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![Page 1: Chapter 5 INCOME AND SUBSTITUTION EFFECTS Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved. MICROECONOMIC THEORY BASIC.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d5d5503460f94a3cb00/html5/thumbnails/1.jpg)
Chapter 5
INCOME AND SUBSTITUTION EFFECTS
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
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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:
X1* = d1(P1,P2,…,Pn,I)
X2* = d2(P1,P2,…,Pn,I)•••
Xn* = dn(P1,P2,…,Pn,I)
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Homogeneity• If we were to double all prices and
income, the optimal quantities demanded will not change– Doubling prices and income leaves the
budget constraint unchanged
Xi* = di(P1,P2,…,Pn,I) = di(tP1,tP2,…,tPn,tI)
• Individual demand functions are homogeneous of degree zero in all prices and income
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Homogeneity• With a Cobb-Douglas utility function
utility = U(X,Y) = X0.3Y0.7
the demand functions are
• Note that a doubling of both prices and income would leave X* and Y* unaffected
XPX
I30.*
XPY
I70.*
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Homogeneity• With a CES utility function
utility = U(X,Y) = X0.5 + Y0.5
the demand functions are
• Note that a doubling of both prices and income would leave X* and Y* unaffected
XYX PPPX
I
/*
1
1
YXY PPPY
I
/*
1
1
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Changes in Income
• An increase in income will cause the budget constraint out in a parallel manner
• 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 X
Quantity of Y
C
U3
B
U2
A
U1
As income rises, the individual choosesto consume more X and Y
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Increase in Income
• If X decreases as income rises, X is an inferior good
Quantity of X
Quantity of Y
C
U3
As income rises, the individual choosesto consume less X and more Y
Note that the indifferencecurves do not have to be “oddly” shaped. Theassumption of a diminishing MRS is obeyed.
B
U2
AU1
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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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Engel’s Law• Using Belgian data from 1857, Engel
found an empirical generalization about consumer behavior
• The proportion of total expenditure devoted to food declines as income rises– food is a necessity whose consumption rises
less rapidly than income
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Substitution & 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
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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
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
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Changes in a Good’s Price
U1
U2
Quantity of X
Quantity of Y
A
B
To isolate the substitution effect, we hold“real” income constant but allow the relative price of good X to change
C
Substitution effect
The substitution effect is the movementfrom point A to point C
The individual substitutes good X for good Y because it is now relatively cheaper
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Changes in a Good’s Price
U1
U2
Quantity of X
Quantity of Y
A
B
The income effect occurs because theindividual’s “real” income changes whenthe price of good X changes
C
Income effect
The income effect is the movementfrom point C to point B
If X is a normal good,the individual will buy more because “real”income increased
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Changes in a Good’s Price
U2
U1
Quantity of X
Quantity of Y
B
A
An increase in the price of good X means thatthe budget constraint gets steeper
CThe 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
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Price Changes forNormal Goods
• If a good is normal, substitution and income effects reinforce one another
– When price falls, both effects lead to a rise
in QD
– When price rises, both effects lead to a drop in QD
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Price Changes forInferior 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 QD, but the income effect leads to a rise in QD
– When price falls, the substitution effect leads to a rise in QD, but the income effect leads to a fall in QD
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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 QD
– An increase in price leads to a drop in real income
– Since the good is inferior, a drop in income causes QD to rise
• Thus, a rise in price leads to a rise in QD
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Summary of Income & Substitution Effects
• Utility maximization implies that (for normal goods) a fall in price leads to an increase in QD
– 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
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Summary of Income & Substitution Effects
• Utility maximization implies that (for normal goods) a rise in price leads to a decline in QD
– The substitution effect causes less to be purchased as the individual moves along an indifference curve
– The income effect causes less to be purchased because the resulting drop in purchasing power moves the individual to a lower indifference curve
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Summary of Income & Substitution Effects
• 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
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The Individual’s Demand Curve
• An individual’s demand for X1 depends on preferences, all prices, and income:
X1* = d1(P1,P2,…,Pn,I)
• It may be convenient to graph the individual’s demand for X1 assuming
that income and the prices of other goods are held constant
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The Individual’s Demand Curve
Quantity of Y
Quantity of X Quantity of X
PX
X2
PX2
U2
X2
I = PX2 + PY
X1
PX1
U1
X1
I = PX1 + PY
X3
PX3
X3
U3
I = PX3 + PY
As the price of X falls...
dX
…quantity of Xdemanded rises.
