Download Demand Curve for “X”

Chapter 5
Income and Substitution
Effects
Demand Functions
• The optimal levels of x1,x2,…,xn can be
expressed as functions of all prices and
income
• These can be expressed as n demand
functions of the form:
x1* = d1(p1,p2,…,pn,I)
x2* = d2(p1,p2,…,pn,I)
•
•
•
xn* = dn(p1,p2,…,pn,I)
Demand Functions
• If there are only two goods (x and y), we can
simplify the notation
x* = x(px,py,I)
y* = y(px,py,I)
• Prices and income are exogenous
– the individual has no control over these parameters
Homogeneity
• If all prices and income were doubled, the
optimal quantities demanded will not change
– the budget constraint is unchanged
xi* = xi(p1,p2,…,pn,I) = xi(tp1,tp2,…,tpn,tI)
• Individual demand functions are homogeneous
of degree zero in all prices and income
Homogeneity
• With a Cobb-Douglas utility function
utility = U(x,y) = x0.3y0.7
the demand functions are
0 .3 I
x* 
px
0 .7 I
y* 
py
• A doubling of both prices and income
would leave x* and y* unaffected
Homogeneity
• With a CES utility function
utility = U(x,y) = x0.5 + y0.5
the demand functions are
1
I
x* 

1  p x / py p x
1
I
y* 

1  py / p x py
• A doubling of both prices and income
would leave x* and y* unaffected
Changes in Income
• An increase in income will cause
the budget constraint out in a
parallel fashion
• Since px/py does not change, the
MRS will stay constant as the
worker moves to higher levels of
satisfaction
Income Consumption Curve
8
Effects of a Rise in Income
• Engel curve - the relationship between
the quantity demanded of a single good
and income, holding prices constant.
• Income-consumption curve shows how
consumption of both goods changes
when income changes, while prices are
held constant.
Engel Curves
I ($)
Engel Curve
92
“X is a normal good”
68
40
0
10
18
24
X (units)
10
Y
Effect of a Budget Increase on an
Individual’s Demand Curve
L1
2.8
Budget Line, L
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $419.
Income goes up!
Price of X
PX
X
PY
12
26.7
I1
X
E1
D1
0
26.7
X
I , Budget
Y=
I PY
0
e1
Y1 = $419
0
E1*
26.7
X
y
L2
Y
Effect of a Budget Increase on
an Individual’s Demand Curve
L1
4.8
2.8
Budget Line, L
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $419.
Income goes up!
$628
Price of X
PX
X
PY
12
I1
26.7 38.2
E1
I2
X
E2
D2
D1
0
26.7 38.2
X
I , Budget
Y=
I PY
0
e2
e1
Y2 = $628
Y1 = $419
0
E2*
E1*
26.7 38.2
X
L3
L2
Y
Effect of a Budget Increase on
an Individual’s Demand Curve
L1
7.1
4.8
2.8
Budget Line, L
PX
X
PY
Initial Values
PX = price of X = $12
PY = price of Y = $35
I = Income = $837.
Price of X
0
12
e2
e1
I1
26.7 38.2 49.1
E1
E2
I3
X
D3
D2
D1
0
26.7 38.2 49.1
E3*
Y2 = $628
Y1 = $419
0
X
Engel curve for X
Y3 = $837
Income goes up again!
I2
E3
I , Budget
Y=
I PY
Income-consumption curve
e3
E2*
E1*
26.7 38.2 49.1
X
Normal and Inferior Goods
• A good xi for which xi/I  0 over some range
of income is a normal good in that range
• A good xi for which xi/I < 0 over some range
of income is an inferior good in that range
Increase in Income
• If x decreases as income rises, x is an
inferior good
As income rises, the individual
chooses to consume less x and
Quantity of y
more y
C
B
U3
U2
A
U1
Quantity of x
Normal and Inferior Goods
Example: Backward Bending
ICC and Engel Curve – a
good can be normal over
some ranges and inferior
over others
16
Price Consumption Curves
Y (units)
PY = $4
I = $40
10
Is the set of optimal baskets for every
possible price of good x, holding all
other prices and income constant.
This is the individual’s demand curve for
good x.
Price Consumption Curve
•
•
•
PX = 1
PX = 2
PX = 4
0
XA=2
XB=10
XC=16
20
X (units)
17
Individual Demand Curve
PX
Individual Demand Curve
For Commodity X
PX = 4
•
PX = 2
PX = 1
XA
•
XB
•
XC
Quantity increasing
X
18
The Individual Demand Curve
