International Tax Competition:
Zero Tax Rate at the Top Re-established
Appendices
Tomer Blumkin* Efraim Sadka Yotam Shem-Tov
September, 2014
*
Department of Economics, Ben-Gurion University, Beer-Sheba 84105, Israel, CesIfo, IZA. E-mail:
tomerblu@bgumail.bgu.ac.il.

The Eitan Berglas School of Economics, Tel Aviv University, Tel-Aviv 69978, Israel, CesIfo, IZA. Email: sadka@post.tau.ac.il

Ph.D. student, Department of Economics, UC Berkeley, USA, E-mail: shemtov@berkeley.edu
Appendix 1: Derivation of Condition (10)
We next turn to solve the optimization program as an optimal control problem employing
Pontryagin’s maximum principle. We choose y   as the control variable and V   as
the state variable. Formulating the Hamiltonian yields:
y  h '( y /  ) 1
H  V ( )  f     y  V  h( y /  )   1  V  V2  /     f   
.


2

(18)

Formulating the necessary conditions for optimality yields (for any    ):
(19)
H
h ' y  h ''/ 
 h'
    1     
 0,
y

2

(20) 

 y  V  h( y /  )     '( ) .
H
   f  1  V  V2  /   

V



In the symmetric equilibrium, by construction, the tax schedules implemented by both
countries are identical and, therefore, no migration takes place. By virtue of the symmetry
property, V  V2 , hence the condition in (20) simplifies to:
(21)

  y  V  h( y /  )  
H
   f  1 
   '( )
V



Integrating the expression in (21) and employing the transversality condition for the
limiting skill level, lim  ( )  0 , yields:
 
(22)
1


 ( )     1 


 y  V  h( y / t )    f (t )dt .



The Hamiltonian in (18) is formulated based on the presumption that |𝑉(𝜃) − 𝑉2 (𝜃)| < 𝛿; namely, that
only a fraction of the population (for each skill level) is migrating.
2
The first order condition for the individual optimization program implies,
(23)
1  T '  y   h '/  .
Denoting the net hourly wage-rate earned by an individual of skill-level
 by
n   (1  T '  y  ) , the first order condition in (23) can be re-written as:
(24)
h '( y /  )  n .
Differentiating the first-order condition in (24) with respect to the net hourly wage-rate,
n , it is straightforward to derive the elasticity of the pre-tax income, which is then given
by:
(25)
y
 
y
y 
 n
.
 n y  h ''
n
Employing (23), (24) and (25) yields:
(26)
h ' y  h ''/    (1  T '  y )   (1  T '  y ) 

1
  (1  T '  y ) 1  
y
  y 
1
Substituting from (22), (23), (24) and (26) into (19), using the symmetry property, which
implies that   f , yields after re-arrangement:
(27)
    y  V  h( y / t )  
  1
T ' y 
1
   1 
 f (t )dt   1   




1 T ' y  


   y  f ( ) 
3
Appendix 2: Proof of the Proposition
We prove the proposition in several steps organized into a series of lemmas.
Lemma 1: The expression in (10) is equivalent to the following condition:
(28)
 T (t )


 y   g (t )dt 
T '( y )
 1 
  B( y ),
1  T '( y ) 
1  G( y)



where
B( y ) 
1   y  y  T ''( y ) / [1  T '( y )] 1  G ( y )
and 𝐺[𝑦(𝜃)] = 𝐹(𝜃),

y
g ( y)  y
with 𝑦(𝜃) denoting the gross income level optimally chosen by an individual of skill level
𝜃 faced with the tax schedule T(y).2
Proof: Re-arranging the expression in (10) yields:
(10’)


T ' y 
1
 T (t )  g (t )dt   1  1   1  F ( ) .
 1   
1  T '  y   
1  F ( )    y  f ( ) 


To prove the claim we need to show that:
(29)

1  1  F ( ) 1   y  y  T ''( y ) / [1  T '( y )] 1  G ( y )
=
,

1   
y
g ( y)  y
  y  f ( ) 
Fully differentiating the identity 𝐺[𝑦(𝜃)] = 𝐹(𝜃) with respect to 𝜃 implies that 𝑔(𝑦) ∙
𝑦 ′ (𝜃) = 𝑓(𝜃). Substitution into (29) and re-arranging yields:
2
By assuming that the second-order conditions for the individual optimization are satisfied; implying,
hence, ‘no bunching’ [see Salanie (2003) for an elaborate discussion] it follows that 𝑦(𝜃) is strictly
increasing and hence invertible. The elasticity term 𝜀𝑦 in B(y) that depends on the skill level of the
individual [see the formula in (A8)] can thus be written as a well-defined function of y. The term B(y) is
therefore well-defined.
4
(30)
1   y  = 1   y  y  T ''( y ) / [1  T '( y)] 
  y '( )
y
.
To establish the condition given in (30), fully differentiate the first order condition for the
individual optimization program [given in (23)] with respect to 𝜃 to obtain:
(31)
1 − 𝑇 ′ (𝑦) − 𝜃 ∙ 𝑦 ′ (𝜃) ∙ 𝑇 ′′ (𝑦) =
𝑦
𝜃
𝜃2
ℎ′′ ( )
[𝜃 ∙ 𝑦 ′ (𝜃) − 𝑦].
Employing the individual first-order condition in (23) and the pre-tax income elasticity
formula in (25) yields upon re-arrangement:
(32)
[𝜀𝑦 ∙ 𝑦]/[1 − 𝑇 ′ (𝑦)] = 𝜃 2 /ℎ′′ (𝑦/𝜃).
Substituting from (31) into (31) following some algebraic manipulations yields the
condition in (30).
Lemma 2: When T '( y )  0 then T ( y) = d .
Proof: Assume by negation that T '( yˆ )  0 and T ( ŷ) < d . By virtue of lemma 1, it follows:
(33)
 T (t )


