Econ 604 Advanced Microeconomics
Davis
Spring 2006
2 March 2006
Reading.
Problems:
Chapter 6 (pp. 152-170) for today
To collect: Ch. 5. 5.6, 5.7, 5.8, 5.9
For next time: Ch. 6. 6.1, 6.3, 6.5
Test next time.
Lecture #7
REVIEW
E. Compensated Demand
1. Relationship between Compensated and Uncompensated
Demand
F. A Mathematical Development of Price Change Responses
1. The Slutsky Equation
2. The Substitution Effect
3. The Income Effect
G. A short introduction to revealed preference
H. Consumer Surplus
1. Consumer Welfare and Expenditure Functions
2. A Graphical Approach
3. Consumer Surplus
4. Welfare Changes and Marshallian Demand Curve
PREVIEW
I. Shephard’s Lemma, Roy’s Identity and Price Indices.
VI.
Chapter 6. Demand Relationships Among Goods
A. An Overview. The Two-Good Case
B. Substitutes and Complements
C. Net Substitutes and Complements
D. Composite Commodities
E. Home Production Attributes of Goods and Implicit Prices
1
Lecture________________________________________________
I. Shephard’s Lemma, Roy’s Identity and Price Indices. In closing chapter 5, it is
instructive to review the relationships between primary problems, dual problems, direct
utility and indirect utility. There are a variety of ways to derive product demand, and we
offer this brief review to clarify the relationships between the concepts.
1) Maximizing utility subject to an income constraint allows for the expression of an
indirect utility function Max U s.t., I = PxX + PyY.
2) An expenditure function E* may be derived by the dual problem, that is minimizing
expenditures subject to a utility level Min PxX + PyY.subject to U(X,Y) = U*
Now the uncompensated demand functions developed by the maximization problems in
(1) allow us also to derive indirect utility by inserting the demand functions (expressed in
terms of prices and incomes) back into the utility function. Our standard method of
finding the expenditure function is simply to solve the resulting expression for I
(expressed in terms of U*, and prices). This is equivalent to solving the minimization
problem in (2)
3) Given indirect utility and expenditure functions we can use the envelope theorem to
express uncompensated and compensated demand, respectively.
In the case of an Expenditure function, we can derive compensated demand as the partial
derivative of the expenditure function w.r.t. Px.
In the case of an indirect utility function, demand dx is the ratio of the partial derivative of
indirect utility with respect to Px, divided by the partial derivative of indirect utilty w.r.t.
I.
Consider these applications of the envelope theorem, in turn.
1. Deriving compensated demand with the Expenditure function. The expenditure
function is derived by minimizing expenditures subject to a utility level, e.g.,
L
=
PXX
+
PYY
+ (U* - U(X, Y))
The Envelope theorem states that at optimal values of X and Y,
L/PX =
E/PX =
X|U*
this application of the Envelope Theorem is known as Shephard’s Lemma.
2
2. Deriving uncompensated demand from the indirect utility function. In the standard
maximization problem, the objective is
L
=
U(X, Y)
+
(I
-
PXX
-
PYY)
Again, the Envelope theorem status that at optimal values of X and Y,
 L/PX
=
U/PX
=
X
and  L/I
=
U/I
=

U/PX
U/I
Thus,
=
X
This application of the Envelope Theorem, is known as Roy’s Identity
Example. These relationships can be illustrated with our standard Cobb-Douglass
function
U(X,Y)
=
X.5Y.5
Recall that indirect utility in this case is
V(PX, PY, I)
=
I/[2PX.5PY.5]
U/PX
=
I/[4PX1.5PY.5]
U/I
1/[2PX.5PY.5]
our well-known expression for uncompensated demand.
Thus,
=
2IPX.5PY.5 =
4PX1.5PY.5
I
2PX
Similarly, we derived the expenditure function as
E
=
2VPX.5PY.5
Applying Shephard’s lemma,
E/PX =
V(PY/PX).5
again, the expression we have developed previously as our compensated utility function.
Importantly, these different concepts allow us alternative ways to estimate
demand. Consider for example, the problem of constructing price indices. When specific
components of a price index increase separately, we overstate the loss in well being if we
compensate consumers with a simple increase in expenditures that offsets the effects of
the single-good price increase on the total bundle (as would be suggested by an
uncompensated demand function). This is the case because consumers may respond to
the increase in the price of a single good by substituting away toward other goods. The
appropriate way to develop the income increase necessary to compensate for an increase
in the price of a single good is compensated demand.
3
Some economists estimate that the failure to appropriately adjust for substitution
effects results that the inflation rate is .2 points higher than it actually is each year.
VI.
Chapter 6. Demand Relationships Among Goods. In chapter 5 we considered the
effects of a change in the price of a particular good on consumption. In this short chapter
we consider the effects of changes in the price of related goods.
