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ECON6021 Microeconomic Analysis

Consumption Theory I

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Topics covered

1. Budget Constraint

2. Axioms of Choice & Indifference Curve

3. Utility Function

4. Consumer Optimum

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Y

X

YA

YB

XA XB

A

B

Bundle of goods

• A is a bundle of goods consisting of XA units of good X (say food) and YA units of good Y (say clothing).

• A is also represented by (XA,YA)

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Convex Combination

B

AC

(xA, YA)

(xA, YB)

x

y))1(,)1((),( BABAcc yttyxttxyx

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BBA

BBA

xxttx

yytty

))1((

))1((CB of Slope

C is on the st. line linking A & B

AB of Slope

BA

BA

BA

BA

xx

yy

txtx

tyty

Conversely, any point on AB can be written as

[0,1] t where))1(,)1(( BABA yttyxttx

Convex Combination

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Slope of budget line

y

x

P

P

dx

dy (market rate of substitution)

Unit: $) ofnt (independejar per loaves loafper $

jarper $

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Example:

jar of beer Px=$4loaf of bread Py=$2

jarper loaves 2loafper 2$

jarper 4$

y

x

P

P

Both Px and Py double,

xP

I0

feasible consumption set

yP

I0|Slope|=

y

x

P

P

jarper loaves 2loafper 4$

jarper 8$'

'

y

x

P

PNo change in market rate of substitution

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Tax: a $2 levy per unit is imposed for each good

2

3

2$2$

2$4$

tP

tP

y

x Slope of budget line changes

y

x

20I

40I

40I

60I

after levy is imposed

80I

After doubling the prices

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Axioms of Choice

& Indifference Curve

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Axioms of Choice

• Nomenclature: : “is preferred to” : “is strictly preferred to” : “is indifferent to”

• Completeness (Comparison)– Any two bundles can be compared and one of the

following holds: AB, B A, or both ( A~B)• Transitivity (Consistency)

– If A, B, C are 3 alternatives and AB, B C, then A C; – Also If AB, BC, then A C.

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Axioms of choice• Continuity

– AB and B is sufficiently close to C, then A C.

• Strong Monotonicity (more is better)– A=(XA , YA), B=(XB , YB) and XA≥XB, YA≥YB with at

least one is strict, then A>B.

• Convexity– If AB, then any convex combination of A& B is

preferred to A and to B, that is, for all 0 t <1,– (t XA+(1-t)XB, tYA+(1-t)YB) (Xi , Yi), i=A or B.– If the inequality is always strict, we have strict

convexity.

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Indifference Curve• When goods are divisible and there are only

two types of goods, an individual’s preferences can be conveniently represented using indifference curve map.

• An indifference curve for the individual passing through bundle A connects all bundles so that the individual is indifferent between A and these bundles.

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Properties of Indifference Curves

• Negative slopes• ICs farther away from

origin means higher satisfaction

X

Y

A

I

II

Not preferred bundles

Preferred bundles

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• Non-intersection– Two indifference

curves cannot intersect

• Coverage– For any bundle, there

is an indifference curve passing through it. X

Y

AQ

P

Properties of Indifference Curves

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• Bending towards Origin– It arises from

convexity axiom

– The right-hand- side IC is not allowed

X

Y

Properties of Indifference Curves

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Utility Function

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Utility Function

• Level of satisfaction depends on the amount consumed: U=U(x,y)

• U0 =U(x,y)– All the combination of x & y that yield U0 (all

the alternatives along an indifference curve)

• y=V(x,U0), an indifference curve U(x,y)/x, marginal utility respect to x,

written as MUx.

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X

Y

A

B

YA

YB

XA XB

U0

dyy

yxUdx

x

yxUdU

),(),(

0

( , ) ( , )0

U x y U x ydU dx dy

x y

(by construction)

Slope: /

0/

xU U

y

dy U x MU

dx U y MU

(if strong monotonicity

holds)

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Bxy

Axy MRSMRS A

B

X

Y

0U

The MRS is the max amount of good y a consumer would willingly forgo for one more unit of x, holding utility constant (relative value of x expressed in unit of y)

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• Marginal rate of substitution

0

/0

/x

xyU U y

dy dy U x MUMRS

dx dx U y MU

DMRS: 0 constant 0U

dMRS

dx

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U=10

U=20

U=30

V=100

V=200V=2001

An order-preserving re-labeling of ICs does not alter the preference ordering.