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The Individual’s Demand Curve
• 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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Shifts in the Demand Curve
• Three factors are held constant when a demand curve is derived– income– prices of other goods– the individual’s preferences
• If any of these factors change, the demand curve will shift to a new position
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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– called a change in quantity demanded
• A shift in the demand curve is caused by a change in income, prices of other goods, or preferences– called a change in demand
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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
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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
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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 two-dimensional representation of the compensated demand function
X* = hX(PX,PY,U)
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hX
…quantity demandedrises.
Compensated Demand Curves
Quantity of Y
Quantity of X Quantity of X
PX
U2
X2
PX2
X2
Y
X
P
Pslope 2
X1
PX1
Y
X
P
Pslope 1
X1 X3
PX3Y
X
P
Pslope 3
X3
Holding utility constant, as price falls...
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Compensated & Uncompensated Demand
Quantity of X
PX
dX
hX
X2
PX2
At PX2, the curves intersect becausethe individual’s income is just sufficient to attain utility level U2
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Compensated & Uncompensated Demand
Quantity of X
PX
dX
hX
PX2
X1*X1
PX1
At prices above PX2, income compensation is positive because the individual needs some help to remain on U2
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Compensated & Uncompensated Demand
Quantity of X
PX
dX
hX
PX2
X3* X3
PX3
At prices below PX2, income compensation is negative to prevent an increase in utility from a lower price
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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
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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
50502 ..),,( utility
YXYX PP
PPVI
I
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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
50
50
.
.
X
Y
P
VPX
50
50
.
.
Y
X
P
VPY
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Compensated Demand Functions
• Demand now depends on utility rather than income
• Increases in PX reduce the amount of X demanded
– only a substitution effect
50
50
.
.
X
Y
P
VPX
50
50
.
.
Y
X
P
VPY
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A Mathematical Examination of a Change in Price
• Our goal is to examine how the demand for good X changes when PX changes
dX/PX
• Differentiation of the first-order conditions from utility maximization can be performed to solve for this derivative
• However, this approach is cumbersome and provides little economic insight
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A Mathematical Examination of a Change in Price
• Instead, we will use an indirect approach• Remember the expenditure function
minimum expenditure = E(PX,PY,U)
• Then, by definition
hX (PX,PY,U) = dX [PX,PY,E(PX,PY,U)]
– Note that the two demand functions are equal when income is exactly what is needed to attain the required utility level
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A Mathematical Examination of a Change in Price
• We can differentiate the compensated demand function and get
hX (PX,PY,U) = dX [PX,PY,E(PX,PY,U)]
X
X
X
X
X
X
P
E
E
d
P
d
P
h
X
X
X
X
X
X
P
E
E
d
P
h
P
d
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A Mathematical Examination of a Change in Price
• The first term is the slope of the compensated demand curve
• This is the mathematical representation of the substitution effect
X
X
X
X
X
X
P
E
E
d
P
h
P
d
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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 necessary expenditure levels
• This is the mathematical representation of the income effect
X
X
X
X
X
X
P
E
E
d
P
h
P
d
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The Slutsky Equation• The substitution effect can be written as
constant
effect onsubstituti
UXX
X
P
X
P
h
• The income effect can be written as
XX
X
P
E
I
X
P
E
E
d
effect income
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The Slutsky Equation
• Note that E/PX = X
– A $1 increase in PX raises necessary expenditures by X dollars
– $1 extra must be paid for each unit of X purchased
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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
I
XX
P
X
P
d
P
d
UXX
X
X
X
constant
effect income effect onsubstituti
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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
