Incentives to tax foreign investors

Abstract

This paper shows that a small country can have an incentive to tax inbound FDI even in a setting with perfect competition and free entry that is compatible with the Diamond–Mirrlees (Am Econ Rev 61(1):8–27, 1971) framework. While firms make no aggregate profits worldwide due to free entry in this setting, they make taxable profits in foreign production locations because their costs are partly incurred in their home countries. These profits are not perfectly mobile because firm productivity varies across locations. Consequently, the host country does not bear the entire burden of a tax on foreign firms, giving rise to an incentive to tax foreign investors. The standard zero optimal tax result can be recovered in this model under an apportionment system that ensures zero economic profits in each location.

A Proofs

1.1 A.1 Profit Function Property

In this appendix, I show that we can write the pre-tax profit function (excluding fixed costs) in the following separable form: $\pi _{ij}(w_{i},r,\tilde{z}_{i})=\tilde{z}_{i}^{1/(1-\lambda )} \pi _{ij}\left( w_{i},r\right)$. First, note that from the homogeneity of the production function, we can use Euler’s rule to obtain:

$$\begin{aligned} \left[ F_{l}(.)l+F_{k}(.)k\right] =\lambda F(.), \end{aligned}$$

where$\lambda <1$ is the returns to scale parameter. The first-order conditions are: $\tilde{z}_{i}F_{l}\left( l^{*},k^{*}\right) =w_{i}$ and $\tilde{z}_{i}F_{k}\left( l^{*},k^{*}\right) =r$, where $l^{*}$ and $k^{*}$ are the optimal choices of l and k, respectively. Using the first-order condition, the firm’s profits (excluding fixed costs) before taxes are:

$$\begin{aligned} \pi _{ij}(w_{i},r,\tilde{z}_{i})= & {} \tilde{z}_{i}F(.) -\tilde{z}_{i}F_{l}(.)l^{*}-\tilde{z}_{i}F_{k}(.)k^{*}\\= & {} \tilde{z}_{i}F(.)-\lambda \tilde{z}_{i}F(.)\\= & {} \tilde{z}_{i}\left( 1-\lambda \right) F(.) \end{aligned}$$

Thus, the firm’s profits (excluding fixed costs) are proportional to its sales. Next, we can differentiate maximized profits, $\tilde{z}_{i}F(.)-wl-rk$, with respect to $\tilde{z}_{i}$ using the envelope theorem to get:

$$\begin{aligned} \frac{\hbox {d}\pi _{ij}(.)}{\hbox {d}\tilde{z_{i}}}\frac{\tilde{z}_{i}}{\pi _{ij}(.)}= & {} F\left( .\right) \frac{\tilde{z}_{i}}{\pi _{ij}(.)}\\ \frac{\hbox {d}\pi _{ij}(.)}{\hbox {d}\tilde{z_{i}}}\frac{\tilde{z}_{i}}{\pi _{ij}(.)}= & {} F\left( .\right) \frac{\tilde{z}_{i}}{\tilde{z}_{i} \left( 1-\lambda \right) F(.)}\\ \frac{\hbox {d}\pi _{ij}(.)}{\hbox {d}\tilde{z_{i}}}\frac{\tilde{z}_{i}}{\pi _{ij}(.)}= & {} \frac{1}{1-\lambda } \end{aligned}$$

The above expression is a separable differential equation and can be solved as follows:

$$\begin{aligned} \frac{1}{\pi _{ij}(.)}\hbox {d}\pi _{ij}(.)= & {} \frac{1}{1-\lambda } \frac{1}{\tilde{z}_{i}}\hbox {d}\tilde{z}_{i}\\ \intop \frac{1}{\pi _{ij}(.)}\hbox {d}\pi _{ij}(.)= & {} \frac{1}{1-\lambda } \intop \frac{1}{\tilde{z_{i}}}\hbox {d}\tilde{z_{i}}\\ \log \pi _{ij}(.)= & {} \frac{1}{1-\lambda }\log \tilde{z}_{i}+c\\ \log \pi _{ij}(.)= & {} \log \tilde{z}{}_{i}^{1/(1-\lambda )}e^{c}\\ \pi _{ij}(w_{i},r,\tilde{z}_{i})= & {} \tilde{z}{}_{i}^{1/(1-\lambda )}e^{c}, \end{aligned}$$

where c is a constant of integration. In order to solve for the constant $e^{c}$, we can set $\tilde{z}_{i}$ equal to 1 (an arbitrary choice) to obtain:

$$\begin{aligned} \pi _{ij}(w_{i},r,1)=e^{c} \end{aligned}$$

If we define $\pi _{ij}\left( w_{i},r\right) \equiv \pi _{ij} (w_{i},r,1)$, then the profits of an individual firm can be expressed as being proportional to a general term that is common to all firms: $\pi _{ij}(w_{i},r,\tilde{z}_{i})=\tilde{z}_{i}^{1/ (1-\lambda )}\pi _{ij}\left( w_{i},r\right)$.

