14 consumer surplus
TRANSCRIPT
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 1
Lecture notes for Intermediate Microeconomics
Part 1: Consumer Theory
Contents
1. Introduction: Microeconomics and the History of Economics
2. The economic agent: Assumptions of the model2.1 Preferences and Axioms of rational choice
2.2 Utility functions
2.3 The Marginal rate of Substitution
3. Utility maximization: Two methods
3.1 Substitution method*
3.2 Lagrangian multiplier method
3.1 Substituting the constraint into the objective function
3.3 Marshallian Demand functions
3.4. Mathematical issues*
3.5 The meaning of the multiplier*
4. Comparative statics
4.1 Change of income
4.2 Change of prices
4.3 Indirect Utility Function
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 2
5. Expenditure minimization and Hicksian demand
5.1 Derivation of the minimum expenditure function
5.2 Hicksian/Compensated Demand Curve
6. Income and Substitution effects
6.1 Graphical analysis
6.2 The Slutsky decomposition*6.3 Computing IE and SE for a discrete price change
7. Labor supply decision of the consumer
7.1 Introduction: Assumptions of the model
7.2 Optimal labor supply decision
7.3 Comparative Statics (for the CD-case)
7.3.1 Change in non-labor income
7.3.2 Change of Labor supply when wage rate changes
7.4 Income and Substitution effects
7. 4 Appendix: analytical derivation of the Slutsky equation*
8. Labor supply and taxes
8.1 A (linear) Income tax
8.2 Per capita tax
8.3 Taxes and Efficiency
Chapters marked with a * do not belong to the core topics and we will deal with them
only if time permits.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 5
2. The economic agent as a consumer: Assumptions of the
model
We will start with neoclassical Consumer Theory, which almost
coincides with the marginalist idea of economics.
In modern economics, especially microeconomics, theory begins
with a set of behavioral assumptions, which are formalized in a
mathematical way. This requires reducing the real world to amodel, in which only (some of) the relevant economic aspects of
behavior are included. The gain is the possibility of building a
formal framework, from which rigorous conclusions can be drawn.
2.1 Preferences and Axioms of rational choice
Indifference Curves and Preferences
In 1101 we modeled a consumer, or his/her taste for goods by so-
called indifference curves (see below for a definition). The
indifference curves represent what a consumer is (for economic
theory). Finding the highest indifference curve subject to a budget
constraint then is a way of describing what a consumerdoes.
As we will shortly see, indifference curves are derived from a
concept that describes the objective of a consumer (maximizing
happiness/utility) by the use of a so-called utility function. Before
we define the idea of utility functions and indifference curves, we
will take a glimpse at a more fundamental (and more general) way
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 6
of defining a consumer. This concept is based on mathematical set
theory; however, their meaning is very intuitive.
Preference orderings:
Consumption set C:
Defines the space of all possible consumption goods and their
combinations for a specific consumer.
Example 1:
C={2 apples, 1 DVD, 1 car, 3 weeks of vacation in Florida}
Example 2:
C={any goods I can buy in the market, given the amount is greater
or equal to zero, 24 hours of time per day,}
Preference relation >
A relation can be any definition of how two objects are related
(e.g. the symbol > relates numbers according to their value).
A preference relation defines how a consumer orders goods or
combinations of goods in terms of utility/benefit.
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Example 1:
1 DVD > 1 apple
This says that the consumer thinks that 1 DVD is as least as good
as 1 apple. It implies, that he or she is either indifferent between 1
DVD and 1 apple or she/he likes 1 DVD more than 1 apple (thats
why people call this is a weak preference ordering)
Axioms of rational choice
To go from the underlying preference orderings to the more
convenient utility functions (which imply indifference curves, see
next section), economists make the following assumptions on how
people choose goods from their consumption sets. Since we are not
claiming that this behavior can be always observed empirically, we
call them axioms.
(Axiom 1) Completeness: The agent is able to compare any two
bundles, i.e. he or she can always say whether bundle A is better,
equal or worse than another bundle B.
Examples: Usually, people know what they like more, or at
least they know that they are indifferent. However, in some
situations, it might not be so easy.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 8
1. Consider another type of ordering (i.e. this is not a
preference ordering), namely the ordering of cities on the
planet by the relationship: City A is west of city B.
Knowing that the earth is a sphere, it is not clear whether
L.A. is west of NYC or the other way round (although
common sense would say it is clear, which comes from the
custom, to draw LA and NYC on a map so that L.A. is inthe west).
2. A more economic problem would be the example of
someone looking for a job, but given a couple of job offers,
which superficially look equally good (say same wage,
location etc.), but the applicant does not know how the
working atmosphere in the prospective company will be and
therefore it might be impossible to say which option is better.
This is a problem of incomplete information that we will rule
out in the models we consider in this class (although it is
important problem that has been analyzed by many
economists) .
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(Axiom 2) Transitivity: If bundle A is better than B and B better
than C then it must be the case that A is better than C. (remember
that this corresponds to the fact that indifference curves must not
cross)
Example: Consider the ordering (not a preference ordering)
of the natural numbers: 1,2,3,Here it is obvious, that the ordering (greater or equal) is
transitive. If 3 1 and 5 3 than 5 1.
(Axiom 3) Continuity: Given that I can assume that all goods are
divisible (i.e. we assume it makes sense, to talk for example about
half of a car, a quarter of an apple etc.), we say preference ordering
is continuous ,
if A > B then for A very similar to A, we have A > B.
REMARK: This last assumption is an important assumption
but purely technical in its nature, so we will ignore it in this
course.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 10
From Preferences to Utility functions
With the above assumptions one can show (the famous economist
Gerard Debreu showed this first) that we can represent the above
defined preference orderings of an agent by a utility function.
2.2 Utility functions
The relationship between indifference curves and Utilityfunctions
In your introductory micro class you have already seen something
that implied the use of utility functions: the indifference curves.
