# Micro 091021

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```					        Micro 091021

Slutsky equation, Consumer surplus
Topics for this week

• Prove that the Hicksian demand function is the partial
derivative of the Expenditure function
• Prove the Slutsky equation
• Introduce the elasticity notation for the Slutsky equation,
and verifying the formula for the utility function discussed
last week
• Discussing how the Slutsky equation holds for cross
elasticities as well as for own elasticities
– requires re-deriving the example discussed last week so that it
includes the prices of both goods x and y (denoted by p and q)
• Introducing the idea of the symmetry of the Slutsky
matrix, and how it imposes restrictions on demand
functions
Topics for this week, cont’d

• Introducing the concept of consumer surplus (CS),
starting with the diagrammatic and intuitive exposition
(the Marshallian measure of consumer surplus)
• Ask the question: is the Marshallian CS an accurate
measure of the consumer’s loss when the price of a
good increases?
– for a good with a substantial income effect, the answer is “no”
• Then ask the question: how can we get an accurate
measure of the consumer’s loss of CS when a price
increases
– the answer turns out to be: using the indirect utility function, or
the expenditure function
Topics for this week, cont’d

• Then we associate the correct measure of the CS loss
associated with a price increase with the area to the left
of the Hicksian (not Marshallian), or compensated
demand curve
• Next we consider the following question. Suppose the
price of good x rises from p1 to p2, and that the
consumer’s CS loss is A dollars. Now suppose the price
of good x falls from p2 to p1. Does this mean that the
consumer gains the same amount (A)?
• It turns out the answer is “no”. This leads to the concepts
of the Equivalent Variation and the Compensating
Variation
More class notes and assignments for next week

• Your assignment for next week is as follows. Consider a
consumer with the budget constraint px+qy=I, where p
and q are the prices of goods x and y, respectively, and
I is the consumer’s income. The consumer’s utility
function is given by

u  ln x  (1 / 3) ln y
• For this consumer, find the own-price elasticities and the
cross-price elasticities of the demand for both goods,
and verify the symmetry condition for this case
• Once you have done this exercise, I will post answers
and further derivation notes
Proving that the Hicksian demand function ..
• Consider x*,p*,u0, where x* is the solution to the
expenditure minimization problem when the price vector
is p*. Denote the expenditure function by E(p,u0)
• Now define G(p,u0)=E(p,u0)-px*. The function G is
negative for all values of p that are different from p*
(why?)
• The function reaches its maximum value (of G=0) at
p=p*. But if it reaches a maximum value for that p-
vector, that means that the derivatives of the G(p,u0)
function with respect to each element of the p-vector
are equal to zero there. This means
G E
 *  xi*  0
pi* pi
Hicksian demand function …

• But since x* is the vector that solves the expenditure
minimization function, the last equation shows that the
partial derivative of the expenditure function is the i’th
element of that vector; that is, the demand function for
the i’th good, as required
• The following three slides are from last week’s set
• Note that in this slide, we use h(j) to denote the Hicksian
demand function for good j, and x(j) to denote the regular
Marshallian demand function
Slutsky equation
• Let’s denote the regular (Marshallian) demand function
for good i by
xi ( p, Y )
• Now the Slutsky equation states:

xi hi        xi
      xj     ,i, j
p j p j      m

• Interpret the Slutsky equation: it divides the total effect of
a price change into a substitution effect and an income
effect
Proving the Slutsky equation
• The following equation is always true:
h j ( p, u0 )  x j ( p, E ( p, u0 ))

• Now take the derivative of this with respect to pi and
evaluate the derivative at u0, Y0

h j ( p, u0 )       x j ( p, Y0 )       x j ( p, Y0 ) E ( p, u0 )
                                                     
pi                  pi                  Y                pi
h j ( p, u0 )       x j ( p, Y0 )       x j ( p, Y0 )
                                        xi
pi                  pi                  Y
The Slutsky equation

• Rearranging the last expression gives the Slutsky
equation. Note that it holds true as well when i and j are
the same
• That is, it holds both when we evaluate the effect of a
change in pi on xi (the own-price effect), but also when
we evaluate the effect of a change in pi on xj, that is, a
cross-price effect
• So the Slutsky equation just gives the subdivision of the
effects of price changes into price and income effects,
and holds both for own-price effects and cross-price
effects
• In the case of 2 goods, it is based on the fact that along
an indifference curve, utility is constant (at u0, say)
Slutsky equation, elasticity version

