Dynamic inconsistency and preferential taxation of foreign capital

Abstract

In a two-period dynamic model in which a single country attempts to attract two large investors endowed with capital with varying rate of returns, we show that the result of Kishore and Roy (Econ Lett 124:88–92, 2014) that a country has incentives to commit to a non-preferential regime to circumvent a dynamic inconsistency problem does not hold. The tax revenue of the government may be higher under a preferential regime compared to a non-preferential regime.

Appendix

This section analyzes the scenario in which investments are only partially sunk and there is a uniform nonnegative cost of capital relocation. A fraction $1-\lambda$ of capital is sunk once the investment is made in the host country, that is, if an investor invests in the host country at period 1 and wants to move the invested capital outside the country in period 2, he can only take away a fraction $\lambda$ of the invested capital and receive a return $\lambda R_{i},\ i=H,L$ and $0\le \lambda \le 1$. Here, $1-\lambda R_{i}$ captures the sunk cost and other expenditures associated with the capital relocation. The cost of capital relocation to the host country is $C\ge 0$ for both investors. Lemma 2 and Propositions 3–4 describe the equilibrium outcomes under full commitment, preferential and non-preferential taxation schemes, respectively.

Lemma 2

Suppose the government can fully commit to future tax rates and, moreover, can commit to different tax rates based on different vintages of capital. When $\alpha \le \frac{1}{2R}+\frac{1}{2}-\frac{C}{4R}$, the tax revenue of the government is equal to $G^{C}\equiv 2\left( 2-2\alpha R-C\right)$. When $\alpha >\frac{1}{2R}+\frac{1}{2}- \frac{C}{4R}$, the tax revenue is equal to $G^{C}\equiv 2-2R-C$.

Proof

The proof is similar to lemma 1. To maximize the tax revenue, the government has to consider the following strategies: $\left( 1\right)$ attract both investors in period 1, $\left( 2\right)$ attract investor L in period 1 and set a forbiddingly high tax rate on foreign capital in period 2, $\left( 3\right)$ attract investor H in period 1 and set a forbiddingly high tax rate on foreign capital in period 2, $\left( 4\right)$ attract investor L in period 1 and investor H in period 2 and $\left( 5\right)$ attract investor H in period 1 and investor L in period 2 . We show that it is always beneficial for the government to keep an investor invested in the country in period 2 if he decides to invest in period 1. We also show that it is not beneficial for the government to not attract any investor in period 1.

$\left( 1\right)$ Attract both investors in period 1 :  The maximum tax rate the government can set in period 2 if he wishes to keep both investors invested in period 2 is $1-\lambda \alpha R$. Because the government can fully commit to future tax rates, only $t_{1}+t_{2}$ matters as long as $t_{2}\le 1-\lambda \alpha R$. Given $t_{2}=1-\lambda \alpha R$, the maximum tax rate with which the government in period 1 induces investment from investor H is $1-2\alpha R-C+\lambda \alpha R$. Hence, the total tax revenue of the government in this case is equal to

$$\begin{aligned} G^{C}=2\left( 2-2\alpha R-C\right) . \end{aligned}$$

(24)

$\left( 2\right)$ Attract investor L in period 1 and set a forbiddingly high tax rate on foreign capital in period 2 :  The maximum tax rate the government can commit to in period 2 is $t_{2}=1-\lambda R$. Given $t_{2}=1-\lambda R$, the maximum tax rate the government can set in period 1 to induce investor L in period 1 is equal to $1-2R-C+\lambda R$. Hence, the total tax revenue of the government is equal to

$$\begin{aligned} G^{C}=2-2R-C. \end{aligned}$$

(25)

$\left( 3\right)$ Attract investor H in period 1 and set a forbiddingly high tax rate on foreign capital in period 2 :  Investor H has a higher outside option. Hence, if it is beneficial for investor H in period 1, it is also beneficial for investor L to invest in period 1. Hence, this outcome is not possible.

