Econ 601 Graduate Microeconomics Lau

Lecture Notes Fall 2014

I. Consumer Theory

Consumer theory is developed to

i) explain the consumption pattern of an individual (household/family) and

ii) carry out welfare analysis of an individual (household/family).

Indifference Curve Analysis

Definition: An indifference curve plots all the consumption bundles which are viewed as

indifference to each other by a consumer.

Definition: An indifference map consists of a set of indifference curves.

Definition: A consumer’s marginal rate of substitution (MRSXY) between X and Y is the

maximum amount of Y a consumer is willing to give up to obtain an additional unit of X.

Remark: MRSXY slope of IC (X on the horizontal axis and Y on the vertical axis)

Remark: On an IC, X Y( , ) constant (MU ) (MU ) 0 X

Y

MUYU X Y X Y

X MU

Properties of indifference curves:

1. There are infinitely many indifference curves. (Each consumption bundle is passed through by

an indifference curve.)

2. No two indifference curves cross each other.

3. The further away from the origin, the higher is the utility.

4. Indifference curves are convex to the origin.

Y

contradiction

U=15

U=10

X

2

2

Y

U1

U0

X

Special case: ( , )U X Y X Y perfect substitutes

Y

10

U=10 U=20

5

X

5 10

Special case: ( , ) min( , )U X Y X Y perfect complements (Leontief Utility Function)

Y

8 U=8

5 U=5

X

5 8

Remark: Different individuals have different indifference maps which are determined by the

preference (taste) of the individuals.

Remark: Preference is assumed to be constant, at least in the short run.

Budget line and budget set

Definition: A feasible consumption bundle is a consumer bundle that a consumer can afford to

purchase with his/her income ( I ) under the current market prices and X YP P .

Definition: A budget set is the set of all feasible consumption bundles.

{( , ) : }X YB X Y P X P Y I

3

3

Definition: A budget line represents the “maximum” combination of and X Y that a consumer can

afford to purchase.

X YP X P Y I

Y

I/PY

feasible set

budget line

X

I/PX

Assumption: A consumer is a utility-maximizer.

Consumer equilibrium

Remark: At equilbrium, in general, the indifference curve is tangent to the budget line. Y

Y

I

P

E

Y*

U0

X

X*

X

I

P

Changes in income

Definition: If ( ) ( ), then is aI X X normal good.

Definition: If ( ) ( ), then is anI X X inferior good.

slope of budget line

Y X

Y

X

I

P P

I P

P

4

4

Y

I’/PY

I/PY

Y

E’

Y’ E

Y* U1

U0

income

consumption

X curve

X* X’ I/PX I’/PX

X

Y

I’/PY

I/PY

E’

Y’

U1

E

Y*

U0

X

X’ X* I/PX I’/PX

Y

I’/PY

I/PY

E

Y* E’

Y’

U1

U0

X

X* X’ I/PX I’/PX

X: normal good

Y: normal good

X: inferior good

Y: normal good

X: normal good

Y: inferior good

5

5

Deriving the ordinary demand curve /Marshallian demand curve

Y PX1

> PX2

PX

I/PY

PX1

E’

Y2 PX2

Y1 E

U1

D

U0

X X

X1 X2 I/PX1 I/PX

2 X1 X2

Y PY1

> PY2 PY

I/PY2

I/PY1

PY1

E’

Y2

PY2

Y1

E U1

U0

D

X Y

X2 X1 I/PX Y1 Y2

6

6

Changes in price and price-consumption curve

Y

I/PY

price consumption curve

E E’ Y*

U1

U0

X

X* I/PX1 I/PX

2

Effect of change in prices and income on welfare

Example: A consumer has an income of $180. When $6 and $4X YP P , her consumption

bundle is ( , ) (20,15)X Y . Suppose her income decreases to $105, XP decreases to $3 and YP

decreases to $3. With the aid of a diagram, determine how would her utility be affected?

Y

15

U1

U0

X

20

Note that with the new prices, the

consumer can “just” afford to

buy the bundle (X,Y)=(20,15).

Hence the new budget line will

pass through the point (20,15)

[($3)(20)+($3)(15)=$105]

new budget line

3slope

3

initial budget line

6slope

4

7

7

Applications of indifference curve analysis

1. Food stamp program OG

food stamp budget line with food stamp

(F#)

I

OG’ E’

OG* E

U1

U0

Food

F# F* F’ I/PF I/PF+F

#

F F

E

2. Cash grant program OG

#

FI' I P F

I’

I

OG’ E’

E

OG* U1

U0

Food

F* F’ I/PF I’/PF=I/PF+F#

i) the consumer will be better off

ii) it is unclear whether the

consumer will consumer more or

less food

i) The consumer will be better off.

ii) It is unclear whether the

consumer will consume more or

less food.

8

8

3. Food stamp program vs cash grant OG

Case 1 UFS=UCG; FFS=FCG I’

budget line with cash grant

I

E’

E

OG* UFS=UCG

U0

Food

F# F* I/PF I/PF+F

#

budget line with food stamp

OG

Case 2 UFS<UCG; FFS>FCG I’

ECG EFS

UCG

I

UFS

E

OG*

U0

Food

F# F* I/PF I/PF+F

#

budget line with food stamp

9

9

4. Sales tax

PX’=PX + t OG

I

tax revenue

OG”

E’ E OG’

U0

U-1

X

X’ I/PX’ I/PX

5. Income tax

OG

income tax

I

I-T

E

OG*

E’

OG’ U0

U-1

X

X’ X* (I-T)/PX I/PX

The consumer will be worse off.