Def: An indifference curve (IC) gives all combinations of goods
for which the consumer is indifferent (given a fixed Utility level,
i.e. U(x,y)=constant, whereas x and y vary).
In other words, the indifference curves are the level sets (or level
curve) of the utility function, i.e. the locus of all combinations of
goods (x and y) that yield the same utility level (see the following
graph for an example).
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Example of a utility function:
First define: N= minutes of taking a nap
T= minutes of watching TV soaps
Then there might be an agent who orders his or her preferences
according to the following Utility function:
U(N,T)=2N+T
Note, that we just made up a utility function forsome consumer;
the form of a utility function for another agent (e.g. you) might
look different. In particular, it depends on the underlying
preference ordering.
Now, take a look at the above mentioned utility function and
observe the following properties:
- The more of either N or T the agent consumes the higher is
the level of utility (i.e. utility is increasing in both goods).
- Consuming one unit of N gives me twice as much utility than
consuming one unit of T. In other words, to achieve the same
level of utility or happiness, the agent needs less N (half)
than T (with this information try to draw an indifference
curve of this agent).
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 12
- To achieve the same level of utility it does not matter
whether the agent consumes both goods, or only one of them
(besides the problem that consuming T is relatively more
costly, but this is another issue, namely that of the time
budget constraint). That means, the agent can (almost
perfectly) substitute one good for the other to get the same
level of utility.
Ordinality of Utility functions
Example: Given the number of minutes sleeping is constant (e.g. =
10), the utility of watching 1 minute TV is:
U=2*10+1=21
And of watching 5 min TV:
U=2*10+5=25
What does the difference in utility (25-21=4) tell us? - Nothing
besides the fact, that the consumer likes watching 5 minutes more
than just watching 1 minute. It does not say, however, by how
much he or she prefers 5 to 1 minute of T, the difference 4 has no
meaning except for its sign (i.e. being bigger or less than zero).
Thats why we call the utility functions we use ordinal. If the
difference of utility would carry information about how much
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 13
better I like some bundle of goods compared to some other bundle,
the utility function would be called cardinal.
Non-Uniqueness of Utility functions
Since Utility functions just represent the orderof preferences (i.e.
they are ordinal as explained above), every preference ordering can
be represented by several (indeed infinitely many) utility functions.
Example:
xyyxyxU
U
==
+5.05.0
2
),(
,:
Now, the utility function xyyxUyxG == 2)],([),( represents the same
preference ordering. In fact, all monotone transformations of a
utility function preserve the preference ordering.
Def.: A monotone transformation of a function U(x,y) is a function
f(U) that is (strictly) increasing.
Examples:
1. f(U) = U
2
2. f(U) = U
3. f(U) = logU
4. f(U)=2U+1
where U is always the original Utility function.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 14
A special Utility function: The Cobb-Douglas Utility function:
Def.(Cobb-Douglas Utility function): = 1),( yxyxU with
10
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Ad 1.: Marginal Utility (MU) is positive:
05.0
05.0
5.05.0
5.05.0
>=
>=
yxy
UMU
yxx
UMU
y
x
Ad 2.: Marginal Utility is decreasing, i.e. the second derivative is
negative
( )
( )05.0)5.0(
05.0)5.0(
5.15.0
2
2
5.05.1
2
2
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 17
dyyUdx
xUyxdU
+
=),(
Assuming that Utility stays constant, i.e. 0),( =xdU , we have:
dyy
Udx
x
U
+
=0 .
Now, solving for the expressiondx
dy , we obtain the formula for
the MRS:
y
U
x
U
dx
dyMRS
= .
Now we have developed the concept of the MRS and we can come
back to our initial question how the MRS relates to the property of
preferring average bundles.Consider some representative ICs of a CD utility function (which
has the property of decreasing MU as shown above). What fact can
we infer about the MRS? It is always decreasing (the absolute
value of the slope goes down, as x, which is on the horizontal axis,
increases). Intuitively speaking, this means, if I have little x and
many of y, the MRS is relative high, so I am willing to give upmany y for one additional (marginal) unit of x. If on the other
hand, I have already a lot of x, the MRS is low, and therefore, I
would be willing to exchange only little y for one more x (or to put
it differently: I would be willing to exchange a lot of x to receive
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 18
one more unit of y), given that utility stays the same (i.e. moving
on the same IC).
Having understood what is going on in economic terms, we will
now formally show, how to prove that the MRS is decreasing
given that MU for both goods is strictly negative (I will use the
short hand notation xU for xU
):
( )
( )
( )
}}
}}} }}
( ){
02
3
0
0
2
00000
2
0
3
22
2
2
>>>>0 are constant coefficients)
2. Linear preferences (perfect substitutes):
U(x,y)=ax+by
x
yIC 1
IC 2
IC 3
x
y
IC 1
IC 2
IC 3
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 21
3. Utility Maximization
In neoclassical theory, we assume that all agents are maximizing.
In particular, consumers are maximizing utility subject to a budget
constraint and firms are maximizing profits given technological
possibilities (in more advanced macro models, there may be also a
government, which could be maximizing aggregated utility of all
agents).Why maximization? Are you doing calculations when you go to
the grocery store?
Economics makes an idealization when considering consumers as
maximizing agents.
A way to justify this idealization was proposed by Paul Samuelson:
He introduced the concept of revealed preferences. That is
economics assumes agents to behave as if they were maximizing.
If an economist analyses demand data, he or she can check whether
these data are coherent with maximizing behavior.
