They are homogeneous of degree zero in income and prices. i.e. if the prices
and income change by the same proportions, the demand remains unchanged.
lackscan demand functions
4.2.1.2. The compensated demand functions
They are derived based on the concept of compensation i.e. following any change in
price, the consumer’s income is adjusted. There are two types of compensations,
(i)
Hicksian compensation: - it assumes that after the price change, the
consumer’s income is adjusted so as to retain the same level of satisfaction.
(remains on the same indifference curve)
(ii)
Slutskys compensation: - after a price change, the consumers income is
minimize
P
Pz
X bundle
t is stillXz
adjusted so that the initial consumption
affordable.
UH X2
I
The compensated demand functions are derived from an expenditure minimization
problem and the quantity demanded is expressed as a function of prices and utility. i.e.
given the same information, the consumer seeks to;
5
min p1 x1 + p 2 x 2
st
U = x1 x 2
L = p1 x1 + p 2 x 2 - l (U - x1 x 2 )
¶L
= p1 + lx 2 = 0...........(i )
¶x1
¶L
= p 2 + lx1 = 0...........(ii )
¶x 2
¶L
= U - x1 x 2 = 0.............(iii )
¶l
Dividing the first two equations;
p1 lx 2
=
p 2 lx 2
p1 x 2
=
p 2 x1
x1 =
p2 x2
p1
x2 =
p1 x1
p2
Substituting into equation (iii)
æpx ö
U = x1 çç 1 1 ÷÷
è p2 ø
p
U = 1 x12
p2
x12 =
x1* =
x 2* =
p2
U
p1
p 2U ü
ï
p1 ï
ý Hicksian compensated demand function
p1U ï
p2 ï
þ
6
4.2.2. The indirect utility function
It represents the highest utility achieved by a consumer given some amount of
income. The function is obtained by substituting the ordinary demand functions into the
direct utility function. i.e.
(
V ( pi , m ) = U x1* , x1*
)
From the previous example, U = x1 x 2 and x1* =
æ m
m
So that V ( pi , m ) = çç
*
è 2 p1 2 p 2
ö
m2
÷÷ =
ø 4 p1 p 2
m
m
, x 2* =
2 p1
2 p2
This is the indirect utility function
from the above direct utility function.
4.2.3. Properties of indirect utility functions
(i)
It is a single valued function of prices and income.
(ii)
It is non-decreasing in income and non-increasing in prices.
(iii)
It is homogenous of degree zero in prices and income.
Consider an indirect utility function V ( pi , m ) . Totally differentiating this function;
7
¶V =
¶V
¶V
¶pi +
¶m
¶pi
¶m
but¶V = 0
¶V
¶V
¶pi +
¶m = 0
¶pi
¶m
m = pi x1
¶m
¶pi
= xi
¶m = xi ¶pi
¶V
¶V
¶pi +
xi ¶pi = 0
¶pi
¶m
¶V
¶V
xi ¶pi = ¶pi
¶m
¶pi
é ¶V
ù
ê ¶p
ú
xi ( pi , m ) = - ê i
¶V úú
ê
¶m úû
êë
This is referrred to as the Roy ' s identity
The Roy’s identity is a tool used to recover the ordinary demand functions from
the indirect utility function.
Example
Given the indirect utility function V ( p, m ) =
m2
4 p1 p 2
8
é ¶V
ù
ê ¶p
ú
x1 = - ê 1
¶V ú
ê
ú
¶m úû
ëê
¶V m 2 p1-1
- m2
=
=
¶p1
4 p2
4 p12 p 2
¶V
2m
m
=
=
¶m 4 p1 p 2 2 p1 p 2
é - m2
m ù
x1 = ê 2 ÷
ú
ë 4 p1 p 2 2 p1 p 2 û
2p p
m2
m
=
´ 1 2 =
2
m
2 p1
4 p1 p 2
¶V
- m2
=
¶p 2 4 p1 p 22
¶V
m
=
¶m 2 p1 p 2
é - m2
m ù
x2 = -ê
÷
ú
2
ë 4 p1 p 2 2 p1 p 2 û
2p p
m2
m
=
´ 1 2 =
2
m
2 p2
4 p1 p 2
4.2.4. The consumer’s expenditure function
It provides the minimum expenditure of obtaining a given level of utility. It is
obtained by substituting the compensated demand functions into the expenditure
equation. i.e.
