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Econ 604 Advanced Microeconomics
Davis Spring 2005, March 24
Lecture 8
Return Examinations
Reading.
Chapter 7 (pp. 172-194)
Next time Chapter 8 (pp. 198-224
Problems: 6.9 7.1, 7.3; 7.5; 7.9
REVIEW
I. Shephard’s Lemma, Roy’s Identity and Price Indices. Recall that the one can
get uncompensated demand from the maximization problem by Roy’s Identity. One can
get compensated demand from the expenditure function via Shepard’s Lemma. Both are
applications of the envelope theorem.
Shepard’s Lemma
E(Px,Py,V)/  Px
=
hx (Px,Py,V)
Roy’s Identity
dx(Px, Py, I)
U/Px /U/I
Finally, constructing price indices on the bases of compensated rather than
uncompensated demand would result in a downward adjustment of cost of living indices.
VI.
Chapter 6. Demand Relationships Among Goods. We considered the
effects of increases in the price of one good on the demand for another. In the 2 good
case, we showed that income and substitution effects again came into play.
A. The two good case. Goods are gross substitutes if dX/PY>0 and
Gross Complements if dX/PY<0. However, the definitions of gross substitutes and
groos complements are a bit clumsy, because income effects make the relations between
the goods asymmetrical. That is, good X may be a gross complement for good Y while
good Y may be a gross substitute for X
B. Net Substitutes and Complements. Goods are net substitutes if
hX/PY>0 and net complements if hX/PY<0.
- Net substitutes and net complements are symmetric (due to Young’s theorem. –
hX and hY are first derivatives of the expenditure function. Young’s theorem states that
the order in which you take cross partial derivatives does not matter)
- Further, because we have diminishing MRS most goods are net substitutes (and,
in a two good world, goods must be net substitutes.
C. Representing Analytically Substitutes and Complements. We showed,
via a Slutsky type decomposition that
X/PY = dX/PY
=
X/PY|U* -
1
Y(X/I)
PREVIEW
Appendix to Chapter 6. Separabililty.
VIII. Chapter 7 Market Demand and Elasticity. Tonight we consider two final
components of standard demand analysis; (a) Converting individual demand curves into
a market demand curve and (b) elasticities. The chapter is organized as follows
A. Market Demand Curves
1. The two consumer case
2. The n consumer case
B. Elasticity
1. Motivation and a general definition
2. Price Elasticity of Demand
3. Income Elasticity of Demand
4. Cross Price Elasticity of Demand
C. Relationships Between Elasticities
1. Sum of income Elasticities for all Goods
2. Slutsky Equation in Elasticities
3. Homogeneity
D. Types of Demand Curves
1. Linear Demand
2. Constant Demand Elasticity
Lecture________________________________________________
3. 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.
2
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. However, one result of such
separability is that it allows spending to be grouped into classes for empirical
applications.
But observe what separabilty does and does not imply
1)
Separability implies cross price elasticities are zero. However, this
does not imply that goods are unrelated, because income effects may
always be exist. For example, in a 2 good world, both products must
necessarily be gross substitutes, just to equate the marginal utility per
dollar spent relationship.
2)
More generally, you can say nothing about gross substitutes or gross
complements for separable goods
3)
Separability is not invariant to a monotonic transformation of utility.
Example
U(X,Y) = XY
Then ln U
=
lnX
+ lnY
Observe that dlnU/dX =
So d2lnU /dXdY
=
1/X
0
However, the same is not true for the orginal function
dU/dX = Y
d2U/dXdY
=1
(Problem 6.9)
VIII. Chapter 8 Market Demand and Elasticity. We have considered in some detail
price and quantity effects for a particular consumer. Suppose now we consider the
effects of aggregating across consumers. We also devote some attention to the elasticity
measures very widely used in empirical work
A. Market Demand Curves
1. The two consumer case. Consider an economy that consists of two
consumers (person 1 and person 2). Their individual (uncompensated) demand curves for
a good X may be written as
X1
=
d1x(PX, PY, I1) and
3
X2
=
d2x(PX, PY, I2)
Market demand is simply the sum of individual demands for good X. Thus
Total X
=
X1
+
X2
=
D(PX, PY, I1,I2)
=
d1x(PX, PY, I1) + d2x(PX, PY, I2)
Observations:
- If each individual’s demand curve for good X (holding PY,I1 and I2 fixed)
is down-sloping, market demand for good X will be down-sloping as well.