We assume not only the consumer to be maximizing but
furthermore maximizing only her or his own utility, .i.e. the
consumer is supposed to be selfish. The consumer in our models
cares only about himself or herself. However, this needs not to be
the case in general. In the theory of externalities the possibility of
some form of altruism is introduced.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 22
Utility Maximization Problem
Mathematical Tools: Lagrangian Optimization
Graphical analysis:
Quantity of y
Quantity of x
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Solving for the OCB analytically
Two methods: I. Substituting the constraint into the
objective function
II. Lagrangian multiplier method
3.1 Substituting the constraint into the objective function
Consider the following example:
The objective function is:
5.05.02 ),(,: yxyxUU +=+
The budget constraint (BC) is: Mypxp yx =+
Solving the BC for x yields:x
y
x p
yp
p
Mx =
Substituting this into the objective function gives:
5.0
5.0
),( yp
yp
p
MyxU
x
y
x
+
=
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 24
Now, to find a maximum, set up the FOC:
0=
y
U,
which is in our example:
05.05.0 5.05.0
5.0
5.0
=+
=
+
yp
yppM
ppy
pyp
pM
y x
y
xx
y
x
y
x
or
)1(
1
22
x
y
yx
y
y
x
xx
y
p
pp
M
p
p
p
p
p
M
p
py
+
=
+
=
To find x, substitute the last expression into the expression for x
(from BC):
)1()1(
))1(
)1()1(
y
xx
y
xyx
x
y
xy
y
xy
x
x
y
y
x
y
xx
y
x
p
pp
M
p
ppp
pp
ppM
p
pp
M
p
M
p
pp
M
p
p
p
My
p
p
p
Mx
+
=
+
+
=
+
=
+
==
.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 25
If our calculations are correct, the quantities of x and y , multipliedby their prices should add up to M:
M
p
p
p
p
p
p
p
p
M
p
p
M
p
p
M
p
pp
Mp
p
pp
Mpypxp
y
x
x
y
y
x
x
y
x
y
y
x
x
y
y
y
y
xx
xyx
=
++
+++
=
+
+
+
=
+
+
+
=+
)1)(1(
)1()1(
)1()1()1()1(
3.2 Lagrangian multiplier method
We start with the same example as above:
The objective function is:
5.05.02 ),(,: yxyxUU += +
The budget constraint (BC) is: Mypxp yx =+
We now define a new function, which incorporates the constraint,
multiplied by a factor, which is called the Lagrangian multiplier:
))((),(),,( ypxpMyxUyxL yx ++=
or ))((),,( 5.05.0 ypxpMyxyxL yx +++=
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 26
To find a maximum, we take FOCs of the Lagrangian function:
0
0
0
=
=
=
L
y
L
x
L
0)(
05.0
05.0
..
5.0
5.0
=+=
==
=
==
=
ypxpML
pypy
U
y
L
pxpx
U
x
L
ei
yx
yy
xx
or
)(
5.0
5.0
5.0
5.0
ypxpM
py
px
yx
y
x
+=
=
=
The first two of the last set of equations can be written as:
y
x
y
x
p
p
p
p
x
y
y
x===
5.0
5.0
5.0
5.0
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You are already familiar with the expressionx
y. It is the MRS of
an indifference curve at the point (x, y):x
y
MU
MUMRS
y
x = .
To finish, we can find optimal consumption of x and y by using the
last equation (the one from the FOC) and the BC which is the same
as the third FOC of the Lagrangian problem. Luckily, we get the
same result as we obtained by the first method).
From the BC:
x
y
x p
yp
p
Mx =
From the FOC:
xp
py
y
x
2
=
Combining:
=
x
y
xy
x
p
yp
p
M
p
py
2
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 28
)1(
*
x
y
yp
pp
My
+=
Using the relationship between x and y given by the FOC, we can
solve for x:
)1()1(
)1()1(
* 2
22
y
xx
y
xy
y
x
x
yy
y
x
x
yy
y
x
y
x
p
pp
M
p
pp
p
p
M
p
pp
p
pM
p
pp
Mppy
ppx
+
=
+
=
+
=
+
=
=
3.3 Marshallian Demand functions
We call the optimal values for x and y, the demands of the
consumer. Furthermore, we define the expressions
),,( Mppx yx and ),,( Mppy yx to be the demand functions,
which give the relationship between optimal consumption of a
good and its price and the income (which is for now a given value,
a parameter). Sometimes, these demand functions are also called
Marshallian demand functions. In our example above the demand
function for x is (similar for y):
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 29
)1(
),,(
y
xx
yx
p
pp
MMppx
+=
Note, that the demands for both goods each depend on both prices
(in the general case at least).
3.4. Mathematical issues
Two additional things have to be checked for our solution to be
complete:
Firstwe note, that this result only holds if we have what is called
an interior solution, i.e. all variables are strictly greater than 0.
To explain this problem, look at the following graph:
The maximum is at the lower right corner of the budget set (i.e.
only x is consumed). In this case, it might be that the MRS, i.e. the
slope of the IC in the optimum is not equal to the price ratio (i.e.
the slope of the budget line). Thus, the condition, given by the
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 30
FOCs as in our example above does not apply although we have amaximum.
In this course, however, we will not deal with these non-interior
cases. They can be solved by a method which is called Kuhn-
Tucker Nonlinear programming. It is a generalization of the
Lagrange method, where inequalities ( ), are involved.
Secondly, in every case we have to make sure, whether secondorder conditions are satisfied. FOCs are only necessary but not
sufficient conditions for a maximum. That is, we need to find out if
our solution to the FOC is a minimum or a maximum. The
theoretical way to find out whether the obtained values
constitute a maximum or not, is to consider the so called bordered
Hessian matrix to find out whether the function is positive definite
or negative definite. Pos./neg. def. is a generalization of the
concept of concavity and convexity respectively to the multi-
variable case.
The applied way to insure that the critical value is a maximum is
to change the value of the variable near the critical point and check
if it increases or decreases the value of the objective function. In
this course, I expect you to have an idea about the meaning and
relevance of this problem, but I am not requiring you to actually
check whether second order conditions hold or not.