I1
1.897367
0.948683
Io
U1
1.048809
1.887856
Uo
2
1
1
2
Y
Y
A. An Overview. The Two-Good Case. It is convenient to start with a two good
case, as we have in other topics. The two panels below illustrate some possible
consequences on the change in the consumption of a good X brought about by a
reduction in the price of a good Y,
0
1
0.707107
I1
Y2
Io
Y1
1
Yo
Y0
U1
Uo
Xo
X1
X
X1
X0
Notice in both panels that the amount of Y consumed increases with the price
reduction (from Yo to Y1). However, the consequences for good X differ. In the panel
shown on the left, there is relatively little substitutability between the goods. As a
consequence the consumer took advantage of the price reduction to increase consumption
of good X. In contrast, as suggested by the very flat indifference curves in the panel on
the right, when the price of Y fell, the consumer increased consumption of Y for two
reasons. First, because of an income effect, and second, because the goods are close
substitutes. As a consequence of the price reduction, consumers shifted away from X and
toward the now-cheaper substitute Y.
The ambiguous effects of a price reduction in a related good can be illustrated
with a Slutsky-type decomposition of uncompensated demand.
X/PY = dX/PY
=
X/PY|U* -
Y(X/I)
Notice that the first term on the right is a substitution effect, and reflects the tradeoffs
between the related good and X. The second term on the right is an income term. It is the
product of the (now cheaper) units of Y purchased, multiplied by the extent to which X
moves with an increase in Income.
In a two good world, these terms have opposite signs. Assuming that indifference
curves are convex, the goods are substitutes, and so X/PY >0. However, if the good is
a normal good, the second term is negative, since both Y>0 and (X/I)>0. Thus the net
4
X
effect is ambiguous. The effects of an increase in PY are positive or negative depending
on whether or not the substitution effect dominates the income effect.
Example 6.1. Consider again the Cobb-Douglas function
U(X,Y)
=
X.5Y.5
We learned previously that
dx(PX, PY, I)
=
I/2PX
=
VPY.5/ PX.5
and that
hx(PX, PY, I)
It is obvious from the uncompensated demand function that
dx/PY
=
0.
Let us decompose this net effect into substitution and income effects. Note first
that
hx/PY
=
V/[2(PYPX).5]
But, recall our indirect utility is given by
V
I/[2PX.5PY.5]
Thus
hx/PY
=
X/PY|U*
=
I/[4PXPY]
To derive the income effect observe that
dx(PX, PY, I)/ I
=
1/2PX
-Ydx/ I
=
-I/[4PYPX]
Thus
Combining terms
X/PY = dX/PY
0
=
=
X/PY|U* I/[4PXPY] -
Y(X/I)
I/[4PXPY]
the desired result. Interestingly, observe that PY is left out of the uncompensated demand
function not because PY exerts no influence on X, but rather because the income and
substitution effects just cancel each other out in this case.
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B. Substitutes and Complements. For the case of many goods, the Slutsky
analysis for cross-price effects mentioned above generalizes to the following expression,
for an too goods i and j
Xi/Pj = di/Pj
=
Xi/Pj|U* -
Xj(Xi/I)
Generally speaking, goods are substitutes if a consumer responds to an increase in the
price of a good j by increasing consumption of good i. Goods are complements if a
consumer responds to an increase in the price of a good j by decreasing consumption of
good i.
There are two ways to make this intuitive relationship precise. One is to combine
substitution and income effects together. Goods are said to be Gross Substitutes if
Xi/Pj = di/Pj
>
0
They are Gross Complements if
Xi/Pj = di/Pj
<
0.
The notion of gross substitutes and gross complements have some undesirable
features. The most important of these is that they are asymmetric. That is, X may be a
gross substitute for Y and, at the same time Y may be a gross complement for X. This
would arise if the substitution effect of an increase in PY dominated the income effect on
X, and, if the income effect of an increase in PX dominated the substitution effect on good
Y.
The following example illustrates
Example 6.2
Let
U(X,Y) =
ln X
+
Y
Then the Lagangrian becomes
L
=
ln X
+
Y + (I – PxX – PyY)
PX
;
From FONC we have
1/X
=
1
=
PY.
Taking the ratio of these terms
X
=
PY/PX
Inserting into the budget constraint renders the demand functions
I
=
PY
+ PYY
6
or
PYY
=
(1
=
I- PY
+ Y)PY
Thus, if PY increases the budget share spent on Y (that is, PYY) must decrease.
Thus, expenditures on X increase, implying that X and Y are gross substitutes. However,
solving the above for Y, note that
Y
=
[I- PY ]/ PY
So
Y/PX =
0, implying that the goods are unrelated.
C. Net Substitutes and Complements. An alternative definition circumvents this
problem with asymmetries. The definition relies on compensated demand effects.