Price-consumption curve Curve tracing the utilitymaximizing combinations of two goods as the price of one
changes.
●
Individual demand curve
Curve relating the quantity of a good that a single
consumer will buy to its price.
●
The individual demand curve has two important
properties:
1. The level of utility that can be attained changes as
we move along the curve.
2. At every point on the demand curve, the consumer
is maximizing utility by satisfying the condition that
the marginal rate of substitution (MRS) of X for Y
equals the ratio of the prices of X and Y.
Demand Curve for “X”
Algebraically, we can solve for the
individual’s demand using the following
equations:
1. pxx + pyy = I
2. MUx/px = MUy/py – at a tangency.
(If this never holds, a corner point may
be substituted where x = 0 or y = 0)
20
Changes in a Good’s Price
• A change in the price of a good alters the
slope of the budget constraint
–it also changes the MRS at the
consumer’s utility-maximizing choices
• When the price changes, two effects come
into play
–substitution effect
–income effect
Changes in a Good’s Price
• Even if the individual remains on the same
indifference curve, his optimal choice will
change because the MRS must equal the new
price ratio
– the substitution effect
• The individual’s “real” income has changed and
he must move to a new indifference curve
– the income effect
Changes in a Good’s Price:
Price of X falls
Suppose the consumer is
maximizing utility at point A.
Quantity of y
B
A
If the price of good x falls,
the consumer will
maximize utility at point B.
U2
U1
Quantity of x
Total increase in x
Changes in a Good’s Price:
Price of X falls
Quantity of y
To isolate the substitution effect, we hold
“real” income constant but allow the
relative price of good x to change
The substitution effect is the movement
from point A to point C
A
C
U1
Substitution effect
The individual substitutes
x for y because it is now
relatively cheaper
Quantity of x
Changes in a Good’s Price :
Price of X falls
The income effect occurs because “real”
income changes when the price of good
x changes
Quantity of y
B
A
The income effect is the movement
from point C to point B
C
U2
U1
Income effect
If x is a normal good,
the individual will buy
more because “real”
income increased
Quantity of x
Total Effect or Price Effect(AB) = Substitution Effect (AC) +
Income Effect (CB)
Changes in a Good’s Price:
Price of X rises
Quantity of y
An increase in the price of good x means
that the budget constraint gets steeper
C
A
The substitution effect is the
movement from point A to point C
B
U1
U2
The income effect is the
movement from point C
to point B
Quantity of x
Substitution effect
Income effect
Total Effect: Price Effect (AB) = Substitution Effect (AC) +
Income Effect (CB)
Price Changes – Normal Goods
• If a good is normal, substitution and income
effects reinforce one another
– when p :
• substitution effect  quantity demanded 
• income effect  quantity demanded 
– when p :
• substitution effect  quantity demanded 
• income effect  quantity demanded 
Price Changes – Inferior Goods
• If a good is inferior, substitution and income
effects move in opposite directions
– when p :
• substitution effect  quantity demanded 
• income effect  quantity demanded 
– when p :
• substitution effect  quantity demanded 
• income effect  quantity demanded 
INCOME AND SUBSTITUTION EFFECTS:
INFERIOR GOOD
Consumer is initially at A on RS.
With a ↓ P of food, the consumer
moves to B.
i) a substitution effect, F1E
(associated with a move from
A to D),
ii) an income effect, EF2
(associated with a move from
D to B).
In this case, food is an inferior
good because the income effect is
negative.
However, because the substitution
effect exceeds the income effect,
the decrease in the price of food
leads to an increase in the quantity
