Choice

Q: What is the Optimal Choice?

Budget constraint

Indifference curves

More preferred bundles

2x

1x

Z

Y

X

A: Optimal Choice is X

Optimal choice: indifference curve tangent to budget line.Does this tangency condition necessarily have to hold at an optimal choice?

2x

1x

*2x

*1x

X

Perfect Substitutes

*X

2x

*2x *

1x 1x

Perfect Complements

2x

1x

*2x

*1x

*X

Q: Is Tangency Sufficient?

2x

1x

X

Y

Z

What is the General Rule?

If: • Preferences are well-behaved. • Indifference curves are “smooth” (no kinks).• Optima are interior.

Then:

Tangency between budget constraint and indifference curve is necessary and sufficient for an optimum.

Multiple Optima

A way to avoid multiplicity of optima, is to assume strictly convex preferences.

This assumption rules out “flat spots” in indifference curves.

2x

1x

bundles

optimal

Economic Interpretation

At optimum: “Tangency between budget line and indifference curve.”

Slope of budget line:

Slope of indifference curve:

Tangency:

2

1

p

p

)( 2,1 xxMRS

)( 2,12

1xxMRS

p

p

Interpretation

At Z:

At Y:

2x

1xY

X

Z

)( 2,12

1xxMRS

p

p

)( 2,12

1xxMRS

p

p

Tangency with Many Consumers

Consider many consumers with different preferences and incomes, facing the same prices for goods 1 and 2.

Q: Why is it the case that at their optimal choice the MRS between 1 and 2 for different consumers is equalized?

)( **2,1 ii xx

Tangency with Many Consumers

A: Because if a consumer makes an optimal choice, then:

Implication: everyone who is consuming the two goods must agree on how much one is worth in terms of the other.

)( **2,1

2

1iii xxMRS

p

p

i

Tangency with 2 Consumers

Indifference curves of consumer 1:

Indifference curves of consumer 2:

Budget line:

2x

1x

X

Y

Finding the Optimum in Practice: a Cobb-Douglas Example

Preferences represented by:

Budget line:

2121 log)1(log),( xcxcxxu

mxpxp 2211

Finding the Optimum in Practice: a Cobb-Douglas Example

Mathematically, we would like to:

such that

212,1

log)1(logmax xcxcxx

mxpxp 2211

Finding the Optimum in Practice: a Cobb-Douglas Example

Replace budget constraint into objective function:

New problem:

12

1

22 x

p

p

p

mx

1

2

1

21

1log)1(logmax x

p

p

p

mcxc

x

Finding the Optimum in Practice: a Cobb-Douglas Example

New problem:

First-Order Condition:

012

1

11

2

1

p

p

xpm

pc

x

c

1

2

1

21

1log)1(logmax x

p

p

p

mcxc

x

Finding the Optimum in Practice: a Cobb-Douglas Example

First-Order Condition:

Rearranging:

012

1

11

2

1

p

p

xpm

pc

x

c

)(1

2,112

11

2

1xxMRS

xp

xpm

c

c

p

p

Finding the Optimum in Practice: a Cobb-Douglas Example

First-Order Condition:

Solve for :

Expenditures share in 1:

1x

12

11

2

1

1 xp

xpm

c

c

p

p

11

*

p

mcx

cm

xp

*11

Finding the Optimum in Practice: a Cobb-Douglas Example

Q: How do I find ?

A: Use the budget constraint:

*2x

12

1

21

2

1

22

**

p

mc

p

p

p

mx

p

p

p

mx

2

1p

mc