Transcript MICROECONOMIC THEORY - University College London

Chapter 5
Consumer Welfare and Elasticities
Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.
1
Consumer Welfare
• An important problem in welfare economics is
to devise a monetary measure of the gains
and losses that individuals experience when
prices change
• This is important for economic policy because
economic policy usually changes prices due
to either taxes or subsidies
• Usually, we would like to design policies that
maximize consumer welfare, so we need to
measure it
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Consumer Welfare
• One way to evaluate the welfare cost of a
price increase (from px0 to px1) would be
to compare the expenditures required to
achieve U0 (initial level of utility) under
these two situations
expenditure at px0 = E0 = E(px0,py,U0)
expenditure at px1 = E1 = E(px1,py,U0)
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Consumer Welfare
• In order to compensate for the price rise, this person would require
a compensating variation (CV) of
V(px0,py,I)= V(px1,py,I+CV),
U0= V(px1,py,I+CV)
Solving for CV…
CV +I= E(px1,py,U0),
CV= E(px1,py,U0) – I,
CV= E(px1,py,U0) – E(p0x,py,U0)
The CV is how much money we would have to give the consumer after
the price change to make him just as well off as he was before the
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price
change
Consumer Welfare
Suppose the consumer is maximizing
utility at point A.
Quantity of y
If the price of good x rises, the consumer
will maximize utility at point B.
A
B
U0
The consumer’s utility
falls from U0 to U1
U1
Quantity of x
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Consumer Welfare
The consumer could be compensated so
that he can afford to remain on U0
Quantity of y
C
A
CV is the amount that the
individual would need to be
compensated
B
U0
U1
I/p1x (I+CV)/p1x I/p0xQuantity of x
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Consumer Welfare
• The derivative of the expenditure function
with respect to px is the compensated
demand function
E ( px , py ,U0 )
px
 x ( px , py ,U0 )
c
This was trick A in the previous set of slides!
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Consumer Welfare
• The amount of CV required can be found
by integrating across a sequence of
small increments to price from px0 to px1
p1x
p1x
px0
px0
CV   dE   x c ( px , py ,U0 )dpx
– this integral is the area to the left of the
compensated demand curve between px0
and px1
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Compensating Variation
px
When the price rises from px0 to px1,
the consumer suffers a loss in welfare
welfare loss
px1
px0
xc(px…U0)
x1
x0
Quantity of x
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Another way to measure changes in consumer
welfare
•
The CV is how much money we would have to give the consumer after the
price change to make him just as well off as he was before the price
changed. The reference of the CV is the initial level of utility
•
An alternative measure of the change in consumer welfare is the
Equivalent Variation (EV)
•
The Equivalent Variation is how much money would have to be taken
away from the consumer before the price changed to leave him as well off
as he would be after the price change. The reference of the EV is the final
level of utility
•
In another words:
– the EV is the amount of money that the consumer will be willing to pay to avoid
the price change
– the CV is the amount of money that the consumer will require to accept the
price change
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Example CV and EV
• CV: compensate individuals for an increase in
the price of air travel due to a green tax
• EV: The government promised to lower the
VAT in their political manifesto to win the
elections. After they are elected, they realise
that EU does not allow them. The government
uses the EV to compensate individuals for that.
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Equivalent Variation
• To avoid the change of prices, the individual will be willing to pay
the EV:
V(px0,py,I-EV)= V(px1,py,I),
V(px0,py,I-EV)= U1
Solving for EV…
I-EV= E(p0x,py,U1),
EV= I-E(p0x,py,U1),
EV= E(px1,py,U1) – E(p0x,py,U1)
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Consumer Welfare
• As in the case of the CV, the EV required can be found
by integrating across the compensated demand curve
from px0 to px1
p1x
p1x
p x0
p x0
EV   dE   x c ( p x , p y ,U1 )dp x
– this integral is the area to the left of the
compensated demand curve between px0
and px1, at the final level of utility
13
Consumer Welfare
• It is unclear which one should be used: CV or
EV
• They are different: notice, areas of different
compensated demand curves (according to
the level of utility: initial or final)
• Do we use the compensated demand curve
for the original target utility (U0) (that is, the
CV) or the new level of utility after the price
change (U1) (that is, the EV)?
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The Consumer Surplus
Concept
• Another way to look at this issue is to
ask how much the person would be
willing to pay for the right to consume all
of this good that he wanted at the
market price of px0
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The Consumer Surplus
Concept
• The area below the Marshallian 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
– changes in consumer surplus measure the
welfare effects of price changes
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Consumer Welfare
px
When the price rises from px0 to px1, the actual
market reaction will be to move from A to C
The consumer’s utility falls from U0 to U1
px1
C
A
px0
x(px…)
xc(...U0)
xc(...U1)
x1
x0
Quantity of x
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Consumer Surplus
px
px1
Is the consumer’s loss in welfare
best described by area px1BApx0
[using xc(...U0)] or by area px1CDpx0
[using xc(...U1)]?
C
B
A
px0
D
xc(...U0)
Is U0 or U1 the
appropriate utility
target?
xc(...U1)
x1
x0
Quantity of x
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Consumer Surplus
px
px1
The change is CS will be between
the CS and the EV
C
B
A
px0
D
x(px…)
xc(...U0)
The area px1CApx0
falls between the
sizes of the welfare
losses defined by
xc(...U0) and
xc(...U1)
xc(...U1)
x1
x0
Quantity of x
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Consumer Surplus
px
px1
The change is CS will be between
the CS and the EV
C
A
px0
D
x(px…)
xc(...U0)
The area px1CApx0
falls between the
sizes of the welfare
losses defined by
xc(...U0) and
xc(...U1)
xc(...U1)
x1
x0
Quantity of x
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Welfare Loss from a Price
Increase
• Suppose that the compensated demand
function for x is given by
x ( px , py ,V ) 
c
0.5
y
0.5
x
Vp
p
• The welfare cost of a price increase
from px = 1 to px = 4 is given by
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CV  Vp p
0 .5
y
1
0.5
x
 2Vp p
0.5
y
p 4
0.5 x
x p 1
X
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Welfare Loss from a Price
Increase
• If we assume that V = 2 and py = 2,
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),
EV = 122(4)0.5 – 122(1)0.5 = 4
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Welfare Loss from Price
Increase
• Suppose that we use the Marshallian
demand function instead
x( px , py , I )  0.5Ipx-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
px 1
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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
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Elasticities
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Elasticity – general formula
• Assume function: H=H(g1,g2,..,gn)
• Elasticity of H wrt. gi:
eH , gi
H / H H gi



