2. 2
Objectives
• How will changes in prices and income
influence influence consumer’s optimal
choices?
– We will look at partial derivatives
3. 3
Demand Functions (review)
• We have already seen how to obtain consumer’s optimal
choice
• Consumer’s optimal choice was computed Max
consumer’s utility subject to the budget constraint
• After solving this problem, we obtained that optimal
choices depend on prices of all goods and income.
• We usually call the formula for the optimal choice: the
demand function
• For example, in the case of the Complements utility
function, we obtained that the demand function (optimal
choice) is:
yx pp
x
25.0
*
+
=
I
yx pp
y
+
=
4
*
I
4. 4
Demand Functions
• If we work with a generic utility function (we do
not know its mathematical formula), then we
express the demand function as:
x* = x(px,py,I)
y* = y(px,py,I)
•We will keep assuming that prices and income is
exogenous, that is:
–the individual has no control over these
parameters
5. 5
Simple property of demand functions
• If we were to double all prices and
income, the optimal quantities demanded
will not change
– Notice that the budget constraint does not
change (the slope does not change, the
crossing with the axis do not change either)
xi* = di(px,py,I) = di(2px,2py,2I)
6. 6
Changes in Income
• Since px/py does not change, the MRS
will stay constant
• An increase in income will cause the
budget constraint out in a parallel
fashion (MRS stays constant)
7. 7
What is a Normal Good?
• A good xi for which ∂xi/∂I ≥ 0 over some
range of income is a normal good in
that range
8. 8
Normal goods
• If both x and y increase as income
rises, x and y are normal goods
Quantity of x
Quantity of y
C
U3
B
U2
A
U1
As income rises, the individual chooses
to consume more x and y
9. 9
What is an inferior Good?
• A good xi for which ∂xi/∂I < 0 over some
range of income is an inferior good in
that range
10. 10
Inferior good
• If x decreases as income rises, x is an
inferior good
Quantity of x
Quantity of y
C
U3
As income rises, the individual chooses
to consume less x and more y
B
U2
A
U1
11. 11
Changes in a Good’s Price
• A change in the price of a good alters the
slope of the budget constraint (px/py)
– Consequently, it changes the MRS at the
consumer’s utility-maximizing choices
• When a price changes, we can decompose
consumer’s reaction in two effects:
– substitution effect
– income effect
12. 12
Substitution and Income effects
• Even if the individual remained on the same
indifference curve when the price changes,
his optimal choice will change because the
MRS must equal the new price ratio
– the substitution effect
• The price change alters the individual’s real
income and therefore he must move to a
new indifference curve
– the income effect
13. 13
Sign of substitution effect (SE)
SE is always negative, that is, if price increases, the
substitution effect makes quantity to decrease and
conversely. See why:
1) Assume px decreases, so: px
1
< px
0
2) MRS(x0,y0)= px
0
/ py
0
& MRS(x1,y1)= px
1
/ py
0
1 and 2 implies that:
MRS(x1,y1)<MRS(x0,y0)
As the MRS is decreasing in x, this means that x
has increased, that is: x1>x0
14. 14
Changes in the optimal choice when a price
decreases
Quantity of x
Quantity of y
U1
A
Suppose the consumer is maximizing
utility at point A.
U2
B
If the price of good x falls, the consumer
will maximize utility at point B.
Total increase in x
15. 15
Substitution effect when a price decreases
U1
Quantity of x
Quantity of y
A
To isolate the substitution effect, we hold
utility constant but allow the
relative price of good x to change.
Purple is parallel to the new one
Substitution effect
C
The substitution effect is the movement
from point A to point C
The individual substitutes
good x for good y
because it is now
relatively cheaper
16. 16
Income effect when the price decreases
U1
U2
Quantity of x
Quantity of y
A
The income effect occurs because the
individual’s “real” income changes
(hence utility changes) when
the price of good x changes
C
Income effect
B
The income effect is the movement
from point C to point B
If x is a normal good,
the individual will buy
more because “real”
income increased
How would the graph change if the good was inferior?
17. 17
Subs and income effects when a price
increases
U2
U1
Quantity of x
Quantity of y
B
A
An increase in the price of good x means that
the budget constraint gets steeper
C
The substitution effect is the
movement from point A to point C
Substitution effect
Income effect
The income effect is the
movement from point C
to point B
How would the graph change if the good was inferior?