E ( p, U ) = p1 x1* + p 2 x 2*
From the previous example
x1* =
p 2U *
x2 =
p1
p1U
p2
9
E = p1
1
p 2U
+ p2
p1
p1U
p2
1
1
1
1
1
E = U 2 p 22 p12 + U 2 p 22 p12
E = 2 Up1 p 2 .......... This is the exp enditure function
4.2.5. Properties of expenditure function
(i)
It is non-decreasing in prices.
(ii)
It is homogenous of degree one in prices.
(iii)
If xi ( pi , U ) is the expenditure minimizing demand that is necessary to
achieve the level of utility U at prices pi , then, xi ( piU ) =
¶E ( pi , U )
¶pi
(shepherds lemma)
Using the shepherd’s Lemma, we can recover the compensated demand functions.
e.g. Consider the expenditure function above E ( p, U ) = 2 Up1 p 2
1
1
1
¶E
= 2U 2 p12 p 22
¶p1
x1 =
1
2
-
1
2
= U p1 p
x1* =
1
2
2
Up 2
p1
1
2
1
2
1
x 2 = 2U p p
1
2
1
2
1
-
= U p p2
x 2* =
1
2
2
1
2
Up1
p2
10
EQuation
Slutsky
4.2.6. Relationship between the compensated and the uncompensated demand
functions
Price
Marshallian
Hicksan
X H (Hicksian demand curve)
A
X M (Masharian demand curve)
At A We have M
Quantity
XA
E
XM
At point A, the income available for consumption will be equal to the minimum
expenditure on the two goods. i.e. the compensated (Hickisian) and the uncompensated
(Marshallian) demand functions yield the same results.
x H ( p iU ) = x M ( p i m )
but m = E ( piU )
x H ( piU ) = x M ( pi E ( piU ))
Partially differentiating with respect to price,
¶x H ( piU ) ¶x M ¶x M ¶E
=
+
´
¶pi
¶pi
¶E ¶pi
XH
Pitt
1 57 age
¶x M ¶x H ( piU ) ¶x M ¶E
=
´
¶pi
¶pi
¶E ¶pi
¶x M ¶x H ¶x M ¶E
=
´
¶pi
¶pi
¶E ¶pi
XIE Ti E Pia
Product rule
but E = m therefore ¶E = ¶m &
¶m
= xi
¶pi
¶x M ¶x H
¶x
=
- xi M ................This is referred to as the slutsky ' s equation
¶pi
¶pi
¶m
7,1
257
t
chain
Ep
11
EI M
D
off
05,1
¶x M
¶pi
Where
2M
JE
Yam Mgp
is the total effect of a price change. It is measured by the slope of the
we
ordinary demand function.
¶x H
¶pi
M
also know that
P X t PaXz tooo t DixitoootPn
Xi
JM pi
is the substitution effect of a price change. It is measured by the slope of
the Hicksian demand function. It is the change in demand resulting from a
É
price change but holding utility constant.
- xi
¶x M
¶m
m
is the income effect of a price change. It may be positive or negative
depending on the nature of the good. e.g.
a)
¶x M
>0
¶m
normal good and income effect is negative
xig
TE
For our
ti M
use
S E t I E
utility function
Pa M
Equation
T IE
S E
T change
slutsky
Map
U xoxo
and
x
P paste
P 051205g
this information to demonstrate theslutsky
equation
12
LH S
R HS
b)
¶x M
<0
¶m
m
x1M =
2 p1
Up2
p1
x1H =
x2M =
inf erior good and income effect is positive
m
2 p2
Up1
p2
x2H =
¶xM - m
=
¶p1 2 p12
¶xH
1 1 -3 1
= - U 2 p1 2 p22
¶p1
2
¶x M
1
=
¶m 2 p1
1
3
1
-m
1
1
m
= - U 2 p1 2 p 22 ´
2
2
2 p1 2 p1
2 p1
1
2
1
2
1
1
2
2
- 2U p p - m
=
4 p12
=
-m-m
4 p12
=
- 2m - m
=
4 p12
2 p12
1
2
1
2
1
1
2
2
but 2U p p = E = m
4.2.7. Four important identities
(i)
The minimum expenditure necessary to reach the maximum utility V ( p, m ) is
the income m
13
E ( p, V ( p, m )) º m
from the previous example, E ( p, u ) = 2 Up1 p 2 .....Exp function
V ( p, m ) =
m2
..........Indirect Utility function
4 p1 p 2
hence,
E ( p, V ( p, U )) = 2
m2
=m
4
=2
(ii)
m2
p1 p 2
4 p1 p 2
The maximum utility from the minimum expenditure E ( p, u ) is U .