10

10

6a. Income tax vs sales tax

OG

budget line under income tax original budget line

I

tax revenue EIT

EST

UIT

UST budget line under sales tax

X

X’ (I-T)/PX I/PX

6b. Income tax vs sales tax

revenuesales tax

revenueincome tax

EST

EIT

sales tax income tax

Revenuesales tax vertical distance between the two parallel budget lines

Revenueincome tax

for the same utility, income tax sales taxRevenue Revenue

11

11

7a. Sales subsidy vs income subsidy OG

budget line with income subsidy

EIS ESS

budget line with sales subsidy

UIS

cost to

government USS

original budget line

X

X’

7b. Sales subsidy vs income subsidy OG

budget line with income subsidy

EIS ESS

budget line with sales subsidy

cost to

government USS=UIS

under income

subsidy

original budget line

X

cost to government under sales subsidy

12

12

Substitution effect (EA) and income effect (AE’)

PX Y X: NORMAL GOOD Y X: INFERIOR GOOD

A

A

E’ E

E E’ U0

U0

U-1

U-1

X X

X’ XA X* XA X’ X*

Substitution effect (EA) and income effect (AE’)

PY Y Y: NORMAL GOOD Y Y: INFERIOR GOOD

Y* E

YA A

U0 Y* E’ E

Y’ A

YA

E’

Y’ U-1 U-1 U0

X

13

13

Normal good, inferior good and Giffen good (XP )

Y

normal good inferior good

Giffen good

A

E

XP

X

A Giffen good must be an

inferior good, but not vice

versa.

14

14

Normal good, inferior good and Giffen good (XP )

Y

inferior good normal good

Giffen good

E

A

XP

X

A Giffen good must be an

inferior good, but not vice

versa.

15

15

What happen if the demand curve is not downward sloping?

Defnition: An equilibrium is Walrasian stable if *P P , then will converge to *P P .

Defnition: An equilibrium is Marshallian stable if *Q Q , then will converge to *Q Q .

P S

E Marshallian stable

P* and

Walrasian stable

D

Q

Q*

P D S

Marshallian unstable

and

Walrasian stable

E

P*

Q

Q*

P S D

Marsahallian stable

E and

P* Walrasian unstable

Q

Q*

16

16

Application of the concept of substitution effect and income effect

1. Will an orange producer consumes more oranges (i.e. sell fewer oranges) when OP ?

income of orange producersOP

substitution effect:

income effect: income (if orange is a normal good)

O O

O

P C

C

same (if orange has no income effect)OC

(if orange is an inferior good)OC

total effect:

if orange is a normal good, then ?

if orange has no income effect, then

if orange is an inferior good, then

O

O

O

C

C

C

Y

PO’O# inferior normal normal

good good good

A

POO#

new budget line

E

initial budget line

oranges

O* O#

# of oranges available

17

17

2. Will a worker work more when the wage rate goes up?

substitution effect: enjoys less leisure, work more

income effect: enjoys more leisure (assume normal good), work less

total effect: ?

w

$

24W’

IE<SE IE>SE

24W

A

E

leisure

24

wage supply curve of labor

labor hours

18

18

3. Inter-temporal consumer choice

1 2

1 2

: income in year 1 : income in year 2 : interest rate

: consumption in year 1 : consumption in year 2

Y Y r

C C

Inter-temporal Budget Constraint: 2 21 1

1 1

Y CY C

r r

Case 1: 1 2 1 1 1 20, 0 0, 0Y Y S Y C S

C2

Y1(1+r)

C2* E

U0

C1

C1* Y1

Case 2: 1 2 1 1 1 20, 0 0, 0Y Y S Y C S

C2

Y2

C2* E

U0

C1

C1* Y2/(1+r)

1

1

(1 )slope (1 )

Y rr

Y

2

2

slope (1 )

1

Yr

Y

r

19

19

Case 3: 1 20, 0 Y Y Saving in the first period

C2

Borrowing in the first period Y2+Y1(1+r)

C2* E

U0

Y2

C2* E

U0 C1

C1* Y1 C1* Y1+Y2/(1+r)

How an increase in the real interest rate affects consumption (assume both C1 and C2 are normal

goods)

borrowing in the first period: incomer

1 2substitution effect ,C C

1 2income effect: ,C C

total effect 1 2, ?C C welfare loss

saving in the first period: incomer

1 2substitution effect ,C C

1 2income effect: ,C C

total effect 1 2?,C C welfare gain

2 1 2 1

2 1 21

(1 ) (1 )slope (1 )

(1 )

1 1

Y Y r Y Y rr

Y Y r YY

r r

20

20

Case 3a: 1 20, 0 Borrowing in the first periodY Y

C2

Y2

C2* E E' U0

(1 )r

(1 ')r U-1

Y1 C1* C1

Case 3b: 1 20, 0 Saving in the first periodY Y

C2

E' C2* E

U1

U0

Y2

C1

C1* Y1

21

21

Deriving the Hicksian Demand Curve/ Compensated Demand Curve

Hicksian demand curve is derived by holding utility constant Y PX’ > PX PX

slope= –PX’/PY

Y’

PX’

PX

Y*

U0 DH

slope= –PX/PY

X X

X’ X* X’ X*

Remark:

1. The Hicksian demand curve is never upward sloping. It is a vertical line if the indifference

curves are of the Lenotief type.