will always be negative
I
XX
P
X
P
d
UXX
X
constant
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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
XX
P
X
P
d
UXX
X
constant
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Revealed Preference & the Substitution Effect
• The theory of revealed preference was proposed by Paul Samuelson in the late 1940s
• The theory defines a principle of rationality based on observed behavior and then uses it to approximate an individual’s utility function
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Revealed Preference & the Substitution Effect
• 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
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Revealed Preference & the Substitution Effect
B
A
I1
I2
I3
Quantity of X
Quantity of Y
Suppose that, when the budget constraint isgiven by I1, A is chosen
A must still be preferred to B when incomeis I3 (because both A and B are available)
If B is chosen, the budget constraint must be similar to that given by I2 where A is not available
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Negativity of the Substitution Effect
• Suppose that an individual is indifferent between two bundles: C and D
• Let PXC,PY
C be the prices at which
bundle C is chosen
• Let PXD,PY
D be the prices at which
bundle D is chosen
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Negativity of the Substitution Effect
• Since the individual is indifferent between C and D– When C is chosen, D must cost at least as
much as C
PXCXC + PY
CYC ≤ PXDXD + PY
DYD
– When D is chosen, C must cost at least as much as D
PXDXD + PY
DYD ≤ PXCXC + PY
CYC
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Negativity of the Substitution Effect
• Rearranging, we get
PXC(XC - XD) + PY
C(YC -YD) ≤ 0
PXD(XD - XC) + PY
D(YD -YC) ≤ 0
• Adding these together, we get
(PXC – PX
D)(XC - XD) + (PYC – PY
D)(YC - YD) ≤ 0
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Negativity of the Substitution Effect
• Suppose that only the price of X changes (PY
C = PYD)
(PXC – PX
D)(XC - XD) ≤ 0
• This implies that price and quantity move in opposite direction when utility is held constant– the substitution effect is negative
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Mathematical Generalization• If, at prices Pi
0 bundle Xi0 is chosen instead of
bundle Xi1 (and bundle Xi
1 is affordable), then
n
i
n
iiiii XPXP
1 1
1000
• Bundle 0 has been “revealed preferred” to bundle 1
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Mathematical Generalization
• Consequently, at prices that prevail when bundle 1 is chosen (Pi
1), then
n
i
n
iiiii XPXP
1 1
1101
• Bundle 0 must be more expensive than bundle 1
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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
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Consumer Welfare
• The expenditure function shows the minimum expenditure necessary to achieve a desired utility level (given prices)
• The function can be denoted as
expenditure = E(PX,PY,U0)
where U0 is the “target” level of utility
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Consumer Welfare
• One way to evaluate the welfare cost of a price increase (from PX
0 to PX1) would be
to compare the expenditures required to achieve U0 under these two situations
expenditure at PX0 = E0 = E(PX
0,PY,U0)
expenditure at PX1 = E1 = E(PX
1,PY,U0)
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Consumer Welfare
• The loss in welfare would be measured as the increase in expenditures required to achieve U0
welfare loss = E0 – E1
• Because E1 > E0, this change would be negative– the price increase makes the person worse
off
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Consumer Welfare• Remember that the derivative of the
expenditure function with respect to PX is the compensated demand function (hX)
),,(),,(
00 UPPh
dP
UPPdEYXX
X
YX
• The change in necessary expenditures brought about by a change in PX is given by the quantity of X demanded
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Consumer Welfare• To evaluate the change in expenditure
caused by a price change (from PX0 to
PX1), we must integrate the compensated
demand function
1
0
1
0
0
X
X
X
X
P
P
P
P
XYXx dPUPPhdE ),,(
– This integral is the area to the left of the compensated demand curve between PX
0 and PX
1
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welfare loss
Consumer Welfare
Quantity of X
PX
hX
PX1
X1
PX0
X0
When the price rises from PX0 to PX
1,the consumer suffers a loss in welfare
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Consumer Welfare
• Because a price change generally involves both income and substitution effects, it is unclear which compensated demand curve should be used
• Do we use the compensated demand curve for the original target utility (U0) or the new level of utility after the price change (U1)?
