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Econ 604 Advanced Microeconomics
Davis
Spring 2005
23 February 2006
Reading.
Problems:
Chapter 5 (pp. 128-150) for today
Chapter 6 (pp. 152-170) for next
time
To collect: Ch. 5. 5.1 5.2 5.4
Next time: Ch. 5. 5.6, 5.7, 5.8, 5.9
Lecture #6
REVIEW
Comments on Homework: Many of you had difficulties with problem 4.6 (The
problem where good Z was not optimally consumed with a budget of $2. Observe that in
this problem, if you take first order conditions, the optimal quantity of z to consume is
negative. Some of you switched the negative sign to positive. This is, of course,
incorrect. The point is that you should be care to attend to such details when you work
with a system. The fact that the optimal amount of Z is negative implies that some
problem exists with the system.
Chapter 4:
V. Income and Substitution Effects. Analysis of demand.
A. Demand Functions
1. Homogeniety
B. Changes in Income
1 .Normal and Inferior Goods.
2. Engel’s Law (A statement about income effects on the food
SHARE rather than quantity of food demanded)
C. Changes in the Price of a Good (with indifference curves).
1. Graphical Analysis – Price Reduction
2. Graphical Analysis – Price Increase
3. Effects of Price Changes for Inferior Goods.
D. Individual’s Demand Curve (Derive from Indifference Curves)
1. Shifts in the Demand Curve
E. Compensated Demand (Reconstruct from Indifference Curves)t
Example #5. Compensated demand functions. (I repeat this, because we will use
the notion of indirect demand further in today’s lecture.) Consider the CobbDouglas utility function U(X,Y) = X.5Y.5. The demand functions for X and Y are
given by
X*
=
I/2Px
and
1
Y*
=
I/2Py
The indirect utility function can be solved by inserting X* and Y* back into the
utility function. This yields
I/(2Px.5Py.5)
Utility = V(I, Px, Py) =
Solving this expression for I and substituting in to X* and Y* yields the
compensated demand functions
X
= VPy.5/Px.5
and
Y
= VPx.5/Py.5
Notice that even though Py did not enter into the uncompensated demand function
for X it does enter into the compensated demand function. This example makes clear
what is being held constant with the two demand forms. With uncompensated demand,
expenditures are held constant, so a rise in the price of X causes a reduction in utility.
With compensated demand utility V is held constant. When the price of X increases,
expenditures must also be raised to keep utility constant. Of course, the price of Y will
affect how this expenditure increase is spent.
PREVIEW
F. A Mathematical Development of Price Change Responses
1. Direct Approach
2. Indirect Approach
3. The substitution Effect
4. The income Effect
5. The Slutsky Equation
G. Revealed Preference and the Substitution Effect
1. Graphical Approach
2. Negativity of the Substitution Effect
3. Mathematical Generalization
H. Consumer Surplus
1. Consumer Welfare and Expenditure Functions
2. A Graphical Approach
3. Consumer Surplus
4. Welfare Changes and Marshallian Demand Curve
Lecture________________________________________________
F. A Mathematical Development of Price Change Responses. Now we go
back to the income and substitution effects developed graphically last lecture, and
develop these results analytically.
1. Direct Approach One way to separate out analytically income and
substitution effects would be to start with our standard constrained optimization problem
and solve for dx/Px and then separate out income and substitution effects. That is, we
would start with the problem
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L = dx(Px, Py, I) + (I - PxX - PyY)
and take FONC w. r. t. Px and I. Solving, we could develop expression for dx/Px and
dx/I that we could use to develop a compensated demand function hx which
incorporates the substation effect, but abstracts out the income effect.
In general, however this solution is rather cumbersome, and not very informative. It is
instructive to take an indirect approach. This indirect approach has the advantage of
allowing us to see how working with the dual to a problem can provide important
insights.
