Choice Continued
Interpretation
At Z:
x2 Z
p1
  MRS ( x1, x 2)
p2
X
At Y:
p1
  MRS ( x1, x 2)
p2
Y
x1
Consumer Demand
Consumer’s demanded bundle: optimal
choice of goods 1 and 2 for given prices and
income.
Consumer’s demand functions:
x1  x1 p1, p2, m
x2  x 2 p1, p2, m
Cobb-Douglas
Demand function for
good 1:
x2
m
x1  c
p1
x2
*
Demand function for
good 2:
*
x1
x1
m
x 2  1  c 
p2
Perfect Substitutes
x2
Demand function for
good 1:
1
x1  m / p1
p1

p2
x1  0
if
p1  p 2
if
p1  p 2
x1  0, m / p1
*
x1
x1
if
p1  p 2
Perfect Complements
Optimal choice:
x2
x2  x1
x2  x1
Budget line:
p1x1  p2 x2  m
x2
*
Demand function for
goods 1 and 2:
*
x1
x1
m
x1  x 2 
p1  p 2
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
x2
Budget line:
p1x1  p2 x2  m
p1

p2
x2
Well-behaved
preferences
o
Consumer chooses:
o
x1
x1
x1 , x2 
o
o
Quantity Tax
Budget line with quantity
tax:
x2
 p1  t x1  p2 x2  m
p1  t

p2
*
x 2o
x2
Tax revenues:
R  tx1
*
*
x1
o
x1
x1
*
Income Tax
New budget line:
p1 x1  p 2 x 2  m  R
where
R  tx1
*
*
*
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 X * when he
is paying the income tax?
Answers
A1: Slope of the budget line with income
tax is:
p1

A2: Lump-sum.
p2
A3: Yes. X
*
Is still Affordable
Budget line with income tax:
p1 x1  p 2 x 2  m  tx
*
1
Substitute
X
*
:
p1x1  p2 x2  m  tx1
*
Rearrange:
*
 p1  t x1
*
*
 p2 x 2  m
*
On the Graph
Budget line with income
tax:
x2
Budget line with quantity
tax:
X
*
X
o
p1

p2
x1
*
Q: Is X Optimal with the
Income Tax?
x2
X
*
X
o
x1
A: No, it is not.
The
x2
budget line is not tangent
X
*
X
to indifference curve at
o
X
X
*
.
Optimal choice:
x1
X
Q: Which Tax Does the
Consumer Prefer?
x2
X
*
X
o
X
x1
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
1  t  p1x1  1  t  p2 x2  m
Rearranging:
m
p1 x1  p 2 x 2 
1  t 
Answer Continued
Rearranging:
m
p1 x1  p 2 x 2 
1  t 
Budget line with income tax:
p1 x1  p 2 x 2  m  R
t
*
*
*
R  tp1 x1  tp2 x 2 
m
1 t
*
Answer Continued
Budget line with income tax:
m
p1 x1  p 2 x 2 
1 t
Same as budget line under value tax:
m
p1 x1  p 2 x 2 
1 t