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Non-convexity of the Preferred SetA more subtle problem might occur, if ICs have weird shapes
(remember, that in general, we are not assuming that every
consumer has the same nice behaved preferences, so anything
could be possible depending on the actual consumer). One
particular freakiness of ICs is called non-convexity. As I
mentioned when I introduced the Cobb Douglas function, if the
preferred set is not strictly convex, several solutions for the OCB,
given income and prices, are possible.
y
x
IC
BL
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 32
3.5 Meaning of the multiplier:Going one step back, we can rewrite our FOCs as:
=
=
==
yx
y
y
x
x
p
y
U
p
x
U
p
MU
p
MU
What does this mean? In an optimum all the marginal utilities per
dollar spent are the same, i.e. I cannot gain additional utility by
increasing the amount of the value of one good without decreasing
any other good by the same (marginal) amount.
The multiplier represents the additional value in utility of
loosening the budget constraint (remember, is the factor for the
BC in the Lagrangian problem). Analytically, this can be shown as
follows:
Remember the Lagrangian function for a general consumer
problem: ((),(),,( ypxpMyxUyxL yx ++=
Now, assume that that we are evaluating L at the optimal
(maximizing) values for x and y and also at the optimal value for, all denoted by a *:
*))*((**)*,(*)*,*,(),,( ypxpMyxUyxLMppV yxyx ++=
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 33
Take first derivative with respect to M:
**)*,(*)*,(
))(1(*
*)*,(*)*,(),,(
FOCby0FOCby0
+
+
=
+
+
+
+
=
==44444 344444 2144444 344444 21M
yp
y
yxU
M
y
M
xp
x
yxU
M
x
M
yp
M
xp
y
yxU
M
y
x
yxU
M
x
M
MppV
yx
yx
yx
Hence: *),,(
=
MppV yx
If income would go up by one unit, the Value of the Lagrangian
objective function evaluated at the optimum would increase by .
Sometimes is called theshadow price (of a constraint). Indeed,
there are economic problems, where you can directly interpret the
multiplier(s) as price(s).
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 34
4. Comparative statics:4.1 Change of income
1. Graphical analysis
2. Mathematical analysis
Remember the example from last time. The expression for the
OCB for the x good is:
)1(y
xx
p
pp
Mx
+
=
Taking the first derivative with respect to (w.r.t.) M, we get:
0
)1(
1
)1(
>
+
=
+
=
y
xx
y
xx
p
pp
p
pp
M
MM
x
A similar result is true for the y-good.
This means, increasing income, the optimal amount for this good is
higher. We call goods with this property normal goods.If the demand goes down when income increases, we call these
goods inferior goods. (graphical example). Analytically speaking,
a good is inferior if the first derivative w.r.t. M of the optimal
demand is negative (or non-positive).
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 37
)1(*
y
xx
p
pp
Mx
+=
)1(
*
x
yy
p
pp
My
+
=
Substituting the optimal demands into the utility function:
),,(
)1()1(
***)*,(
5.05.0
5.05.0
MppV
p
pp
M
p
pp
M
yxyxU
yx
x
yy
y
xx
+
+
+
=
+=
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 38
5. Expenditure minimization and Hicksian demand
[Graphical Analysis]
Why is this useful?
- for analyzing price changes (Slutsky decomposition)
- empirical work: expenditures are observable, utility is hard to
determine directly
(Remark: In mathematical language, the expenditure minimization
problem is the dual problem to utility maximization. The two
problems are closely related.).
The problem of minimizing expenditure looks as follows:
UyxU
ts
ypxpE yx
=
+=
),(
..
min
In our example:
Uyx
ts
ypxpE yx
=+
+=
5.05.0
..
min
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 39
5.1 Derivation of the minimum expenditure functionThere are two ways of solving the problem for the expenditure
function. The first way starts from the previous result of the
Marshallian demand functions and uses the indirect utility
function. Although more intuitive it requires more work in terms of
computations. The second approach is just using the Lagrangian
procedure to solve the above minimization problem.
First Approach (via demand functions):
In this approach we use the fact that we know the demand
functions for the given utility function from the Utility
maximization problem.
)1(
*
y
xx
p
pp
Mx
+
=
)1(
*
x
yy
p
pp
My
+
=
In contrast to the utility maximization problem, M is now treated
as a variable instead of a given parameter. We indicate this writing
E (for expenditure) instead of M:
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 40
)1(
*
)1(*
x
yy
E
y
xx
E
p
pp
Ey
p
pp
Ex
+
=
+=
The problem is now, that we dont know what the optimal E (givenprices and fixed utility level) will be.
One way to derive the minimum expenditure function ),,( yx ppUE
is to substitute the optimal demands (given the unknown E), i.e.
x*E
and y*E
into the constraint 5.05.0 yxU += .
So we get:
5.05.0
)1()1(
+
+
+
=
x
yy
y
xx
p
pp
E
p
pp
EU
Solving for E gives:
++
+
= 5.05.0
)1()1(x
yy
y
xx
p
pp
p
pp
UE
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 41
Now we have an expression of E in terms of the parameters ofthisproblem. Substituting E into the expressions for Ex * and Ey * gives
the optimal, i.e. expenditure minimizing bundle, given prices and a
fixed level of Utility:
=
+
+
+
=
+
=2
2
)1(
)1(
1
)1(
*
x
yy
y
xxy
xx
E
p
pp
p
pp
U
p
pp
Ex
2
2
2
2
2
2
1
)1(
)1(
1
*
++
++
=
+
+
+
=
yxy
xyx
y
x
x
y
y
x
y
x
E
ppp
ppp
p
p
U
p
p
p
p
p
p
Ux
[cont.]