Specifically, if
Xi/Pj|U*
>
0 goods i and j are said to be net substitutes, and if
Xi/Pj|U*
<
0 goods i and j are said to be net complements.
Where U* is a constant.
These terms are sometimes called “Hicksian” substitutes and complements. Symmetry
exists in these definitions by simple application of Young’s Theorem, that is
Xi/Pj|U*
=
Xj/Pi|U*
[Recall from Shephard’s Lemma, Xi = E/Pi. Thus, Xi/Pj = 2E/[PiPj], which would
render the same result were the order of differentiation reversed]
D. Composite Commodities. To this point, our discussion of price effects has
been confined to the effects of a price change on a single related good. More generally, a
price change affects all other goods available to a consumer, and the analysis quickly
becomes overwhelming. (If there are n goods, there are (n+1)n/2 possible substitution
effects. To simplify the analysis, it is conventional to aggregate goods, and consider the
effects of price changes using some good X and a composite “other goods” Y. This
section considers the assumptions implicit in such an analysis.
Consider an economy that consists of n goods, X1, and n-1 other goods, X2, to Xn
If X1 is available for price P, and the other goods are available for prices Po2 …Pon . Write
the total expenditure on other goods as Y
Y
=
Po2X2+ …+Pon Xn .
Thus, the budget constraint is
7
I
=
P1X1
+
Y
Now consider the effect of a constant percentage markup in all other goods Y. Denote
this markup as t.
I
=
P1X1
+
tY
We can analyze the effects of changes in t just as we did any of the other “price effects”
previously. Note the following
a) The analysis does require that all other goods increase by the same relative
amount
b) This analysis does not require that the change in consumption of other goods
X2 … Xn be equivalent. What is necessary is that the MUi/Pi be equivalent for all goods.
This leads to the following definition
Composite Commodity: A composite commodity is a group of goods for which all
prices move together. These goods can be treated as a single “commodity” in that the
individual behaves as if he or she were choosing between other goods and total spending
on the entire composite group.
This definition and the Composite Commodity Theorem, this result, which makes
it possible to aggregate goods, allows the simplification of a number of very complex
problems into a two dimensional space.
Importantly, however, the general result that price changes for n-1 other goods
may be analyzed as a single price change requires strict proportionality in price increase.
Even slight departures from strict proportionality may negate the composite commodity
theorem if cross- substitution effects are large.
Example: Housing Costs as a Composite Commodity. Suppose that an individual
receives utility from 3 goods; food (X), housing services (square feet Y) and household
services (energy use Z). Suppose utility takes the CES form
Utility =
U(X,Y,Z)
=
-1/X
-1/Y
-1/Z
Using the Lagrangian Technique demand functions for the three goods are
X
=
I/[PX+ ( PXPY).5 + ( PXPZ).5]
Y
=
I/[PY+ ( PYPX).5 + ( PYPZ).5]
Z
=
I/[PZ+ ( PZPX).5 + ( PZPY).5]
Letl I = 100, PX = 1, PY = 4 and PZ = 1 then optimal levels of the goods become
8
X*
Y*
Z*
=
=
=
25
12.5
25
Hence 25 (1) = 25 is spent on food, and 12.5*4 + 25*1 = 75 is spent on housingrelated needs.
Let’s assume that housing services and housing operations move together,
Define a composite commodity H = 4Y+1Z. Here define the initial PH =1
(arbitrarily) The initial quantity of housing is simply the total dollars spent on housing
H
=
4(12.5) + 1(25)=
75
Further, since all price increases are by assumption proportional, the following
price relationship holds
PH
=
PZ
=
.25PY
Now we can rewrite the demand function for X as
X
=
=
I/[PX+ ( 4PXPH).5 + ( PXPH).5]
I/[PX+ 3(PXPH).5]
As before, I = 100, PX = 1 and PH = 1 so X* = 25. Housing is most easily
calculated from the budget constraint as everything but food. H = 75.
An increase in housing costs. Suppose the prices of Y and Z increase proportionally to
PY = 16 and PZ = 4. Then PH also rises proportionally to PH =4.
Then demand for X would fall to
X
=
=
100/[1+ 3(4).5]
100/7
Housing purchases are given by
PHH
=
100
-
100/7 =
600/7
Or, with PH = 4
H*
=
150/7
This is exactly the level of housing purchases that arises from inserting PX = 4 and
PY = 16 into the original demand equations
X*
=
100/7
Y*
=
100/28
Z*
=
100/14
9
So the total amount of “housing” consumed is
H*
=
4Y* + 1Z*
=
150/7
Hence, we obtain the same responses regardless of whether we use the composite
approach, or look at proportional changes in the housing related goods separately.
E. Home Production Attributes of Goods and Implicit Prices.
To this point, we have skirted the issue of why goods are complements and substitutes. In
closing this chapter, we outline briefly some models that economists have developed to
gain insight into these issues.