V ( p, E ( p, U )) º U
V ( p, m ) =
m2
ºU
4 p1 p 2
where m = E ( p, U ) = 2 Up1 p 2
so that ,
=
(iii)
2 Up1 p 2
4 p1 p 2
=
(2 Up1 p 2 ) 2
4 P1 P2
=U
The Marshallian demand function at an income level m yields similar results
to the Hickisian demand function at the highest level of utility V ( p, m ) .
xiM ( p, m ) º xiH [ p, V ( p, m )]
x1M =
m
2 p1
x1H =
m
º
2 p1
=
Up 2
p1
p
m2
´ 2
4 p1 p 2 p1
m2
m
=
2
2 p1
4 p1
14
(iv)
The Hicksian demand function at the utility level U , yields the same results
E ( p, u )
as the Marshallian demand function at the minimum expenditure
xiH ( p, U ) º xiM [ p, E ( p, U )]
x1 =
m
2 p1
Up 2
p1
x1 =
m = E ( p, U ) = 2 Up1 p 2
x1 =
2 Up1 p 2
x1 =
2 p1
up 2
p1
Note
A point of tangency between the consumer’s budget line and the indifference curve is
an equilibrium only if the indifference curve is strictly convex.
Question
In a two goods world, the marginal rate of substitution is -
1
. If the two goods cost
2
$1 each and the consumer’s income is $100 , how many units of each good will maximize
the consumer's utility? Suppose the price of one good changes to $2 , how would the
answer change.
15
Activity
Using a utility function of your choice, demonstrate the slutsky’s equation.
Summary
In this lesson, the consumer is depicted as an agent who allocates his/her income in
such a manner that maximum utility is derived by consuming goods and services. A
consumer who seeks to maximize utility subject to a given budget also succeeds in
minimizing expenditure subject to achieving highest possible utility.
U Xi x2
P X t Paxz
Maximize
s t
L
Xi X2 X
X2
242
242
Xz
XP
Xz
M
X PX
t
PzXz
M
i
0
0
XP
ri
2
Yay
From
x
X
I
i
Pix
t
and di
PaXz M
X
0
E
Iii
16
P
P x2
Pix
D
M
2P X
X
til I
er
M
Pix
t
X
P Pa M
Yp
Ip
I m
u
05,1
Y'p
x
05nF
o
I
1 2 3
ooo
Properties of Marshathan
i
They are Hob 0 in
demand functions
Pi and
m
Xi ftp tea ootitm
I e
Xi P Pa
ooo
to x
Pa Pa
o
Pn
M
Pn M
Changing income and prices by the samescalar
leaves demand unchanged
Example
x
th th th
to 05,7
a
They are non
0541,1 0.5411
05,1
increasing in Pi
if
P
P
xi Pi
then
axiff.MIL
ni
o
M
then
jm
y
Pi
x
Pi Pym
negative
They are non decreasing
M
X
Pa m
in
M
X Pi Pa M
Pa M
o
Tx
jig
Domain
Randge
X
to
D
many
has one and
value
range
in
the
y 5
5
one
many
consumers
with the
same
economic environment
will
make
similar
0.5 P M
Xi Pi Pa M
an Pi
same
utility function and
choices
o g pi'm
Pa m
05,11 LO
OP
V
Hickscan Demand functions
minimize
I
s t
P X t PzXz
ucx x2
h
ti
ti
t
ha
x
ta
t
ki ta
X
and
Pix
constraint
Xix
t
Pax
Heck scan compensated
demand functions
X
x
Xz Te
JEY
Xi Pi B E
X2 Pi
Pa
0
p 5120
Sy O 5
demand
or compensated
Hickscan
the
suppose
in
5
pO p
T
0
to
5
functions
function was
utility
ICA
5
X2
as
The consumers problem
to
Minimize P X
I
s t
da Xz
X X
min
t
PaXz
X X da Xi
min
The functional form implies that
x
and Xz are perfect
complements
statements
using logical
a positive price
