Review of Previous Lecture
1
1. Main goal: Derive consumer demand.
2. Definition of a bundle (or basket)
3. Many factors influence choices, we start by focusing on tastes or preference
Preferences satisfy 3 assumptions….
Review of Previous Lecture
2
4. Expressed preferences using indifference curves,
Properties of indifference curves
Units of Clothing
•
•
•
•
•
B
A
0
C
F
D
1.
Direction of improvement is “north east”
2.
Each basket lies on a single indifference curve
3.
Indifference curves have negative slope
4.
Indifference curves do not cross
5.
Indifference curves are not “thick”
Units of Food
The Utility Function
3
Another way to express preferences is through utility functions.
Definition:
A Utility Function is a function that attaches a number to each bundle.
We say that a utility function u(x) represents preferences when
Bundle A is preferred to bundle B if and only if u(A)>u(B)
That is, a more preferred bundle gets a higher number than a less preferred
one.
Types of Ranking
4
Definition:
- A ranking is ordinal when only the order is important
- A ranking is cardinal when the order and the absolute performance are both
important
Example:
- Ordinal ranking: At the exam, Harry did best, Joe did second best, Betty did third
best, and so on.
- Cardinal ranking: Harry got 80, Joe got 75, Betty got 74 and so on.
The ordinal ranking is the important ranking in Consumer Theory.
– Any transformation of a utility function that preserves the original ranking of bundles
is an equally good representation of preferences.
The Utility Function: Example
5
Example:
1.
U(x,y) = x*y2
2.
3.
U(x,y)= x*y2
x
U(x,y)= x2 +y2
U(x,y) = (xy) 0.5
y
7
4
112
5,29
65
U(x,y) = (xy) 0.5
5
5
125
5,00
50
U(x,y)= x2+y2
4
5
100
4,47
41
3
7
147
4,58
58
1
12
144
3,46
145
Q: What is each consumer’s favorite bundle? What is each
consumer’s least favorite bundle?
The Utility Function: Example
6
Indifference curves:
Example:
20
18
1.
U(x,y) = x*y2
16
2.
U(x,y) = (xy) 0.5
12
14
Series2
10
3.
U(x,y)= x2+y2
Series3
8
Series4
6
4
2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Marginal Utility
7
Definition:
The marginal utility of a good, is the additional utility that the consumer gets from
consuming a little more of x when the consumption of all the other goods in the
consumer’s basket remains constant.
Marginal Utility
8
Example:
x*y2
1.
U(x,y) =
2.
U(x,y) = (xy) 0.5
3.
U(x,y)= x2+y2
Does the marginal utility diminish?
Marginal Rate of Substitution
9
Along the indifference curve, the utility does not change.
Technically:
Marginal Rate of Substitution
10
Example:
1.
U(x,y) = x*y2
2.
U(x,y) = (xy) 0.5
3.
U(x,y)= ax2+by2
Is MRS diminishing?
Preferences and Utility Functions
11
There are many types of goods and many types of preferences.
Some goods are substitutes (examples?)
Some goods are complements (examples?)
Some goods are neither.
Special Functional Forms
12
Cobb-Douglas: U = axy
y
Key Property:
The MRS diminishes, indifference curves are
convex
Preference Direction
IC2
IC1
Special Functional Forms
13
Perfect Substitutes: U = ax + by
Key Property:
The consumer is willing to substitute a/b units of y for
1 more unit of x, everywhere on the indifference curve
y
Slope = -a/b
IC1
IC2
IC3
Special Functional Forms
14
Perfect Complements: U = min{x,y}
y
Key Property:
The consumer can only enjoy one good when it is
consumed with the other.
IC2
IC1
Special Functional Forms
15
Key property:
MRS increases: the more x you have, the
more y you are willing to trade for it.
Examples?
Special Functional Forms
16
Quasi linear preferences: U = v(x) + ay
Key property:
MRS depends only on the quantity of x
IC2
IC1
•
•
IC’s have same
slopes on any
vertical line