1.2 A.2 Small country assumption

This appendix elaborates on the small country assumption introduced in Sect. 3.1. Given $L_{1}=\epsilon L_{2}$ and $K_{1}=\epsilon K_{2}$, the small country assumption captures the equilibrium as $\epsilon \rightarrow 0$. First, note that as $\epsilon$ approaches zero, the mass of entrants in country 1, $m_{1}$, must also approach zero. This is because a part of the fixed costs that enable entry is paid in terms of labor—specifically, the part captured by the fixed cost $f_{1}$—and so as $L_{1}$ approaches zero, the portion of country 1 labor used to pay these entry costs must also go to zero. Second, the equilibrium measure of country 2 firms that site in country 1 must also approach zero so that $\Theta _{22}\rightarrow U$, where U is defined as the set of all country 2 firms. This is because otherwise, labor demand, which is the right-hand side of (2.4), would not go to zero for country 1 even as the labor supply, $L_{1}$ does.

With these two points in mind, the labor market clearing condition for country 2, (2.4), the capital market clearing condition, (2.5), and the domestic entry condition for country 2, (2.2), become independent of the terms that depend on $\Theta _{12}$ and $m_{1}$ and, respectively, take the following forms:

$$\begin{aligned}&L_{2}=m_{2}\intop _{U}l_{22}\left( w_{2},r,z_{2}\right) g_{2}(z)\hbox {d}z+m_{2}f_{2}\\&K_{2}=m_{2}\intop _{U}k_{22}\left( w_{2},r,z_{2}\right) g_{2}(z)\hbox {d}z+m_{2}\phi _{2}\\&\intop _{U}z_{2}\pi _{22}\left( w_{2},r\right) g_{2}(z)\hbox {d}z =f_{2}w_{2}+\phi _{2}r \end{aligned}$$

Intuitively, as country 1’s size become negligible, country 1 variables cease to effect country 2’s labor market and domestic entry conditions as well as the global market clearing condition for capital. These three conditions now determine $w_{2}$, r and $m_{2}$, which are therefore independent of country 1’s policies.²⁰ Hence, we can think of the small country assumption made in this paper as capturing the limiting equilibrium as country 1’s share of labor and capital become small.

1.3 A.3 Proof of $\hbox {d}R_{12}/\hbox {d}\tau _{1}<0$

This appendix proves that $\hbox {d}R_{12}/\hbox {d}\tau _{1}<0$, which, as discussed in the main text, will imply that the optimal tax is positive. To see this, we can start with the definition of inframarginal profits:

$$\begin{aligned} R_{12}=m_{2}\intop _{\Theta _{12}}\left[ \left( 1-\tau _{1}\right) z_{1}\pi _{12}\left( w_{1},r\right) -z_{2}\pi _{22}\left( w_{2},r\right) \right] g_{2}(z)\hbox {d}z \end{aligned}$$

Differentiating this term, we obtain:

$$\begin{aligned} \frac{\hbox {d}R_{12}}{\hbox {d}\tau _{1}}=m_{2}\intop _{\Theta _{12}}\left\{ z_{1} \frac{\left[ d\left( 1-\tau _{1}\right) \pi _{12}\left( w_{1},r\right) \right] }{\hbox {d}\tau _{1}}\right\} g_{2}(z)\hbox {d}z \end{aligned}$$

When differentiating this term, we are again using the fact that firms on the boundary set make no inframarginal profits and so the derivative of the integral is simply the integral of the derivative. A firm that is on the boundary set, i.e., $z\in \partial \Theta _{12}$, will be indifferent between locating in country 1 and country 2:

$$\begin{aligned} \left( 1-\tau _{1}\right) z_{1}\pi _{12}\left( w_{1},r\right)= & {} z_{2}\pi _{22}\left( w_{2},r\right) \nonumber \\ \left( 1-\tau _{1}\right) \pi _{12}\left( w_{1},r\right)= & {} a_{12}\pi _{22}\left( w_{2},r\right) , \end{aligned}$$

(A.1)

where $a_{12}$ is the cutoff value of $z_{2}/z_{1}$ that defines the indifferent firms. For later use, note that (A.1) implies a function $a_{12}=\gamma \left( w_{1},\tau _{1}\right)$, with $\partial \gamma /\partial w_{1}<0$ and $\partial \gamma /\partial \tau _{1}<0$.

