Transcript Income-Sustitution Effects

INCOME AND SUBSTITUTION
EFFECTS
1
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)
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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
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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
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Increase in Income
• If both x and y increase as income rises,
x and y are normal goods
Quantity of y
As income rises, the individual chooses
to consume more x and y
B
C
A
U3
U1
U2
Quantity of x
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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 more y
Quantity of y
Note that the indifference
curves do not have to be
“oddly” shaped. The
assumption of a diminishing
MRS is obeyed.
C
B
U3
U2
A
U1
Quantity of x
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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
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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
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Changes in a Good’s Price
• 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
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Changes in a Good’s Price
Suppose the consumer is maximizing
utility at point A.
Quantity of y
If the price of good x falls, the consumer
will maximize utility at point B.
B
A
U2
U1
Quantity of x
Total increase in x
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Changes in a Good’s Price
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
The individual substitutes
good x for good y
because it is now
relatively cheaper
Quantity of x
Substitution effect
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Changes in a Good’s Price
Quantity of y
The income effect occurs because the
individual’s “real” income changes when
the price of good x changes
B
A
The income effect is the movement
from point C to point B
C
U2
U1
If x is a normal good,
the individual will buy
more because “real”
income increased
Quantity of x
Income effect
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Changes in a Good’s Price
Quantity of y
An increase in the price of good x means that
the budget constraint gets steeper
The substitution effect is the
movement from point A to point C
C
A
B
U1
The income effect is the
movement from point C
to point B
U2
Quantity of x
Substitution effect
Income effect
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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
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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
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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
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The Individual’s Demand Curve
• An individual’s demand for x depends on
preferences, all prices, and income:
x* = x(px,py,I)
• An individual 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
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The Individual’s Demand Curve
Quantity of y
As the price
of x falls...
px
…quantity of x
demanded rises.
px’
px’’
px’’’
U1
x1
I = px’ + py
x2
x3
I = px’’ + py
U2
U3
Quantity of x
I = px’’’ + py
x
x’
x’’
x’’’
Quantity of x
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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
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Demand
• The Cobb-Douglas utility function is
U(x,y) = xy
(+=1)
• The demand functions for x and y are
I
x
px
I
y
py
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Demand Functions and Curves
0 .3 I
x* 
px
0.7I
y* 
py
• If the individual’s income is $100, these functions
become
• Any change in income will shift these demand curves
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x* 
px
70
y* 
py
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Uncompensated 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
• know as an uncompensated or Marshallian Demand
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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 constrain the individual to remain on the same
indifference curve
– reactions to price changes include only substitution
effects
• Also know as Hicksian Demand
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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)
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Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of y
px
p '
slope   x
py
slope  
…quantity demanded
rises.
px ' '
py
px’
px’’
slope  
px ' ' '
py
px’’’
xc
U2
x’
x’’
x’’’
Quantity of x
x’
x’’
x’’’
Quantity of x
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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
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A Mathematical Examination
of 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
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The Slutsky Equation
• The substitution effect can be written as
x c
x
substituti on effect 

px px
U constant
• The income effect can be written as
x E
x E
income effect  

 

E px
I px
• Note that E/px = x
– a $1 increase in px raises necessary expenditures by x dollars
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The Slutsky Equation
• The utility-maximization hypothesis
shows that the substitution and income
effects arising from a price change can be
represented by
x
 substituti on effect  income effect
px
x
x

px px
U constant
x
x
I
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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 ,p x
x / x
x px



px / px px x
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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)
ex , py
x / x
x py



py / py py x
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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
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Homogeneity
• Demand functions are homogeneous of
degree zero in all prices and income
• Euler’s theorem for homogenous
functions shows that
x
x
x
0  px 
 py 
I
px
py
I
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Homogeneity
• Dividing by x, we get
0  ex,px  ex,py  ex,I
• Any proportional change in all prices
and income will leave the quantity of x
demanded unchanged
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Engel Aggregation
• We can see this by differentiating the
budget constraint with respect to
income (treating prices as constant)
x
y
1  px 
 py 
I
I
x xI
y yI
1  p x    py  
 s x e x , I  s y ey , I
I xI
I yI
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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
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