Economics 101A Lecture 05 Revised

7/25/2019 Economics 101A Lecture 05 Revised

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Economics 101A

(Lecture 5, Revised)

Stefano DellaVigna

September 9, 2003


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Outline

1. Properties of Preferences (continued)

2. From Preferences to Utility (and viceversa)

3. Common Utility Functions

4. (Utility maximization)


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2 From preferences to utility

Nicholson, Ch. 3

Economists like to use utility functions u:X R

u(x) is liking of good x

u(a)> u(b) means: I prefer ato b.

Def. Utility function u represents preferences if,

for all x and y in X, x y if and only ifu(x)

u(y).

Theorem. If preference relation is rational andcontinuous, there exists a continuous utility function

u:X Rthat represents it.


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Doesu represent?

xyimplies(u(x), u(x)) xy (u(y), u(y))

[by transitivity](u(x), u(x))(u(y), u(y)) =

[by monotonicity]u(x) u(y)

Similarly can prove other direction (exercise!)

(We do not prove continuity ofu(x))


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Utility function representing is not unique

Take exp(u(x))

u(a)> u(b) exp(u(a))>exp(u(b))

Ifu(x) represents preferences and f is a strictly

increasing function, then f(u(x)) represents as

well.

If preferences are represented from a utility function,

are they rational?

completeness

transitivity


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3 Common utility functions

Nicholson, Ch. 3, pp. 8084

1. Cobb-Douglas preferences: u(x1, x2) =x1 x

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MRS=

x

a1

1 x

1

2 /(1

a)x

1 x

2 =

1

x2

x1

2. Perfect substitutes: u (x1, x2) =x1+x2

MRS= /


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3. Perfect complements: u (x1, x2) = min (x1, x2)

MRSdiscontinuous at x2= x1

4. Constant Elasticity of Substitution: u (x1, x2) =

x

1+x

2

1/

MRS=

x1x2

1

if= 1, then...

if= 0, then...

if +, then...

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Economics 101A Lecture 05 Revised