Modeling Tradeoffs
x2(Things)
x2=Income
Lots of stuff

X
What else do we need in order to
build a theory that is useful for
analyzing decisions such as these?
X2
?
1
2
x
x

1
1
1

x
,
x
X
1
2

X
3
0
0
0


x
,
x
X

1
2  (An “Endowment”)
0
2
x11
24 hrs.

x10
x1 =Time
(Leisure)
Budget Constraints
0
m
x20
x2
Budget Line
(efficient
consumption)
Feasible
consumption
set
p1

p2
x1
m0
x10
Budget Constraints
Price Changes
Income Changes
x2
x2
m1
p2
m
p2
m0
p2
p11  p10
p10

p2
p11

p2
m
p10
m1  m0

m
p11
x1
p1
p2
m0
p1
m1
p1
x1
Modeling Preferences

Fundamental axioms:
1. Completeness
∀[A, B]∈X , A  B, B  A, or A ~ B
2. Transitivity
If A  B, and B  C, then A  C
Necessary for rank
ordering and
functional
representation.
3. Continuity

Supplemental axioms:
4. Monotonicity
5. Convexity
If 0 < t < 1, then [tA + (1 t )B]  A ~ B
Defines “wellbehaved
preferences.
Indifference Curves, Preferences, and Rational Ordering
x2
x20

X0
X2
x22
x12
X 2  X 0 ~ X1  X 3


x10
X3

x12
X1
x11
x1
Indifference Curves, Preferences, and Rational Ordering
x2
x20

X0
X2
x22
x12


x10
X3

x12
X1
x11
x1
Indifference Curves and Utility
x2
U x1 , x2   x10.5 x20.5
16
U = 16
8
4
0
0
4
8
16
U=8
U=4
x1 U = 2
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 is
considered …
“The interest of the community then is what? -- the sum
of the interests of the several members who compose it.
Jeremy Bentham
1748 – 1832
“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.
ux1 , x2 
2 x1  3x2
ax1  bx2
x1  x2 2
2 x1  x2
ln x1  x2
min ax1 , bx2 
x1 x2
x1a x2b
x1  2x2  1
MU1 
u x1 , x2 
u x1 , x2 
MU 2 
x2
x1
MRS1, 2  
MU 1
MU 2
Individual Demand
x2
m
p2
p1
m
p12
m
p11
p12
p11
p10
D
x1
m
p10
x1
Substitution & Income Effects: Normal Good
The Slutsky Approach – “Pivot and Shift”
Price Decrease for x1 (p1↓)
x2
A
B
A’
I1
I0
0
1
1
1
x
2
1
x
SE
x
IE
x1
Substitution & Income Effects: Normal Goods
The Slutsky Approach – “Pivot and Shift”
x1 (p1, p2, m) = 10 + m/10p1
p10 = 3
p11 = 2
m0 = 120
x2
Original Demand (A): {x10, x20} = {14, 78}
Compensated Income, m’ = 106
Compensated Demand (A’): {x1’, x2’} = {15.3, 75.4}
B
88
SE
IE
75.4
78
A
A’
New Demand (B): {x11, x21} = {16, 88}
I1
I0
14
x1
15.3 16
1.3 = SE IE = 0.7
Substitution & Income Effects: x1,Normal Good; x2, Inferior Good
The Slutsky Approach – “Pivot and Shift”
x2
SE
IE
A
A’
B
I1
I0
x1
SE
IE
Substitution & Income Effects: Inferior Good
The Slutsky Approach – “Pivot and Shift”
Price Decrease for x1 (p1↓)
x2
B
I1
A
A’
I0
0
1
2
1
x x
SE
IE
1
1
x
x1
Substitution & Income Effects: Giffen Good
Slutsky Approach – “Pivot and Shift”
Price Decrease for x1 (p1↓)
x2
B
I1
A
A’
I0
2
1
x
0
1
1
1
x
x
SE
IE
x1
Substitution & Income Effects: Normal Good
The Hicksian Approach – “Rotate and Shift”
Price Decrease for x1 (p1↓)
x2
A
B
A’
I1
I0
0
1
1
1
x
2
1
x
SE
x
IE
x1
Buying & Selling
x2
x2
Initial
Endowment
x 20
Net Demander
of good x
2

I0
0
2
Initial
Endowment
Net Supplier
of good x
 20
2
x 20
0
1
x

0
1
Net Supplier
of good x1
x1
 10
x10
Net Demander
of good x1
I0 x
1
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.
x2
 20
x20
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.
•
E = { 10 , 20 }
•
•
{
}
A= x ,x
0
1
0
2
•
I0
 10
x1
x10
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 demanded.
x2
 20
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.
•
•
E = { 10 , 20 }
x20
•
•
 10
•
A = {x10 , x20 }
x10
OIE
EIE
SE
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.
x1
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.
x2
•
x
0
2
A = {x10 , x20 }
•
•
•
 20
• E = {
x10
 10
SE
EIE
OIE
0
1
, 20 }
x1
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.
x2
•
x
0
2

0
2
•
A = {x10 , x20 }
•
•
E = { 10 , 20 }
•
x10
OIE
EIE
SE
 10
x1
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.
x2
An outcome illustrating perfectly inelastic
demand for x1 …
•
•
x
A = {x10 , x20 }
0
2
•
•

E = { 10 , 20 }
•
0
2
x10
OIE
EIE
SE
 10
x1
Labor-leisure tradeoffs and labor supply
x2 = consumption, c
C0 + (w/p)L0
•
c0
A
•E
C0
l0
Leisure
L0
Labor
x1 = leisure, l