18. 18
Price Changes for
Normal Goods
• If a good is normal, substitution and
income effects reinforce one another
– when price falls, both effects lead to a rise in
quantity demanded
– when price rises, both effects lead to a drop
in quantity demanded
19. 19
Price Changes for
Inferior Goods
• If a good is inferior, substitution and
income effects move in opposite directions
• The combined effect is indeterminate
– when price rises, the substitution effect leads
to a drop in quantity demanded, but the
income effect is opposite
– when price falls, the substitution effect leads
to a rise in quantity demanded, but the
income effect is opposite
20. 20
Giffen’s Paradox
• If the income effect of a price change is
strong enough, there could be a
positive relationship between price and
quantity demanded
– an increase in price leads to a drop in real
income
– since the good is inferior, a drop in income
causes quantity demanded to rise
21. 21
A Summary
• Utility maximization implies that (for normal goods)
a fall in price leads to an increase in quantity
demanded
– the substitution effect causes more to be purchased as
the individual moves along an indifference curve
– the income effect causes more to be purchased
because the resulting rise in purchasing power allows
the individual to move to a higher indifference curve
• Obvious relation hold for a rise in price…
22. 22
A Summary
• Utility maximization implies that (for inferior
goods) no definite prediction can be made
for changes in price
– the substitution effect and income effect move
in opposite directions
– if the income effect outweighs the substitution
effect, we have a case of Giffen’s paradox
23. 23
Compensated Demand Functions
• This is a new concept
• It is the solution to the following problem:
– MIN PXX+ PYY
– SUBJECT TO U(X,Y)=U0
• Basically, the compensated demand functions are the
solution to the Expenditure Minimization problem that we
saw in the previous chapter
• After solving this problem, we obtained that optimal
choices depend on prices of all goods and utility. We
usually call the formula: the compensated demand
function
• x* = xc
(px,py,U),
• y* = yc
(px,py,U)
24. 24
Compensated Demand Functions
• xc
(px,py,U0
), and yc
(px,py,U0
) tell us what quantities
of x and y minimize the expenditure required to
achieve utility level U0
at current prices px,py
• Notice that the following relation must hold:
• pxxc
(px,py,U0
)+ pyyc
(px,py,U0
)=E(px,py,U0
)
– So this is another way of computing the expenditure
function !!!!
25. 25
Compensated Demand Functions
• There are two mathematical tricks to obtain the
compensated demand function without the need to solve
the problem:
– MIN PXX+ PYY
– SUBJECT TO U(X,Y)=U0
• One trick A) (called Shepherd's Lemma) is using the
derivative of the expenditure function
• Another trick (B) is to use the marshaling demand and
the expenditure function
26. 26
Compensated Demand Functions
• Sheppard’s Lemma to obtain the compensated
demand function
y
yx
x
yx
dp
uppE
y
dp
uppE
x
),,(
),,(
∂
=
∂
=
Intuition: a £1 increase in px raises necessary
expenditures by x pounds, because £1 must be paid
for each unit of x purchased.
Proof: footnote 5 in page 137
28. 28
Trick (B) to obtain compensated demand functions
• Suppose that utility is given by
utility = U(x,y) = x0.5
y0.5
• The Marshallian demand functions are
x = I/2px y = I/2py
• The expenditure function is
IU2 5.05.0
== yx ppE
29. 29
• Substitute the expenditure function into
the Marshallian demand functions, and
find the compensated ones:
5.0
5.0
x
y
p
Up
x = 5.0
5.0
y
x
p
Up
y =
Another trick to obtain compensated demand
functions
30. 30
Compensated Demand
Functions
• Demand now depends on utility (V) rather
than income
• Increases in px changes the amount of x
demanded, keeping utility V constant. Hence
the compensated demand function only
includes the substitution effect but not the
income effect
5.0
5.0
x
y
p
Vp
x = 5.0
5.0
y
x
p
Vp
y =
31. 31
Roy’s identity
• It is the relation between marshaling demand
function and indirect utility function
( , , ) ; ( , , ) yx
x y x y
VV
dpdp
x p p I y p p I
V V
dI dI
∂∂
= − = −
∂ ∂
Proof of the Roy’s identity…
32. 32
Proof of Roy’s identity
x
( , , )
( , , ( , , ))
Taking derivatives wrt p :
'(.) '(.) ' 0
Using previous trick:
'
Substituting:
'(.) '(.) 0
and solving for x, we find the Roy's identity
x x
x
x
x y
x y x y
p I p
p
x
p I
V p p I u
V p p E p p u u
V V E
E
E x
dp
V V x
=
=
+ =
∂
= =
+ =
33. 33
Demand curves…
• We will start to talk about demand
curves. Notice that they are not the
same that demand functions !!!!