Y

U0

X

2. Each point on the DH corresponds to a different level of money income. Hence the Hicksian

demand curve is also called an (Income)-Compensated Demand Curve.

22

22

II. General Equilbrium I

Two-person exchange economy

Let X#

and Y#

be the goods available in an economy.

Definition: A feasible allocation is an array of consumption bundles # #{( , ),( , ) : , }A A B B A B A BX Y X Y X X X Y Y Y

Edgeworth Box XB OB

XA+XB=X#

YA+YB=Y# YB

YA

OA XA

Remark: We only consider feasible allocation in our discussion.

Definition: An allocation {( *, *),( *, *)}A A B BX Y X Y is Pareto Superior to another allocation

{( ', '), ( ', ')}A A B BX Y X Y if either

i) ( *, *) ( ', ') and ( *, *) ( ', ')A A A A B B B BU X Y U X Y U X Y U X Y or

ii) ( *, *) ( ', ') and ( *, *) ( ', ') B B B B A A A AU X Y U X Y U X Y U X Y

The welfare of one person increases without hurting the welfare of the other individual.

Definition: An allocation is Pareto optimal (efficient) if there is no allocation which is Pareto

Superios to it.

Definition: An allocation is inefficent if it is not efficient.

Definition: The set of all Pareto Optimal allocations is called the contract curve.

Remark: A contract curve may not look like a curve.

Note: At the efficient point, the ICs of the two individuals are (usually) tangent to each other.

efficient (cannot increase the utility of one without

lowering the utiltiy of the other)

U1A

inefficient

(can increase the utility of A without

lowering the utility of B) U0B U0

A

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23

Edgeworth Box XB OB

XA+XB=X#

YA+YB=Y# YB

UA1

UB2 UA

2

UB1

YA

OA XA

Note: From the contract curve, we can plot the utility frontier which is the “best” combination of UA

and UB which can be attained by the 2 individuals.

Utility frontier UB

social optium

social indifference curve

UA

Proposition: If the endowment is not on the contract curve, there will be gain from trade. XB

OB

core YB

YA contract curve

OA XA

efficient point

24

24

Two person competitve economy XB OB

XA+XB=X#

YA+YB=Y# YB

UA

2

contract curve

UB1

YA

OA XA

Proposition: (Fundamental Theorem of Welfare Economcis)

A competitve equilibrium is Pareto optimal.

Proprosition: A Pareto optimal allocation is a competitive equilibrium with lump sum transfer.

competitive equilibrium

25

25

Production economy: 2 2 2 model (2 agents, 2 goods and 2 inputs) LB OB

LA+LB=L#

KA+KB=K#

(K/L)B KB

efficient allocation

contract curve

(K/L)A

KA

OA LA

Determining the output in a production economy

closed economy QB

social optium

Production

Possibility

Frontier

social indifference curve

QA

open economy QB

production point (P)

Production

Possibility

Frontier

C

–PX/PY

QA

26

26

Arrow’s Impossibility Theorem

Proposition:

There is no social welfare function which satisfies the following “reasonable” assumptions:

1. Pareto rule

If every one prefers X1 to X2, then X1 is preferable by the society. It is true for any other pair

(Xi, Xj).

2. Independence of irrelveaant alternatives

Whether a society is better off with X1 or X2 should depend only on individual preference

between X1 or X2, but not on any consumption bundle X3 or X4.

3. Unrestricted domain

The rule must hold for all logically possible sets of preferences.

4. Nondictatorship

We do not allow a rule whereby the social ordering is automatically taken to be the same as one

particular individual’s prefereences, irrespective of the preferences of the others.

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27

III. Consumer Theory II

Indirect utility function and Marshallian demand functions

1

1 1,...,

1

( , ) max ( ,..., ) s.t. 0 ( ,..., )n

n

n i i nX X

i

v P M U X X M P X P P P

Proposition: The utility function is unique up to a monotonic transformation, i,e. if 1 2( , )U X X is an

utility function representing the underlying preference, then 1 2[ ( , )], ' 0F U X X F is also an appropriate

utility function.

Proof:

This proposition can be proved by the following theorem.

Theorem: Let 1 2 1 2( , ) [ ( , )]W X X F U X X where (.)F is an increasing function.

1 2 1 2 1 2

1 2 1 2 1 2

If ( *, *) maximizes ( , ) s.t. ( , ) 0,

then ( *, *) maximizes ( , ) s.t. ( , ) 0.