gi / gi gi H
*eH,gi= how much H changes in %, if gi increases by 1%
It is useful because it is unit-free, we do not have to specify
the units… a 1% can be easily understood independently that
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we use £ or $
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
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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
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Iso-elastic demand function
• Usually used in econometrics
• Ln(x)=a*ln(px)+b*ln(py)+c*ln(I)
• Elasticities are:
– if ex,px =a
– if ex,py =b
– if ex,I = c
• Called iso-elastic because the
elasticities are fixed numbers
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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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Price Elasticity and Total
Spending
• Total spending on x is equal to
total spending =pxx
• Relation between total spending and
changes of the price:
( p x x )
x
 px 
 x  x[ex,px  1]
px
px
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Price Elasticity and Total
Spending
( p x x )
x
 px 
 x  x[ex,px  1]
px
px
• The sign of this derivative depends on whether
ex,px is greater or less than -1
– if ex,px > -1, quantity changes less than price in %
terms, so total spending and price move in the same
direction
– if ex,px < -1, quantity changes more than price in %
terms, so total spending and price move in the same
direction
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Compensated Price Elasticities
• It is also useful to define elasticities
based on the compensated demand
function
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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)
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Compensated Price Elasticities
• The compensated own price elasticity of
demand (exc,px) is
e
c
x ,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
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Compensated Price Elasticities
• The relationship between Marshallian and
compensated price elasticities can be shown using
the Slutsky equation:
x x c
x