- Market demand here is created from uncompensated individual demands.
Compensated Market demand could be constructed in the same way.
Graphically, market demand is simply the horizontal summation of individual
demands
P
P
P
P1
X1
Individual 1
X2
Individual 2
X1 +
Market
X2
Factors that shift individual demands would generally shift market demand in a
similar manner.
- A change in the price of a related good can affect all individual demands
uniformly.
- Income effects, however, are a bit more complicated, since incomes can
change differently for different individuals. The way that the distribution
of income changes can affect demand. (This is often overlooked)
Example:
Consider two consumers with the following simple linear demand curves for
Oranges
X1
X2
=
=
10
17
-
2PX
PX
+
+
PX
=
Price of oranges
Where
4
.1I1
.05I2
+
+
.5PY
.5PY
Ii
PY
=
=
Individual i’s income (in thousands of dollars)
Price of Grapefruit (a gross substitute for Oranges)
Market Demand becomes
DX
=
X1 + X2
=
27 - 3PX
+ .1I1 + .05I2 +
PY
(Notice that we can sum across PX and PY, assuming the law of one price. On the other
hand, we cannot sum across individual incomes.)
To graph DX, we need values for the variables other than own price. Let I1 = 40, I2 = 20
and PY = 4. Then
DX
=
=
=
27 - 3PX
27 +9
36
+ .1(40) + .05(20)
3PX
3 PX
+
4
If the price of grapefruit were to rise to $6, then demand would shift out to
DX
=
=
=
27 - 3PX
27 + 11
38
+ .1(40) + .05(20)
3PX
3 PX
+
6
On the other hand, setting PY = 4 again, impose a redistributive income tax that takes 10
from 1 and transfers it to 2. The following results:
DX
=
27 - 3PX
+ .1(30) + .05(30)
+
4
=
27 + 8.5
3PX
=
35.5
3 PX
Notice that none of these changes affects the market coefficient on own prices.
2. The n consumer case This simple analysis with two consumers extends readily to the
case of n consumers. Given a representative consumer j with demand for good i
Xij
=
dij(P1, … , Pn, Ij)
Then the market demand for m consumers would be
Xi
=
 Xij
=
Di(P1, … , Pn, I1, I2,…,In )
B. Elasticity
1. Motivation and a general definition As we have seen, economists are
often interested in the way that one variable A affects another variable B. Economists,
for example, are often interested in the way that changes in various prices affect the
quantity demanded of a good. An important problem with such comparisons is that the
variables are not measured in readily comparable terms (for example, steak is measured
5
in pounds, the price is in dollars. Oranges may be sold by the dozen, and price changes
may be on the order of dimes).
One coherent way to address these different units of measure is to denominate all
these changes in percentage terms. This of elasticity
eB,A
=
percentage change in B
percentage change in A
=
B/B =
A/A
BA
AB
Notice from the definition that elasticity is a(n inverse) slope coefficient, weighted
by a location. We know, for example, that apple consumption will decrease with an
increase in the price of apples. However, the location allows us to speak more
meaningfully of the magnitude of the response.