X X U X X g X X

X X W X X g X X

Proof:

Suppose 1 2 1 2 1 2( *, *) maximizes ( , ) s.t. ( , ) 0X X U X X g X X

1 2 1 2 1 2 1 2( *, *) ( , ) ( , ) fulfilling ( , ) 0U X X U X X X X g X X

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2

Since (.) is an incresing function

[ ( *, *)] [ ( , )] ( , ) fulfilling ( , ) 0

( *, *)] ( , )] ( , ) fulfilling ( , ) 0

( *, *) maximizes ( , ) s.t. g( , ) 0

F

F U X X F U X X X X g X X

W X X W X X X X g X X

X X W X X X X

Example:

The following 2 problems are equivalent.

1) 1 2

1 2 1 2,

1 1 2 2

max ( , )

s.t. 0

a b

X XU X X X X

M P X P X

2) 1 2

1 2 1 2,

1 1 2 2

max ( , ) ln ln

s.t. 0

X XU X X a X b X

M P X P X

28

28

Example:

1 2

1 2 1 2 1 1 2 2,

max ( , ) s.t. 0a b

X XU X X X X M P X P X

Perform a montonic transformation:

1 2

1 2 1 2,

1 1 2 2

max ( , ) ln ln

s.t. 0

X XU X X a X b X

M P X P X

1 2 1 2 1 1 2 2( , , ) ln ln [ ]L X X a X b X M PX P X

FONC:

1

2

1

1 1

2

2 2

1 1 2 2

0 (1)

0 (2)

0 (3)

X

X

L aL P

X X

L bL P

X X

L M P X P X

2 2 1 1

1 1 2 2

(1) and (2) (4)a b b

P X P XP X P X a

1 1 1 1 1 1(4) (3) ( ) 0 ( ) 0b a b

M P X P X M P Xa a

1

1

*a M

Xa b P

Marshallian demand function

1 1 1

2 2 1 1 2

2 2 1 2

bP X bPb a M b MP X P X X

a aP aP a b P a b P

Marshallian demand function

SOSC:

1 2

1 1 1 1 2

2 2 1 2 2

1 2

2 2

2 11 2 2 2

21 1

2 2

2

0

0 0 (maximum)

0

X X

X X X X X

X X X X X

P PL L L

aP bPaH L L L P

XX XL L L

bP

X

Indirect utility function: 1 2

1 2 1 2

( , , ) ( ) ( ) ( ) ( ) ( )a b a b a ba M b M a b Mv P P M

a b P a b P P P a b

Income consumption curve:

1

2 1

2

bPX X

aP

29

29

Example:

1 2

1 2 1 2 1 2 1 1 2 2,

max ( , ) s.t. 0X X

U X X X X X X M P X P X

1 2 1 2 1 1 2 2[ ]L X X X X M P X P X

FONC:

1

2

2 1

1

1 2

2

1 1 2 2

1 0 (1)

1 0 (2)

0 (3)

X

X

LL X P

X

LL X P

X

LL M P X P X

2 1 2 2

1

1 2 1

1 1 ( 1)(1) and (2) 1

X X P XX

P P P

2 2

1

1

( 1)1 (4)

P XX

P

income consumption curve

2 2

1 2 2 2 2 1 2 2

1

( 1)(3) : [ 1] 0 ( 1) 0

P XM P P X M P X P P X

P

1 2

1 2 2 2 2

2

2 * (5) 2

M P PM P P P X X

P

Marshallian demand function

2 2 2 1 2 2 1 2 2

1

1 1 2 1 2

( 1) 2(4) : * 1 ( 1) 1 ( ) 1

2 2

P X P M P P P M P P PX

P P P P P

1 2 1 2 1 2 1

1

1 1 1

21 * (6)

2 2 2

M P P M P P P M P PX

P P P

Marshallian demand function

Note that 1 2* and * must be non-negativeX X

1 2 1

2 1 2

(5) 0 0

(6) 0 0

X M P P

X M P P

If 2 1 0M P P , then 1 2

2

* 0 and *M

X XP

.

If 1 2 0M P P , then 2 1

1

* 0 and *M

X XP

.

X2

U

X1

U

1 2( , )U X X U

The slope of the IC is

1 2

2 1

1

1

MU X

MU X

30

30

X2

X1

Assume we have interior solution.

2 1 1 2 2 1 1 2

1 2 1 2 1 2

1 2 1 2

( , , ) ( )( )2 2 2 2

M P P M P P M P P M P Pv P P M X X X X

P P P P

Assume 2 1 0M P P

1 2 1 2 1 2

2 2 2

( , , ) (0)( ) 0M M M

v P P M X X X XP P P

Assume 1 2 0M P P

1 1 1

1 2 1 2 1 2

1 1 1

( , , ) ( )(0) 0M M M

v P P M X X X XP P P

2

(0, )M

AP

1

( , 0)M

BP

@A, slope of IC2

2

1

( 1)0 1

M

P M

P

If the price line is steeper than the IC at that point, then

there will be a corner solution.

1 1 2

1 2

2 2 2 2

2 1

1

0

P P M PMP M P

P P P P

M P P

@B, slope of IC

1 1

0 1 1

1 1M M

P P

If the price line is flatter than the IC at this point, there

will be a corner solution.