$\left( 4\right)$ Attract investor L in period 1 and investor H in period 2 :  The maximum tax rate the government can set on foreign capital in period 2 to induce investment from investor H is $1-\alpha R-C$. If the government commits to $t^{N}=1-\alpha R-C$, then if investor L invests in period 2, his gain from relocating to the host country is equal to $\left( \alpha -1\right) R$. Suppose the government commits to $t_{2}=1-\lambda R$. The maximum tax rate the government can set in period 1 is $t_{1}=1-2R-C+\lambda R-\left( \alpha -1\right) R$. Hence, the total tax revenue of the government is equal to

$$\begin{aligned} G^{C}=3-2C-R-2\alpha R. \end{aligned}$$

(26)

$\left( 5\right)$ Attract investor H in period 1 and investor L in period 2 :  Note that if it is beneficial for investor H to invest in period 1, then it is also beneficial for investor L to invest. The minimum gain to investor L if he also chooses to invest in period 1 is $2\left( \alpha -1\right) R$. Hence, investor L will wait until period 2 only if $t^{N}\le 1-R-C-2\left( \alpha -1\right) R$. However, if $t^{N}\le 1-R-C-2\left( \alpha -1\right) R$, investor H can also gain from waiting until period 2. Hence, the government can earn more by attracting both investors in period 1. Note that when both investors invest in period 1, investor H is indifferent between making an investment in period 1 and staying out in both periods. Additionally, the gain to investor L is $2\left( \alpha -1\right) R$.

Now the equilibrium outcomes can be obtained by comparing the tax revenues given by (24), (25) and (26). When $\alpha \le \frac{1}{2R}+\frac{1}{2}-\frac{ C}{4R}$, we have 2 $\left( 2-2\alpha R-C\right) \ge 2-2R-C$. Additionally, $2-2R-C\ge 3-2C-R-2\alpha R$ when $\alpha \ge \frac{1}{2R}+ \frac{1}{2}-\frac{C}{2R}$. Note that $\frac{1}{2R}+\frac{1}{2}-\frac{C}{2R} \le \frac{1}{2R}+\frac{1}{2}-\frac{C}{4R}$ when $C\ge 0$ and $R>0$. We need to show that when $\alpha \le \frac{1}{2R}+\frac{1}{2}-\frac{C}{4R}$, it is beneficial for the government to keep both investors invested if both invest in period 1. Note that $2\left( 1-\lambda \alpha R\right) \ge 1-\lambda R$ when $\alpha \le \frac{1}{2\lambda R}+\frac{1}{2}$. Once we observe that $\frac{1}{2\lambda R}+\frac{1}{2}>\frac{1}{2R}+\frac{1}{2}- \frac{C}{4R}$, the proof is obvious. $\square$

Note that $\frac{C}{4}-\frac{C}{2R}<0$ because $R\le 1$. Compared to the outcome with no cost of capital relocation, the government is less willing to offer large tax discounts in period 1 to attract both investors. The outcome does not depend on $\lambda$ because it does not change the outside options of the investors. When investments are only partially sunk, the government has to offer a lower tax discount in period 1,  which compensates for a relatively lower tax rate in period 2.

Proposition 3

Under a preferential taxation scheme, when $\alpha \le \max$ $\left\{ \frac{1}{3R}+\frac{2}{3} ,\ \frac{1}{2}+\frac{1}{2R}-\frac{C}{2R}\right\}$, in a unique subgame-perfect Nash equilibrium the government’s tax revenue is equal to $G^{NC}\equiv 2\left( 2-2\alpha R-C\right) ,\ e.g,\ G^{NC}=G^{C}$. When $\alpha >\max \left\{ \frac{1}{3R}+\frac{2}{3},\ \frac{1}{2 }+\frac{1}{2R}-\frac{C}{2R}\right\}$, the government’s tax revenue is equal to $G^{NC}\equiv 3-2R-\alpha R-2C$ , $e.g,\ G^{NC}>G^{C}$.

Proof

First, let us look at the outcome in period 2. There are four different situations that the government can encounter in period 2: $\left( 1\right)$ both investors invest in period 1, $\left( 2\right)$ only investor L invests in period 1, $\left( 3\right)$ only investor H invests in period 1 and $\left( 4\right)$ none of the investors invest in period 1 . When both investors invest in period 1 and the government sets $t_{2}=1-\alpha \lambda R,$ then both investors remain invested in period 2 as well. The total tax revenue of the government is equal to $2\left( 1-\lambda \alpha R\right)$. If the government sets $t_{2}=1-\lambda R,$ then only investor H remains invested in period 2 and the total tax revenue of the government is equal to $1-\lambda R$. Note that $2\left( 1-\lambda \alpha R\right) \ge 1-\lambda R$ when $\alpha \le \frac{ 1+\lambda R}{2\lambda R}$. Hence, the tax revenue of the government in period 2 when both investors invest in period 1 is

$$\begin{aligned} G_{2}^{PC}=\left\{ \begin{array}{c} 2\left( 1-\lambda \alpha R\right) \ if\ \alpha \le \frac{1+\lambda R}{ 2\lambda R} \\ 1-\lambda R\ if\ \alpha >\frac{1+\lambda R}{2\lambda R} \end{array} .\right. \end{aligned}$$

