Download Econ 604 Advanced Microeconomics

Econ 604 Advanced Microeconomics
Davis
Spring 2005
16 February 2006
Reading.
Chapter 5 (pp. 116-130) for today
Chapter 5 (pp. 130-144) for next
time
Problems:
To collect: Ch. 4: 4.2, 4.4, 4.6, 4.7
4.9
Next time: Ch. 5 5.1 5.2 5.4
Observe: 5.1 and 5.2 don’t fit into the
standard Lagrangian setup, because, in each case, the products are perfect substitutes.
Below a given price level, the consumer will devote all income to only one product 5.1 is
a case of perfect substitutes. 5.2 pertains to perfect complements.
Lecture #5
REVIEW
IV. Utility Maximization and Choice
A. An Introductory Illustration. The two good case. (This is largely a graphical
representation)
1. The Budget Constraint
2. First Order Conditions for a Maximum
Px/Py =
Ux/Uy, which implies
Ux/Px =
Uy/Py
Intuitively: Purchase goods until the MU per dollar spent on each good is the
same.
3. Second Order Conditions for a Maximum. The intersection of first
order conditions will be a maximum if the quasi-convexity conditions are satisfied (that
is, for any order pairs on an indifference curve U, (xo, yo), (x1, y1), it is the case that
U(xo + (1-)x1, yo + (1-)y1)> U
This condition will be satisfied if the quasi-concavity condition in 2.107 is met. It will
alternatively be satisfied if d2Y/dX2.>0
Example, Suppose U* = XY. Differentiating,
dU = YdX + XdY.
On a level curve, this expression equals zero, so
dY/dX = -Y/X
1
Now suppose that you wanted to check the concavity of this level curve.
What if we just took the derivative of the first order condition w.r.t. X? That is,
d2Y/dX2 = Y/X2 > 0.
Does this imply convexity in XY space? No! We must recognize that Y is
implicitly a function of X. Rather, we have two (ultimately equivalent ways) to
check for the concavity of the indifference curve. One is to totally differentiate U
twice, subject to the constraint that utility is constant. We developed the sufficient
condition for concavity of the constrained utility function in equation 2.107
[U22U11 -2 U1U2U12 + U12 U22]/ U22 <0
For U = XY,
U1 = Y,
U2 = X,
U11 = 0,
U22 = 0,
U12
=
1
Inserting,
-2YX/X3
= -2Y/X2 <0 (Note the sign is reversed relative to the standard
second order condition. This is due to the way 2.107 is set up)
The alternative approach is to solve the expression YX = U directly for Y and then
differentiate twice. For example, let U = U* a constant. Then
Y = U*/X
dY/dX = -U*/X2 (with U* =XY, dY/dX = -Y/X)
d2Y/dX2 = 2U*/X3 = 2Y/X2 <0
4. Corner Solutions
B. The n-Good Case
1. First Order Conditions.
For n goods, an optimizing consumer purchases all products so that the MU per
dollar spent is equal for all goods.
2. Implications of First Order Conditions
3. Interpreting the LaGrangian Multiplier
C. Indirect Utility Function
We can express U(x1, x2,….xn) +(I –p x1 - …- p xn) as
U(x1(p1…pn,I),……. xn(p1…pn,I)
or
V((p1…pn,I)
The advantage of this is that indirect utility is expressed in terms of observables.
Thus, given, of course our assumptions about the functional form of utility) we can talk
about change in utilities in terms of changes in observable variables. We used the
example of analyzing the effects of a lump sum tax to analyze this last time.
D. Expenditure Minimization
Associated with any utility maximization problem is a ‘dual’ problem where the
agent minimizes expenditures needed to achieve a given level of utility. That is
E = p1x1 + p2x2 + …pnxn
Subject to
U = U(x1,x2,….,xn)
L
=
p1x1+p2x2 +…. + pnxn + ( U - U(x1, x2, …., xn))
This is important from a practical level because the dual yields exactly the same first
order conditions as the original problem, and because it allows us to assess the
price/income tradeoffs necessary to maintain a status quo level of utility. For example,
given an assumption about the nature of utility, as well as prices and incomes, we can
compute the change in, say, income necessary to offset the effects of a price increase. As
we will see later, expenditures are directly observable, while utilities are not.