1 1

2 1

2 1

1

1 2

1

1

0

P PP M P

MP M P

P

M P P

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31

Definition: 1( ,..., )nf x x is homogeneous to degree k

1 1if for any 0, ( ,..., ) ( ,..., )k

n nf x x f x x

Example:

1 2 1 2( , ) is homogemeous to degree a b

f x x x x a b

1 2 1 2 1 2 1 2( , ) ( ) ( ) ( , )a ba b a b a bf x x x x x x f x x

Example:

11 2 2

2

( , )x

f x xx

is homogeneous to degree 1

11 11 2 2 2

2 2

( )( , )

( )

x xf x x

x x

Example:

1 2 1 2

0

1 2

*( , , ) is homogeneous to degree 0 in ( , , ).2

Note: this is a Marshallian demand function or ordinary demand function.

*( , , )2 2 2

i

i

i

i i i

Mx P P M P P M

P

M M Mx P P M

P P P

Envelop Theorem for Constrained Optimization

1 2

1 2 1 2 1 2, ,...,

1 2 1 2

1 2 1 2 1 2 1 2

*( , ,..., ) ( , ,..., ; , ,..., )

s.t. ( , ,..., ; , ,..., ) 0

Let ( , ,..., , ; , ,..., ) ( , ,..., ; , ,...

n

m n mx x x

n m

n m n

z optimize f x x x

g x x x

L x x x f x x x

1 2 1 2, ) ( , ,..., ; , ,..., )

* *Then

m n m

i i

g x x x

z L

ix : choice variables

j : parameters

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32

Properties of Indirect Utility Function

Let 1( ,..., )nP P P

1. ( , ) is nondecreasing in and nonincreasing in iv P M M P

2. ( , ) is homogeneous to degree 0 in ( , ).iv P M P M

3.

( , )

( , ) ( , )

i

i

v P M

Px P M

v P M

M

Roy's Identity

4. ( , ) is quasi-convex in v P M P ; i.e. { : ( , ) }A P v P M is a convex set

Proof:

1. By the Envelop Theorem, 1 1 1[ ( ,..., ) ( ... )( , ) ** 0n n nU x x M Px P xv P M L

M M M

When we carry out the differentiation, ,ix etc. are treated as constants!!!

Also,

1 1 1[ ( ,..., ) ( ... )( , ) ** * 0n n n

i

i i i

U x x M Px P xv P M Lx

P P P

2. 2-goods case: X2

M

P2

X2* E

X1

X1* M

P1

When 1 2( , , )P P M are replaced by 1 2( , , )P P M , the budget line

remains the same.

1 1 2 2

;M M M M

P P P P

If the budget line remains the same, E will be the same. Hence

1 2( *, *)X X will be the same.

In general, *( , )iX P M is obtained by solving the problem:

1

1,...

1

max ( ,..., ) s.t. 0n

n

n i iX X

i

U X X M P X

When ( , ) is replaced by ( , )P M P M , there is no change in the

objective function and in the constraint. Hence same solution.

is the marginal

utility of income

33

33

3. Problem: 1

1,...,

1

max ( ,..., ) s.t. 0n

n

n i iX X

i

U X X M PX

1

1

( ,..., ) [ ]n

n i i

i

L U X X M PX

By the Envelop Theorem:

( , ) ** * (1)

( , ) ** (2)

( , ) ( , )

* *(1)* ( , )

( , ) ( , )(2) *

i

i i

i i i

i

v P M LX

P P

v P M L

M M

v P M v P M

X P PX P M

v P M v P M

M M

4. Given so that ( , )P A v P M and ' so that ( ', )P A v P M .

Let " (1 ) 'P P P . 1 1 1( ,..., ); ' ( ',..., '); " ( ",..., ")n n nP P P P P P P P P

Want to show: " , i.e. ( ", )P A v P M

Define the budget sets { : }i iB X PX M

' { : ' }i iB X P X M

" { : " }i iB X P X M

Want to show: " or 'X B X B B

Assume not. i.e. [ (1 ) ']i i iP P X M , but and 'i i i iPX M P X M .

' (1 ) ' (1 )

i i i i

i i i i

PX M PX M

P X M P X M

" [ (1 ) ']i i i i iP X P P X M

which contradicts the orginal assumption.

Note that

( ", ) max ( ) s.t. "

max ( ) s.t. or ' since " '

since ( , ) and ( ', )

X

X

v P M U X X B

U X X B X B B B B

v P M v P M

34

34

Example: 2

1 2

1 2

2 2

1 2 1 2 1 2

2 2

1 2 1 21 2 1 2

2

1 2

2

1 1 2

1

1 2 1

1 2

2

1 2

2 1 22

1 2

( , , )4

( , , ) ( , , ) ( , , ), ,

24 4

( , , )

4*

( , , ) 2

2

( , , )

4*

( , , )

Mv P P M

P P

v P P M v P P M v P P MM M M

P P M P PP P P P

Mv P P M

P P P MX

v P P M M P

P PM

Mv P P M

P P PX

v P P M

M

2

2

1 2

2

2

M

M P

P P

35

35

Expenditure function and Hicksian demand function

11

,...,1

( , ) min s.t. ( ,..., )n

n

i i nX X

i

e P U P X U U X X

Example:

1 21 1 2 2 1 2 1

,min s.t. 0X X

PX P X U X X X

1 1 2 2 1 2 1[ ]L P X P X U X X X

FONC:

1

2

1 2

2 1

1 2 1

( 1) 0 (1)

0 (2)

0 (3)

X

X

L P X

L P X

L U X X X

1 2 2

1 2

2 1 1

(1) and (2) ( 1) (4)1

P P PX X

X X P

2

1 2 1 1 2 2 2

1

(3) : 0 ( 1) [ ( 1)]( 1) 0P

U X X X U X X U X XP

2 22 1 12 2 2

1 2 2

( 1) 0 ( 1) 1 (5)hP UP UP

U X X XP P P

Hicksian demand function

2 2 1 21 2

1 1 2 1

( 1) = (6)h P P UP UP

X XP P P P

Hicksian demand function

Note that 2

hX must be non-negative.