2. Price Elasticity of Demand Perhaps the most important elasticity
measure is own price elasticity, or the responsiveness of changes in a price on the
quantity of that good consumed.
eQ,P
=
a. Definition
percentage change in Q
percentage change in P
=
Q/Q =
P/P
QP
PQ
Barring a Giffen Good relationship, own price elasticity measures are always negative
numbers (since Q/P<0). However, economists often divide goods by the magnitude of
the quantity response. If |eQ,P|< 1, then consumers are said to be insensitive, or inelastic
consumers of a good. On the other hand, if |eQ,P|> 1 then consumers are said to be
sensitive or price elastic.
b. Price Elasticity and Total Expenditures. A common way to
explain these notions of “sensitivity” and “insensitivity” is in terms of the effects of a
price change on total expenditures. Recall, that total expenditures equal PQ. Write Q as
a function of P (for example Q = 10 –P; to raise quantity one must lower price). Then
take the partial derivate w.r.t. P.
PQ(P)
P
=
Q
+
Q(P)
P
+
eQ,P )Q
Pulling Q out of the RHS
TR
P
=
(1
Notice that TR will move with price if demand is inelastic ( |eQ,P|<1) and TR will move
inversely with price if demand if elastic (|eQ,P|>1).
Graphically, this can easily be seen by considering price changes at different points along
a linear demand curve.
6
P
P
P
Price box
P
1
Qty Box
Qty Box> Price Box
Qty Box = Price Box
Qty Box < Price Box
Elastic Segment
Unitary Elastic Segment
Inelastic Segment
In the leftmost panel, observe that when price changes, the effects on total revenue can be
divided into a “price box” and a “quantity box”. In the case of a price reduction, for
example, the price box is the revenues lost from units that would have sold at the higher
price (Dimension PQ). The quantity box denotes the extra revenues realized from lower
the price (Dimension QP). The left panel illustrates a situation where TR moves
inversely with the price change. This is an elastic segment of the demand curve (recall
|eQ,P| = |(Quantity box)/ (Price Box)| = |QP /QP| >1). People are price sensitive in the
sense that total revenue increases when price falls.
The right most panel illustrates an inelastic segment (|eQ,P| = |(Quantity box)/
(Price Box)| = |QP /QP| <1), Here consumers are price insensitive, in the sense that
TR falls with a price reduction – or, equivalently, TR increases with a price increase.
More revenues are gained from consumers staying in the market and paying the higher
price than are lost from consumers leaving the market.
The middle panel illustrates a unitary elastic segment, where the price box just
equals the quantity box (and eQ,P| = |(Quantity box)/ (Price Box)| = |QP /QP| =1). Here
the price and quantity boxes indicate that price and quantity effects are exactly offsetting.
3. Income Elasticity of Demand Another standard type of elasticity
assesses the responsiveness of consumers to a change in income levels. This is termed
income elasticity of demand
eQ,I
=
percentage change in Q
percentage change in I
=
Q/Q =
I/I
QI
IQ
Unlike own price elasticity, income elasticity can be positive or negative. The sign and
the magnitude of income elasticity is important.
eQ,I > 1
0
<
eQ,I
eQ,I
<
<
1
0
goods are luxury goods or cyclical normal
goods (e.g., automobiles)
goods are normal goods (e.g., food)
goods are inferior goods
7
Notice: Recall here the discussion at the outset of this chapter. We can
only calculate income elasticities across individuals if we invoke some assumption
about the way that a change in income is distributed across market participants.
(We might, for example, and if reasonable, assume that all participants realize the
same percentage change incomes. See, for example problem 7.1)
4. Cross Price Elasticity of Demand Another standard elasticity deals with
the response of one good to the change in the price of a related good. This is
termed cross price elasticity
eQ,P’
=
percentage change in Q
percentage change in P’
=
Q/Q =
P’/P’
QP’
P’Q
As with income elasticities, cross price elasticities can be positive or negative.
The sign is important.
eQ,P’>0 implies goods are substitutes
eQ,P’<0 implies goods are complements (the same as own price changes)
Example: Elasticities are easily understood in terms of a linear demand function.
For example, consider the market demand function
X
=
10
-
2PX
+
.1I1
+
.5PY
Obviously, the positive coefficient on the income parameter indicates that the
good is a normal good, while the positive coefficient on PY indicates that the products are
substitutes.