of food demanded.
A Special Case: The Giffen Good
● Giffen
good Good whose demand curve slopes upward
because the (negative) income effect is larger than the
substitution effect.
UPWARD-SLOPING DEMAND
CURVE: THE GIFFEN GOOD
•Food: an inferior good
• Income effect dominates over
the substitution effect,
• Consumer is initially at A
• After the ↓P of food falls,
moves to B and consumes less
food.
• The income effect F2F1 > the
substitution effect EF2,
• The ↓P of food leads to a lower
quantity of food demanded.
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 decrease in price leads to a increase in real
income
– since the good is inferior, a increase in income causes
quantity demanded to fall
The Individual’s 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 i.e (py) are held
constant
The Individual’s Demand Curve
Quantity of y
As the price
of x falls...
px
…quantity of
x
demanded rises.
px’
px’’
px’’’
U1 U2
x1
I = px’ + py
x2
x3
I = px’’ + py
U3
Quantity of x
I = px’’’ + py
x
x’
x’’
x’’’
Quantity of x
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
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
Compensated Demand Curve
• The demand curves shown thus far have
all been uncompensated, or
Marshallian, demand curves.
• Consumer utility is allowed to vary
with the price of the good.
• Alternatively, a compensated, or
Hicksian, demand curve shows how
quantity demanded changes when price
increases, holding utility constant.
Compensated 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
Compensated Demand Curves
• An alternative approach holds real income
(or utility) constant while examining
reactions to changes in px
–the effects of the price change are
“compensated” so as to force the
individual to remain on the same
indifference curve
–reactions to price changes include only
substitution effects
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 twodimensional representation of the
compensated demand function
x* = xc(px,py,U)
Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of y
px
p '
slope   x
py
…quantity
demanded
rises.
slope  
px ' '
py
px’
px’’
slope  
px ' ' '
py
px’’’
Xc
U2
x’
x’’
x’’’
Quantity of x
x’
x’’
x’’’
Quantity of x
Compensated & Uncompensated Demand
px
At px’’, the curves intersect because
the individual’s income is just sufficient to
attain utility level U2
px’’
x
xc
x’’
Quantity of x
Compensated & Uncompensated Demand
px
At prices above px’, income
compensation is positive
because the individual needs
more income to remain on U2
px’
px’’
x
xc
x’
x*
Quantity of x
Compensated & Uncompensated Demand
px
At prices below px’”, income
compensation is negative to
prevent an increase in utility
from a lower price
px’’
px’’’
X
xc
x***
x’’’
Quantity of x
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
Compensated Demand Functions
• Suppose that utility is given by
utility = U(x,y) = x0.5y0.5
• The Marshallian demand functions are
x = I/2px
y = I/2py
• The indirect utility function is
utility  V ( I, px , py ) 
I
2 px0.5 py0.5
Compensated Demand Functions
• To obtain the compensated demand
functions, we can solve the indirect utility
function for I and then substitute into the
Marshallian demand functions
x
0 .5
y
0 .5
x
Vp
p
0 .5
x
0 .5
y
Vp
y
p
Compensated Demand Functions
x
0 .5
y
0 .5
x
Vp
p
0 .5
x
0 .5
y
Vp
y
p
• Demand now depends on utility (V)
rather than income
• Increases in px reduce the amount of x
demanded
–only a substitution effect
The Response to a Change in Price
• We will use an indirect approach using the
expenditure function
minimum expenditure = E(px,py,U)
• Then, by definition
xc (px,py,U) = x [px,py,E(px,py,U)]
The Response to a Change in Price
xc (px,py,U) = x[px,py,E(px,py,U)]
• We can differentiate the compensated
demand function and get
x c
x
x E