No consumer will waste any good with
consume at the points
reason
they'll
this
for
42 2
where
I X X
we optimize
I
Xi Pi Pa T
U
Xz Pi Pa
For every
SB
as his
s
2
UH X2
Min
Xi
2 2
and
I
mdot a teaspoonful of butter
L
x x
suppose the utility function was u
what would be the Hickscan demand functions
The
consumers problem
x
tax
is to
Minimize P X t Pa Xz
X X t haXz
I
st
Given the functional
substitutes
we
should
conditions
use logical
P
Case 1
form ox
and Xz are
statements or
perfect
Kuhn Tucker
Pa
substitutes
Since x and Xz are perfect
Xz
consumer will opt to consume
P Pa T
X
X2
Pa
P
E
Since x and
consumer will
Xz are
Xz P Pa E
Case 3
LI
P
perfect substitute
the
X
opt to consume
P Pa T
X
o
0
PL P
Case 2
the
Pa
I
0
the
substitute
perfect
either consume
Since x and Xz are
They
indifferent
be
will
consumer
combination
x alone Xz alone 08 consume some
of X and x2
0
I
EO
I
I
Xi Pi Pa
X2 Pi Pa
t
conclusion
P 7 Pa
O
p
L
O
E
P
P
L
Pa
P
and
P L P
O
xp
L
O
E
I
P
Pa
R P
Expenditure Function
on good 9 and
expenditure
the
is
aggregate
Pix t Pax
does
it
function
2 However It is not an expenditure
the minimum expenditure of attaining X
not
we call it a dummy or Pseudo expenditure
and x
function
a
expenditure function for a cobb boughtas utility
function
Recall that for cobb doughlas utility function
demand curves
the
can
hicks
X
to
u xoxo
were
i
Fay
pjo.su
Xi Pi B E
X2 Pi
Pa
For this
e
Pi Pa
PO
E
5
reason the
I
P
I
P 05150
2
X
0
p 5120
5
to
5
O5
expenditure function
Pi Pa t
5
5
as
T P Xz Pi Pa T
05
yo
pi
Pi Pi v05 a Expenditure function
pao
y
is cobb doughlas the
doughlas
expenditure function yincobb
when the utility function
A x
1
I
L
n
For perfect complements the Hickscan demand were
and
Xi Pi Pa T
I
I
U
X2 Pi Pa
and so the expenditure function is
e
P
Pi Pa Vi
G
t P
I
I
th
Pity
Pya t Pym
perfect
utility function i's Leontief for
is linear
complements the expenditure function
when the
n
expenditure function for a linear utility function
Recall that the Hickscan demand functions are
P
O
to
pi p ie
E
P
Pi Pa
ya
PI
e
R BE
I
I
PJ
is
P
P
Pay
J
P L P2
aa
P P
Pi p
to Eal
Y
PLP
the expenditure function
PL P
O
P
Par
ELP
min
Pa it
min
244
PE
ya
ya
t
is
expenditure
linear
function
the
when the utility
Leontief
of an expenditure function
Properties
1 it
in prices
as non decreasing
If P
P
2 e Pi
Pat
JP
e go
then
RT
yo positive
I7
e Pi Pa
217,05120
t
t i scalar
th th I
El th th E
2
5
40
5
5
20 Pi Pa t
OP
it noD 1 in P
e
E Pi
P Pa t
e
pyo5120 40
e P Pa
th
2 to
5
yo
I
541210
54
5
to p
05
o 5120.5
405
t 2 Pio51205005
t e Pi Part
Doubling prices
doubles the
expenditures
Positive prices and positive a
it
Positive
Pitt
El
Pi
70
free lunch
j
expenditure
a
must
there
positive
it
For any positive
Je Pite
No
in
shephards Lemma
J ELP Prot
P Pa
X
t
OP
xz.lk
2etyjd
For our example
Hessian matrix
v
its as negative semi
Hessian matrix
ELP Pa T
definite
is a
matrix
Pa
t
217,0512054
HESSIAN
of second order
derivatives
p
Pa
l