Suppose PX = 5, I = 40 and PY = 4. Then
X
=
10
-2(5) +
.1(40) +
=
6
Then eQ,P
= -2(5)/6
=
eQ,I
= .1(40)/6
=
eQ,P’
= .5(4)/6
=
.5(4)
-1.67, the firm is on the elastic portion its
demand curve
0.67, the product is a normal, noncyclical
good.
0.33, good Y is a substitute.
C. Relationships Between Elasticities
1. Sum of Income Elasticities for all Goods. We have developed elasticity
in terms of market demand. If we further treat individuals uniformly as representative
consumers, then we can derive some important relationships between elasticities.
Consider the case of homogeneous customers with two goods, forming the market budget
constraint is
8
PXX + PYY =
I.
Market demands for X and Y are
X
=
dX(PX, PY, I)
and
Y
=
dY(PX, PY, I).
Further, assume these demand functions are homogeneous of degree zero in prices
and income. Differentiating the budget constraint w.r.t. I,
PX(X/I) + PY(Y/I)
=
1
This expression can be readily converted into elasticities .
(PXX/I)(X/I)I/X + (PYY/I )(Y/I)I/Y
writing PXX/I = sx
sxeX,I
=
1
and PYY/I = sy yields
+ sYeY,I
=
1
Thus, the share weighted sum of the income elasticities for all good equals 1.
(e.g., if income increases 10%, expenditures must increase 10%). Thus, for every
“luxury” good (with income elasticity greater than 1) there must be offsetting goods with
income elasticities less than 1.
2. Slutsky Equation in Elasticities Recall in chapter 5 the Slutsky equation
X
PX
=
X |
PX |U = constant
-
X X
I
This expression may be converted into elasticities by multiplying by PX/X
X PX =
PX X
X PX |
PX X |U = constant
eXP
=
eSXP
eSXP
denotes the compensated price elasticity of demand
and
denotes the share of income spent on X.
-
-
X[ X I ] PX
IX I
eXI sx
Where
sx
Notice that the above (bolded expression provides an additional reason to use
uncompensated demand: When sx is small uncompensated and compensated elasticities
are very similar.
9
3. Homogeneity As a final example of relationships between elasticities, we
exploit the fact that demand functions are homogeneous of degree zero in prices and
income.
Prior to developing this relationship, we review Euler’s Theorem. A function that
is homogeneous of degree m, means that, for any t>0.
f(tX1, tX2, …. , tXn) = tmf(X1, X2, …. , Xn).
Thus a function that is homogenous of degree zero implies
f(tX1, tX2, …. , tXn) = f(X1, X2, …. , Xn).
Euler’s theorem states that a function that is homogeneous of degree m, then
f1X1+ f2X2 +… fnXn = mf(X1, X2, …. , Xn)
Example: f(X,Y) = 10X + 20Y. Notice that increasing X and Y by k inflates the function
by k.
k10X + k20Y =
k(10X + 20Y)
=
kf(X,Y).
Thus, this function is homogenous of degree 1. Trivially,
fxX + fyY
=
10X + 20Y
=
f(X,Y)
Example: Demand relationships are typically homogeneous of degree zero. Recall, for
example, in the case of a Cobb-Douglas utility function in X and Y
dx
dy
=
=
X
Y
=
=
I/2Px,
I/2Py
Increase I and Px by k will recover not affect demand. So this function is
homogeneous of degree 0. (Note, you can take the quantity-weighted derivatives of dx or
dy and you will find that they sum to zero)
When m=0, then the Euler’s theorem states that the sum of the quantity weighted
first derivatives equals zero. Consider a demand function X = dx(Px, Py, I)
By Euler’s Theorem+
(X/PX) PX + (X/PY)PY
+(X/I)I
=
0
Convert to elasticities by dividing by X,
(X/PX) PX /X+ (X/PY)PY/X
+(X/I)I/X
eX,Px + e X,Py + e X,,I = 0
10
=
0/X
This is another way to state the homogeniety of degree zero property of demand
functions. An equal percentage change in all prices and incomes will leave the quantity
demanded of X unchanged.
