Download Econ 604 Advanced Microeconomics

Econ 604 Advanced Microeconomics
Davis
Spring 2004
10 February 2004
Reading.
Problems:
Chapter 5 (pp. 116-144) for today
Chapter 6 (pp. 152-170) for next
time
To collect: Ch. 4: 4.2, 4.4, 4.6, 4.9
Next time: Ch. 5 5.1 5.2 5.4 5.5
Lecture #5
REVIEW
IV. Utility Maximization and Choice
A. An Introductory Illustration. The two good case.
1. The Budget Constraint
2. First Order Conditions for a Maximum
3. Second Order Conditions for a Maximum
4. Corner Solutions
B. The n-Good Case
1. First Order Conditions
2. Implications of First Order Conditions
3. Interpreting the LaGrangain Multiplier
4. Corner Solutions
C. Indirect Utility Function
D. Expenditure Minimization
PREVIEW
V. Income and Substitution Effects
Some final observations about utility maximization
1) The lump sum principle. Last time, I failed to make an observation regarding the
indirect utility function that will be useful later. Recall, the indirect utility function
expresses Utility for a consumer in terms of “observables”, prices and income levels.
Consider, for example the problem of maximizing U = X.5Y.5 subject to the constraint
that I = pxX + pyY. Then the Lagrangian expression becomes
L
=
X1/2Y1/2
+
(I- PxX -PyY)
Taking FONC
L/ X =
1/2X-1/2Y1/2
-
px
=
0
L/ Y =
1/2X1/2Y-1/2
-
py
=
0
1
L/  =
I-
pxX
-
pyY
=
0
Taking the ratio of the first two expressions yields
Y/X =
px/py
Solving this expression for Y (or X) and inserting into the budget constraint allows for
expression of the individual demand functions
X*
= I/2px
Y*
=
I/2py.
Inserting these into the utility function yields our indirect utility
V
=
(I/2px)1/2(I/2py)1/2
=
I/(2px1/2py1/2)
In the case where I = 2, px = .25 and py = 1. V = 2/(2(1/4)).51.5)=2
Now, consider the effects of a tax that collected $.50 of revenue. This tax could be
collected in one of two ways. One option would be to tax income by $.50. Another would
be to tax a good, say X by $.25 (Recall that proportional expenditures are constant in a
Cobb Douglas function. The consumer spends 50% of his income on X in this case. At a
price of $.25 (s)he consumes 4 units. At a price of $.50 (s)he consumes 2 units.
An important point in the economics of taxation is that utility falls less with an income
tax (a lump sum tax) than with a tax on goods. To see this, observe that with an income
tax I falls to 1.5 and
V = 1.5/[2(.25) .5(1).5 ] = 1.5
With an increase in the price of good X, income stays the same, but px increases
V = 2/[2(.5) .5(1).5 ] = 1.41
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.
Outline
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 Fall
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
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
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. Homogeniety 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.
Y
Y
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 expansions in utility. The question is whether the
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
An Inferior Good
X
I1
I2
I3
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 n
2. Engel’s Law: One of the most robust findings regarding inferior goods 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 = Px(I)[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.
C. Changes in the Price of a Good. Now let’s consider a third comparative statics
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 Fall – 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.
I1=Px 1X+Py Y
I1=Px 2X+Py Y
X1
X2
U2
U1
Substitution Effect
Income Effect
X
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.
Y
2. Graphical Analysis – Price Increase
I1=Px 2X+Py Y
I1=Px 1X+Py Y
X2
X1
U2
U1
Income Effect
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.
Y
3. Effects of Price Changes for Inferior Goods. For inferior goods,
substation and income effects work in the opposite directions. Most generally, the net
result will still be an inverse relationship between price and quantity.
The panel on the left illustrates (or
I1=Px 1X+Py Y
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!
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.
Y
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.
I1 =Px 1X + Py Y
2
2
I =Px X + Py Y
I3 =Px 3X + Py Y
U3
U2
U1
X2
X3
X1
X2
X3
X
P
X1
Notice in the figure to the right, that
as px falls, reflected by the
progressively flatter indifference
curves, the quantity of x increases.
This give 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.
P1
P2
P3
X
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*
=
Setting I = 100 generates the demand curves
.7I/Py
X*
70/Py
=
30/Px
and
Y*
=
Notice that a new I would cause an outward shift in each demand curve.
Notie 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) =
.5
.5
X + Y . The demand functions are given by
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
=
(100/Px)[1 + Px] -1= 100/[ Px + Px2]
Notice that a new higher I would shift the demand curve out, and a new highe 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 utilty are held constant. The curve therefore illustrates only
substitution effects. Mathematically, the curve is a two-dimensional representation of the
compensated demand function.
Px 1/Py
Px 2/Py
Px 3/ Py
P1
P2
P3
U
X1
X2
X3
X
X1
X2
X3
X
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.
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 utilty.
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
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