 g (t )dt 


T '( y )
y

 1 
  B( y ),
1  T '( y ) 
1  G( y)



where
B( y ) 
1   y  y  T ''( y ) / [1  T '( y )] 1  G ( y )

>0 .
y
g ( y)  y
Notice that the B(y)>0 by virtue of (29).
Fully differentiating the expression in (33) with respect to y yields,
5
T ''( y )
(34)
1  T '( y) 
2

B( y )




T (t )  g (t )dt  B '( y )  T (t )  g (t )dt

g ( y) 


 T ( y )  y
 y
1  G( y) 
1  G( y) 

1  G( y)


 B '( y )
Substituting from (33) into (34) yields,
(35)
T ''( y )
1  T '( y) 
2
  B( y ) 
 B '( y )  T '( y )

g ( y)  T ( y)
T '( y )
1
 


 1 

 B( y )   B '( y )
1  G( y)  
1  T '( y ) B( y )  B( y )  1  T '( y )

Substituting T '( y )  0 into (35) and re-arranging yields,
(36)
T ''( y )   B( y ) 
g ( y)  T ( y) 
 
 1
1  G ( y )  

It follows from (36) and by our presumption that T ( ŷ) < d , that T ''( yˆ )  0 ; hence, by
continuity (invoking a first-order approximation), T '( yˆ   )  0 for sufficiently small
  0 . That is, the marginal tax rate is negative within a small neighborhood to the right
of ŷ . As the marginal tax rate is zero at ŷ , it follows from the condition in (33) that:
(37)


yˆ
T (t )  g (t )dt   1  G ( yˆ )  .
It follows by virtue of (37) and our presumption that T ( ŷ) < d that there exists some
income level y '  yˆ for which T ( y ') > d . Hence, there exists some income level
yˆ  y ''  y ' for which T '( y '')  0 . Then, by the intermediate value theorem there exists an
income level for which the marginal tax rate is zero within the interval ( ŷ , y''). Let A
denote the (non-empty and bounded) set of all income levels within the interval ( ŷ , y'')
for which the marginal tax rate is zero, and further denote by y the greatest lower-bound
6
of the set A. By construction, it follows that y > ŷ . By virtue of (33) and the definition of
y , it follows that:
(38)
ò
¥
y
(
)
T (t) × g(t)dt = d 1-G( y) .
It further follows that T '( y )  0 for all ŷ < y < y . Hence, it follows that T ( y) < d for all
ŷ < y < y , which, by virtue of (38), implies that:
(39)


yˆ
T (t )  g (t )dt   1  G ( yˆ )  .
Thus we obtain a contradiction to (37).
In exactly the same manner (the formal steps are therefore omitted) one can prove by
negation that it cannot be the case that T '( y )  0 and T ( y) > d . This concludes the proof.
*
*
Lemma 3: If T '( y )  0 and y*   then y, y  y, T '( y )  0 .
*
Proof: Suppose by negation that for some y, ŷ  y , T '( yˆ )  0 . For concreteness, we
assume that T '( yˆ )  0 (the other case can be proved by symmetric arguments and is
hence omitted). We first turn to show that T ( ŷ) > d . Suppose by negation that T ( ŷ) £ d .
As T '( yˆ )  0 , it follows by virtue of (33) that:
(40)


yˆ
T (t )  g (t )dt   1  G ( yˆ )  .
By our presumption that T ( ŷ) £ d it necessarily follows that T ( y) > d for some y  yˆ .
Hence, there exists some y, y  yˆ , for which T '( y )  0 . By virtue of the intermediate
value theorem, it follows that there exists an income level y, y  yˆ , for which the
marginal tax rate is zero. Let A denote the (non-empty and bounded from below) set of all
7
income levels within the interval ( ŷ ,  ) for which the marginal tax rate is zero, and
further denote by y the greatest lower-bound of the set A. By construction, it follows that
y > ŷ . By virtue of (33) and the definition of y , it further follows that:
(41)
ò
¥
y
(
)
T (t) × g(t)dt = d 1-G( y) .
It further follows that T '( y )  0 for all ŷ < y < y . Hence, it follows that T ( y) < d for all
ŷ < y < y , which, by virtue of (41), implies that:
(42)