Y
Pz
app
2122P
g
OP OP
05
OP
For this matrix to be negative definite main
that
diagonal items need to be negative Recall
Ip
so
Xi R R E
o l
225Gt
B ti
OP
Ofp I Xi
Pi Pa I
definitely negative
The slope of a demand curve
is negative
which
is
22 e Pi Pa t
L o
oPi
Ip
so
Xi R R E
o l
225,1ft
Ba
2Pz
J
x2 Pi Pa I
Ofp
definitely negative
The slope of a demand curve
is negative
which
is
22 e Pi B
o
u
g
op
since the main diagonal items are negative then
the expenditure function as negative semi
definite
In
in
the longrun
expenditures are
fixed
P
43 P
PI t 43Pat
I
is no D1
in
zero
I
b and f
obtain the values of a
to
function
expenditure
an
we use the properties of
obtain these values
Pi
e
Pa
Expenditure function
which means that e t Pi tea I
e.g
on our case
e
to
ta
Pit
lg
tr tea I
of the function
ta t
to
tf
our
4
is to
bet
t
P
be Hod 1
I
D a
1
6
0.4
b
0 6
t
1
expenditure function is
t e Pi Pa t
tb
then
P
Psb t
Pat t
tf
2,3
13 P t
El Pi Pa I
P O41206
t
213 Pa
Indirect utility function
to
said
A utility function such as
since we derive
function
be a direct utility
which in this
services
and
from consuming goods
as
U xoxo
satisfac ion
X and x2
case are
of
expressed as a function
to as
it
referred
then
environment
utility function
a consumer
of
environment
economic
when
when
V
the economic
an
Indirect
P andM
Pi
Indirect utility function
P Pa M
obtained by substituting Marshall an
demand functions in the direct utility function
U X Xz the
the
instance
it
as
For
function
for
Marshall can demand functions were
X
P
Pa M
X2 Pi Pa M
ME
V Pi Pa M
Map
it tells the maximum amount
be obtained at Pi Pa and M
characteristics of
an
maps
Mgp
YEP
of utility
that can
Indirect utility function
in prices and income
Em
v1 EP tea tm
it is homogeneous degree o
For
example
our
as a
t
to
its
HOD
when income
utility gained
4 EP Pa
scalar
to
M
V
Pi Pa M
4P P
O
and prices change by the same scalar
remains unchanged
valued function in Pi and M
single
in
consumers with similar economic environment
and utility functions have similar choices
iii
it
as a
it
is
non
M
increasing
Up
O
OY m
0
For our case
2
p
dug
in Pi and
MIp
ftp
LO
co
non decreasing
in
0
Map
m
identity Duality in consumption
Roy's
retrace the utility
Given the demand functions can we
function and vice versa
M
V6
Pi
function
the
consider
Indirect utility
we have
function
this
different
Totally
ing
de
Jyp
du
but Vl Pi m
Pi and M so
M
I
Px
dy
t
dm
at
attainable
utility
as the maximum
du 0
Pax
t
o o o
Pi Xi t
t
which implies that dyp
o oo
t Pn X n
xi
which further implies that 2m
Plugging these facts
0
0
o
d pi
0
0
1
d Pit
t Yp
1
t
in our
total derivative implies that
ofmaid
t
Oyama
Kid
of msci
2
x
For our
Jyp
mxi
Pi
Hopi
M
example
VIP
gym
Roy's
identity
M
Pe M
4PP
by Roy's Identity
2
x
Pi
Up
i
Pa M
gym
Um Yip
YEP
Xi Pi Pa
1
Ifp PII
M
2P
M
Jp
X2 Pi Pa M
Y
2Pa
Find the marshallian demand functions
X P Pa M
Mp
Slutsky Equation
Derivation
X2 Pi Pa M
My