34. 34
The Marshallian Demand Curve
• An individual’s demand for x depends
on preferences, all prices, and income:
x* = x(px,py,I)
• It may be convenient to graph the
individual’s demand for x assuming that
income and the price of y (py) are held
constant
35. 35
x
…quantity of x
demanded rises.
The Marshallian Demand Curve
Quantity of y
Quantity of x Quantity of x
px
x’’
px’’
U2
x2
I = px’’ + py
x’
px’
U1
x1
I = px’ + py
x’’’
px’’’
x3
U3
I = px’’’ + py
As the price
of x falls...
36. 36
The Marshallian Demand Curve
• The Marshallian demand curve shows the
relationship between the price of a good
and the quantity of that good purchased
by an individual assuming that all other
determinants of demand are held constant
• Notice that demand curve and demand
function is not the same thing!!!
37. 37
Shifts in the Demand Curve
• Three factors are held constant when a
demand curve is derived
– income
– prices of other goods (py)
– the individual’s preferences
• If any of these factors change, the
demand curve will shift to a new position
38. 38
Shifts in the Demand Curve
• A movement along a given demand
curve is caused by a change in the price
of the good
– a change in quantity demanded
• A shift in the demand curve is caused by
changes in income, prices of other
goods, or preferences
– a change in demand
39. 39
Compensated Demand Curves
• An alternative approach holds utility
constant while examining reactions to
changes in px
– the effects of the price change are
“compensated” with income so as to constrain
the individual to remain on the same
indifference curve
– reactions to price changes include only
substitution effects (utility is kept constant)
40. 40
Marshallian Demand Curves
• The actual level of utility varies along
the demand curve
• As the price of x falls, the individual
moves to higher indifference curves
– it is assumed that nominal income is held
constant as the demand curve is derived
– this means that “real” income rises as the
price of x falls
41. 41
Compensated Demand Curves
• A compensated (Hicksian) demand curve
shows the relationship between the price
of a good and the quantity purchased
assuming that other prices and utility are
held constant
• The compensated demand curve is a two-
dimensional representation of the
compensated demand function
x* = xc
(px,py,U)
42. 42
xc
…quantity demanded
rises.
Compensated Demand Curves
Quantity of y
Quantity of x Quantity of x
px
U2
x’’
px’’
x’’
y
x
p
p
slope
''
−=
x’
px’
y
x
p
p
slope
'
−=
x’ x’’’
px’’’
y
x
p
p
slope
'''
−=
x’’’
Holding utility constant, as price falls...
43. 43
Compensated & Uncompensated
Demand for normal goods
Quantity of x
px
x
xc
x’’
px’’
At px’’, the curves intersect because
the individual’s income is just sufficient
to attain utility level U2
44. 44
Compensated & Uncompensated Demand for
normal goods
Quantity of x
px
x
xc
px’’
x*x’
px’
At prices above p’’x, income
compensation is positive because the
individual needs some help to remain
on U2
As we are looking at normal goods, income and substitution effects go
in the same direction, so they are reinforced. X includes both while Xc
only the substitution effect. That is what drives the relative position of
both curves
45. 45
Compensated & Uncompensated
Demand for normal goods
Quantity of x
px
x
xc
px’’
x*** x’’’
px’’’
At prices below px2, income
compensation is negative to prevent an
increase in utility from a lower price
As we are looking at normal goods, income and substitution effects go
in the same direction, so they are reinforced. X includes both while Xc
only the substitution effect. That is what drives the relative position of
both curves
46. 46
Compensated &
Uncompensated Demand
• For a normal good, the compensated