Measurability of Utility

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22

66

222''''

'''

''

222'

yxUU

xyUU

xyUU

yxUU

xyU

Positive monotonic (order-preserving) transformation

• They are called positive monotonic transformation

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Positive Monotonic Transformation

What is the MRS of U at (x,y)?

How about U’?

xy

U

yx

U

/MRS

/

U x y

U y x

yxy

U

xyx

U

2'

2'

2

2

2

2

'/ 2MRS'

'/ 2

U x xy y

U y x y x

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Positive Monotonic Transformation

• IC’s of order-preserving transformation U’ overlap those of U.

• However, we have to make sure that the numbering of the IC must be in same order before & after the transformation.

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Positive Monotonic Transformation

• Theorem: Let U=U(X,Y) be any utility function. Let V=F(U(X,Y)) be an order-preserving transformation, i.e., F(.) is a strictly increasing function, or dF/dU>0 for all U. Then V and U represent the same preferences.

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Proof

Consider any two bundles and

Then we have:

( , )A AA x y

( , ).B BB x y

( , ) ( , )

( ( , )) ( ( , ))

( , ) ( , )

U

A A B B

A A B B

A A B B

V

A B

U x y U x y

F U x y F U x y

V x y V x y

A B

Q.E.D.

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Consumer Optimum

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Constrained Consumer Choice Problem

• Preferences: represented by indifference curve map, or utility function U(.)

• Constraint: budget constraint-fixed amount of money to be used for purchase

• Assume there are two types of goods x and y, and they are divisible

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Consumption problem• Budget constraint

– I0= given money income in $– Px= given price of good x– Py= given price of good y

• Budget constraint: I0Pxx+Pyy• Or, I0= Pxx+Pyy (strong monotonicity)

dI0= Pxdx+Pydy=0 (by construction)

Pxdx=-Pydy

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D

BYB

YD

XB XA XD

C

A

BA

DBxy XX

yyMRS

Psychic willingness to substitute

At B, my MRS is very high for X. I’m willing to substitute XA-XB forYB-YD. But the market provides me more X to point D!

y

xxy P

PMRS

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Consumer Optimum

• Normally, two conditions for consumer optimum:

• MRSxy = Px/Py (1)

• No budget left unused (2)

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Y

X

A

C

U1U0

Both A & C satisfy (1) and (2)

Problem: “bending toward origin” does not hold.

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coffee

teaU0

U1

U2

Generally low MRS

tea

coffee

Generally high MRS

y)(x, allfor y

xxy P

PMRS y)(x, allfor

y

xxy P

PMRS

Special Cases

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Quantity Control

• Max U=U(x,y)

Subject to (i) I ≥Pxx+Pyy

(ii) R≥x

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y

x

(1)

(2)

(3)

(4)

(1) Corner at x=0(2) Interior solution 0<x<R(3) “corner” at R(4) “corner” at R

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,max

subject to (1)

,

MRS market rate of sub. (2)

or

Hence ,

x y

x y

x y

x x

y y

y x

x y

U xy

I P x P y

MU y MU x

MU y P

MU x P

P y P x

I Ix y

P P

( (1))

An Example: U(x,y)=xy

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C

D

B

A

A satisfies (1) but not (2)B, C satisfy (2) but not (1)Only D satisfies both (1) &(2)

1 11

2 2 / 2x x x

x I I I

I x P x P I P

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Other Examples of Utility Functions

},min{),(

),(

yxyxU

yxyxU

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An application: Intertemporal Choice

• Our framework is flexible enough to deal with questions such as savings decisions and intertemporal choice.

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Intertemporal choice problem

C1

C2

1600

1000

500 Slope = -1.1

u(c1,c2)=const

Income in period 2

Income in period 2

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1000-C1=S (1)

500+S(1+r)=C2 (2)Substituting (1) into (2), we have

500+(1000-C1)(1+r)=C2

Rearranging, we have

1500+1000r-(1+r) C1=C2 > C

Using C1=C2=C, we finally have 1600 1500 1000 500

10002.1 2 2

rC

r r

r C (S )