(27)

If only investor L invests in period 1, then it is optimal for the government to set $t_{2}=1-\lambda R$ and $t^{N}=1-\alpha R-C$. The total tax revenue of the government in period 2 in this case is equal to

$$\begin{aligned} G_{2}^{PC}=2-\lambda R-\alpha R-C. \end{aligned}$$

(28)

If only investor H invests in period 1, then it is optimal for the government to set $t_{2}=1-\lambda \alpha R$ and $t^{N}=1-R-C$. The total tax revenue of the government in period 2 in this case is equal to

$$\begin{aligned} G_{2}^{PC}=2-\lambda \alpha R-R-C. \end{aligned}$$

(29)

If none of the investors invest in period 1, then the government can either set $t^{N}=1-\alpha R-C$ and attract both investors or set $t^{N}=1-R-C$ and attract only investor L. It is optimal for the government to attract both investors in period 2 when $2\left( 1-\alpha R-C\right) \ge 1-R-C$, i.e., $\alpha \le \frac{1+R-C}{2R}$. Hence, the total tax revenue of the government in period 2 is

$$\begin{aligned} G_{2}^{PC}=\left\{ \begin{array}{c} 2\left( 1-\alpha R-C\right) \ if\ \alpha \le \frac{1+R-C}{2R} \\ 1-R-C\ if\ \alpha >\frac{1+R-C}{2R} \end{array} .\right. \end{aligned}$$

(30)

Now let us look at the outcome in period 1. If the government wishes to attract both investors in period 1, the maximum tax rate it can levy depends on the outcome in period 2. In period 2, the net return on investment by investor H is $\lambda \alpha R$. To induce an investment from investor H,  the government has to compensate for the cost of capital relocation C and the outside option $2\alpha R-\lambda \alpha R$. Hence, the maximum tax rate the government can levy in period 1 is equal to $1-C-2\alpha R+\lambda \alpha R$. Using (27), the total tax revenue of the government is equal to

$$\begin{aligned} G^{PC}=\left\{ \begin{array}{c} 2\left( 2-C-2\alpha R\right) \ if\ \alpha \le \frac{1+\lambda R}{2\lambda R} \\ 2\left( 1-C-2\alpha R+\lambda \alpha R\right) +1-\lambda R\ if\ \alpha > \frac{1+\lambda R}{2\lambda R} \end{array} \right. . \end{aligned}$$

(31)

If the government only wants to attract investor L in period 1, then the maximum tax rate it can levy in period 1 is equal to $1-C-2R+\lambda R$. Using (28), the total tax revenue of the government is equal to

$$\begin{aligned} G^{PC}=3-2C-2R-\alpha R. \end{aligned}$$

(32)

If the government only wants to attract investor H in period 1, then the maximum tax rate it can levy in period 1 is equal to $1-2\alpha R-C+\lambda \alpha R$. In period 2, investor L is indifferent between making an investment in the host country and staying out. Hence, it is not possible for the government to attract only investor H in period 1. If the government sets a high tax rate so that none of the investors invests in period 1, then the total tax revenue of the government is given by (30). The equilibrium outcome can now be obtained from (31) and (32). It is obvious that the equilibrium is unique, because equilibrium strategies are dominant strategies. $\square$

Proposition 4

Under a non-preferential taxation scheme, when $\alpha \le \frac{1}{2R}+\frac{1}{2}-\frac{C}{4R}$, in a unique subgame-perfect Nash equilibrium, the government’s tax revenue is equal to $G^{PC}\equiv 2\left( 2-2\alpha R-C\right)$. When $\alpha >\frac{1}{2R}+\frac{1}{2}-\frac{C}{4R}$, the government’s tax revenue is equal to $G^{PC}\equiv 2-2R-C$. The government’s tax revenue is equal to what it can obtain under full commitment, e.g., $G^{PC}=G^{C}$.