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 42
2
2
2
2
1
*
similarlyand
1)(
)(1
*
+
=
+
=
+
++
=
x
y
E
y
x
xyy
xyx
y
x
E
p
p
Uy
p
p
U
ppp
ppp
p
p
Ux
Second Approach (via Lagrangian)
I introduced the above solution approach because it gives you
some idea of the relationship between M and E. This, however,
was very complicated in terms of computations (in fact we were
moving back and forth between different expressions). Next we
will use our Lagrangian recipe which turns out to be much easier.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 43
1. Step: The Lagrangian is:Take the objective function (not the same as in the Utility max
problem!) and add (RHS of the constraint minus LHS of
constraint) times the Lagrangian multiplier ( in this case):
))(()y,L(x, 5.05.0 yxUypxp yx +++= ,
2. Step: Taking FOCs:
0)(
0)5.0(
0)5.0(
5.05.0
5.0
5.0
=+=
=+=
=+=
yxUL
ypy
L
xpx
L
y
x
3. Step: Solve the first two FOCs for y (or x):
y
x
p
p
x
y
y
x
==
5.0
5.0
5.0
5.0
Which gives the same condition (surprise!?) for the MRS as the
utility maximization problem, we discussed before.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 44
xp
p
p
p
x
y
y
x
y
x
2
yor
==
4. Step: Solve the third FOC (the constraint) for y (or x):
( )25.0
5.05.0
5.05.0
x
or
or
0)(
yU
xyU
yxU
=
=
=+
5. Step: Substitute the first of the last two results in the second (or
vice versa):
( )2
5.0
25.0225.0
x
x
=
==
xp
pU
xp
p
UyU
y
x
y
x
[cont. next page]
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 45
2
E
5.0
5.0
5.05.0
1
*x
1
x
)1(x
x
+
=
+
=
=+
=
y
x
y
x
y
x
y
x
p
p
U
p
p
U
Up
p
xppU
Notice, this is the same result as above.
Final Result: the minimum expenditure functionWe are not done yet. We know the expenditure minimizing
bundles, but we have to combine them to obtain the minimum
expenditure function:
22
11
**),,(
+
+
+
=
+
x
yy
y
x
x
Ey
Exyx
p
p
Up
p
p
Up
ypxpUppE
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 46
5.2 Hicksian/Compensated Demand CurveThe so called Hicksian or compensated demand
x*=hx(px,py,U)
considers demand behavior for changing price but constant utility.
Graphical analysis of the Hicksian demand for good x:
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 47
Example CD Utility:
5.05.02
),(,: yxyxUU=
+ By solving the Lagrangian problem (max U s.t. BC), the optimal
demands are:
y
x
p
My
p
Mx
2*
2*
=
=
Then the Indirect Utility function is:
),,(2
22***)*,(
5.05.05.05.0
MppVpp
M
p
M
p
MyxyxU
yxyx
yx
=
==
To get the compensated demand, i.e. the bundle which is
demanded if we could compensate the Income such that Utility is
not affected by the variation of the prices, we take the indirect
utility function and solve for the income M
yx
yxyx
ppVM
MppVpp
M
2
),,(2
=
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 48
and substitute the expression for M into the solutions for the
optimal consumption for the original optimum problem:
y
x
y
yx
yyxy
x
y
x
yx
xyxx
p
pU
p
ppV
p
VMppUh
p
pU
p
ppV
p
VMppUh
===
===
2
2
2
)(),,(
2
2
2
)(),,(
Compare this with the other demand, which we first developed, the
so called Marshallian demand (the ones we already know):
yy
xx
pMyd
p
Mxd
2*
2*
=
=
The difference is that now with the Hicksian or compensated
demand the price of the other good is in the argument of the
function (which is a specific feature of the CD utility function),
and more important, instead of income M, demand depends on the
level of U (which is a general property of the Hicksian demand).In ),,( yxx ppUh only substitution effects appear (i.e. no income
effects), that is why it is steeper than the Marshallian demand
),,( yxx ppMd (cf. next chapter).
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 49
6. Income effect (IE) and Substitution effect (SE)
So far we have developed different demand concepts and talked
about comparative statics. We analyzed how demand reacts, when
prices change. Usually, if price goes up one expects demand to
decrease. But why is this so? It turns out that two effects contribute
to the total effect of a change in demand: One is called Substitution
effect (SE) and is concerned with the change of demand when
relative prices change but utility (or alternatively: income) is kept
constant. There is however, a second effect: If one good becomes
more expensive, the overall purchasing power to buy any good
decreases. This is called the income effect. A priori, it is not clear
in which direction the income effect goes. The IE depends on what
kind of good we are analyzing (normal or inferior good). The
overall effect depends therefore both on the sign (direction) and the
relative magnitude of the two effects in combination.
We will see an interesting application of IE and SE when we will
talk about the labor supply decision later in the course.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 50
6.1 Graphical analysis: Effects of a decrease in pY:
Definitions:1
Substitution effect: Geometrically, the SE is the movement from A
(initial OCB) to B [the dotted line is a parallel of BC2, which is
tangent to the initial optimal IC (I1)].]. Economically, it is the
amount of y I would substitute for x if prices change, given that the
level of utility is kept constant.
Income effect: Geometrically, the IE is the movement from B to C
(OCB after price change). Economically, it is the change in
demand due to a change in purchasing power (income) keeping
prices constant. Note, that a price increase for one good implies
that there is less money left to buy any of the other goods
(provided I want to buy at least some of the good with the
increased price).
1Remark: There are two different concepts of IE and SE which are called Slutsky and
Hicks decomposition respectively. We will focus only on the Slutsky version.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 51
6.2 The Slutsky decomposition
In the following, we want to describe IE and SE in analytical
terms. To do this, we will use all of the concepts we have learned
so far: Marshallian demand, Hicksian demand, the indirect Utility
function and Minimum expenditure function. In general, there are
two approaches to obtain the formulas for IE and Se: A direct and
an indirect (dual) approach. We will only cover the latter one (For
a direct approach, i.e. solving the system of the derivatives of the
FOCs of the utility maximization problem: see Eugene Silberberg,
The Structure of Economics (3rd Ed.), p. 276 ff).
First Step: To start with, we state a relation between the Hicksian
demand yxx ppUh ,, and the Marshallian demand
yxx pMd ,, :2
yxyxxyxx ppppUEdppUh ,),,,(,, =
Why can we set them equal that way? Because we set the income
M in the Marshallian demand equal to the amount which is needed
to obtain the utility level U. Remember that the minimumexpenditure function gives the optimal spending subject to a fixed
level of utility (show this in the graph of the two demand curves).
