Modeling Tradeoffs

24 hrs.

(An “Endowment”)

x1 =Time (Leisure)

x2=Income (Things)

0X02x

01x

11x

12x

1X—

02

01 , xx

12

11 , xx

—

?

2X

3X

What else do we need in order to build a theory that is useful for analyzing decisions such as these?

x2

X—Lots of stuff

Budget Constraints

x1

x2

01

0

x

m

02

0

x

m

2

1

p

p

Feasibleconsumption

set

Budget Line(efficient

consumption)

Budget Constraints

1x

2x

2p

m

01p

m2

01

p

p

2

11

p

p

11p

m

01

11 pp

1x

2x

2

0

p

m

2

1

p

p

01 mm

2

1

p

m

1

0

p

m

1

1

p

m

Price Changes Income Changes

Modeling Preferences

Fundamental axioms:

1. Completeness

2. Transitivity

3. Continuity

[ ] BAorABBAXBA ~ , ,,∈,∀

CACBBA then , and , If

5. Convexity( )[ ] BABttAt ~1+ then ,1<<0 If

4. Monotonicity

Supplemental axioms:

Necessary for rank ordering and functional representation.

Defines “well-behaved preferences.

Indifference Curves, Preferences, and Rational Ordering

1x

2x

—0X

—1X

—3X

01x 1

1x

12x

02x

—

2X

21x

22x

3102 ~ XXXX

Indifference Curves, Preferences, and Rational Ordering

1x

2x

—0X

—1X

—3X

01x 1

1x

12x

02x

—

2X

21x

22x

0

0 1x

2x 502

50121

.., xxxxU

U = 16

U = 8U = 4U = 2

Indifference Curves and Utility

4 8 16

4

8

16

Bentham and Utilitarianism

“By utility is meant that property in any object,whereby it tends to produce benefit, advantage, pleasure, good, or happiness (all this in the present case comes to the same thing) or (what comes again to the same thing) to prevent the happening of mischief, pain, evil, or unhappiness to the party whose interest isconsidered …

“The interest of the community then is what? -- the sum of the interests of the several members who compose it.

“An action then may be said to be conformable to the principle of utility* … when the tendency it has to augment the happiness of the community is greater than any it has to diminish it.”

An Introduction to the Principles of Morals and Legislation, 1780

*Elsewhere, the greatest happiness principle.

Jeremy Bentham 1748 – 1832

2

12,1 MU

MUMRS

21 32 xx

21xxba xx 21

21 bxax

212 xx

12 21 xx

21, xxu

21ln xx

1

211

,

x

xxuMU

2

212

,

x

xxuMU

21,min bxax

221 xx

Individual Demand

1x

2x

2p

m

01p

m

1x

1p11p

m21p

m

01p

11p

21p

D

1x

2x

1I

0I

01x 1

1x 21x

SE IE

Substitution & Income Effects: Normal GoodThe Slutsky Approach – “Pivot and Shift”

AA’ B

Price Decrease for x1 (p1↓)

Substitution & Income Effects: Normal GoodsThe Slutsky Approach – “Pivot and Shift”

1x

2x

1I

0I

1.3 = SE IE = 0.7

A A’

B

14 15.3 16

78

Original Demand (A): {x10, x2

0} = {14, 78}

Compensated Income, m’ = 106

Compensated Demand (A’): {x1’, x2’} = {15.3, 75.4}

New Demand (B): {x11, x2

1} = {16, 88}88

75.4

x1 (p1, p2, m) = 10 + m/10p1

p10 = 3

p11 = 2

m0 = 120

SE

IE

Substitution & Income Effects: x1,Normal Good; x2, Inferior GoodThe Slutsky Approach – “Pivot and Shift”

1x

2x

1I0I

SE IE

A A’B

SEIE

1x

2x

1I

0I

01x 1

1x21x

SE

IE

Substitution & Income Effects: Inferior GoodThe Slutsky Approach – “Pivot and Shift”

A’

B

A

Price Decrease for x1 (p1↓)

1x

2x

1I

0I

01x 1

1x21x

SE

IE

Substitution & Income Effects: Giffen GoodSlutsky Approach – “Pivot and Shift”

AA’

B

Price Decrease for x1 (p1↓)

1x

2x

1I0I

01x 1

1x 21x

SE IE

Substitution & Income Effects: Normal GoodThe Hicksian Approach – “Rotate and Shift”

A

A’B

Price Decrease for x1 (p1↓)

1x

2x

0I

01

02

Buying & Selling

01x

02x

Net Supplierof good 1x

Net Demanderof good

2xInitial

Endowment

1x

2x

0I01

02

01x

02x

Net Demanderof good 1x

Net Supplierof good

2x

Initial Endowment

Buying and Selling: Consumer Choice Theory with Endowments Decomposition of effects of a price change into substitution, ordinary income and endowment income

effects: Price decrease of a good currently being demanded.

0I

•

•

{ }02

01 ,= xxA

1x

2x

02x

{ }02

01 ,= E

02

01x0

1

•

OIE

•

SE

•

EIE

The agent is a net demander of x1; thus, when its price falls, the agent’s purchasing power goes up – the ordinary income effect is positive.

The value of the agent’s endowment is dependent upon the price of x1; thus, when its price falls, the agent’s purchasing power goes down – the endowment income effect is negative.

Buying and Selling: Consumer Choice Theory with Endowments Decomposition of effects of a price change into substitution, ordinary income and endowment income

effects: Price increase of a good currently being demanded.

•

{ }02

01 ,= xxA

1x

2x

02x

{ }02

01 ,= E

02

01x0

1

•

OIE

•

SE

••

EIE

The agent is a net demander of x1; thus, when its price increases, the agent’s purchasing power goes down – the ordinary income effect is negative.

The value of the agent’s endowment is dependent upon the price of x1; thus, when its price increases, the agent’s purchasing power goes up – the endowment income effect is positive.

Buying and Selling: Consumer Choice Theory with Endowments Decomposition of effects of a price change into substitution, ordinary income and endowment income

effects: Price decrease of a good currently being supplied.

•{ }0

201 ,= xxA

1x

2x

02x

{ }02

01 ,= E

02

01x 0

1

•

•

SE OIE

EIE

• •

Buying and Selling: Consumer Choice Theory with Endowments Decomposition of effects of a price change into substitution, ordinary income and endowment income

effects: Price increase of a good currently being supplied.

•{ }0

201 ,= xxA

1x

2x

02x

{ }02

01 ,= E0

2

01x 0

1

•

•

SEOIE

•

•

EIE

Buying and Selling: Consumer Choice Theory with Endowments Decomposition of effects of a price change into substitution, ordinary income and endowment income

effects: Price increase of a good currently being supplied.

•

•{ }0

201 ,= xxA

1x

2x

02x

{ }02

01 ,= E0

2

01x 0

1

•

•

SEOIE

EIE

•

An outcome illustrating perfectly inelastic demand for x1 …

Labor-leisure tradeoffs and labor supply

•

1x

2x

•

Leisure

= leisure, l

= consumption, c

l0 L0

C0

c0

C0 + (w/p)L0

Labor

E

A