Choice Continued

Interpretation

At Z:

At Y:

2x

1xY

X

Z

)( 2,12

1xxMRS

p

p

)( 2,12

1xxMRS

p

p

Consumer Demand

Consumer’s demanded bundle: optimal choice of goods 1 and 2 for given prices and income.

Consumer’s demand functions:

mppxx ,, 2111

mppxx ,, 2122

Cobb-Douglas

Demand function for good 1:

Demand function for good 2:

2x

1x

*2x

*1x

11

p

mcx

2

2 1p

mcx

Perfect Substitutes

2x

1x

1

2

1

p

p

*1x

Demand function for good 1:

if

if

if

21 pp 11 / pmx

11 /,0 pmx

01 x 21 pp

21 pp

Perfect Complements

Optimal choice:

Budget line:

Demand function for goods 1 and 2:

*1x

*2x

1x

2x12 xx

12 xx

mxpxp 2211

2121

pp

mxx

Example: Choosing a Tax

Q: Suppose that the government wants to raise a certain amount of revenue. Is it better to raise it via a quantity tax or an income tax?

Before the Tax

Budget line:

Well-behaved preferences

Consumer chooses:

2x

1x

ox2

ox1

mxpxp 2211

oo xx 21 ,

2

1

p

p

Quantity Tax

Budget line with quantity tax:

Tax revenues:

2x

1x

*2x

*1x

mxpxtp 2211

*1

* txR

2

1

p

tp

ox1

ox2

Income Tax

New budget line:

where

*2211 Rmxpxp

*1

* txR

Questions

Q1: What is the slope of the budget line with the income tax?

Q2: Which kind of tax is this income tax?

Q3: Can the consumer afford when he is paying the income tax?

*X

Answers

A1: Slope of the budget line with income tax is:

A2: Lump-sum.

2

1

p

p

A3: Yes. Is still Affordable

Budget line with income tax:

Substitute :

Rearrange:

Evaluate at : Rearrange:

*12211 txmxpxp

*X***

12211 txmxpxp

mxpxtp **2211

*X

On the Graph

2x

1x

oX*X

2

1

p

p

Q: Is Optimal with the Income Tax?

2x

1x

oX*X

*X

A: No, it is not.

2x

1x

oX*X *X

X X

Q: Which Tax Does the Consumer Prefer?

2x

1x

oX*X

X

A: Income Tax Gives Higher Utility to the Consumer

Caution:

A uniform income tax for all consumers is not necessarily better than a uniform quantity tax.

Assumption that income tax is lump-sum is key.

Another Question

Suppose that the government can use a uniform value tax on goods 1 and 2 to raise revenue (e.g. a sales tax). Does the consumer still prefer an income tax?

Answer

With a uniform value tax, the budget consumer line reads

Rearranging:

mxptxpt 2211 11

tm

xpxp

1

2211

Answer Continued

Rearranging:

Budget line with income tax:*

2211 Rmxpxp

tm

xpxp

1

2211

mt

txtpxtpR

1**

2211*

Answer Continued

Budget line with income tax:

Same as budget line under value tax:

t

mxpxp

12211

t

mxpxp

12211