x
px px
I
px x
px x c
p
x

 ex , p x  c 
 x x
x px
x
px
x
I
• If sx = pxx/I, then
ex,px  exc,px  sx ex,I
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Compensated 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
• So, the different measures of welfare
will be very similar if:
– the share of income devoted to x is small
– the income elasticity of x is small
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Another interesting relation
• If we increase prices and income by the same
percentage, optimal quantities will not
change. So:
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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Why is the previous relation useful?
• The previous relation could be used to either:
– Test the assumptions embedded in econometric
models
– Incorporate restrictions in econometric models to
gain efficiency
– Find out the value of some elasticities when we
have information about other elasticities
• The following expressions can also be used
for the same purposes
• Notice that econometrics is very important in
our context… we need numbers to compute
the welfare measures !!!
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Engel Aggregation
• The relation between the income elasticity of
several goods
• We can obtain it by differentiating the budget
constraint with respect to income:
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
• This relation is useful because it might
be easier to estimate the elasticities of
some goods rather than others…
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Cournot Aggregation
• The relation between own and cross-price
effects
• Useful because we might have information
about the elasticity of some goods, but we
want to find out about the elasticity of
others…
• We can obtain this relation by differentiating
the budget constraint with respect to px
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Cournot Aggregation
I
x
y
 0  px 
 x  py 
px
px
px
x px x
px
y px y
0  px 
  x
 py 
 
px I x
I
px I y
0  s x ex,px  s x  sy ey ,px
s x ex,px  sy ey ,px  s x
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Demand Elasticities
• 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 Elasticities
• Calculating the elasticities, we get
x px
I p x
ex ,px 

 2 
 1
px x
p x  I 
 
 px 
py
x py
ex , py 

 0
0
py x
x
e x ,I
x I 
I

 

1
I x px  I 
 
 px 
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Demand Elasticities
• We can also show
– homogeneity
ex,px  ex,py  ex,I  1 0  1  0
– Engel aggregation
s x ex,I  sy ey ,I    1    1      1
– Cournot aggregation
s x ex,px  sy ey ,px    ( 1)    0    s x
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Demand Elasticities
• We can also use the Slutsky equation to
derive the compensated price elasticity
exc,px  ex,px  sx ex,I  1 (1)    1  
• The compensated price elasticity
depends on how important other goods
(y) are in the utility function
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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
48
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
• no unambiguous prediction is possible
50
Important Points to Note:
• 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
51
Important Points to Note:
• The Marshallian demand curve
summarizes the total quantity of a good
demanded at each possible price
– changes in price prompt movements
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
– they are constructed by holding utility
constant and exhibit only the substitution
effects from a price change
– their slope is unambiguously negative (or
zero)
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Important Points to Note:
• Demand elasticities are often used in
empirical work to summarize how
individuals react to changes in prices
and income
– the most important is the price elasticity of
demand
• measures the proportionate change in quantity
in response to a 1 percent change in price
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Important Points to Note:
• There are many relationships among
demand elasticities
– own-price elasticities determine how a
price change affects total spending on a
good
– substitution and income effects can be
summarized by the Slutsky equation
– various aggregation results hold among
elasticities
55
Important Points to Note:
• Welfare effects of price changes can
be measured by changing areas below
either compensated or ordinary
demand curves
– such changes affect the size of the
consumer surplus that individuals receive
by being able to make market transactions
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Important Points to Note:
• The negativity of the substitution effect
is one of the most basic findings of
demand theory
– this result can be shown using revealed
preference theory and does not
necessarily require assuming the
existence of a utility function
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