Tax competition and the efficiency of “benefit-related” business taxes

Abstract

We construct a tax competition model in which local governments finance business public services with either a source-based tax on mobile capital, such as a property tax, or a tax on production, such as an origin-based value-added tax, and then assess which of the two tax instruments is more efficient. Many taxes on business apply to mobile inputs or outputs, such as property taxes, retail sales taxes, and destination-based VATs, and their inefficiency has been examined in the literature; however, proposals from several prominent tax experts to utilize a local origin-based VAT have not been analyzed theoretically. Our primary finding is that the production tax is less inefficient than the capital tax under many—but not all—conditions. The intuition underlying this result is that the efficiency of a user fee on the public business input is roughly approximated by a production tax, which applies to both the public input and immobile labor (in addition to mobile capital). In marked contrast, the capital tax applies only to mobile capital and is thus likely to be relatively inefficient.

Appendix: Log (sub/super) modularity

Before discussing log (sub/super) modularity, we first review (sub/super) modularity and then point out the differences between these two related concepts.

1.1 A.1 Definition and examples

To define (sub/super) modularity of B and K, consider any two input vectors with the same amount of labor, but different amounts of B and K, with one input vector containing strictly less of both $\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} ,L} \right)$ than the other $\left( {\bar{B},\bar{K},L} \right)$. Since we keep labor the same in all our analysis, L is suppressed in our analysis. By definition, modularity holds if and only if

$$ F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right) + F\left( {\bar{B},\bar{K}} \right) = F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\bar{K}} \right) + F\left( {\bar{B},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right). $$

(A1)

Under modularity, a given increase in one input results in the same increase in output regardless of how much output is being produced. Another way of thinking about the implications of modularity is by choosing the input combinations on the RHS of Eq. (A1) such that

$$ F\left( {\underline{{B,\bar{K}}} } \right) = F\left( {\bar{B},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right) = Y^{0} . $$

(A2)

With modularity, the arithmetic mean of $F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)$ and $F\left( {\bar{B},\bar{K}} \right)$ must equal $Y^{0}$. For super (sub) modularity, the arithmetic mean of $F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)$ and $F\left( {\bar{B},\bar{K}} \right)$ must be greater (less) than $Y^{0}$. For example, in the case of submodularity, total output of $Y^{0}$ can be produced with the arithmetic mean of $F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)$ and some other output level generated with larger amounts of both inputs, but the output level $F\left( {\bar{B},\bar{K}} \right)$ is not large enough to achieve this; instead, inputs larger than $\left( {\bar{B},\bar{K}} \right)$ are necessary to achieve an arithmetic mean of $Y^{0}$.

An example of a production function that is modular in B and K is one for which the functions involving inputs B and K enter additively, i.e.,

$$ F\left( {B,K,L} \right) = g\left( B \right)h\left( L \right) + i\left( K \right)j\left( L \right) + m\left( B \right) + o\left( K \right) + p\left( L \right), $$

where all the individual functions are increasing in their inputs. By comparison, a function of the form

$$ F\left( {B,K,L} \right) = g\left( B \right)h\left( L \right)i\left( K \right)j\left( L \right) + m\left( B \right) + o\left( K \right) + p\left( L \right) $$

is supermodular.

Log modularity, in contrast, compares relative marginal products rather than the absolute marginal products of inputs. That is, log modularity in B and K holds if and only if

$$ F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)F\left( {\bar{B},\bar{K}} \right) = F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\bar{K}} \right)F\left( {\bar{B},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right). $$

(A3)

As in the discussion of (sub/super) modularity, another way to think about the implications of log modularity is by choosing the input combinations on the RHS of Eq. (A3) so that (A2) holds. For log modularity, the geometric mean of $F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)$ and $F\left( {\bar{B},\bar{K}} \right)$ must equal $Y^{0}$. For log super (sub) modularity, the geometric mean of $F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)$ and $F\left( {\bar{B},\bar{K}} \right)$ must be greater (less) than $Y^{0}$. For example, in the case of log submodularity, total output of $Y^{0}$ can be produced with the geometric mean of $F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)$ and some other output level generated with higher amounts of both inputs, but the output level $F\left( {\bar{B},\bar{K}} \right)$ is not large enough to achieve this; instead, inputs larger than $\left( {\bar{B},\bar{K}} \right)$ are necessary to achieve a geometric mean of $Y^{0}$.

An example of a production function that is log modular in B and K is one for which the inputs enter multiplicatively, e.g.,

$$ F\left( {B,K,L} \right) = g\left( B \right)h\left( K \right)i\left( L \right), $$

where all the individual functions are increasing in their inputs. Note that such functions are simultaneously supermodular in B and K. An example of a log supermodular function is

$$ F\left( {B,K,L} \right) = g\left( B \right)h\left( K \right)i\left( L \right) + j\left( L \right), $$

which is simultaneously supermodular in B and K.

An example of a log submodular function is

$$ F\left( {B,K,L} \right) = g\left( B \right) + h\left( K \right) + i\left( L \right), $$

which is simultaneously modular in B and K. However, log submodular functions can also be simultaneously supermodular; an example is

$$ F\left( {B,K,L} \right) = \left[ {g\left( B \right) + h\left( K \right) + i\left( L \right)} \right]^{\alpha } , $$

where $\alpha > 0$.

1.2 A.2 Proof of Lemma 1: Expressing log (sub/super) modularity in terms of elasticities

Note that log modularity can be expressed as

$$ \frac{{F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\bar{K}} \right)}}{{F\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)}} = \frac{{F\left( {\bar{B},\bar{K}} \right)}}{{F\left( {\bar{B},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{K} } \right)}}. $$

(A4)

Hence, a log modular production function must exhibit the same percentage change in output due to a given increase in K, regardless of the level of B, or

$$ \frac{{\partial \varepsilon_{K} }}{\partial B} = 0, $$

(A5)

where $\varepsilon_{K} = F_{K} K/F$, which holds if and only if

$$ K\frac{{F_{KB} F - F_{B} F_{K} }}{{F^{2} }} = 0. $$

(A6)

Since by Young’s theorem $F_{KB} = F_{BK}$, and $\varepsilon_{{F_{B} ,K}} = F_{BK} K/F_{B}$, (A6) can be expressed as

$$ \varepsilon_{{F_{B} ,K}} = \varepsilon_{K} . $$

(A7)

Similarly, log submodularity requires $\varepsilon_{{F_{B} ,K}} < \varepsilon_{K}$ and log supermodularity requires $\varepsilon_{{F_{B} ,K}} > \varepsilon_{K}$.