General equilibrium analysis:
More welfare results
Lectures in Microeconomic Theory
Fall 2010, Part 15
07.07.2010
G.B. Asheim, ECON4230-35, #15
1
Does welfare increase?
Suppose we are in a competitive
equilibr. ( x 1 , x 2 , p ) that maximizes
u2
(u1 (x 1 ), u 2 (x 2 ))
W ( u1 ( x 1 ), u 2 ( x 2 ))

1 1
U

2
 a u ( x1 )  a u 2 ( x 2 )
1
1

so that a i   
 v i ( p , px i )
i
m i
( a1 , a2 )
u1
If we consider moving to a new allocation ( x 1 , x 2 , p ) ,
how to decide whether welfare will be improved?
07.07.2010
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G.B. Asheim, ECON4230-35, #15
W ( u1 ( x 1 ), u 2 ( x 2 ))  W ( u1 ( x 1 ), u 2 ( x 2 )) 
a1 D u1 ( x 1 )( x 1  x 1 )  a 2 D u 2 ( x 2 )( x 2  x 2 )

 a1 1p ( x 1  x 1 )  a 2  2 p ( x 2  x 2 ) where D u i ( x i ) 
 p ( x 1  x 1 )  p ( x 2  x 2 )
  u i ( x i )  u i ( x i ) 

, 1 2 
1
since
  xaii  λ  x i 
i
Because distribution is already optimal,
we are left with a simple criterion:
A small project increases welfare if
national income (at original prices) increases.
07.07.2010
G.B. Asheim, ECON4230-35, #15
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1
Optimal taxation

Untaxed good
Tax

Tax
burd
den
Without possibilities for
lump-sum taxation or
taxation of all goods, the
tax burden exceeds the tax.
Problem:
Substitution effect:
Untaxed good
Taxed
good
Taxed good
07.07.2010
G.B. Asheim, ECON4230-35, #15
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Why not lump-sum taxation?


A lump-sum tax that does not vary between
the individuals is unfair.
A fair lump-sum tax must vary between the
individuals according to criteria that the individuals cannot influence through their actions.
Such criteria are not observable.
Why not tax all goods?

Leisure cannot be taxed.
Optimal tax rates on the taxed goods?
07.07.2010
G.B. Asheim, ECON4230-35, #15
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Consider two taxed goods, 1 and 2,
and one untaxed good, 0.
Direct u-fn: Assume that production prices are fixed.
u ( x 0 , x1 , x 2 ) Assume one representative consumer.
Problem
Indirect u-fn with 0 as numeraire: v ( p1  t1 , p 2  t 2 , m )
Government revenue: R ( t )  t  x ( p  t , m )
Maximize the consumer’s utility w.r.t. the tax rates,
s.t. the contraint that the tax system raises R.
max v ( p  t , m ) s.t. t  x ( p  t , m )  R
t
L  v ( p  t , m )   t  x ( p  t , m )  R 
07.07.2010
G.B. Asheim, ECON4230-35, #15
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2
First-order conditions

v
x
x
   x1  t1 1  t 2 2
 p1
 p1
 p1

R
 t1

  0


v
x
x 
   x 2  t1 1  t 2 2   0
p 2

p

p2 

2
R
t 2
1 v
1 v
R
 

x1  0
  p1
 m
 t1
1 v
1 v
R
 

x2  0
 p 2
 m
t 2
07.07.2010

By Roy’s
Identity.
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G.B. Asheim, ECON4230-35, #15
2
2
h j
x j
1 v
xi  xi   t j
 xi  t j
 0 By the
 m
p

m
j 1
j 1
i
Symmetry
h
tj 1


pj
j 1
Slutsky
equation
Does not depend on i.
2
x1
2
2
x j

1 v

  1   t j
 m
m
j 1


 


t
j 1
j
 h2
p j
x2
Relative
Increased Increased tax burden
reduction in
tax
Conclusion: It is comcompensated
pensated quantities (not prices) that
demand for 1.
should be changed as little as possible.
07.07.2010
G.B. Asheim, ECON4230-35, #15
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3