L04

Choice

Big picture

Behavioral Postulate:A decisionmaker chooses its most preferred alternative from the set of affordable alternatives.

Budget set = affordable alternatives To model choice we must have

decisionmaker’s preferences.

MRS and Utility Function

xx22

xx11

MRS is a slope of the RS is a slope of the indifference curve at x indifference curve at x

xx Q: Can we recover MRS from knowledge of U(x)?

Utility and Marginal Utility

)(xU235.3

1 2 3

1 2 3

)(xMU

x

U

x

MU

MRS and Marginal Utility

Yes, but we have to find marginal utilities for both goods first (MU)

MU: How much utility we gain by adding an extra unit of good i

MU U

xii

.2

1

MUMUMRS

Example U(x1,x2) = x1

x2

Example V(x1,x2) = ln(x1)+ln(x2)

Cobb Douglass utility function

Log function

Cobb-Douglass utility

baxxxxU 2121 ),(

Conclusion: preferences

yxxy lnlnln lnln xaxa

are Cobb-Douglass

Problem: We know preferences (utility function)

and

We want to know optimal choice

10,1,1 21 mpp

2121 lnln),( xxxxU

),( *2

*1 xx

MRS of Monotonic transformations V=f(U) and f strictly increasing Can we say something about MRS?

Choice 10,1,1 21 mpp2121 lnln),( xxxxU

$ $ $ $ $ $ $ $ $ $

1MU 2MU

How can we modify our argument if

Marginal utility of a dollar

Different prices

We should equalize MU of a $!

1 ,1 21 porp

Choice 10,2,1 21 mpp2121 lnln),( xxxxU

$ $ $ $ $ $ $ $ $ $

1

1

pMU

2

2

pMU

Problem (thinking on the margin!)Two secrets of happiness:1. Spend your total income

2. Equalize marginal utility of a $

Rearranging:

Choice: Calculation 10,2,1 21 mpp2121 lnln),( xxxxU

Choice: geometric solution

x1

x2

10,2,1 21 mpp2121 lnln),( xxxxU

SOH for Well-behaved preferences

But: Perfect Complements (Right and Left

shoe) Perfect Substitutes (Example: French

and Dutch Cheese)

SOH for other preferences

Perfect Complements (Shoes)

L

R10,2,1 21 mpp),min(),( 2121 xxxxU

More generally

L

R10,2,1 21 mpp),2min(),( 2121 xxxxU

),min(),( 2121 bxaxxxU

Choice: Calculation 10,2,1 21 mpp),2min(),( 2121 xxxxU

Perfect Substitutes (F & D cheese)

F

10,2,1 21 mpp2121 ),( xxxxU

D

More generally

X1

2121 3),( xxxxU

2121 ),( bxaxxxU

10,2,1 21 mpp

X2

Perfect Substitutes 10,6,2 21 mpp2121 84),( xxxxU