Proof

First, we will look at the outcome in period 2. There are four different situations that the government can face in period 2: $\left( 1\right)$ both investors invest in period 1, $\left( 2\right)$ only investor L invests in period 1, $\left( 3\right)$ only investor H invests in period 1 and $\left( 4\right)$ none of the investors invest in period 1 . When both investors invest in period 1,  then the tax revenue of the government in period 2 is given by (27). Additionally, if none of the investors invest in period 1,  then the tax revenue of the government is given by (30). If only investor L invests in period 1, then in period 2, the government can either set $t^{N}=1-\lambda R$ and obtain taxes only from investor L or set $t^{N}=1-\alpha R-C$ and attract investor H as well. Hence, the tax revenue of the government in period 2 is

$$\begin{aligned} G_{2}^{PC}=\left\{ \begin{array}{c} 2\left( 1-\alpha R-C\right) \text { }if\ \alpha \le \frac{1-2C+\lambda R}{2R} \\ 1-\lambda R\ if\ \alpha >\frac{1-2C+\lambda R}{2R} \end{array} \right. . \end{aligned}$$

(33)

Similarly, if only investor H invests in period 1, then in period 2, the government can either set $t^{N}=1-\lambda \alpha R$ and receive taxes only from investor H or set $t^{N}=1-R-C$ and attract investor L as well. Hence, the tax revenue of the government in period 2 is

$$\begin{aligned} G_{2}^{PC}=\left\{ \begin{array}{c} 2\left( 1-R-C\right) \ if\ \alpha \ge \frac{2R+2C-1}{\lambda R} \\ 1-\lambda \alpha R\ if\ \alpha <\frac{2R+2C-1}{\lambda R} \end{array} \right. . \end{aligned}$$

(34)

Now let us look at the outcome in period 1. When both investors invest in period 1 and none of the investors invest in period 1, then the tax revenue of the government is given by (31) and (30), respectively. If the government wishes to attract only investor L in period 1, then if $\alpha >\frac{1-2C+\lambda R}{2R}$, the maximum tax rate the government can set in period 1 is equal to $1-2R-C+\lambda R$. Using (33), the total tax revenue of the government is

$$\begin{aligned} G^{PC}=2-2R-C. \end{aligned}$$

(35)

The equilibrium outcome can be obtained by comparing (31) and (35). We can see that $2\left( 2-2\alpha R-C\right) \ge 2-2R-C$ when $\alpha \le \frac{1}{2R}+\frac{1}{2}- \frac{C}{4R}$. Now we need to show that neither investor nor the government has the incentive to deviate unilaterally. When $\frac{1}{2R}+\frac{1}{2}- \frac{C}{4R}$, investor H is indifferent between making an investment in period 1 and staying out in both periods. If he decides not to invest in period 1,  then in period 2,  the tax rate is given by (33). Whether or not the government chooses to attract investor H in period 2, investor H cannot do better. Hence, he has no incentive to deviate. Similarly, if investor L decides against making investment in period 1, the tax rate in period 2 is given by (34). Whether or not the government chooses to attract investor L in period 2, he cannot gain from making an investment in period 2. Hence, both investors have no incentive to deviate unilaterally. We need to show that the government has no incentive to deviate unilaterally. Note that when $\alpha \le \frac{1}{2R}+\frac{1}{2}-\frac{C}{4R}$, the tax rate in period 2 on domestic capital is $1-\lambda \alpha R$. When $\alpha \le \frac{1}{2} +\frac{1}{2\lambda R}$, then $2\left( 1-\lambda \alpha R\right) \ge 1-\lambda R,~$and it is better for the government to receive taxes from both investors (if both invest in period 1). Note that $\alpha \le \frac{1}{2}+ \frac{1}{2\lambda R}$ implies that $\alpha \le \frac{1}{2R}+\frac{1}{2}- \frac{C}{4R}$. Hence, the proof is complete. $\square$

We can see that the tax revenue of the government does not depend on whether investments are fully or partially sunk. Investors are compensated up-front for their loss of returns in period 2. The tax revenue of the government only depends on the investor’s outside option.