Differentiating (A.1), we obtain:

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}\tau _{1}}\left[ \left( 1-\tau _{1}\right) \pi _{12} \left( w_{1},r\right) \right] =\frac{\hbox {d}a_{12}}{\hbox {d}\tau _{1}} \pi _{22}\left( w_{2},r\right) \end{aligned}$$

Thus:

$$\begin{aligned} \frac{\hbox {d}R_{12}}{\hbox {d}\tau _{1}}= & {} m_{2}\intop _{\Theta _{12}} \left\{ z_{1}\frac{\left[ d\left( 1-\tau _{1}\right) \pi _{12} \left( w_{1},r\right) \right] }{\hbox {d}\tau _{1}}\right\} g_{2}(z)\hbox {d}z\\= & {} m_{2}\frac{\left[ d\left( 1-\tau _{1}\right) \pi _{12} \left( w_{1},r\right) \right] }{\hbox {d}\tau _{1}} \intop _{\Theta _{12}}z_{1}g_{2}(z)\hbox {d}z\\= & {} m_{2}\left[ \frac{\hbox {d}a_{12}}{\hbox {d}\tau _{1}}\pi _{22} \left( w_{2},r\right) \right] \intop _{\Theta _{12}}z_{1}g_{2}(z)\hbox {d}z\\= & {} \frac{\hbox {d}a_{12}}{\hbox {d}\tau _{1}}\times m_{2}\intop _{\Theta _{12}} \left[ z_{1}\pi _{22}\left( w_{2},r\right) \right] g_{2}(z)\hbox {d}z \end{aligned}$$

Hence, the sign of $\hbox {d}R_{12}/\hbox {d}\tau _{1}$ will be the same as the sign of $\hbox {d}a_{12}/\hbox {d}\tau _{1}$. Since higher taxes will cause firms to leave country 1, it follows that the new marginal firm will be one that is relatively more productive in country 1, i.e., $\hbox {d}a_{12}/\hbox {d}\tau _{1}<0$. To show this formally, we need to use the labor market clearing condition.

With no domestic firms, the labor market clearing condition can be written as:

$$\begin{aligned} L_{1}= & {} m_{2}\intop _{\Theta _{12}}l_{12} \left( w_{1},r,z_{1}\right) g_{2}(z)\hbox {d}z\\= & {} m_{2}\intop _{0}^{z_{1}^{\max }}\intop _{0}^{a_{12}z_{1}}l_{12} \left( w_{1},r,z_{1}\right) g_{2}(z)\hbox {d}z_{2}\hbox {d}z_{1}, \end{aligned}$$

where $z_{1}^{\max }$ is the upper-bound on productivity for $z_{1}$. The right-hand side above is decreasing in $w_{1}$ and increasing in $a_{12}$. Thus, this expression defines a positive relationship between $w_{1}$ and $a_{12}$. Intuitively, at a fixed wage, the presence of more firms means that labor supply exceeds labor demand, necessitating an increase in the wage to restore equilibrium. We can express this relationship as a function: $a_{12}=\delta (w_{1})$ with $\partial \delta /\partial w_{1}>0$. Note that $\delta \left( .\right)$ , unlike $\gamma \left( .\right)$, does not depend on $\tau$ because $\tau$ does not directly enter into the labor market clearing condition. Of course, $a_{12}$ and $w_{1}$ do both depend on $\tau$ in equilibrium because $\tau$ enters into $\gamma \left( .\right)$ and it is $\gamma \left( .\right)$ and $\delta \left( .\right)$ that together determine the equilibrium values for $a_{12}$ and $w_{1}$.

