ECON 5020: Microeconomic Theory
Lecture 6: Consumer Choice
–Preference-based approach (Part III)
1
How to Find the Marshallian Demands and the Hicksian Demands?
What are the Marshallian Demands and the Hicksian Demands?
De…nition 1
1. The Marshallian demand x (p; w) is the point(s) that maximizes an
agent’s wellbeing within the feasible budget set Bp;w : That is, the Marshallian demand
x (p; w) satis…es
x (p; w) y for any y 2 Bp;w :
2. The Hicksian demand h (p; ) is the point(s) that minimizes an agent’s cost within
the upper counter set x : That is, the Hicksian demand h (p; ) satis…es
ph (p; )
py for any y 2
x
:
From the de…nitions, we can see they are totally di¤erent. So, the conditions for existing
the two demands are also quite di¤erent. How to …nd them? There are two cases needed to
discuss:
Case I: There exist utility functions
In this case, the question becomes very easy. The Marshallian demand is the solution to
the UMP
x (p; w) 2 arg max u (x) for any pair (p; w)
x2Bp;w
and the Hicksian demand is the solution to the EMP
h (p; u) 2 arg
min
u(x) u and x2RL
+
p x for any pair (p; u)
Case II: There do not exist utility functions
To …nd the Marshallian demand x (p; w), it is not necessary to solve the UMP. Generally,
for a given rational preference relation and feasible budget constraint Bp;w ; the agent tries
to solve the following problem:
Find a Marshallian demand x (p; w) from Bp;w such that
x (p; w)
y for any y 2 Bp;w .
1
That is,
x (p; w) 2 C ( ; Bp;w ) = fx : x
y for any y 2 Bp;w g for any pair (p; w) :
Similarly, to …nd a Hicksian demand, for any given buddle x ; de…ne the upper contour
set
U (x ) = y 2 RL+ : y
Then try to …nd a bundle h p;
x
such that h (p;
x
x
:
) is a solution to
min p y for any pair p;
y2U (x )
x
:
Next, we consider an example:
Example 1 Consider the following lexicographical preference relation
(x1 ; x2 ) = x
:
y = (y1 ; y2 ) () x1 > y1 or x1 = y1 and x2 > y2 :
1. Given price (p1 ; p2 ) >> 0 and w > 0; …nd the Marshallian demand x (p; w) :
From the picture, we can see
x (p; w) =
which is a corner solution.
2
w
;0
p1
2. Find the o¤er curve
The o¤er curve is
x (p; w) =
w
;0
p1
3. Find the Engle curve
3
for any p >> 0:
The Engle curve is
w
;0
p1
x (p; w) =
4. Find the Hicksian demand h p;
x
for any w
0:
:
For any point (x1 ; x2 ), the upper contour set U (x1 ; x2 ) is
U (x1 ; x2 ) = f(x1 ; x2 ) : (x1 ; x2 ) (x1 ; x2 )g
= f(x1 ; x2 ) : x1 > x1 g [ f(x1 ; x2 ) : x1 = x1 and x2
To …nd the Hicksian demand h p;
x
; we consider two cases:
Case I: x2 > 0:
4
x2 g
From the above picture, there is no solution to the EMP
min p x:
x2U (x )
That is,
h p;
x
= ?:
Can you imagine why? The reason is the upper contour set U (x1 ; x2 ) is not closed and
accordingly the extreme value theorem cannot used here!
Case II: x2 = 0
5
From the above picture, there is a solution to the EMP
min p x:
x2U (x )
The solution is x . That is,
h p;
x
=x
and the expenditure is
e (p; ) = p x :
Can you imagine why? The reason is the upper contour set U (x1 ; x2 ) is closed and
accordingly the extreme value theorem can used here!
Thus, combining the above two cases, we have
h p;
x
=
? if x2 > 0
:
x if x2 = 0
For this particular preference, the Marshallian demand x (p; w) and h (p; ) have no more
the “dual” relationship. It is surprising and amazing! That is, the Marshallian demand
x (p; w) always exists as long as the person is rational. However, the Hicksian demand
h p; x may not exist even the person is rational! That is the key di¤erence between the
two demands.
6
2
Some Applications
Fixing income w; changing prices from p0 to p1 ; is the consumer better o¤ or worse o¤?