Example: Cobb-Douglas Elasticities Consider the Cobb Douglas demand function
U(X,Y)
=
XY where  +  = 1.
Demand functions are
X
=
I/PX
Y
=
I/PY
The elasticities are easy to calculate. For example,
eX,Px
=
=
=
(X/PX) PX /X =
(-I/PX2)( PX2 /I )
-1
=
1
eX,Py
=
0
eY,Py
=
-1
eY,I
=
1
eY,Px
=
0
Similarly
eX,I
(-I/PX2)( PX /X)
Hence these demand functions have elementary elasticity values. Further
sX
=
PXX/I =
sY
=
PYY/I
PXI/PXI
=

=

The constancy of income shares provides another way of shown the unitary
elasticity of demand.
Homogeneity holds trivially,
eX,Px
+
e X,Py +
e X,,I = 0
-1
+
0
-1
+
=
0
Finally, consider the elasticity version of the Slutsky equation.
eXP
=
eSXP
-
11
sx eXI
-1
=
eSXP
=
-(1 - ) = -
-
(1)
Thus
eSXP
In words, the compensated price elasticity of demand for one good is the income
share for the other good. This is a special case of the more general result that
eSXP
=
-(1 - sx)
where  is the elasticity of substitution (between two goods) in chapter 3 (note 6).
(To see this, note that in the case of a Cobb Douglas function =1.)
D. Types of Demand Curves. Economists consider various types of demand
forms. Here in closing we consider some of the problems associated with two of these
functions.
1. Linear Demand Consider a demand function of the form
Q = a + bP + cI + dP’
Where a, b, c and d are demand parameters, and .
b<0 (the good is not a Giffen good)
c>0 (the good is a normal good)
d< > 0 if the related good is a gross substitute or a gross complement.
Holding I and P’ constant
Q = a’ + bP
Where a’
=
a + cI + dP’. Clearly this describes a linear demand curve.
Further, changes in a’ will shift demand. Despite the simplicity of this demand
statement, linear demand has the deficiency that elasticity changes as one moves along
the demand function. To see this notice that
eX,Px
=
(X/PX) PX /X = bP/Q
Obviously as P rises Q falls, and demand becomes more elastic.
Example: Linear Demand. Consider a demand function
Q
=
36 – 3P.
Price elasticity of demand is
eX,Px
=
-3P/Q =
-3P/(36 – 3P)
12
Notice demand is unit elastic when P = 6. For P>6 demand is elastic. For P<6
demand is inelastic. Because elasticity varies with price on a linear demand curve, one
must be very careful to specify the point at which elasticity is being measured. In
empirical work with linear demand curves, it is conventional to report the arc price
elasticity, that is, the average elasticity over the average price prevailing for a period.
2. Constant Elasticity Demand If one want to assume that elasticities are constant
use an exponential function.
Q
=
aPbIcP’d
Where a>0, b<0, c>0 (a normal good) and d<>0, as for the linear good. Notice that one
can easily “linearize” such a function by taking natural logarithms (ln)
lnQ
=
lna
+ blnP + clnI + dlnP’.
Notice that one can estimate the parameters of such a function with ordinary least
squares. Notice also that
eQ,P = (Q/P) P /Q =
=
b aPb-1IcP’d P/(aPbIcP’d)
b
Thus, the price elasticity of demand is constant.
Example: Elasticities, Exponents and Logarithms. Notice in the above example
that income and cross price elasticities are also directly read from the exponents of the
demand functions
eQ,I
=
c
;
eQ,P’
=
d
Therefore, from a linear regression, one can read elasticities without having to
make any mathematical computations. For example, if one estimated
lnQ = 4.61
We know that
eQ,P
=
and
eQ,P’ =
-
1.5lnP + .5ln(I) + ln(P’)
-1.5
1
;
eQ,I
=
13
.5