PREVIEW
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
C. Changes in the Price of a Good.
1. Graphical Analysis – Price Reduction
2. Graphical Analysis – Price Increase
3. Effects of Price Changes for Inferior Goods.
D. Individual’s Demand Curve
1. Shifts in the Demand Curve
E. Compensated Demand
1. Relationship between Compensated and Uncompensated
Demand
LECTURE_____________________________________________
V. Income and Substitution Effects. In this chapter we will use the utility
maximization model to examine how a consumer responds optimally to the change in the
price of a good. Our analysis will allow insights into the components underlying a
quantity response (e.g., income and substitution effects) as well as insights into the
ceteras paribus conditions that underlie the analysis of demand.
A. Demand Functions. As we saw in the last chapter, we can write a constrained
optimization problem involving n goods as a system of n+1 equations in n+1 unknowns.
These expressions, at least in principle, can be solved for each of the individual goods xi*
e.g.,
x1*
=
d1(p1, p2, …., pn, I)
x2*
=
d2(p1, p2, …., pn, I)
.
.
.
xn*
=
dn(p1, p2, …., pn, I)
These expressions d1 to dn are the demand functions for the individual. In this chapter, we
consider the effects of a change in prices and income on the consumption of the good.
Generally, these exercises are known as comparative statics analysis.
1. Homogeneity One of the most straight-forward of these comparative
statics exercises regards the effects of an equal percentage increase in all prices,
as well as the income. For such increases to affect preferences, consumers would
have to suffer “money illusion” (or perhaps money disillusion”). We presume
that consumers are not affected by such offsetting changes. Formally, we assume
that demand functions are homogeneous of degree 0 in all prices and income.
That is, for any constant t, we assume that
x1*
=
d1(p1, p2, …., pn, I) =
d1(tp1, tp2, …., tpn, tI)
Example #1: Consider the Cobb-Douglas utility function U(X,Y) = X.3Y.7.
Following our analysis from the last chapter, we can show that
X*
=
.3I/Px
and
Y*
=
.7I/Py
Clearly, doubling I and each of the prices would leave unaffected both X*
and Y*. Thus, Cobb-Douglas functions are homogeneous of degree 0.
Example #2: The CES utility function. Suppose that the demand function
was given by U(X,Y) = X.5 + Y.5. Following our analysis from the last chapter, the
demand functions are given by
X*
=
(I/Px)[1 + Px/Py] -1
Y*
=
(I/Py)[1 + Py/Px] -1
Again, doubling y, x and I do not affect X* and Y*, making these functions
also homogeneous of degree 0.
B. Changes in Income. Now let’s move on to more general comparative statics
effects. We start with an analysis of a change in income. In terms of an indifference
map, income increases will lead to a utility expansion. The question is whether the
Y
Y
consumption of a good X will increase (as shown in the left panel) or decrease, (as shown
in the right panel).
U3
U3
U2
U2
U1
X1 X2 X3
X
I1
I2
A Normal Good
I3
U1
X3
X2 X1
X
I1
I2
I3
An Inferior Good
Notice in the above chart that there’s nothing unusual about the indifference
curves in the panel on the right. Either relationship is possible. For some goods (such as
automobiles, education and housing) consumption usually increases with income. For
other goods (macaroni and cheese in a box, second hand clothing and generic beer)
consumption diminishes with income increases. We define these relationships as follows
1 .Normal and Inferior Goods. When consumption of a good increases with
income (x/I >0) the good is said to be a normal good. When (x/I <0) the good is said
to be an inferior good.