2. Indirect Approach. Consider an expenditure function. Recall that the
expenditure function reflects the minimum amount that must be spent by an individual in
order to achieve a given level of utility U*.
minimum expenditure =
E(Px, Py, U*)
Then, by definition at reference prices Px and Py, compensated demand function hx equals
the normal (uncompensated) demand given a budget I equal to expenditures.
hx(Px, Py, U*) =
dx(Px, Py, E(Px, Py, U*))
Now, we can isolate the (compensated) substitution effect simply by taking the
partial derivative of hx with respect to Px.
hx/Px
=
dx/Px
+
dx/E  E/Px
=
hx/Px
-
dx/E  E/Px
Rearranging
dx/Px
The expression to the right of the equality reflects a combination of substitution
and income effects associated with a price change on uncompensated demand. Further,
the left of the two terms is the substitution effect. The sign on this term is negative.
The rightmost term reflects an income effect. Consider the sign of this term.
E/Px is clearly greater than zero, since a consumer must be compensated for a price
increase in order to be indifferent between the new higher price and an old lower one. If
a good is a normal good, dx/E>0 as well. Thus, the sign on the expression is negative
only because of the ‘– ‘sign appearing to the left of it. On reflection that should be
appealing. For a normal good, a price increase reduces income which will cause the
consumer to purchase less of the good.
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5. The Slutsky Equation. The above decomposition can be stated a bit
more clearly with some notational changes. First, rewrite the substitution effect
hx/Px
=
X/Px|U* = constant
Next rewrite the income effect. First, observe that E and I refer to the
same thing, and that compensated demand dx = an amount X consumed. Thus,
dx/E =X/I. Next, E/Px =X. (recall E(U*) = PxX+PyY). Thus,
-dx/E E/Px
=
-X/I X
Combining, we write what is termed the Slutsky equation as
dx/Px
=
X/Px|U* = constant
-
X/I X
Intuitively, this equation says that the effects of a $1 increase in the price of a
good X can be divided into a substitution effect prompted by the lower relative price of
other goods, and an income effect that arises because the consumer is poorer as a
consequence of the price reduction.
Notice further, as we illustrated with indifference curves for the 2 good
case, that when a good is normal, X/I>0, so the income effect reinforces the
substitution effect. On the other hand if a good is inferior X/I<0 and the income effect
damps the effect of substitution.
In the extreme, it is possible (at least as an analytical matter) that the
income effect term dominates the substitution effect. In this case, the overall slope of the
demand curve would be positive. Such goods are called Giffen Goods.
Example. Slutsky Decomposition with a Cobb Douglas Utility function
Perhaps the ideas of this section would be made more concrete with an example.
Consider again the Utility Function U(X,Y)
=
X.5Y.5. Suppose further that
PX = .25, PY = 1 and I = 2.
We have, in several past examples, derived the uncompensated demand functions
for this function under these conditions as
X
=
dx(Px, Py, I)
=
I/(2Px)
=
dy(Px, Py, I)
=
I/(2Py)
and
Y
By the Slutsky equation
dx/Px
=
X/Px|U* = constant
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-
X/I X
Let us verify that the left and right sides are equivalent. On the left side,
dx/Px
=
-I/ 2Px2
To develop the first term on the right side, we need to find the indirect demand
function for X. Given X and Y,
V
Thus
I
=
(I/(2Px)).5(I/(2Py)).5
=
2VPx.5Py.5
=
I/(2Px.5Py.5)
Substituting for I in the income constraint, and solving yields the expenditure function
2VPx.5Py.5
=
I
Substituting I into the direct demand functions yields
X
=
hx(Px, Py, I)
=
2VPx.5Py.5/(2Px)
=
VPy.5/Px.5
Thus,
hx(Px, Py, I)/ Px
=
-VPy.5/(2Px15)
Using the indirect utility function again, we can express compensated demand in
terms of income
-IPy.5/[(4Px15) Px.5Py.5]
=
-I/(4Px2)
To calculate the income effect, -
X/I X, simply take the deriviative of dx
X/I =
=
w.r.t.
dx(Px, Py, I)/ I
Thus, - X/I X
=
-
1/(2Px)
X/2Px =
-I/(4Px2)
Combining terms, we have
dx/Px
=
X/Px|U* = constant
-I/ 2Px2
=
-I/(4Px2)
-
-I/(4Px2)
X/I X
=
-I/(2Px2)
Thus, for example, given the above utility function, I = 2, Py=1 and Px = .25, an
increase in Px from .25 to 1 changes quantity demanded from 2/2(.25) = 4 to 2/2(1) = 1.