1. Household Production Model. Assume that individuals do not
derive utility directly from goods they purchase in the market, but rather,
only from those goods, used as inputs, along with time. Raw hamburger,
doesn’t provide utility, for example, rather raw hamburger, along with
cooking and cleanup time causes utility.
This problem can be analyzed by considering these goods as a
household production function. Specifically. Consider three goods, X, Y
and Z, each of which can be purchased in the market. Utility comes from a
combination of these goods. For example, suppose
U(a1, a2) where
a1
=
f1(X, Y, Z)
=
f2(X, Y, Z)
and
a2
As before, the individual maximizes utility subject to the standard budget
constraint
PXX + PYY + PZZ
=
I
We won’t derive results here in detail, but two interesting observations are
possible from the structure of the problem.
First, such a model clarifies the relationship between market goods.
Essentially, the production relationships a1 and a2 are, at least in principle
observable. Thus, this approach allows us to treat households as multi-product
firms
Second, and more importantly, this setup provides a framework for
establishing “implicit” or “shadow” prices of home-produced goods a1 and a2. We
illustrate this with a very simple variant of the home production model, that of
linear attributes.
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2. The linear attributes model K. J. Lancaster developed a particularly
simple variant of the household production model to examine the attributes of
goods that generate utility for a consumer. Consider, for example food
consumption. Rather than considering separately the utlity of a very wide array of
foods, consider the attributes of the different foods, for examples, calories (a1) and
vitamins (a2) that the foods provide.
In simple form, suppose that aX1 denotes the caloric content of food X.
Then we may write.
a1
=
aX1X+ aY1Y+ aZ1Z
a2
=
aX2X+ aY2Y+ aZ2Z
Now consider the amounts of Calories and vitamins available from each of the
goods. If a person consumed only X, then their budget would be
I = pXX
so
X
=
I/pX.
a2
Thus, caloric intake becomes a1*= aX1(I/pX) and vitamins becomes a2*= aX2(I/pX), as
illustrated at point X* along ray X
X
in the figure below. Similarly for
points Y* and Z* along the
Y
respective rays Y and Z.
Now the ray X* to Z*
illustrates combinations of a1 and
Z
a2 that may be purchased with
a2*
X*
budget I and combinations of X and
Y*
Z. Similarly for the cords X and Y
and Y and Z.
Z*
One thing immediately obvious
from the figure is that rational
consumers would never consume
a1
a1*
all three goods. Rather, individual
with a preference for a1 would
consume Y and Z, while consumers with a preference for a2 would consume X and Y.
Thus the attributes model predicts the following
a. Consumers will purchase fewer goods than the maximum available
(particularly if there are fewer attributes than goods)
b. Second if preferences or prices change, consumption patterns may shift
abruptly. This is interesting in that, as is observed with households, discontinuous shifts
in consumption bundles occur in response to small price changes. A small increase in
wages for females, for example may cause women with children to leave the house and
work full time (rather than part time) because the attributes of work include may include
11
social relations and the satisfaction associated with maintaining out of the house
responsibilities in addition to income.
Appendix: Separable Utility and the Grouping of Goods. One shortcoming of
general utility theory is that it says relatively little about demand relationships between
goods. Other than the result that net substitution effects are symmetric, virtually any type
of relationship is possible. Stronger results can be obtained only by placing more
restrictions on the utility function. One particularly useful restriction is the assumption
of separability, or an assumption that consumption decisions about one good (or group of
goods) do not affect the utility associated with consuming other goods. For example, in
the public goods literature it is typically assumed that the amount consumers decide to
give to charity does not affect directly the utility of other goods derive from other goods.
Simple Separability. Consider the following simple case. Suppose an individual
consumes three goods, X1, X2 and X3, and that his or her utility is additively separable
U(X1, X2, X3)
=
U(X1) +
U(X2) +
U(X3)
Where Ui’>0 and Ui”<0 for i = 1, 2, and 3.
Under these conditions, it is easy to show that X2/P1 and X3/P1 must have the
same sign (either gross substitutes or gross complements). Since MUi/P is the same for
all goods, a rise in P1 must cause X2 and X3 to move in the same direction.
Separability into Groups and Two-Stage Budgeting. More generally, a utility function
for some n goods
U(X1, X2, … XN) may be partitioned in to k mutually exclusive groups
U[U1(Xg1), U2(Xg2),… , UK(XgK)]
(Such as food, clothing, shelter, etc.)
An individual with such a utility function will engage in “two-stage” budgeting.
That is the individual will decide on a clothing allowance, a food allowance, a housing
allowance, etc. based on the relative composite prices of these good classes, prior to
deciding how much to spend on each product within classes. The proof of this is
analogous to the composite commodity theorem. However, one result of such
separability is that it allows spending to be grouped into classes for empirical
applications.
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