px px
E px
x
x c
x E



px px
E px
The Response to a Change in Price
x x c
x E



px px
E px
• The first term is the slope of the compensated
demand curve
– the mathematical representation of the substitution
effect
The Response to a Change in Price
x x
x E



px px
E px
c
• 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
The Slutsky Equation
• The substitution effect can be written as
x c
x
substitution effect 

px px
U constant
• The income effect can be written as
x E
x E
income effect  

 

E px
I px
The Slutsky Equation
• The utility-maximization hypothesis shows that
the substitution and income effects arising
from a price change can be represented by
x
 substitution effect  income effect
px
x
x

px px
U constant
x
x
I
The Slutsky Equation
x
x

px px
U constant
x
x
I
• The first term is the substitution effect
– always negative
x
x

px px
U constant
x
x
I
• The second term is the income effect
– if x is a normal good, income effect is negative
– if x is an inferior good, income effect is positive
Marshallian Demand Elasticities
• Most of the commonly used demand elasticities
are derived from the Marshallian demand
function x(px,py,I)
• Price elasticity of demand (ex,px)
ex ,px
x / x
x px



px / px px x
Marshallian Demand Elasticities
• Income elasticity of demand (ex,I)
e x ,I
x / x x I



I / I I x
• Cross-price elasticity of demand (ex,py)
e x ,py
x / x
x py



py / py py x
Price Elasticity of Demand
• The own price elasticity of demand is always
negative
– the only exception is Giffen’s paradox
• The size of the elasticity is important
– if ex,px < -1, demand is elastic
– if ex,px > -1, demand is inelastic
– if ex,px = -1, demand is unit elastic
Price Elasticity and Total Spending
• Total spending on x is equal to
total spending =pxx
• Using elasticity, we can determine how total
spending changes when the price of x changes
( p x x )
x
 px 
 x  x [ex ,px  1]
px
px
Price Elasticity and Total Spending
( p x x )
x
 px 
 x  x[ex,px  1]
px
px
• If ex,px > -1, demand is inelastic
– price and total spending move in the same
direction
• If ex,px < -1, demand is elastic
– price and total spending move in opposite
directions
Compensated Price Elasticities
• It is also useful to define elasticities based on
the compensated demand function
Compensated Price Elasticities
• If the compensated demand function is
xc = xc(px,py,U)
we can calculate
– compensated own price elasticity of demand
(exc,px)
– compensated cross-price elasticity of demand
(exc,py)
Compensated Price Elasticities
• The compensated own price elasticity of
demand (exc,px) is
exc,px
x c / x c x c px


 c
px / px px x
• The compensated cross-price elasticity of
demand (exc,py) is
x / x
x py