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Consumer Welfare
Quantity of X
PX
hX(U0)
PX1
X1
When the price rises from PX0 to PX
1, the actual market reaction will be to move from A to C
hX(U1)
dX
A
C
PX0
X0
The consumer’s utility falls from U0 to U1
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Consumer Welfare
Quantity of X
PX
hX(U0)
PX1
X1
Is the consumer’s loss in welfare best described by area PX
1BAPX0 [using hX(U0)]
or by area PX1CDPX
0 [using hX(U1)]?
hX(U1)
dX
A
BC
DPX
0
X0
Is U0 or U1 the appropriate utility target?
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Consumer Welfare
Quantity of X
PX
hX(U0)
PX1
X1
We can use the Marshallian demand curve as a compromise.
hX(U1)
dX
A
BC
DPX
0
X0
The area PX1CAPX
0 falls between the sizes of the welfare losses defined by hX(U0) and hX(U1)
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Loss of Consumer Welfare from a Rise in Price
• Suppose that the compensated demand function for X is given by
50
50
.
.
),,(X
YYXX P
VPVPPhX
the welfare loss from a price increase from PX = 0.25 to PX = 1 is given by
1
250
50501
25050
50
2
X
X
P
PXYX
XY PVPP
dPVP.
..
..
.
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Loss of Consumer Welfare from a Rise in Price
• If we assume that the initial utility level (V) is equal to 2,
loss = 4(1)0.5 – 4(0.25)0.5 = 2• If we assume that the utility level (V)
falls to 1 after the price increase (and used this level to calculate welfare loss),
loss = 2(1)0.5 – 2(0.25)0.5 = 1
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Loss of Consumer Welfare from a Rise in Price
• Suppose that we use the Marshallian demand function instead
XYXX P
PPdX2
I ),,( I
the welfare loss from a price increase from PX = 0.25 to PX = 1 is given by
1
250
1
250 22
X
X
P
P
XX
X
PdP
P ..
lnI
I
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Loss of Consumer Welfare from a Rise in Price
• Because income (I) is equal to 2,
loss = 0 – (-1.39) = 1.39• This computed loss from the Marshallian
demand function is a compromise between the two amounts computed using the compensated demand functions
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Important Points to Note:• Proportional changes in all prices and
income do not shift the individual’s budget constraint and therefore do not alter the quantities of goods chosen– demand functions are homogeneous of degree
zero in all prices and income
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Important Points to Note:• When purchasing power changes (income
changes but prices remain the same), budget constraints shift– for normal goods, an increase in income
means that more is purchased– for inferior goods, an increase in income
means that less is purchased
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Important Points to Note:• A fall in the price of a good causes
substitution and income effects– For a normal good, both effects cause more of
the good to be purchased– For inferior goods, substitution and income
effects work in opposite directions
• A rise in the price of a good also causes income and substitution effects– For normal goods, less will be demanded– For inferior goods, the net result is ambiguous
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Important Points to Note:• The Marshallian demand curve summarizes
the total quantity of a good demanded at each price– changes in price prompt movemens along the
curve– changes in income, prices of other goods, or
preferences may cause the demand curve to shift
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Important Points to Note:• Compensated demand curves illustrate
movements along a given indifference curve for alternative prices– these are constructed by holding utility constant– they exhibit only the substitution effects from a
price change– their slope is unambiguously negative (or zero)
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Important Points to Note:• Income and substitution effects can be
analyzed using the Slutsky equation
• Income and substitution effects can also be examined using revealed preference
• The welfare changes that accompany price changes can sometimes be measured by the changing area under the demand curve