International Tax Competition:
Zero Tax Rate at the Top Re-established
Appendices
Tomer Blumkin[*] EfraimSadka Yotam Shem-Tov
September, 2014
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 as the control variable and as the state variable. Formulating the Hamiltonian yields:
(18).[1]
Formulating the necessary conditions for optimality yields (for any ):
(19) ,
(20) .
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, , hence the condition in (20) simplifies to:
(21)
Integrating the expression in (21) and employing the transversality condition for the limiting skill level,, yields:
(22).
The first order condition for the individual optimization program implies,
(23).
Denoting the net hourly wage-rate earned by an individual of skill-level by, the first order condition in (23) can be re-written as:
(24).
Differentiating the first-order condition in (24) with respect to the net hourly wage-rate, , it is straightforward to derive the elasticity of the pre-tax income, which is then given by:
(25) .
Employing (23), (24) and (25) yields:
(26)
Substituting from (22), (23), (24) and (26) into (19), using the symmetry property, which implies that ,yields after re-arrangement:
(27)
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)
where
and,
with denoting the gross income level optimally chosen by an individual of skill level faced with the tax schedule T(y).[2]
Proof: Re-arranging the expression in (10) yields:
(10’).
To prove the claim we need to show that:
(29)=,
Fully differentiating the identity with respect to implies that. Substitution into (29) and re-arranging yields:
(30) =.
To establish the condition given in (30), fully differentiate the first order condition for the individual optimization program [given in (23)] with respect to to obtain:
(31).
Employing the individual first-order condition in (23) and the pre-tax income elasticity formula in (25) yields upon re-arrangement:
(32).
Substituting from (31) into (31) following some algebraic manipulations yields the condition in (30).
Lemma 2: When then.
Proof: Assume by negation that and. By virtue of lemma 1, it follows:
(33)
where
.
Notice that the B(y)>0 by virtue of (29).
Fully differentiating the expression in (33)with respect to yields,
(34)
Substitutingfrom (33) into (34) yields,
(35)
Substituting into (35) and re-arranging yields,
(36)
It follows from (36) and by our presumption that,that; hence, by continuity (invoking a first-order approximation), for sufficiently small . That is, the marginal tax rate is negative within a small neighborhood to the right of .As the marginal tax rate is zero at, it follows from the condition in (33) that:
(37).
It follows by virtue of (37) andour presumption that that there exists some income levelfor which. Hence, there exists some income levelfor which.Then, by the intermediate value theorem there exists an income level for which the marginal tax rate is zero within the interval (, y''). Let A denote the (non-empty and bounded) set of all income levels within the interval (, y'') for which the marginal tax rate is zero, and further denote by the greatest lower-bound of the set A.By construction, it follows that. By virtue of (33) and the definition of, it follows that:
(38).
It further follows that for all. Hence, it follows that for all, which, by virtue of (38), impliesthat:
(39) .
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 and . This concludes the proof.
Lemma 3: If and then.
Proof: Suppose by negation that for some y, , .For concreteness, we assume that (the other case can be proved by symmetric arguments and is hence omitted). We first turn to show that. Suppose by negation that. As , it follows by virtue of (33) that:
(40) .
By our presumption that it necessarily follows that for some. Hence, there exists some y, , for which . By virtue of the intermediate value theorem, it follows that there exists an income level y, , for which the marginal tax rate is zero. Let A denote the (non-empty and bounded from below) set of all income levels within the interval (,) for which the marginal tax rate is zero, and further denote by the greatest lower-bound of the set A. By construction, it follows that. By virtue of (33) and the definition of, it further follows that:
(41).
It further follows that for all. Hence, it follows that for all, which, by virtue of (41), impliesthat:
(42) .
Thus we obtain a contradiction to (40). Thus we have established that:
(43).
By virtue of lemma 2 and as , it follows that . It therefore follows that there exists some level of income y', , for which . Hence by our presumption that, it follows that there exists some level of income y, , for which .
Let A denote the (non-empty and bounded) set of all income levels within the interval (y',) for which the marginal tax rate is zero, and further denote by the leastupper-bound of the set A.By construction, it follows that. By virtue of the definition of, it follows that:
(44).
It further follows by the definition of that for all. By virtue of lemma 2, , hence for all, whichimpliesthat:
(45).
Thus we obtain a contradiction to (43).
This establishes the claim.
Lemma 4: 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) =>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 the greatest lower-bound of the set A.By construction, it follows that. By virtue of (33) and the definition of, it follows that:
(47).
By virtue of lemma 3, it follows that for all. By construction,for all . Thus we obtain the desired contradiction.
Lemma 5:.
We first establish that for all y. To see this, suppose by negation that there exists some income level, y', for which. Then, as the marginal tax is non-negative for all y (by lemma 4) it follows that for all .It follows that,
(48).
By virtue of (B6) it then follows that , which contradicts lemma 4.We conclude that is bounded from above by.As T(y) is non-decreasing, it follows thatT(y) converges to some finite limit.Let . We turn next toexamine the marginal tax rate as . Taking the limit of the expression in (33) implies:
(49)
By our earlier assumptions (see the discussion in footnote 5 in the main text),.ApplyingRule then implies:
(50).
As both T(y) and T'(y) converge to a finite limit when y goes to infinity, it follows that. This concludes the proof.
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)
where
Fully differentiating the expression in (51) with respect to yields,
(52)
With a Pareto skill distribution and an iso-elastic disutility from labor,B’(y)=0, hence, substitution into (52) yields:
(53)
Now consider some level of income y’, for which T’(y’)>0. By virtue of lemma 4, for all y,hence, as T’(y’)>0, it follows that:
(54).
Substituting into (53) implies that . This concludes the proof.
References
Salanie, B.(2003) "The Economics of Taxation", MIT Press.
1
[*]Department of Economics, Ben-Gurion University, Beer-Sheba 84105, Israel, CesIfo, IZA. E-mail: .
The EitanBerglas School of Economics, Tel Aviv University, Tel-Aviv 69978, Israel, CesIfo, IZA. E-mail:
Ph.D. student, Department of Economics, UC Berkeley, USA, E-mail:
[1] The Hamiltonian in (18) is formulated based on the presumption that ; namely, that only a fraction of the population (for each skill level) is migrating.
[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 is strictly increasing and hence invertible. The elasticity term 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.