Commodity taxes and rent extraction

Abstract

It is difficult for WTO member countries to raise tariffs unilaterally under current WTO regulations. Therefore, given a constant tariff rate, we examine the impacts of two commodity taxes, an ad valorem tax and a specific tax, on the rent-extracting effect regarding the foreign firm and on the protection effect regarding the domestic firm. We obtain two main results. First, the government can extract more profits from the foreign firm by imposing an ad valorem (a specific) tax, when the tariff rate is low (high); and second, when the tariff rate is low, an ad valorem tax is welfare superior to a specific tax while the reverse may occur when the tariff rate is high. This demonstrates that the magnitude of the tariff rate is crucial when the government chooses the commodity tax scheme.

Appendices

Appendix 1

The proof of ${{\varvec{T}}}_{1}\in ({{\varvec{c}}}_{{\varvec{h}}}-{{\varvec{c}}}_{{\varvec{f}}}, \overline{{\varvec{T}} })$

(1) ${R}^{\tau }-{R}^{t}>0$ when $T={c}_{h}-{c}_{f},$ and ${R}^{\tau }-{R}^{t}<0$ when $T=\overline{T }$.

(2) By (4), we can derive the influences of T on the equilibrium ${p}^{\tau }$, ${q}_{f}^{\tau },$ and ${q}_{f}^{t}$ as follows:

$$\frac{{dp^{\tau } }}{{dT}} = \frac{1}{{\left( {1 - \tau } \right)}}\frac{{p'\left( {Q^{\tau } } \right)}}{{3p^{\prime}\left( {Q^{\tau } } \right) + p''\left( {Q^{\tau } } \right)Q^{\tau } }} > 0,$$

(12)

$$\frac{{dq_{f}^{\tau } }}{{dT}} = \frac{1}{{\left( {1 - \tau } \right)}}\frac{{2p^{\prime}\left( {Q^{\tau } } \right) + p''\left( {Q^{\tau } } \right)q_{h}^{\tau } }}{{\left( {3p^{\prime}\left( {Q^{\tau } } \right) + p^{\prime\prime}\left( {Q^{\tau } } \right)Q^{\tau } } \right)p^{\prime}\left( {Q^{\tau } } \right)}} < 0,$$

(13)

$$\frac{{dq_{f}^{t} }}{{dT}} = \frac{1}{{\left( {1 - \tau } \right)}}\frac{{2p^{\prime}\left( {Q^{t} } \right) + p^{\prime\prime}\left( {Q^{t} } \right)q_{h}^{t} + \left( {\frac{\tau }{{2\left( {1 - \tau } \right)}}} \right)\left( {p^{\prime}\left( {Q^{t} } \right) + p^{\prime\prime}\left( {Q^{t} } \right)\left( {q_{h}^{t} - q_{f}^{t} } \right)} \right)}}{{\left( {3p^{\prime}\left( {Q^{t} } \right) + p^{\prime\prime}\left( {Q^{t} } \right)Q^{t} } \right)p^{\prime}\left( {Q^{t} } \right)}} < 0$$

(14)

Differentiating (8) with respect to T gives:

$$\frac{d({R}^{T}-{R}^{t})}{dT}=\frac{d(\tau {p}^{\tau }-t)}{dT}{q}_{f}^{\tau }+(\tau {p}^{\tau }-t)\frac{d{q}_{f}^{\tau }}{dT}-t\frac{d({q}_{f}^{t}-{q}_{f}^{\tau })}{dT}$$

(15)

By using (12, 13 ,14) and ${Q}^{t}={Q}^{\tau },$ we can obtain $\frac{{d\left( {\tau p^{\tau } - t} \right)}}{{dT}} = \left( {\frac{{ - \tau }}{{1 - \tau }}} \right)\frac{{p^{\prime } \left( {Q^{\tau } } \right) + p^{{\prime \prime }} \left( {Q^{\tau } } \right)Q^{\tau } }}{{3p^{\prime } \left( {Q^{\tau } } \right) + p^{{\prime \prime }} \left( {Q^{\tau } } \right)Q^{\tau } }}\; < 0,\;\frac{{dq_{f}^{\tau } }}{{dT}}{\text{ }} < 0\;and~\frac{{d\left( {q_{f}^{t} - q_{f}^{\tau } } \right)}}{{dT}} = \frac{1}{{3p^{\prime } \left( {Q^{\tau } } \right) + p^{{\prime \prime }} \left( {Q^{\tau } } \right)Q^{\tau } }}\left[ {\frac{{ - 3\tau p^{\prime } \left( {Q^{\tau } } \right)}}{{2\left( {1 - \tau } \right)}} + \frac{{p^{{\prime \prime }} \left( {Q^{\tau } } \right)(\left( {2\left( {q_{h}^{t} - q_{h}^{\tau } } \right) - \tau \left( {\left( {q_{h}^{t} + q_{f}^{\tau } } \right)} \right)} \right)}}{{2\left( {1 - \tau } \right)}}} \right] > 0$.¹⁵ By substituting these inequalities into (15), we obtain that $\frac{d({R}^{T}-{R}^{t})}{dT}<0.$

(3) We can find from (1) and (2) that there exists a threshold value ${T}_{1}\in ({c}_{h}-{c}_{f}, \overline{T })$ such that $T<(\ge ){T}_{1}$ if and only if ${R}^{\tau }-{R}^{t}$> ($\le$) 0.

Appendix 2

The impact of an increase in the specific tax rate t on social welfare

Given a linear demand function, $p=a-Q$, the equilibrium outputs and price can be obtained as follows:

$${q}_{f}^{\tau }=\frac{a-t-2\left({c}_{f}+T\right)+{c}_{h}}{3}, {q}_{h}^{\tau }=\frac{a-t-2{c}_{h}+\left({c}_{f}+T\right)}{3}, {Q}^{t}=\frac{2\left(a-t\right)-\left({c}_{f}+T\right)-{c}_{h}}{3}, {p}^{t}=\frac{a+2t+\left({c}_{f}+T\right)+{c}_{h}}{3}.$$

(16)

The social welfare under a specific tax can be described as ${SW}^{t}={TR}^{t}+{R}^{t}+{PS}_{h}^{t}+{CS}^{t}$, where ${CS}^{t}$ is the consumer surplus under a specific tax. From (16), we can obtain:

$$\frac{{dSW^{t} }}{{dt}} = \frac{{\left( { - 2\left( {t + T} \right) - c_{f} + c_{h} } \right)}}{3} < 0,\,~for\,~T \ge c_{h} - c_{f} ,$$

(17)

Given the social welfare under an ad valorem tax and by (17), we can find that the value of ${T}_{3}$ in Proposition 4 rises.