The change from 4 to 2 is driven by the substitution effect. A change from 2 to 1 is the
income effect (Note: How do you get these changes? .. By inserting parameter values into
the indirect demand function.)
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But observe, the fact that income shares are constant in this example, implies that
income and substitution effects are equal. The effects of the price change in X on
consumption and income are equal. For good Y these effects are also equal, but opposite
in sign. Thus the amount of Y consumed does not change.
Prior to proceeding, consider again our assertion that E/Px =X. Intuitively, this
says that when the price of X increases by a dollar, $X extra dollars of expenditures are
needed to maintain utility at the same level. We can demonstrate this formally by solving
the dual problem to utility maximization
L = PxX + PYY +( U -U(X,Y))
Applying the envelope theorem to this problem
L/Px =E/Px = X
Notice, we are recovering in this way the demand function. This result called Shepard’s
Lemma, is important in empirical work because it implies that one can find the demand
function for a good simply by taking the derivative of the expenditure function. Since, by
construction, the expenditure function holds Utility constant, this is a compensated
demand curve.
To illustrate, recall that in the above example,
E
Thus
V(2Px.5Py.5),
=
E/Px = VPy.5 Px-.5
We’ll return to this in section I, below.
G. Revealed Preference and the Substitution Effect. The principal unambiguous
prediction that can be derived from the utility maximization model is that the demand
curve has a negative slope. The proof of this assertion requires a diminishing MRS
(which makes necessary conditions for a maximum also sufficient). Some economists
consider basing demand on unobservable utility considerations is undesirable. This
section outlines an alterative
approach.
1. Graphical Approach. In the
panel to the left, suppose that with
the budget line I1 allocation A is
selected over B. We will say that A
is revealed preferred to B, and the
fact that A is preferred to B implies
that B will only be selected when A
is unaffordable (e.g, with budget
Y
C
A
B
I2
I1
X I3
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line I3.) Similarly given more is preferred to less, bundle C must be revealed preferred
A. This implies that B will be selected over C if C is unaffordable. (That is, at relative
prices of Y less than those implied by I2.) Given an observed preference over alternative
bundles A and B, we are able to rank a series of alternative bundles as well. This turns
out to be a fairly powerful observation.
2. Negativity of the Substitution Effect. Using the principle of rationality, we can
show why the substitution effect must be negative (or zero). Suppose an individual is
indifferent between two bundles C (XC and YC) and D (XD and YD). Let PxC and PyC
denote the prices at which C is chosen and PxD and PyD the prices at which D is chosen.
Then when C is chosen, if the individual is indifferent, D must cost at least as much as C
(and probably more). That is
PxC XC + PyC YC
< PxC XD + PyC YD
Similarly, when D is chosen, C must cost at least as much as D (or probably more)
PxD XD + PyD YD < PxD XC + PyD YC
Rewriting
Pxc(Xc -XD) + PyC (YC - YD)<0 and
PxD(XD -XC) + PyD (YD- YC) <0
Adding together yields
(PxC- PxD)(XC -XD) + (PyC - PyD)(YC- YD)<0
Now, suppose that only the price of X changes; PyC = PyD. Thus
(PxC- PxD)(XC -XD) <0
But this expression states only that price and quantity must move in the opposite
directions. (If the first term above is negative, the other must be positive, and vice versa).
This is precisely a statement about the non-positive nature of the substitution effect.
X/Px|U* = constant<0
Notice, that we needed neither a utility function nor an assumption of diminishing MRS.
In terms of the above graph, this simply says that if there exists a pair of bundles between
which a consumer is indifferent, then it must be the case that prices which make one
unaffordable must imply lower prices for some goods in the preferred bundle. Another
bundle will be preferred only if the relative prices of those previously preferred goods
increases.
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3. Mathematical Generalization. Generalizing is straightforward. If at prices Pi0
bundle Xi0 is preferred to bundle Xi1 and bundle Xi1 is affordable, then
 Pi0 Xi0< Pi0 Xi1
But prices when bundle Xi1 is bought, say Pi1, bundle Xi0 must be more expensive
 Pi1 Xi0> Pi1 Xi1
We summarize this as follows:
Strong Axiom of Revealed Preference. If a commodity bundle 0 is revealed
preferred to bundle 1, and if bundle 1 is reveled preferred to bundle 2, and if bundle 2 is
revealed preferred to bundle 3, …., and if bundle K-1 is reveled preferred to bundle K,
then bundle K cannot be revealed preferred to bundle 0.