12 1 2

2

(5) 0 1 0h UPX UP P

P

If 1 2UP P , then 2 10 and h h

X X U

X2

X1

( , 0)A U

The slope of the IC is 1 2

2 1

1MU X

MU X

@ A , slope of IC2

1

1 0 1 1X

X U U

If the price line is flatter than the IC at this point, there

will be a corner solution.

1

2 1

2

1PP UP

P U

36

36

Definition:

1( ,..., ) is a nf x x concave function 0 0 0 1 1 1

1 1 if ( ,..., ) and ( ,..., ), (0,1),n nx x x x x x we have 0 1 0 1 0 0 1 1

1 1 1 1ˆ( ) [ (1 ) ,..., (1 ) ] ( ,..., ) (1 ) ( ,..., )n n n nf x f x x x x f x x f x x

where 0 1 0 1 0 1

1 1ˆ (1 ) ( (1 ) ,..., (1 ) )n nx x x x x x x

Properties of the Expenditure Function:

1. ( , )

( , )h

i

i

e P Ux P U

P

2. ( , ) is nondecreasing in .ie P U P

3. ( , ) is concave in ( ).e P U P

4. ( , ) is homogeneous to degree 1 in ( ).e P U P

Proof:

1) and 2)

1 1 1[ ... ( ( ,..., )]( , ) *By the Envelop Theorem, 0hn n n

i

i i i

P x P x U U x xe P U Lx

P P P

3) 0 1 0 1

0 0 0 1 1 1

1 1

Want to prove: (0,1), [ (1 ) , ] ( , ) (1 ) ( , )

where ( ,..., ) and ( ,..., )n n

e P P U e P U e P U

P P P P P P

0 0 0 0

1

1 1 1 1

1

1

Let ( ,..., ) be the cheapest bundle to attain when

( ,..., ) be the cheapest bundle to attain when

ˆˆ ˆ ˆ ( ,..., ) be the cheapest bundle to attain when

n

n

n

x x x U P P

x x x U P P

x x x U P P

0 1 0 1 0 1

1 1 1ˆ ˆ ˆwhere ( ,..., ) (1 ) ( (1 ) ,..., (1 ) )n n nP P P P P P P P P

0 1ˆ x x x

0( )f x

1( )f x

0 1[ (1 ) ]f x x

0 1( ) (1 ) ( )f x f x

37

37

Note that

0 0 0 0

1

1 1 1 1

1

ˆ ˆ( , ) (1) when , is the cheapest bundle, not

ˆ ˆ( , ) (2) when , is the cheapest bundle, not

n

i i

i

n

i i

i

P x e P U P P x x

P x e P U P P x x

0 0

1

1 1

1

0 1 0 1

1 1

0 1 0

1

ˆ(1) ( , ) (3)

ˆ(2) (1 ) (1 ) ( , ) (4)

ˆ ˆ(3)+(4) (1 ) ( , ) (1 ) ( , )

ˆ[ (1 ) ] ( , ) (1

n

i i

i

n

i i

i

n n

i i i i

i i

n

i i i

i

P x e P U

P x e P U

P x P x e P U e P U

P P x e P U

1

0 1 0 1

) ( , )

ˆ( , ) [ (1 ) , ] ( , ) (1 ) ( , )

e P U

e P U e P P U e P U e P U

4. 2-goods case

When 1 2 1 2( , ) becomes ( , )P P P P , the slope of the iso-cost line remains the same ( 1 1

2 2

P P

P P

),

hence the consumer will buy the same bundle 1 2( , )h hx x

1

1 1 2 2 1 1 2 2new cost ( ) ( ) ( ) (cost)P x P x Px P x

1

1

In general, we have ( , ) min s.t. ( ,..., ) 0n

i i n

i

e P U Px U U x x

1 1Now suppose ( ,..., ) becomes ( ,..., ),n nP P P P , the constraint will be the same. On the other hand, the

objective function becomes 1 1

n n

i i i i

i i

Px Px

. Clearly the new objective function is a monotonic

transformation on the original one. Hence after the change in prices, we will still have the same cost-

minimization bundle.

1

1 1

new cost ( ) (cost)n n

i i i i

i i

P x Px

38

38

Example: ( ( , ) is a concave function in e P U P )

A consumer wants to attain U by consuming good 1 and good 2. He is facing an uncertainty on the

prices of the goods. He knows that there is a 50% probability that 1 2( , ) ($2,$2)P P , and a 50%

probability that 1 2( , ) ($4,$4)P P . Instead of facing the uncertainty, the consumer can sign a contract

allowing him to buy the goods at 1 2( , ) ($3,$3)P P without uncertainty.