This$\delta \left( w_{1}\right)$function together with $\gamma (w_{1},\tau _{1})$ defined earlier implies that an increase in $\tau _{1}$ will shift down $\gamma (.)$ and cause a movement along $\delta (.)$ corresponding to a lower wage. Consequently, $\hbox {d}w_{1}/\hbox {d}\tau _{1}<0$ and $\hbox {d}a_{12}/\hbox {d}\tau _{1}<0$.

1.4 A.4 Positive optimal tax in the multiple country case

This appendix shows that the optimal tax rate is positive in the multiple country case, assuming that there are no domestic firms in operation. The steps followed are substantially identical to those in Appendix A.3. When there are domestic firms, the argument is identical to the one made in the text in Sect. 3.2.2.

$$\begin{aligned} \tau _{i} = \frac{\hbox {d}R_{iF}/\hbox {d}\tau _{i}}{\frac{\hbox {d}}{\hbox {d}\tau _{i}} (1-\tau _{i})\Pi _{iF}} \end{aligned}$$

Note that the derivative of inframarginal profits with respect to the tax rate is:

$$\begin{aligned} \frac{\hbox {d}R_{iF}}{\hbox {d}\tau _{i}}=\sum _{j\ne i}m_{j} \intop _{\Theta _{ij}}\left[ z_{i}\frac{d\left( 1-\tau _{i}\right) \pi _{ij}\left( w_{i},r\right) }{\hbox {d}\tau _{i}}\right] g_{j}(z)\hbox {d}z \end{aligned}$$

The sign of the $d\left( 1-\tau _{i}\right) \pi _{ij}\left( w_{i}, r\right) /\hbox {d}\tau _{i}$ term thus determines the sign of $\hbox {d}R_{iF}/\hbox {d}\tau _{i}$. A firm that is on the boundary, i.e., $z\in \partial \Theta _{ij}$, will be indifferent between locating in i and locating in some country h:

$$\begin{aligned} \left( 1-\tau _{i}\right) z_{i}\pi _{ij}\left( w_{i},r\right)= & {} \left( 1-\tau _{hj}\right) z_{h}\pi _{hj}\left( w_{h},r\right) \nonumber \\ \left( 1-\tau _{i}\right) \pi _{ij}\left( w_{i},r\right)= & {} \left( 1-\tau _{hj}\right) a_{ihj}\pi _{hj} \left( w_{h},r\right) , \end{aligned}$$

(A.2)

where $a_{ihj}$ the cutoff value of $z_{h}/z_{i}$ for a firm from j that is indifferent between locating in h and i. We will have a maximum of $C-1\times C-1$ such conditions. Subtracting the right-hand side of (A.2) from both sides, we have an equation $\psi _{ihj}\left( a_{ihj},w_{i},\tau _{i}\right) =0$ such that $\psi _{1}<0$, $\psi _{2}<0$ and $\psi _{3}<0$.

The labor market clearing condition, in notation using $a_{ihj}$ instead of $\Theta _{ij}$, and denoting country i as country 1—without loss of generality—is the following:

$$\begin{aligned} L_{i}=\sum _{j\ne i}m_{j}\intop _{0}^{z_{1}^{\max }} \intop _{0}^{a_{12j}z_{1}}\ldots \intop _{0}^{a_{1Cj}z_{1}}l_{ij} \left( w_{i},r,z_{i}\right) g_{j}(z)\hbox {d}z, \end{aligned}$$

(A.3)

where $z_{1}^{\max }$ is the upper-bound on productivity for $z_{1}$. The labor market clearing condition defines an equation $\Psi (a,w_{i})=0$—where a is a vector of $a_{ihj}$—such that $\Psi (.)$ is decreasing in each $a_{ihj}$ and increasing in $w_{i}$. An small increase in $\tau _{i}$ will increase the value of $a_{ihj}$ for a given $w_{i}$. This corresponds to a movement along the (A.3) curve toward a higher $a_{ihj}$ and lower $w_{i}$. This is intuitive: an increase in $\tau _{i}$ leads to fewer firms (i.e., a higher $a_{ihj}$) and a lower wage.

Differentiating both sides of (A.2), we obtain:

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}\tau _{i}}\left( 1-\tau _{i}\right) \pi _{ij} \left( w_{i},r\right) =\frac{\hbox {d}a_{ihj}}{\hbox {d}\tau _{i}} \left( 1-\tau _{h}\right) \pi _{hj}\left( w_{h},r\right) <0 \end{aligned}$$

Thus, $\hbox {d}R_{iF}/\hbox {d}\tau _{i}<0$.