2 Rmk: To avoid confusion, I will not use the shorthand for derivatives by using subscripts anymore. Herethe subscribt x denotes the demand for x
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 52
This last equation is very useful to analyze the income and
substitution effect.
Second Step: Partially differentiating this equation w.r.t. px, gives:
x
x
x
x
x
x
p
E
E
d
p
d
p
h
+
=
Rearranging:
{ { 43421ctIncomeEffe
x
x
onEffectSubstituti
x
x
x
x
p
E
E
d
p
h
p
d
=
changeTotal
Where the substitution effectx
x
p
h
could be written as
constant =
Uxp
x, which says we are looking at the change of the
optimal x w.r.t. a price change, when the level of utility is kept
constant. To get what is called the Slutsky equation or Slutsky
decomposition one further step is needed.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 53
Third Step: We want to determine the termxp
E
.
To find the derivative of the expenditure function w.r.t. to xp , we
go back to the well known expenditure minimization problem to
get the minimum expenditure function:
UyxU
tsypxpE yx
=
+=
),(
..min
Set up the Lagrangian:
)),(()y,L(x, yxUUypxp yx ++=
The FOCs are:
3)(FOC0)),((
2)(FOC0)),(
(
1)(FOC0)),(
(
==
=
+=
=
+=
yxUUL
y
yxUp
y
L
x
yxUp
x
L
y
x
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 54
The trouble is, without specifying a utility function we cannot go
on solving for the expenditure function. However, we can use a
little trick, - we use the Lagrangian function and evaluate it at the
optimal points for x and y, which then becomes:
),,())*,*((**
),*,*L(
3FOCbyOptimumin0
UppEyxUUypxp
yx
yxEEE
yE
x
EE
++=
=
=444 3444 21
Notice that we have shown that the expenditure function can be
expressed as the Lagrangian function evaluated at the expenditure
minimizing values of x and y.
:w.r.t.E()ofderivativefirstthecomputecanweNow, xp
( )
x
EEEy
Ex
x
EE
x
yx
p
yxUUypxp
p
yx
p
UppE
++=
=
=
))*,*((**
),*,*L(
),,(
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 55
),,(*),*(
),,(*),*(
),,(*
),,(*),,(*
x
yxEE
x
yxEE
x
yxE
y
x
yxE
xyxE
p
Uppx
x
yxU
p
Uppy
y
yxU
p
Uppyp
p
UppxpUppx
+
+=
),,(*
)*,*(),,(*
)*,*(),,(*
),,(*
2FOCby0
1FOCby0
Uppx
y
yxUp
p
Uppy
x
yxU
pp
Uppx
Uppx
yxE
EE
xx
yxE
EE
xx
yxE
yx
E
=
+
+=
=
=
4444 34444 21
4444 34444 21
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 56
For the derivation two things were used:
First at the optimum values for x and y, i.e. *Ex and Ey * , the
Lagrangian function equals the minimum expenditure function
E(.).
Second, at the optimum values, we can use the FOC of the
expenditure minimum problem. From this, the result follows. The
procedure we used is also called Envelope Theorem (see
Nicholson p 46 f).
Final Step: With our results we can now express the demand
effect of changing prices as:
{32144 344 2143421
IE
x
SE
Ux
ctIncomeEffe
x
x
onEffectSubstituti
x
x
x
x xMd
px
pE
Ed
ph
pd
=
=
= constant
This says, that a marginalincrease in the price of good x can be
decomposed into a change of demand due to a movement along the
IC (the SE), which is the change of the Hicksian demand minus the
change due to an income change multiplied with the initial amount
of x. Note that the Slutsky decomposition only tells us something
about marginal IE and SE. In the second example below, I will
show how to compute IE and SE as a response to a discrete price
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 57
change (i.e. how to compute the actual IE and SE we have
observed in the 2-goods diagram). We will use the concept of the
Slutsky decomposition again when we will talk about the labor
supply decision of the consumer.
Example The Slutsky decomposition for a CD Utility function
Assume the Utility function has the form:
xyyxU =),(
As we have shown the Marshallian and Hicksian demands, are,
respectively (we do the analysis for the x-good, for y the procedure
is similar):
xx
p
Mxd
2* =
x
yyxx
p
pUppUh =),,(
Substitution effect:
5.05.1
2
1yx
x
x
y
x
x pUpp
p
pU
p
hSE
=
=
We can eliminate U by using the indirect utility function:
( ) 5.05.05.0
222),,(
yxyxyx
pp
M
p
M
p
MMppV =
=
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 58
( )
2
5.05.15.05.05.1
4
1
22
1
2
1
x
yx
yx
yx
p
M
pppp
MpUpSE
=
==
As expected, the SE is negative, i.e. if the price of good increases,
the other becomes relatively cheaper, and I want to buy more of
the other good.
Income effect:
xp
xM
p
M
xM
dIE
x
xx
2
12=
=
Since, by the Marshallian demand function: xp
M
x 2= , the
previous expression becomes:
2422
1
2
1
xxxx p
M
p
M
px
pIE ===
Now we can combine the two effects to obtain the Slutsky
equation:
222 2
1
44
1
x
ctIncomeEffe
x
onEffectSubstituti
xx
x
p
M
p
M
p
M
p
d==
32143421
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 59
First, observe that in this example both IE and SE are negative and
of the same magnitude (this is due to the special form of the utility
function).
Second, compare the total effect to the direct comparative statics
result, i.e. how demand changes, when the price changes:
22
12
xx
x
x
x
p
M
p
p
M
p
d=
=
.
They are the same, as it should be. However, the Slutsky
decomposition includes more information, it says how much of the
demand change can be accounted for by a change of relative prices
(SE) and how much is due to a change in purchasing power (IE).
6.3 Computing IE and SE for a discrete price change
We will now analyse the compution of a positive, i.e. non marginal
price change (the Slutsky decomposition above only deals with
marginal effects, thats why there are only derivatives in the
equation). We will explain the computation of discrete IE and SE
by the use of an example. Consider again the Utility function
xyyxU =),( . Assume that income is $ 100 and that the price of x
is 1=xp and 1=yp and that price of x increases to 2=new
xp .