(p0 ; w)
+
max u (x)
s.t. p0 x w …xed
+
x (p0 ; w)
The Marshallian Demand
+
v (p0 ; w) = u (x (p0 ; w)) = u0
The Indirect Utility Function
To the same consumer
!
(p1 ; w)
+
max u (x)
s.t. p1 x w …xed
+
x (p1 ; w)
The Marshallian Demand
+
v (p1 ; w) = u (x (p1 ; w)) = u1
The Indirect Utility Function
Thus, we have
1: If v (p0 ; w) > v (p1 ; w) ; then the consumer is worse o¤;
2: If v (p0 ; w) < v (p1 ; w) ; then the consumer is better o¤; and
3: If v (p0 ; w) = v (p1 ; w) ; then the consumer is indi¤erent.
It is well known that the expenditure function e (p; u) is increasing in utility level u.
Thus, for …xed price p;
u1 u2 () e p; u1
e p; u2 :
Thus, by the de…nition of utility function, for …xed price p; the expenditure function e (p; u)
is also a utility function representing the preference relation ; but in dollar units!
The question is: Why don’t we use it often in economics? The reason is
e (p; u) = p h (p; u) ;
and the Hicksian demand h (p; u) cannot be observed. As a result, the expenditure function
e (p; u) is not observable.
However, theoretically, it is still very useful. We can use it to study welfare problems.
Question: Suppose the Government was going to change prices from p0 to p1 ; but the Government has changed its mind later and is going to give the consumer some amount of
money (could be negative!) as a compensation such that the two policies are indi¤erent
to the consumer.
How much money should the Government give to the consumer as a compensation?
Assume the consumer’s income endowment is w:
7
Policy 1: Fixing income w; changing prices from p0 to p1 :
(p0 ; w)
+
max u (x)
s.t. p0 x w …xed
+
0
x (p ; w)
The Marshallian Demand
+
v (p0 ; w) = u (x (p0 ; w)) = u0
The Indirect Utility Function
To the same
consumer
!
(p1 ; w)
+
max u (x)
s.t. p1 x w …xed
+
1
x (p ; w)
The Marshallian Demand
+
v (p1 ; w) = u (x (p1 ; w)) = u1
The Indirect Utility Function
Policy 2: Fixing price p0 ; changing incomes from w to w0 :
In order to have the standard of living level u1 ; how much income does the consumer
need exactly?
e p0 ; u1 :
That is, to replace policy 1, how much more income should the Government give to the
agent?
To have the utility level u1 ; he needs e (p0 ; u1 ) : But he has income w already. Thus, the
consumer just needs
e p0 ; u1
w
more. At price p0 and income w; what is the agent’s utility level? It is u0 : At price p0 ; to
keep utility level u0 , how much income does the consumer need exactly?
e p0 ; u0 = w:
Thus, we have
e p0 ; u1
w = e p0 ; u1
e p0 ; u0 :
which is called equivalent variation (EV for short). That is,
EV p0 ; p1 ; w = e p0 ; u1
w = e p0 ; u1
e p0 ; u0 :
In other words, to replace policy 1, the consumer just needs EV (p0 ; p1 ; w) more income
as a compensation for policy 2.
For example,
p0 = p01 ; p02 ; and p1 = p11 ; p12 = p11 ; p02 with p11 < p01 and p02 = p12
8
How to calculate
EV p0 ; p1 ; w ?
By the de…nition,
EV p0 ; p1 ; w
= e p0 ; u1
w
0
0
1
= p h p ;u
w:
Thus, we only need to …nd h (p0 ; u1 ) from solving the following EMP
min p0 x:
u(x) u1
Or from the following
EV p0 ; p1 ; w
= e p0 ; u1
w = e p0 ; u1
e p1 ; u1
R p0
R p0 PL @e (p; u1 )
dpl
= p1 de p; u1 = p1
l=1
@pl
PL R p0l @e (p; u1 )
P R p0
=
dpl = Ll=1 p1l hl p; u1 dpl :
l=1 p1l
l
@pl
9
A special case: L = 2 and
p0 = p01 ; p02 ; and p1 = p11 ; p02 with p11 < p01 and p02 = p12 :
By using the above formula, we have
EV p0 ; p1 ; w
=
=
=
=
See the picture as follows:
P2 R p0l
l=1
R p01
p11
R p01
p11
R p01
p11
h1
h1
hl p; u1 dpl
R p0
p; u1 dp1 + p12 h2 p; u1 dp2
2
R p12
1
p; u dp1 + p1 h2 p; u1 dp2
p1l
2
1
h1 p; u
dp1 :
Now, we ask next question:
Question: Suppose the prices have been changed from p0 to p1 already. The consumer
may be better o¤ or worse o¤. For example, suppose that the consumer is better o¤.