Question: Should the consumer sign the contract?

Signing the contract: cost ($3,$3, )e U

Facing the uncertainty: expected cost 0.5 ($2,$2, ) 0.5 ($4,$4, )e U e U

Note that

($3,$3, ) [(0.5)($2) (0.5)($4), (0.5)($2) (0.5)($4), ] 0.5 ($2,$2, ) 0.5 ($4,$4, ),

hence the consumer SHOULD NOT sign the contract

e U e U e U e U

Important identities of duality

1. [ , ( , )]e P v P M M

2. [ , ( , )]v P e P U U

3. ( , ) [ , ( , )]h

i iX P M X P v P M

4. ( , ) [ , ( , )]h

i iX P U X P e P U

Example: Let 1

1 2 1 2( , , )a a

e P P U P P U

By the Envelop Theorem, we have

1 11 1 2 2 1 2 1 21 1 2

1 1 1

1 1 2 2 1 2 1 2

2 1 2

2 2 2

[ ( ( , )) ( , , )*

[ ( ( , )) ( , , )*(1 )

h a a

h a a

P X P X U U X X e P P ULX aP P U

P P P

P X P X U U X X e P P ULX a P P U

P P P

By the duality identity, we have

1 1

1 2 1 2 1 2 1 2 1 2 1

1 2

( , , ) ( , , ) ( , , )a a a a

a a

Me P P U P P U M P P v P P M v P P M

P P

By the Envelop Theorem, we have

1 1 1 21 2 1 2 1 2

1 2 1 2 1 2

1 2

( , , ) ( , , ) ( , , ), , ( 1)

a a a a a av P P M v P P M v P P MP P aMP P a MP P

M P P

1 1

1 1 2

1 1

11 2

2

2 1 22 1

21 2

Roy's identity * ,

( 1) (1 ) *

a a

a a

a a

a a

v P aMP P aMX

v M PP P

v P a MP P a MX

v M PP P

39

39

Example: Let

1

1 2 1 2( , , ) ( )r r rv P P M P P M

We have 1 1 1

1 11 11 2 1 2

1 2 1 2 1 1 2 1

1

1 11 11 11 2

1 2 2 1 2 2

2

( , , ) ( , , ) 1( ) , ( ) ( ) ,

( , , ) 1( ) ( )

r r r r r r r rr r r

r r r r r rr r

v P P M v P P MP P P P rP M P P P M

M P r

v P P MP P rP M P P P M

P r

11

1 1

1 1 2 1 1

1 1

1 2

1 2

11

1 1

2 1 2 2 2

2 1

1 2

1 2

( )Roy's identity * ,

( )

( ) *

( )

r r r rr

r r

r r r

r r r rr

r r

r r r

v P P P P M MPX

v M P PP P

v P P P P M MPX

v M P PP P

By the duality identity, we have 1 1 1

1 2 1 2 1 2 1 2 1 2 1 2( , , ) ( ) ( ) ( , , ) ( , , ) ( )r r r r r rr r rv P P M P P M U P P e P P U e P P U U P P

1 1

1 11 11 2

1 1 2 1 1 2 1

1

1 11 11 11 2

2 1 2 2 1 2 2

2

( , , ) 1( ) ( )

( , , ) 1( ) ( )

h r r r r r rr r

h r r r r r rr r

e P P UX U P P rP U P P P

P r

e P P UX U P P rP U P P P

P r

40

40

Example: min[2 ,3 ]U X Y

With this utility function, we know that we always buy 2 3X Y . [i.e. more X than Y]

Hicksian demand function:

In order to attain U , the consumer needs to buy

(3 2 ), ( , , )

2 3 2 3 6

h h X YX Y X Y

U P PU U U UX Y e P P U P P

( , , )(3 2 ) 6By duality ( , , )

6 3 2

X Y X YX Y

X Y

v P P M P P MM v P P M

P P

Marshallian demand function:

2

1

2

1

6 (3 2 ) (3) 3*

6(3 2 ) 3 2

6 (3 2 ) (2) 2*

6(3 2 ) 3 2

X X Y

X Y X Y

Y X Y

X Y X Y

v

P M P P MX

v P P P P

M

v

P M P P MY

v P P P P

M

Note that we can construct baskets of goods like this: ( , ) (3,2)X Y This basket will give the consumer

6 utils. Each basket costs 3 2X YP P

3* 3

3 2 3 2

2* 2

3 2 3 2

X Y X Y

X Y X Y

M MX

P P P P

M MY

P P P P

SAME AS BEFORE

41

41

Numerical example:

U XY 2

* , * and ( , , )2 2 4

X Y

X Y X Y

M M MX Y v P P M

P P P P

Let $100, $4, $5X YM P P , then 2 2100 100 100

* = =12.5, * = =10, ( , , ) 125 12.5 102 2 4 2 2 5 4 4 4 5

X Y

X Y X Y

M M MX Y v P P M

P P P P

Question: Suppose ' $5XP , how much extra money is needed to maintain the same utility?