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 60
Before you read the following calculations, you may draw a sketch
of the IE and SE and compare each step of the computations to
points in the graph.
Substitution effect:
1. We want to get the initial bundle first (i.e. the OCB with prices
all equal to 1). The first step is to compute the Marshallian
demands from the Utility maximization problem. By our familiar
procedure (using the Lagrangian), we get:
501*2
100
2*
501$*2
100$
2*
===
===
yy
xx
p
Myd
p
Mxd
2. To get the bundle, where the SE leads to, we have to compute
the Hicksian demand, using the old Utility level (IC) but changing
the price of x to $2 (Remember that the Hicksian demand keeps U
constant when price changes, that is, we move along an IC).
First we need the Indirect Utility function:
( ) 5.0
5.05.0
222),,(*)*,(
yxyxyx
ppM
pM
pMMppVyxU =
==
Solving for M as a function of U gives:
yxppUUM 2)( =
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 61
Taking the Marshallian demand and using the relation between M
and U gives the Compensated or Hicksian demands:
y
x
y
yx
yyxy
x
y
x
yx
xyxx
p
p
Up
ppU
p
UM
ppUh
p
pU
p
ppU
p
UMppUh
===
===
2
2
2
)(
),,(
2
2
2
)(),,(
3. To calculate the actual values of the Hicksian demands, we need
to know the Utility level of the original IC:
Take the Utility function and plug in the values for the original
demands (where the prices are all=1):
( )50
112
100*)*,(
5.0=
=yxU
This is the Utility level of the original demanded bundle for the
price for x of $1.
The Hicksian demands for the new price of x, which is $2 are then:
7.701
250*)*,(),,(
4.352
150*)*,(),,(
==
==
y
newx
ynew
xSE
y
newx
yy
newx
SEx
p
pyxUppUh
p
pyxUppUh
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 62
Thus, the Substitution effect for x is 014.6-50-35.4*x-==SEyh .
Income effect:
4. The easiest way to compute the IE is to first compute the new
optimal consumption bundle, given by the Marshallian demand for
the new prices:
501*2
100
2**),,(
252*2
100
2**),,(
===
===
yy
newxy
newx
ynew
xx
p
MyppMd
p
MxppMd
Then the IE is just the difference of the new bundle and the bundle
reached by the SE, i.e. 04.104.3525-**x
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 63
7. Labor supply decision of the consumer
7.1 Introduction: Assumptions of the model
The basic trade-off for the labor supply decision is whether to have
more free time and fewer earnings and therefore less consumption
or more consumption, but also more time to spend on working (i.e.
less leisure time).
Similarly as in the 2-goods problem, the consumers decision
problem is based on his or her preferences (utility function) and a
budget constraint.
Let utility be given by: U(c, F), where c symbolizes consumption
(i.e. a basket of consumption goods), and F is the symbol for
leisure (Free time). We assume that F is less or equal to 24 hours:
240 F . The remaining time (24-F) is used for earning money
(which is spent on c), called labor time L. Thus we have F+ L = 24
or L =24 F.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 64
Assumptions on preferences
For our analysis we assume that an increase in either of the goods
(leisure or consumption) yields an increase in utility, or formally:
0),(
0),(
>
>
F
FcU
c
FcU
Further, we assume that the marginal utilities are decreasing:
( )
( )0
),(
0),(
2
2
2
2
=
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 75
Three cases:
1. IESE: In this case the Income effect outweighs the increase of
the relative price of leisure, we will supply less labor although the
wage is increasing.
Illustration of case 3:
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 76
If this last case applies for some range of the labor supply, we
might obtain the following backward bending (aggregated) labor
supply function:
Empirical studies claim that for men, the labor supply is slightly
backward bending and for women it is always increasing. The
latter is the case, so it is argued, because if household production(as part of leisure time) is taken into account, then the
substitution effect will always outweigh the income effect.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 77
Discussion:
Certainly people are not changing their labor leisure choice day by
day. (why?) So the concept we just developed has more appeal for
long term decisions (if I choose a high earning job which usually
implies a long working day, may be I plan to retire earlier.)
Other issues:
Hours of work versus Participation
Unemployment Insurance/Protection
Search for a job
Bargaining for wages
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 78
7. 4 Appendix: analytical derivation of the Slutsky equation
The problem with the analytical solution is that income wT+
does itself depend on one of the prices, namely w. This implies that
there are two income effects.
Marshallian demand for leisure is ),wwTdF +43421(M)incomefull
(
Differentiating w.r.t to w:
constM
effectincomefull
w
dT
M
d
w
d FFF+
=
321
Where the last expression now can be decomposed into the usual
income and substitution effects. Using the relation:
( ) ( )wwUEdwUh FF ),,(, =
Taking first derivative w.r.t. w and rearranging:
{ 434214342143421IE
F
SE
U
IE
F
SE
FF FM
d
w
F
w
E
M
d
w
h
w
d
=
=
==
const
constM
Combining with the first result gives the Slutsky equation::
44 344 2144 344 21
IEcombinedSE
constant
FM
dT
M
d
w
F
w
d FF
U
F
+
=
=
LM
d
w
FFT
M
d
w
F F
U
F
U
+
=
+
=
== constantconstant )(
Note, that we wrote everything in terms of leisure demand. We could transform this into
a statement in terms of labor supply using the relationship: L=T-F.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 79
8. Labor supply and taxes
There are two stylized types of taxes concerning the labor supply:
per capita tax and (linear) income tax. We will first look at the
general set-up of the tax models and then later compute demand
functions given taxes for a specific example.
8.1 A (linear) Income tax
The framework:
Utility: U(c, F)
Budget constraint: +w)F-24(=c
Assume = 0, then: w)F-24(=c
Now introduce a tax t:
wt)-)(1F-24(=c
Budget constraint pivots downward if t is introduced (1>t>0).