2. Engel’s Law: One of the most robust findings regarding consumer expenditures
regards the relationship between food expenditures and income. As first studied
by Ernst Engle (1857) the percentage of income devoted to food tends to fall as
income increases. This relationship is known as Engel’s Law. It has been
established over time and across cultures. For this reason, the percentage of
income spent on food is sometimes taken as a poverty measure.
Question: Does a conclusion that the share of income going to food falls as
income increases imply that food is an inferior good?
Answer: Not necessarily. Engel’s law implies that (PxX/I)/ I = PxI[(X/I) –
PxX]/I2 <0. Obviously, if X/I <0, the law holds. However, if PxX is large
enough, the relationship could be negative, even with X/I >0. (Thus, Engel’s
law does not imply that food is necessarily an inferior good.
C. Changes in the Price of a Good. Now let’s consider a third comparative static effect:
the effects of a price change. As will be obvious momentarily, the effects of price
adjustments are a bit more involved than income effects.
Y
1. A Price Reduction– A Graphical Analysis. Consider a demand function
for just two goods, X and Y. If the price of a good falls, then the Budget Constraint Y =
I/Py + (Px/Py)X will “flatten out. The new optimal consumption bundle is the point of
tangency between the highest attainable indifference curve consistent with the new
budget constraint.
1
I1=Px X+Py Y
I1=Px 2X+Py Y
X1
X2
U2
U1
X
Income Effect
Y
Substitution Effect
2. Graphical Analysis –
Price Increase
2
I1=Px X+Py Y
I1=Px 1X+Py Y
X2
X1
U2
U1
Income Effect
The increase in the consumption of
good X can be divided into two
parts, a substitution effect
attributable to the change in relative
prices along the original
indifference curve, and an income
effect which reflects the increase in
effective income available as a
result of the price reduction.
Observe in the chart on the left that
these two effects are reinforcing in
this case. This need not always be
true.
Substitution Effect
X
The quantity reduction due to an
increase in the price of a good X
may similarly be decomposed into
income and substitution effects.
Starting at original income I1, the
substitution effect associated with
increasing the price of a good is
found by rotating the budget
constraint back along the original
indifference curve. The income
effect then is the reduction in
effective income due to the price increase.
3. Effects of Price Changes for Inferior Goods. For inferior goods, substitution
and income effects work in the opposite directions. Most of the time, the net result will
still be an inverse relationship between price and quantity.
Y
The panel on the left illustrates (or
tries to illustrate the effects of a
price reduction on an inferior good.
Rotating the budget constraint
along the original indifference
curve generates a fairly sizable
2
substitution effect. However, the
I2=Px X+Py Y
parallel upward shift of the budget
line results in negative income
X1
X2
effect.
Still, in general, we would
U2
expect
that,
as shown here, the
U1
effects of a price reduction would
X
be positive, on net. It is not a
Income
Effect
Substitution Effect
logical impossibility that the
income effect might dominate a substation effect. Such goods are termed “Giffen Goods.
However, it is doubtfully the case that such goods exist. To see this intuitively, reflect
for a moment on what it would imply for a good to be a Giffen Good: As a consequence
of a price reduction, income increased enough that consumers no longer wanted the good!
I1=Px 1X+Py Y
D. Individual’s Demand Curve. It often convenient to express quantity as a function of
price, holding other things constant. This price/quantity relationship, for example,
represents the underpinnings of much of the graphical analysis in elementary economics.
Given the demand function for a good x, x* = d(px, py, I), we can write a demand curve
by holding all variables other than px constant.
Formally,
Individual Demand Curve: An individual demand curve shows the relationship
between the price of a good and the quantity of that good purchase by an individual
assuming that all other determinants of demand are held constant.
A demand curve can be readily derived from an indifference map, simply by considering
the quantities of X optimally chosen as the price of X, px, changes.
P
Y
I1 =Px 1X + Py Y
I2 =Px 2X + Py Y
P1
I3 =Px 3X + Py Y
P2
P3
U3
U2
U1
X1
X2
X3
X
X1
X2
X3
Notice in the figure to the right, that as px falls, reflected by the progressively flatter
indifference curves, the quantity of x increases. This gives rise to the standard inverse
relationship between price and quantities, as shown, for example, in the graph below.