22 ( , , )

( , , ) ( , , ) 44 4

X YX Y X Y X Y

X Y X Y

e P P uMv P P M u e P P u P P u

P P P P

If 125, ' $5, $5X Yu P P , then ( , , ) 4 4 5 5 125 12500 111.80X Y X Ye P P u P P u

Also 1

21 5 125

4 ( ) 125 11.182 5

h YY X

X X

P ueX P u P

P P

1

21 5 125

4 ( ) 125 11.182 5

h XX Y

Y Y

P ueY P u P

P P

42

42

Example

A household has a utility function

1 1

2 2( , )U H G H G where H is the housing consumption in square

feet and G is the amount of money spent on other goods.

a) Calculate the indirect utility function of this household.

Let $10/ square footHP and Income $10000 .

Calculate the optimal level of and H G . How many "utils" does it enjoy?

b) Suppose the government provides a 50% rent subsidy for this household. so that the rent goes

down to $5/ square foot (from the market rent of $10/square foot).

i) Calculate the optimal level of and H G . How many "utils" does it enjoy?

ii) What is the cost to the government?

c) Instead of rent subsidy, the government provides a cash subsidy to this household, how much

cash subsidy is needed to make the household as happy as enjoying the rent subsidy?

d) If your answer in c) is less than the cost to the government in b), the difference is the dead

weight loss (DWL) of the rent subsidy program. How large is this DWL?

e) If the government provides a cash subsidy to the household which is equal to your answer in b)

ii), how much utility will the household enjoy?

Solution This is a Cobb-Douglas utility function.

a)

1 1

2 2* , *1 1 1 12 2 2

2 2 2 2

H G

H G

M M MP H M H P G M G

P P

1 1

2 2( , ) ( ) ( )2 2 2

H

H H

M M Mv P M

P P

10000 10000* 500, * 5000

2 (2)(10) 2 2H

M MH G

P

10000( , ) 1581.14 500 5000

2 2 10H

H

Mv P M

P

b) When the subsidized rent $5 , 10000 10000

* 1000, * 50002 (2)(5) 2 2H

M MH G

P

1000 5000 2236.07U

cost to governmentmarket price subsidized price ($10)(1000) ($5)(1000) $5000

c) Let X be the amount of cash subsidy needed.

From b) we have 10000

2236.07 $4142.152 10

XX

d) DWL $5000 $4142.15 $857.85

e) From the indirect utility function, 5000 10000 5000

2371.712 2 10H

MU

P

43

43

Proposition (Slutsky Equation) ( , ) ( , ) ( . )j j j

i

i i

X P M h P U X P MX

P P M

Proof:

Let *X be the utility-maximizing bundle at ( *, *)P M and let * ( *)U U X .

It is identically true that ( , *) [ , ( , *)]j jh P U X P e P U

Differentiate with respect to iP and evaluate the derivative at *P :

( *, *) ( *, *) ( *, *) ( *, *) ( *, *)( , *)*

j j j j j

i

i i i i

h P U X P M X P M X P M X P Me P UX

P P M P P M

( *, *) ( *, *) ( *, *)*

j j j

j i i i i

i i

X P M h P U X P MX P P X P

P P M

in income to keep utility constant

total effect substitution effect income effect

1 1 1 11 2

1 21 1 1

2 2 22 2 2 21 2

1 2

h h X XX X

P PX P PM M

X P Ph h X XX X

P P M M

44

44

Lancaster’s characteristic approach to consumer theory

(“A New Approach to Consumer Theory”, Journal of Political Economy, 1966, pp. 132-57)

Motivation:

We observe that people will (suddenly) quit buying a good when its price goes up. This cannot be

explained by traditional theory.

i) Y

X

ii) Y

X

iii) Y

X

In this case, a consumer will always buy all

goods.

In this case, a consumer will always buy all

goods.

In this case, a consumer will only buy one and

one good only.

45

45

Assumption of Lancaster’s model:

People consume “characteristics” which are embodied in the goods.

Protein

C

B

C’ P

Vitamin

B: the amount of protein and vitamin embodied in the beef purchased by all the money one has.

P: the amount of protein and vitamin embodied in the pork purchased by all the money one has.

C: the amount of protein and vitamin embodied in the chicken purchased by all the money one has.

C’: the amount of protein and vitamin embodied in the chicken purchased by all the money one has

when the price of chicken goes up

At the initial prices, the consumer is going to buy chicken and pork.

When price of chicken goes up, C will move towards the origin. At first, the consumer will continue to

buy chicken and pork. Once the price of chicken rises beyond a certain “critical” level, then the

consumer will no longer buy any chicken, it will only buy beef and pork.

E E’

46

46

Irrational behavior and economic theory (Becker, Gary (1962): “Irrational Behavior and Economic Theory”, Journal of Political Economy,

February.)

Motivation:

It is hard to believe people are rational. In this paper, Becker argued that

i) the empirical results are consistent with the main implication of utility theory––a downward

sloping demand curve;

ii) households can be said to behave as if they are rational.

Y

C’

C

X

Assume a person is completely irrational, she/he will randomly pick a consumption bundle, then on

average, she/he will consume at C.

When or X YP P , the budget line becomes the dotted line. The budget set tilts towards Ythe

center of the new budget set is C’. Hence or ,X YP P X Y

The same result as is under the rational behavior assumption.

initial budget line

new budget line