What is the revenue the state will get?
From the BC:{ {
nconsumptiotaxbeforeincomerevenueTax
c-wLtwLor
wt)-L(1c
wt)-)(1F-24(=c
=
=
321
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 80
8.2 Per capita tax
Lets compare this tax scheme with a so called lump sum tax, a per
capita tax.
The Budget constraint is in this case:
F)w(c = 24
or
Lwc =
where denotes the amount of tax everyone (regardless of
income) has to pay.
Budget constraint shift down for 0> .
Question: How could someone come up with such an idea?
The advantage of this kind of tax is that it is more efficient.
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 81
8.3 Taxes and Efficiency
The concept of efficiency4
The official definition of (economic) efficiency, is the so called
Pareto Efficiency (for an economy with N agents and L goods):
An allocation of goods among N individuals:
X={( )x,...x,(x),..,..x,...x,(x),x,...x,(x NLN2
N1
2L
22
21
1L
12
11 }
is called efficient, if
1. the allocation X isfeasible, i.e. the allocation is available (either
by endowment or in the sense that they can be produced by the
firms given the resources of the economy),
and if
2. there is no otherfeasible allocation
Y={ ),...y,(y),..,..,...y,(y),,...,(y NLN2
N1
2L
22
21
1L
12
11 yyyy }
such that Y is preferred by all consumers, i.e.
forall consumers i=1,2.N,
)x,...x,U(x)y,...y,U(y iLi2
i1
iL
i2
i1
and for at least one consumer j :
4 We will discuss this issue more in detail later in the course, the following is for those who are curious toknow.
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 82
)x,...x,U(x)y,...y,U(y j
L
j
2
j
1
j
L
j
2
j
1>
In words: an allocation is Pareto Optimal (PO) if no consumer
can be made better of without making at least one other
consumer worse of.Note that this concept doesnt imply anything
about fairness: efficient does not mean good.
This definition will be of interest when we will talk about generalequilibrium analysis later in the course.
In our single consumer model with taxes, the concept of efficiency
boils down to the statement, that, given the tax revenue the
government wants to raise, the consumer is maximizing (why?).
Lets look at the model for the per capita tax:
[Graphical analysis]
What can you say concerning the MRS at the new optimal bundle?
With the lump sum tax , the MRS will be the same since the
relative prices do not change (like a pure IE).
In comparison with the income tax, discussed above, the choices
are not distorted between the two goods (c and F). The purchasing
power decreases, though. But the ratio of the consumed goods
remains constant. This is implicitly the reason why the per capita
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 83
tax (assuming standard preferences) is more efficient: it does not
distort choices.
To see the difference in efficiency, let us consider a specific
example.
We will assume that in both tax scenarios, the total amount of the
tax incidence (government revenue) is the same, and we will
compare the level of utility the consumer has under both tax
regimes.
Example:
We will now compute utilities ex post for the two tax regimes,
assuming that the lump sum tax will be equal to the tax revenue of
the income tax problem. That is we compare the tax systems in
terms of the achieved utility by the consumer, given that the
government gets the same absolute value of tax revenues.
I. Linear Income tax
The objective function is (note, the U fcn is not CD):
5.05.02 ),(,: cFFcUU += +
The budget constraint:
)24()1( Fwty = The Lagrangian:
))24()1((),,( 5.05.0 cFwtcFFc ++= L
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 84
FOCs of the Lagrangian function:
0)24()1(
05.0
0)1(5.0
5.0
5.0
==
==
==
cFwt
cc
wtFF
L
L
L
From the first two FOCs:
( ) Fwtc
or
wtF
c
2)1(
)1(
=
=
Combining with the BC (3rd
FOC) yields:
( )
wtF
wt
wtc
t
t
)1(1
24
)1(1
)1(24
*
2*
+=
+
=
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 85
Now we compute the tax revenue. For convenience we set
w = 0.5 and t = 0.5:
2.1)1(1
)1(24
)1(1
2424)
2* =
+
=
+==
wt
wt
wttwFtwTR t-(24
II. Per capita tax
We use, of course, the same utility fcn:
5.05.0),( cFFcU +=
However, the budget constraint now looks like:
= )24( Fwc
Lagrangian:
))24((),,(L 5.05.0 cFwcFFc ++=
FOCs:
0)24(L
05.0L
05.0L
5.0
5.0
==
==
==
cFw
cc
wFF
3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 86
The first two FOCs give:
( ) Fwc
or
wF
c
2=
=
Again, combining with the BC:
ww
wF
w
wwc
t
+
=
+
=
2
2*
24
1
24
Now set equal to TR, so that we can compare the efficiency of
the two tax regimes by computing the Utilities in both cases (note
that we set w = 0.5 and t = 0.5):
Utility in the first case:
( )48.5
)1(1
)1(24
)1(1
24),(
2**
+
+
+=
wt
wt
wtcFU tt
And Utility in case of the per capita tax:
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3101 Spring 2005 Edgar Preugschat Lecture notes Part 1 page 87
2.1)-(24where
70.51
2424),(
*
2
2
**
==
+
+
+
=
tFtw
w
ww
ww
wcFU
This says that raising the same revenue, the per capita tax is more
efficient (b/c it implies a higher Utility for the consumer):
),(48.570.5),( **** tt cFUcFU =>= .
Hence, the utility is higher if the same amount of revenue is raised
by a per-capita tax, than with a linear income tax. Economically,
this means that a per-capita tax is more efficient. In terms of
equality, on the other hand, a per-capita tax would be very unfair.
Everyone has to pay the same, be he/she a factory worker or a
CEO of a big company. Neoclassical Economics as a positive
science tries to provide tools to evaluate the efficiency costs of
equality (or inequality some research suggests that under some
specified conditions, a high degree of inequality might be
inefficient). Neoclassical Economics does not, however, analyze or
give reasons for or against equality/inequality per se. That is both a
philosophical and political question.