• Remember that if z=f(x,y), the elasticity of z with respect to
x and y can be written

f ( x, y ) x         f ( x, y ) y
 zx                  , zy 
x       z            y       z
• After multiplying through by pi/xj we can therefore rewrite
the Slutsky equation as

x j pi              h j pi      pi x j I
  x j pi           xi
pi x j              pi x j      x j I I
Slutsky equation, elasticity version

• But the right-hand side can be rewritten, if we use hj to
denote the Hicksian demand function for the jth good,
and si to denote the share of the consumer’s income
that she spends on good i:
x   j pi
 h j pi  six j I

• Again, note that this holds for i=j, as well as when i
and j are different.
• Next we show a restriction that must hold for cross-
elasticities
Slutsky equation, cross-elasticities

• Cross elasticities of Hicksian demand curves must
satisfy the following restriction:
s jh j ps  sshs p j ,s, j
• Note carefully what this says: the cross-elasticity of the
Hicksian demand function for good j with respect to the
price of good s, multiplied by the budget share of good j
in total spending, must equal the cross-elasticity of the
Hicksian demand function for good s with respect to the
price of good j, multiplied by the budget share of good s
in total spending
Restriction on cross elasticities

• The reason for this is that the cross-price effects of the
Hicksian demand functions must satisfy the following
restrictions:

h j hs
     ,j, s
ps p j
• This restriction, in turn, follows because h(j) is the partial
derivative of the expenditure function with respect to the
price of j. Thus the derivative on the left in the equation
above can also be written

h j   2 E ( p, u )  2 E ( p, u ) hs
                             
ps     p j ps       ps p j      p j
Restriction on cross elasticities

• The reason why the cross-price effects are the same,
therefore, is that both of them are second cross-partial
derivatives of the expenditure function. But for
continuous differentiable functions, the order in which
you perform the differentiation when finding cross-partial
derivatives doesn’t matter, so the two cross-partial
derivatives in the middle of the above formulae are the
same. Therefore, the cross-price effects are the same
• The restriction on the cross-elasticities can then be
derived by using the definition of elasticities
Consumer surplus

• The concept of consumer surplus arises because with
convex preferences, consumers in competitive markets
are able to buy inframarginal units of individual goods at
a price that is lower than she would be willing to pay for
them
• Definition of “inframarginal” units: all units that the
consumer would hypothetically buy even if the price of
the good was higher than it actually is
• In equilibrium, the consumer buys enough units so that
her willingness to pay for the last (marginal) unit is equal
to the price.
Consumer surplus, cont’d

• Since (with convex preferences) her willingness to pay
declines as she buys more units, the willingness to pay
for earlier (inframarginal) units must therefore be higher
than the market price
• The diagram on the next page illustrates an individual
consumer’s demand curve for a good, assuming the
consumer has a given income and the prices of other
goods remain constant
• If the market price of the good is p0, the consumer buys
the quantity indicated at Q. If the price were higher, the
consumer would buy smaller quantities, but she would
still buy at least some units as long as the price were
less than p*
Consumer surplus, cont’d

• The conventional way to measure consumer surplus now
is as area A in the diagram on the following slide. Note
• First, it can also be interpreted as the sum of the
surpluses that the consumer enjoys on discrete units of
the good (see the textbook). If each unit is very small,
this sum is approximately the same as the area under
the curve. It can then be evaluated as an integral
– Note that there are two ways of evaluating the integral
• Note also that the area that measures the consumer
surplus is measured in money (dollars), not “utility”
P        Consumer Surplus (CS) is then
given by area A in the figure,
which in turn can be written as:
P*

CS  0.5Q( P *  P 0 )
A

P0

D

Q
Consumer surplus, cont’d

• Note that the consumer surplus is most interesting when
we consider something that changes it
• Specifically, suppose the price of the good in the
previous diagram rises from P0 to P1. Then the
consumer surplus changes to area A’<A. The difference
is area B+C. It can be measured in dollars, and can be
interpreted as the consumer’s loss from the price
increase
• But on the earlier slides, we asked the question: is this
area an accurate measure of the consumer’s loss?
• In general, the answer is “no”. Let’s try to see why
When price rises from P0 to P1,
Consumer Surplus (CS), changes to area A’,
P
which is smaller than A. It is given by:

P*
CS  0.5Q1 ( P*  P1 )

The change in CS is equal to the
A’                          difference, area (B+C)

P1

B                  C
P0

D

Q1               Q0
Consumer surplus, cont’d

• To see this, note that if the price of the good rises from
p0 to p1, the consumer’s utility is reduced
• An accurate measure of the consumer’s loss could then
be based on an answer to the following question: if the
price of this good were to rise, how much money would it
take to compensate the consumer for this loss?
• One way to measure the consumer surplus loss is on the
basis of the indirect utility function V(p,I)
• Specifically, we can ask the following question: by how
much would we have to increase the consumer’s income
when the price of this good rises from P0 to P1, in order
to leave the consumer at the same level of utility as
before?
Consumer surplus, cont’d

• But when we define it this way, we can in fact compute
the loss of consumer surplus directly from the
expenditure function
• But here comes an interesting point: if we interpret the
consumer surplus measure this way, it will in fact be
measured correctly by area B+C only if the demand
curve D in the diagram is the Hicksian (compensated)
demand curve
• To see this, consider the CS loss from a very small price
increase from P0; call it dP
• The loss from a small price increase dP is just the dP
times the derivative of the expenditure function
• But the derivative of the expenditure function is the
compensated demand function!
Consumer surplus and compensated demand functions

• So the number we get if we take the (horizontal) integral
of the area between the old and the new price, to the left
of the compensated (Hicksian) demand curve, is in fact
the change in the expenditure function that is necessary
to compensate the consumer for a price increase from
p0 to p1
• But here comes one further question: suppose we ask
the question: if we start at p1, and then reduce the price
of the good to p0, what is an accurate measure of the
consumer’s gain from this price decrease?
• At first glance, it would seem that the answer should be:
“the same as the loss from a price increase from p0 to
p1”
Consumer surplus, cont’d

• Unfortunately, this answer turns out to be wrong. In the
next little while, we will discuss why it is wrong
• (Note one thing first: If we measure consumer surplus
using the area to the left of the Marshallian demand
curve between the old and the new price, the two are the
same. Thus, the fact that the loss from a price increase,
when correctly measured, is not of the same size as the
gain from the same change in the other direction, must
have something to do with the fact that we are
measuring the gains and losses using Hicksian demand
curves)
• First, we do an example
Hicksian and Marshallian demand curves

• From the example, we can see that in this case, the
consumer’s loss from a price increase is larger than the
consumer’s gain from a price decrease
• The reason in this case is as follows. There is a different
Hicksian demand curve through each point on the
Marshallian demand curve
– The different Hicksian demand curve don’t cross. Why?
• Since the CS measure is the area to the left of the
Hicksian demand curve between the old and the new
price, when the demand curve shifts the CS measure
changes
Consumer surplus, concluded

• In the example, the Hicksian demand curve is steeper
than the Marshallian demand curve. Is that always true
(for other utility functions as well)?
• Ans: No. If the good that we are illustrating is an inferior
good, the Hicksian demand curve is less steep than the
Marshallian demand curve
• Finally, some people measure the consumer’s loss from
a price increase by the area to the left of the Hicksian
demand curve that goes through the point on the
Marshallian demand curve at the higher price
– We have called that measure “the consumer’s gain from a price
decrease”
Consumer surplus, concluded

• This alternative measure of the consumer’s loss from a
price increase is referred to as the “Equivalent Variation”,
whereas our measure of the loss is called the
“Compensating Variation”
– The equivalent variation is the answer to the question: at the
initial price, how much would the consumer be willing to pay for a
promise that the price would not increase?
– The compensating variation, in turn, is the answer to the
question: how much money would we have to give the consumer
after the price has increased in order to keep her at the same
utility as at the lower price?
– So the difference between the two measures is that we hold the
consumer’s utility constant at two different levels

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 views: 35 posted: 1/19/2012 language: pages: 28