It means the agent has more “income.” But the Government also needs money to do
something else. Thus, the agent should pay some tax to the Government. How much
should the agent pay to the Government such that the agent’s wellbeing is exactly the
10
same as before after paying tax?
To the same
consumer
!
(p0 ; w)
+
max u (x)
s.t. p0 x w …xed
+
0
x (p ; w)
The Marshallian Demand
+
v (p0 ; w) = u (x (p0 ; w)) = u0
The Indirect Utility Function
(p1 ; w)
+
max u (x)
s.t. p1 x w …xed
+
1
x (p ; w)
The Marshallian Demand
+
v (p1 ; w) = u (x (p1 ; w)) = u1
The Indirect Utility Function
Assume
u1 > u0 :
In order to keep the original utility level u0 ; how much income does the agent exactly
need at the new price p1 ?
e p1 ; u0 !
However, the agent has income w. Thus, should pay
e p1 ; u0
w
to the Government. If
w
e p1 ; u0 < 0;
then it means the Government gives the agent
w
e p1 ; u0
as a compensation for the price change.
Thus,
w
= e p1 ; u0
w>0
e p1 ; u0
is called compensation variation (CV (p0 ; p1 ; w) for short.)
CV p0 ; p1 ; w = w
11
e p1 ; u0 :
How to calculate CV (p0 ; p1 ; w)?
From
e p1 ; u0 = p1 h p1 ; u0 ;
to calculate CV (p0 ; p1 ; w) ; we only need to …nd h (p1 ; u0 ) by solving the following EMP
min p1 x:
u(x) u0
Or from the following
CV p0 ; p1 ; w
= w e p1 ; u0 = e p0 ; u0
e p1 ; u0
R p0
R p0 P @e (p; u0 )
dpl
= p1 de p; u0 = p1 Ll=1
@pl
PL R p0l
PL R p0l @e (p; u0 )
0
=
dp
=
dpl :
1
l
l=1 pl
l=1 p1l hl p; u
@pl
A special case: L = 2 and
p0 = p01 ; p02 ; and p1 = p11 ; p02 with p11 < p01 and p02 = p12 :
12
By using the above formula, we have
EV p0 ; p1 ; w
=
=
For example,
P2 R p0l
l=1
R p01
p11
h1
R p0
R p0
hl p; u0 dpl = p11 h1 p; u0 dp1 + p12 h2 p; u0 dp2
1
2
R p12
R p01
0
0
p; u dp1 + p1 h2 p; u dp2 = p1 h1 p; u0 dp1 :
p1l
2
1
p0 = p01 ; p02 ; and p1 = p11 ; p02 with p11 < p01 and p02 = p12 ;
we have the following picture
I think most of you are very family with consumer surplus (CS(p0 ; p1 ; w) for short).
(p0 ; w)
+
max u (x)
s.t. p0 x w …xed
+
0
x (p ; w)
The Marshallian Demand
To the same consumer
!
13
(p1 ; w)
+
max u (x)
s.t. p1 x w …xed
+
1
x (p ; w)
The Marshallian Demand
Consumer surplus is de…ned as follows
R p0
R p 0 PL
CS p0 ; p1 ; w = p1 x (p; w) dp = p1
l=1 xl (p; w) dpl
PL R p0l
=
l=1 p1 xl (p; w) dpl :
l
A special case: L = 2 and
p0 = p01 ; p02 ; and p1 = p11 ; p02 with p11 < p01 and p02 = p12 :
By using the above formula, we have
CS p0 ; p1 ; w
=
=
See the picture as follows:
P2 R p0l
l=1
R p01
p11
p1l
xl (p; w) dpl =
x1 (p; w) dp1 :
R p01
p11
x1 (p; w) dp1 +
R p02
p12
x2 (p; w) dp2
Do you understand the meaning of CS (p0 ; p1 ; w)?