 c
py / py py x
c
e
c
x ,py
c
c
Price Elasticities
• The Slutsky equation shows that the
compensated and uncompensated
price elasticities will be similar if
–the share of income devoted to x is
small
–the income elasticity of x is small
Relationship among demand
elasticities
• 1) Homogeneity
• 2) Engel aggregation
• 3) Cournot aggregation
Homogeneity
• Demand functions are homogeneous of
degree zero in all prices and income
• Any proportional change in all prices and
income will leave the quantity of x
demanded unchanged
Engel Aggregation
• Engel’s law suggests that the income elasticity
of demand for food items is less than one
– this implies that the income elasticity of demand for
all nonfood items must be greater than one.
Income elasticity of demand for various goods
Automobiles
2.98
Books
1.44
Restaurant Meals
1.40
Tobacco
0.64
Public Transportation
−0.36
Cournot Aggregation
• The size of the cross-price effect of a change in
px on the quantity of y consumed is restricted
because of the budget constraint.
• Differentiate the budget constraint with
respect to Px.
Consumer Surplus
• Suppose we want to examine the change in an
individual’s welfare when price changes
Consumer Welfare
• If the price rises, the individual would have to
increase expenditure to remain at the initial level of
utility
expenditure at px0 = E0 = E(px0,py,U0)
expenditure at px1 = E1 = E(px1,py,U0)
• In order to compensate for the price rise, this person
would require a compensating variation (CV) of
CV = E(px1,py,U0) - E(px0,py,U0)
Consumer Welfare
Quantity of y
Suppose the consumer is maximizing
utility at point A.
If the price of good x rises,
the consumer will maximize
utility at point B.
The consumer’s utility
falls from U2 to U1
A
B
U2
U1
Quantity of x
Consumer Welfare
Quantity of y
The consumer could be compensated so
that he can afford to remain on U2
CV is the amount that the individual
would need to be compensated
CV
C
A
B
U2
U1
Quantity of x
Consumer Welfare
• The derivative of the expenditure function with
respect to px is the compensated demand
function
c
x ( p x , p y ,U ) 
E( p x , p y ,U )
p x
• The amount of CV required can be found by
integrating across a sequence of small
increments to price from one price to another.
Consumer Welfare
px
When the price rises from px0 to px1,
the consumer suffers a loss in welfare
welfare loss
px1
p1x
p1x
p x0
p x0
CV   dE   x c ( px , py ,U 0 )dpx
px0
xc(px…U0)
x1
x0
Quantity of x
Consumer Welfare
• A price change generally involves both
income and substitution effects
–should we use the compensated
demand curve for the original target
utility (U0) or the new level of utility
after the price change (U1)?
The Consumer Surplus Concept
• The area below the compensated
demand curve and above the market
price is called consumer surplus
–the extra benefit the person receives
by being able to make market
transactions at the prevailing market
price
Consumer Welfare
px
px1
Is the consumer’s loss in welfare best described
by area px1BApx0 [using xc(...,U1)] or by area
px1CDpx0 [using xc(...,U0)]?
C
B
A
px0
Is U1 or U0 the
appropriate utility target?
D
xc(...,U1)
xc(...,U0)
x1
x0
Quantity of x
Consumer Welfare
px
px1
We can use the Marshallian demand curve as
a compromise
The area px1CApx0 falls between
the sizes of the welfare losses
defined by xc(...,U1) and xc(...,U0)
C
B
A
px0
D
x(px,…)
xc(...,U1)
xc(...,U0)
x1
x0
Quantity of x
Consumer Surplus
• We will define consumer surplus as the area
below the Marshallian demand curve and
above price
– shows what an individual would pay for the
right to make voluntary transactions at this
price
– changes in consumer surplus measure the
welfare effects of price changes
Welfare Loss from a Price Increase
• Suppose that the compensated demand
function for x is given by
x c ( px , py ,V ) 
Vpy0.5
px0.5
• The welfare cost of a price increase from px
= $1 to px = $4 is given by
4
CV   Vp p
0 .5
y
1
 0 .5
x
 2Vp p
0 .5
y
p 4
0 .5 x
x p 1
X
Welfare Loss from a Price Increase
• If we assume that V = 2 and py = 4,
CV = 222(4)0.5 – 222(1)0.5 = 8
• If we assume that the utility level (V) falls to
1 after the price increase (and used this
level to calculate welfare loss),
CV = 122(4)0.5 – 122(1)0.5 = 4
Welfare Loss from a Price Increase
• Suppose that we use the Marshallian
demand function instead
x ( px , py , I )  0.5 Ipx-1
• The welfare loss from a price increase from
px = $1 to px = $4 is given by
4
Loss   0.5 Ip dpx  0.5 I ln px
-1
x
1
px  4
p x 1
Welfare Loss from a Price Increase
• If income (I) is equal to 8,
Loss = 4 ln(4) - 4 ln(1) = 4 ln(4) = 4(1.39) = 5.55
– this computed loss from the Marshallian
demand function is a compromise between
the two amounts computed using the
compensated demand functions
Revealed Preference Hypothesis
• The theory of revealed preference was
proposed by Paul Samuelson in the year 1938.
• Assumptions:
– 1) Rationality
– 2) Consistency
– 3) Transitivity
– 4) Revealed Preference axiom
Revealed Preference Axiom
• Consider two bundles of goods: A and B
• If the individual can afford to purchase
either bundle but chooses A, we say that
A had been revealed preferred to B
• Under any other price-income
arrangement, B can never be revealed
preferred to A
Revealed Preference Hypothesis: Derivation of
Demand Curve
• Initially consumer at
B with budget line RS.
• A ↓ PF shifts budget
line from RS to RT.
• The new market
basket chosen must lie
on line segment BT' of
budget line R′T' (which
intersects RS to the
right of B), and the
quantity of food
consumed must be
greater than at B.
Revealed Preference Theory
• Preferences  predict consumer’s purchasing behavior
• Purchasing behavior  infer consumer’s preferences
Strong Axiom of Revealed Preference
• If commodity bundle 0 is revealed preferred
to bundle 1, and if bundle 1 is revealed
preferred to bundle 2, and if bundle 2 is
revealed preferred to bundle 3,…, and if
bundle K-1 is revealed preferred to bundle K,
then bundle K cannot be revealed preferred
to bundle 0