Most of the other properties we have developed can also be developed from the
revealed preference axiom. For example, it is easy to show that demand functions are
homogenous in prices and Income. As shown by Houthakker (1950) revealed preference
and utility theory are equivalent conditions. The revealed preference approach is widely
used in the construction of price indices.
H. Consumer Surplus. One important applied area in economics involves
developing a monetary measure of the gains or losses individual experiences as a result of
price changes. In order to make such calculations, we need a measure of the welfare
consequences of a price change. This is the notion of consumer surplus
1. Consumer Welfare and Expenditure Functions. An easy way to
develop the notion of consumer surplus is to consider again the notion of an expenditure
function
expenditure =
E(Px, Py, Uo)
where Uo is a target level of utility
A monetary measure of the consequences of a price increase from Px0 to Px1is the change
in expenditures necessary to maintain the target utility level
Eo(Pxo, Py, Uo) vs.
E1(Px1, Py, Uo)
With Px1 > Px0 E1 > E0., and
Welfare change
=
E1
-
E0
2. A Graphical Approach
We can make further headway using the envelope theorem on the Expenditure function
dE(Px, Py, Uo) /dPx
=
hx(Px, Py, Uo).
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Thus, we can measure the change in welfare from a price increase as the integral of the
expenditures function over the price change
Change in Expenditures
=
 dE
=
 hx(Px, Py, Uo)dPx.
P
Graphically, this is just the area under the compensated demand curve,
between the two prices
P1
P0
X
. Thus, consumer surplus is simply the area under the compensated demand curve
between the two prices.
3. Welfare Changes and Marshallian Demand Curve. One problem with
this notion of consumer surplus for applied purposes is that compensated demand is not
observable. Rather we can see only standard, uncompensated demand. In most empirical
work analysts use uncompensated demand to estimate consumer surplus. Fortunately, the
costs of this imprecision are not great, since the consumer surplus estimate from
uncompensated demand falls between two rival consumer surplus estimates arising via
compensated demand.
Any time price rises, there are two possible reference points. Compensated
demand at the higher price (with terminal utility U1), or compensated demand at the
lower price (with initial utility U0). The compensated demand curves at each utility level
are different (with utility being lower in the former case.) By construction,
uncompensated demand must cross these two compensated demand functions at the
reference points. Thus, the consumer surplus for the compensated demand will be about
halfway between the consumer surplus measures for the two compensated demand
functions.
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P
P1
dx
P0
hx(Uo)
As illustrated in the figure on the
left, for uncompensated demand at
U0 the welfare loss is the sum of all
three areas shaded. For
uncompensated demand at U1 the
welfare loss is the striped area.
With uncompensated
demand, the estimated welfare loss
is almost the average of these two
measures, the striped area plus the
shaded area.
hx(U1)
X
Example: Loss of Consumer Surplus from a Price Rise. Continuing with out
Cobb-Douglas Demand example.
For purposes of reference, recall the U = (XY).5 and thus, compensated demand for the
products was
X
=
I/(2Px) and
Y
=
I/(2PY)
V
=
I/[2(PxPy).5]
(With Px = .25, PY=1 and I=2, U* = V = 2. With Px =1, Py =1 and I=2, U*=V=1.)
Let’s examine this within the context of compensated demand. Form above,
X = hx(Px, Py, V)
=
VPy.5/Px.5
The welfare loss of a change in price from Px =.25 to Px = 1.00 is
1.00
 VPy.5/Px.5 dPx
0.25
Px = 1.00
=
2VPy.5Px.5
|
Px=0.25
Let’s examine this in the context of the in, , and With V = 2 and PY =1 the C.S.
loss is 4(1).5 – 4(.25).5 = 2.
If we started with the post change price V = 1, and loss would be
2(1).5 -2(.25).5 = 1
Finally if we start with uncompensated demand
1.00
 I/(2Px) dPx
0.25
Px = 1.00
= 0 – (-1.39) = 1.39,
= (I/2)ln Px |
Px=0.25
Which is about halfway between the other measures.
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