Theoretically speaking, it has no any meaning at all. But in reality, we use CS (p0 ; p1 ; w) ;
rather than use both EV (p0 ; p1 ; w) and CV (p0 ; p1 ; w)? Why?
Since
14
1: Compensated demand h (p; u) is not observable, thus, EV (p0 ; p1 ; w) and CV (p0 ; p1 ; w)
are not observable; and
2. The Marshallian demand x (p; w) is observable; Therefore, CS (p0 ; p1 ; w) is observable.
From the above picture, we can see: Either
EV p0 ; p1 ; w
CS p0 ; p1 ; w
CV p0 ; p1 ; w
CV p0 ; p1 ; w
CS p0 ; p1 ; w
EV p0 ; p1 ; w :
or
That is, CS (p0 ; p1 ; w) is an approximation of EV (p0 ; p1 ; w) and CV (p0 ; p1 ; w) :
Example 2 (Lump-sum tax or good tax) Let L = 2 and the original good prices are
p0 = p01 ; p02 :
Suppose that the government taxes commodity 1, setting a tax on the consumer’s purchases
of good 1 of t per unit. That is, the new prices are
p1 = p01 + t; p02 :
Now we ask: How much money does the Government collect from a consumer?
(p0 ; w)
+
max u (x)
s.t. p0 x w …xed
+
0
x (p ; w)
The Marshallian Demand
+
v (p0 ; w) = u (x (p0 ; w)) = u0
The Indirect Utility Function
To the same consumer
!
(p1 ; w)
+
max u (x)
s.t. p1 x w …xed
+
1
x (p ; w)
The Marshallian Demand
+
v (p1 ; w) = u (x (p1 ; w)) = u1
The Indirect Utility Function
Thus, the consumer pays tax
T = tx p1 ; w
(< w) :
The only purpose of the Government is to get T amount of income from the consumer. Then,
the following question becomes very interesting:
Is there another alternative to the good tax such that 1: the Government still
gets the same T amount of income from the consumer; and 2: the consumer
becomes better o¤?
15
For example, the Government uses the lump-sum tax rather than the good tax.
With the original price p0 ; to keep the utility level u1 ; how much income does the agent
need exactly?
e p0 ; u1 :
How much income does the agent have in his pocket?
w!
With the good tax, to keep the utility level u1 ; the agent needs w exactly. However, with
the lump-sum tax, to keep the utility level u1 ; the agent just needs e (p0 ; u1 ) : Thus, as long
as we can prove
w e p0 ; u1 > T;
then we can claim that the lump sum tax is much better than the good tax and
w
T
e p0 ; u1
is called the dead-weight.
w e p0 ; u1
e p0 ; u1
w+T
0
1
1
1
e p ;u
e p ;u + T
EV p0 ; p1 ; w + T
>
<
<
<
T ()
0 ()
0 ()
0
EV p0 ; p1 ; w + T = e p0 ; u1
e p1 ; u1 + T
R p0
P R p0
= p1 de p; u1 + T = 2l=1 p1l hl p; u1 dpl + T
l
R p02
R p01
1
= p0 +t h1 p; u dp1 + p0 h2 p; u1 dp2 + T
1
2
R p01 +t
1
= tx1 p ; w
h1 p; u1 dp1
p01
R p01 +t
= th1 p1 ; u1
h1 p; u1 dp1
p01
R p0 +t
= p01 h1 p1 ; u1
h1 p; u1 dp1 < 0
1
from that
p1 < p11 = p01 + t =) h1 p; u1 > h1 p1 ; u1
due to the substitution e¤ect or from the following picture
16
3
How to Calculate EV and CV?
According to the de…nitions, we have
EV p0 ; p1 ; w
CV p0 ; p1 ; w
= e p0 ; u1
w
1
0
= w e p ;u
and
e p0 ; u1
e p1 ; u0
= p0 h p0 ; u1
= p1 h p1 ; u0 :
Thus, to calculate e (p0 ; u1 ) and e (p1 ; u0 ) ; we have to calculate the two Hicksian demands
h p0 ; u1 and h p1 ; u0 :
To calculate them, we have to calculate u0 and u1 …rst.
