14 Consumer Surplus

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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


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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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


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.


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.


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).


- 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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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.


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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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


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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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.


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

+

=


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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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 +++=


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


)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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)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


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


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 ++=


17/44


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).


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).


18/44


19/44


)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

+

+

+

=

+=


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


20/44


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:


)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


21/44


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.]


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.


22/44


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.


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]


23/44


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


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:


24/44


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

=


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).


25/44


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.


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.


26/44


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


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.


27/44


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


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(

),,(


28/44


),,(*),*(

),,(*),*(

),,(*

),,(*),,(*

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


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


29/44


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 =

=


( )

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


30/44


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 .


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)( =


31/44


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


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


32/44


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.


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

=


38/44


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:


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.


39/44


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


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.


40/44


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


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.


41/44


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.


)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


42/44


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


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*

+=

+

=


43/44


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


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:


44/44


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.

Date post:	30-May-2018
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14 Consumer Surplus

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