yˆ
T (t )  g (t )dt   1  G ( yˆ )  .
Thus we obtain a contradiction to (40). Thus we have established that:
(43) T ( ŷ) > d .
*
By virtue of lemma 2 and as T '( y )  0 , it follows that T ( y* ) = d . It therefore follows
*
that there exists some level of income y', y  y '  yˆ , for which T '( y ')  0 . Hence by our
presumption that T '( yˆ )  0 , it follows that there exists some level of income y,
y*  y '  y  yˆ , for which T '( y )  0 .
Let A denote the (non-empty and bounded) set of all income levels within the interval
(y', ŷ ) for which the marginal tax rate is zero, and further denote by y the least upperbound of the set A. By construction, it follows that y < ŷ . By virtue of the definition of y ,
it follows that:
(44) T '( y) = 0 .
8
It further follows by the definition of y that T '( y )  0 for all y < y £ ŷ . By virtue of
lemma 2, T ( y) = d , hence T ( y) < d for all y < y £ ŷ , which implies that:
(45) T ( ŷ) < d .
Thus we obtain a contradiction to (43).
This establishes the claim.
Lemma 4: T '( y )  0 for all y.
Proof: By virtue of (33) the marginal tax rate faced by the individual with the lowest
income level is given by:
(46)
T '( y)
1- T '( y)
=𝐵(𝑦)>0.
Suppose by negation that there exists an income level for which the marginal tax rate is
negative. By the intermediate value theorem there exists an income level for which the
marginal tax rate is zero. Let A denote the (non-empty and bounded from below) set of all
income levels for which the marginal tax rate is zero, and further denote by y the greatest
lower-bound of the set A. By construction, it follows that y > y . By virtue of (33) and the
definition of y , it follows that:
(47)
ò
¥
y
(
)
T (t) × g(t)dt = d 1-G( y) .
By virtue of lemma 3, it follows that T '( y )  0 for all y ³ y . By construction, T '( y )  0
for all y < y . Thus we obtain the desired contradiction.
9
Lemma 5: lim y  T '( y )  0 .
We first establish that T ( y) £ d for all y. To see this, suppose by negation that there exists
some income level, y', for which T ( y ') > d . Then, as the marginal tax is non-negative for
all y (by lemma 4) it follows that T ( y) > d for all y  y ' . It follows that,
(48)


y'
T (t )  g (t )dt   1  G ( y ')  .
By virtue of (B6) it then follows that T '( y ')  0 , which contradicts lemma 4.We conclude
that T ( y ) is bounded from above by d . As T(y) is non-decreasing, it follows that T(y)
converges to some finite limit. Let lim T ( y )  T   . We turn next to examine the
y 
marginal tax rate as y   . Taking the limit of the expression in (33) implies:
(49)
 T (t )


 g (t )dt 


𝑇′(𝑦)
y

lim
= lim 𝐵(𝑦) × lim 1 

𝑦→∞ 1−𝑇′(𝑦)
𝑦→∞
𝑦→∞
1  G( y)




By our earlier assumptions (see the discussion in footnote 5 in the main text),
ˆ
lim B( y )  B   . Applying L ' Hopital
' s Rule then implies:
y 
(50)
lim
𝑇′(𝑦)
𝑦→∞ 1−𝑇′(𝑦)
𝑇
= B ∙ [1 − 𝛿 ] < ∞.
As both T(y) and T'(y) converge to a finite limit when y goes to infinity, it follows that
lim y  T '( y )  0 . This concludes the proof.
10
Appendix 3: The marginal tax rate is declining with respect to income under a
Pareto skill distribution and an iso-elastic disutility from labor
Re-arranging the expression in (10) yields,
(51)
 T (t )


 y   g (t )dt 
T '( y )
 1 
  B( y ),
1  T '( y ) 
1  G( y)



where
B( y ) 
1   y  y  T ''( y ) / [1  T '( y )] 1  G ( y )

0
y
g ( y)  y
Fully differentiating the expression in (51) with respect to y yields,
(52)
T ''( y )
1  T '( y) 
2

B( y )




T (t )  g (t )dt  B '( y )  T (t )  g (t )dt

g ( y) 
y
y


 T ( y ) 


1  G( y) 
1  G( y) 

1  G( y)


 B '( y )
With a Pareto skill distribution and an iso-elastic disutility from labor, B’(y)=0, hence,
substitution into (52) yields:
(53)
T ''( y)
1  T '( y) 
2


T (t )  g (t ) dt 

B( y ) g ( y) 
y



 T ( y ) 
 1  G( y) 
1  G( y) 


Now consider some level of income y’, for which T’(y’)>0. By virtue of lemma 4,
T '( y )  0 for all y, hence, as T’(y’)>0, it follows that:
11
(54)


y'
T (t )  g (t )dt
1  G( y ') 
 T ( y ') .
Substituting into (53) implies that T ''( y )  0 . This concludes the proof.
References
Salanie, B. (2003) "The Economics of Taxation", MIT Press.
12