(p0 ; w)
+
0
v (p ; w) = u (x (p0 ; w)) = u0
The Indirect Utility Function
To the same consumer
!
17
(p1 ; w)
+
1
v (p ; w) = u (x (p1 ; w)) = u1
The Indirect Utility Function
To calculate u0 and u1 ; we need to calculate the two Marshallian demands
x p0 ; w and x p1 ; w :
How to calculate the two Marshallian demands? By the de…nition,
max u (x)
s.t. p0 x w …xed
+
x (p0 ; w)
The Marshallian Demand
max u (x)
s.t. p1 x w …xed
+
x (p1 ; w)
The Marshallian Demand
Finally, the two Hicksian demands
h p0 ; u1 and h p1 ; u0
are the solutions to
min p0 x
s.t. u (x)
u1
min p1 x
s.t. u (x)
…xed
+
h (p0 ; u1 )
The Hicksian Demand
u0
…xed
+
h (p1 ; u0 )
The Hicksian Demand
respectively.
p
Example 3 John is working in Ottawa and his utility function is u(x1 ; x2 ) = x1 x2 . John
has income $2500 per month and the prices of goods 1 and 2 are both $10 in Ottawa. John’s
boss is thinking of sending him to Toronto where the price of good 1 is $10 and the price of
good 2 is $40. The boss o¤ers no raise in pay. John, who studied EV (equivalent variation)
and CV (compensation variation) from course ECON 5020–Advanced Microeconomic Theory
at Carleton a few years ago and understands them perfectly, complains bitterly. He says that
although he does not mind moving for its own sake and Toronto is just as pleasant as Ottawa
for him, having to move is as bad as a cut in pay $A for still working and living in Ottawa.
He also says he wouldn’t mind moving if he got a raise of $B when he moved.
1. Find the value of A? Is it CV or EV?
2. Find the value of B? Is it CV or EV?
Solution: Based on the de…nition, A is EV since A is the amount of money as a cut in
pay $A for still working and living in Ottawa. That is, he still faces the original prices in
Ottawa.
B is CV since B is the more amount of money he needs to keep his original standard
living level in Ottawa. That is, he will face the new prices in Toronto.
18
Based on the discussion before, we have to calculate u0 (Ottawa’s utility level) and u1
(Toronto’s utility level) and h (p0 ; u1 ) and h (p1 ; u0 ) :
p0 = (10; 10); p1 = (10; 40) and w = 2500: For any p;
max u(x1 ; x2 )
s.t p x 2500
w
2p1
w
2p2
x(p; w) =
Thus
!
2500
2p01
2500
2p02
x(p0 ; w) =
x(p1 ; w) =
2500
2p11
2500
2p12
!
2500
20
2500
20
=
u0 = u(x1 ; x2 ) =
and
2500
2p1
2500
2p2
=
p
125
=
u1 = u(x1 ; x2 ) =
p
2500
20
2500
80
125
:
=
125
125
125 = 125
=
125
31:25
31:25 = 62:5:
To calculate e(p; u); we derive Hicksian demands h(p; u) :
min p h
s.t u(h)
From u(h) =
p
h1 h2 = u; we have h1 =
p1 h1 + p2 h2 =
d
p1 u2
h2
+ p2 h2
dh2
=
e(p; u) =
e(p0 ; u1 ) =
e(p1 ; u0 ) =
u2
:
h2
u
Substituting h1 =
u2
h2
in p h;
p1 u2
+ p2 h2
h2
r
r
p1 u 2
p1
u2
p2
p2
= 0 =) h2 =
u and h1 =
=
u
2
h2
p2
h2
p1
r
r
p2
p1
p
p1 h1 + p2 h2 = p1
+ p2
u = 2 p1 p2 u:
p1
p2
q
2 p01 p02 u1 = 20 62:5 = 1250
q
2 p11 p12 u0 = 40 125 = 5000
1. According to the de…nition, A is
A =
=
EV and
EV (p0 ; p1 ; w) = e(p0 ; u1 ) + w
1250 + 2500 = 1250:
19
2. According to the de…nition, B is
CV and
B =
CV (p0 ; p1 ; w) = e(p1 ; u0 )
= 5000 2500 = 2500:
20
w

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