demand curve is less responsive to price
changes than is the uncompensated
demand curve
– the uncompensated demand curve reflects
both income and substitution effects
– the compensated demand curve reflects only
substitution effects
47. 47
Relations to keep in mind
• Sheppard’s Lema & Roy’s identity
• V(px,py,E(px,py,Uo
)) = U0
• E(px,py,V(px,py,I0
)) = I0
• xc
(px,py,U0
)=x(px,py,I0
)
48. 48
A Mathematical Examination
of a Change in Price
• Our goal is to examine how purchases of
good x change when px changes
∂x/∂px
• Differentiation of the first-order conditions
from utility maximization can be
performed to solve for this derivative
49. 49
A Mathematical Examination
of a Change in Price
• However, for our purpose, we will use an
indirect approach
• Remember the expenditure function
minimum expenditure = E(px,py,U)
• Then, by definition
xc
(px,py,U) = x[px,py,E(px,py,U)]
– quantity demanded is equal for both demand
functions when income is exactly what is needed to
attain the required utility level
50. 50
A Mathematical Examination
of a Change in Price
• We can differentiate the compensated
demand function and get
xc
(px,py,U) = x[px,py,E(px,py,U)]
xxx
c
p
E
E
x
p
x
p
x
∂
∂
⋅
∂
∂
+
∂
∂
=
∂
∂
xx
c
x p
E
E
x
p
x
p
x
∂
∂
⋅
∂
∂
−
∂
∂
=
∂
∂
51. 51
A Mathematical Examination
of a Change in Price
• The first term is the slope of the
compensated demand curve
– the mathematical representation of the
substitution effect
xx
c
x p
E
E
x
p
x
p
x
∂
∂
⋅
∂
∂
−
∂
∂
=
∂
∂
52. 52
A Mathematical Examination
of a Change in Price
• The second term measures the way in
which changes in px affect the demand
for x through changes in purchasing
power
– the mathematical representation of the
income effect
xx
c
x p
E
E
x
p
x
p
x
∂
∂
⋅
∂
∂
−
∂
∂
=
∂
∂
53. 53
The Slutsky Equation
• The substitution effect can be written as
constant
effectonsubstituti
=
∂
∂
=
∂
∂
=
Uxx
c
p
x
p
x
• The income effect can be written as
x
x
p
Ex
x
p
E
p
Ex
p
E
E
x
x
x
xx
II
I
∂
∂
−=
∂
∂
⋅
∂
∂
−=
=
∂
∂
∂
∂
⋅
∂
∂
−=
∂
∂
⋅
∂
∂
−=
effectincome
:AtrickUsing
effectincome
54. 54
The Slutsky Equation
• A price change can be represented by
I∂
∂
−
∂
∂
=
∂
∂
+=
∂
∂
=
x
x
p
x
p
x
p
x
Uxx
x
constant
effectincomeeffectonsubstituti
55. 55
The Slutsky Equation
• The first term is the substitution effect
– always negative as long as MRS is
diminishing
– the slope of the compensated demand curve
must be negative
I∂
∂
−
∂
∂
=
∂
∂
=
x
x
p
x
p
x
Uxx constant
56. 56
The Slutsky Equation
• The second term is the income effect
– if x is a normal good, then ∂x/∂I > 0
• the entire income effect is negative
– if x is an inferior good, then ∂x/∂I < 0
• the entire income effect is positive
I∂
∂
−
∂
∂
=
∂
∂
=
x
x
p
x
p
x
Uxx constant
57. 57
A Slutsky Decomposition
• We can demonstrate the decomposition
of a price effect using the Cobb-
Douglas example studied earlier
• The Marshallian demand function for
good x was
x
yx
p
ppx
I
I
5.0
),,( =
58. 58
A Slutsky Decomposition
• The Hicksian (compensated) demand
function for good x was
5.0
5.0
),,(
x
y
yx
c
p
Vp
Vppx =
• The overall effect of a price change on
the demand for x is
2
5.0
xx pp
x I−
=
∂
∂
59. 59
A Slutsky Decomposition
• This total effect is the sum of the two
effects that Slutsky identified
• The substitution effect is found by
differentiating the compensated demand
function
5.1
5.0
5.0
effectonsubstituti
x
y
x
c
p
Vp
p
x −
=
∂
∂
=
60. 60
A Slutsky Decomposition
• We can substitute in for the indirect utility
function (V)
25.1
5.05.05.0
25.0)5.0(5.0
effectonsubstituti
xx
yyx
pp
ppp II −
=
−
=
−−
61. 61
A Slutsky Decomposition
• Calculation of the income effect is easier
2
25.05.05.0
effectincome
xxx ppp
x
x
II
I
−=⋅
−=
∂
∂
−=
• By adding up substitution and income
effect, we will obtain the overall effect