1. Shifts in the Demand Curve. A change in either the price of good Y or income
I will cause a new demand curve to be constructed. It is important to keep in mind that a
demand curve is simply a two dimensional representation of an n dimensional
relationship.
Example #3: Consider the Cobb-Douglas utility function U(X,Y) = X.3Y.7.
Recall that the demand functions for X and Y are given by
X*
=
.3I/Px
and
Y*
=
.7I/Py
=
70/Py
Setting I = 100 generates the demand curves
X*
=
30/Px
and
Y*
Notice that a new I would cause an outward shift in each demand curve.
Notice also, however, that in this case, Py does not affect X and vice versa.
Example #4: Consider again the CES utility function given by U(X,Y) =
X + Y . The demand functions are given by
.5
.5
X*
=
(I/Px)[1 + Px/Py] -1
Y*
=
(I/Py)[1 + Py/Px] -1
Suppose we set I = 100 and Py = 1. Then the demand curve for X becomes
X
X
=
(100/Px)[1 + Px] -1=
100/[ Px + Px2]
Notice that a new higher I would shift the demand curve out, and a new higher Py
would shift the curve in (indicating that the goods are substitutes)
E. Compensated Demand Curves. Notice in the development of the above demand
curve, that the actual level of utility varies as price changes. This occurs, of course,
because income effects impact utility as well as substitution effects. Thus only nominal
income is held constant as the price falls, for example.
Although this is the most conventional way to construct a demand curve, it is not
the only way. Sometimes it is useful to construct a demand curve holding real income
constant. The idea here would be to “compensate” individuals with income increases or
reductions as prices change, so that they stay on the same indifference curve. These
compensated demand curves thus, illustrate pure substitution effects..
P
Y
Compensated Demand Curve: A compensated (or Hicksian) demand curve shows the
relationship between the price of a good and the quantity purchased on the assumption
that the other prices and utility are held constant. The curve therefore illustrates only
substitution effects. Mathematically, the curve is a two-dimensional representation of the
Px 1/Py
Px 2/Py
Px 3/ Py
P1
P2
P3
U
X1
X2
X3
X
X1
X2
X3
compensated demand function.
The panel above to the left illustrates the development of a compensated demand curve
from an indifference map. Notice that as the price of x falls, income is adjusted
(implicitly) so that the budget line remains tangent to the initial indifference curve. The
inverse relationship between price and quantity generate the standard demand curve,
shown above on the right.
The use of compensated or uncompensated demand is a matter of choice. In most
empirical work uncompensated demand curves are estimated, because they are the curves
for which data are available. However, compensated demand curves are useful in
theoretical work of welfare analyses, because it is often desirable to evaluate the effects
of changes that hold utility constant.
X
Example #5. Compensated demand functions. Consider the Cobb-Douglas utility
function U(X,Y) = X.5Y.5. The demand functions for X and Y are given by
X*
=
I/2Px
and
Y*
=
I/2Py
The indirect utility function can be solved by inserting X* and Y* back into the
utility function. This yields
Utility = V(I, Px, Py) =
I/(2Px.5Py.5)
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 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.
1. Relationship between Compensated and Uncompensated Demand
The same reference price vector, income level and initial level of utility, compensated
and uncompensated demand functions must, by construction intersect at one point.
However, at all other prices they differ. Intuitively the compensated demand is less
sensitive than uncompensated demand to price increases above the point of intersection,
because income supplements enable the consumer to purchase more of the good. On the
other hand at prices below the point of intersection, compensated demand is steeper,
since income must be taken away from compensated demand as price falls in order to
keep utility constant. This relationship is illustrated graphically below.
Price
h(x) - compensated
demand
dx - uncompensated
demand
hx > dx
Quantity
hx < dx