Antidumping protection and welfare in a differentiated duopoly

Abstract

This paper employs a two-country Cournot model to investigate the protection and welfare effects of an antidumping (AD) duty and a price-undertaking policy under different dumping measures (i.e., injury margin and dumping margin) in a differentiated duopoly. We show that the welfare levels of the host country and the world as a whole are lower under a price-undertaking policy than an AD-duty policy. However, the former is superior to the AD-duty policy in terms of protection. These results are robust even if the firms engage in Bertrand competition.

Appendix

We use this appendix to supplement the mathematical calculations in Sect. 5. We shall first show the derivations of results in Sect. 5.1 followed by those in Sect. 5.2.

By substituting (10), the equilibrium outputs under the DM regime, and (15), the equilibrium outputs under the IM regime, into the corresponding profit and welfare functions, we can compare the profits and welfare between DM and IM under the AD-duty policy as follows:

$$ \pi_{x}^{DM} - \pi_{x}^{IM} = \frac{{k(c_{x}^{D} - c_{x} )\Delta_{1} }}{{4(2 - k)^{4} (2 + k)^{4} }}, $$

$$ \pi_{y}^{DM} - \pi_{y}^{IM} = - \frac{{(c_{x}^{D} - c_{x} )\Delta_{2} }}{{(2 - k)^{4} (2 + k)^{4} }}, $$

$$ {\text{SW}}^{DM} - {\text{SW}}^{IM} = \frac{{(c_{x}^{D} - c_{x} )\Delta_{3} }}{{8(2 - k)^{3} (2 + k)^{3} }}, $$

$$ {\text{WW}}^{DM} - {\text{WW}}^{IM} = - \frac{{(c_{x}^{D} - c_{x} )\Delta_{4} }}{{8(2 - k)^{4} (2 + k)^{4} }}, $$

where,

$$ \Delta_{1} \equiv (32 - 16k - 6k^{2} + 3k^{3} )(a - c_{x} ) - (12 + 2k - k^{2} )kc_{x} , $$

$$ \Delta_{2} \equiv (16 - 10k - 3k^{2} + 2k^{3} )(a - c_{x} ) + (12 + 2k - k^{2} )c_{x} , $$

$$ \Delta_{3} \equiv (16 - 6k - k^{2} )(a - c_{x} ) - (12 - 12k - 6k^{2} )c_{x} , $$

$$ \Delta_{4} \equiv (64 - 56k - 4k^{2} + 10k^{3} - k^{4} )a + (16 + 48k - 20k^{2} - 4k^{3} + 6k^{4} )c_{x} . $$

Since $c_{x} \, < \, (2 - k)a/2$ by (4) and $k \in \left( {0,1} \right)$, $\Delta_{1}$, $\Delta_{2}$, $\Delta_{3}$, and $\Delta_{4}$ are all positive. As a result, we can obtain that $\pi_{x}^{DM} \, > \, \pi_{x}^{IM}$, $\pi_{y}^{DM} \, < \, \pi_{y}^{IM}$, $SW^{DM} \, > \, SW^{IM}$, and $WW^{DM} \, < \, WW^{IM}$ if $c_{x} \, < \, c_{x}^{D}$, where ${{c_{x}^{D} \equiv ka ( 2- k )} \mathord{\left/ {\vphantom {{c_{x}^{D} \equiv ka ( 2- k )} {2(2 - k^{2} )}}} \right. \kern-0pt} {2(2 - k^{2} )}}$. This is the result in Sect. 5.1.1.

Similarly, we can derive the counterparts under the price-undertaking policy as follows:

$$ \pi_{x}^{DM} - \pi_{x}^{IM} = \frac{{k(c_{x}^{U} - c_{x} )\Delta_{5} }}{{4(2 - k^{2} )^{2} (2 + k)^{2} }}, $$

$$ \pi_{y}^{DM} - \pi_{y}^{IM} = - \frac{{(c_{x}^{U} - c_{x} )\left( {(1 + k)ka + (2 - k^{2} )c_{x} } \right)}}{{2(2 - k^{2} )(2 + k)^{2} }}, $$

$$ {\text{SW}}^{DM} - {\text{SW}}^{IM} = - \frac{{(c_{x}^{U} - c_{x} )\Delta_{6} }}{{8(2 - k^{2} )^{2} (2 + k)}}, $$

$$ {\text{WW}}^{DM} - {\text{WW}}^{IM} = - \frac{{(c_{x}^{U} - c_{x} )\Delta_{7} }}{{8(2 - k^{2} )^{2} (2 + k)^{2} }}, $$

where

$$ \Delta_{5} \equiv (8 - 3k^{2} )(a - c_{x} ) - (2 + k)kc_{x} , $$

$$ \Delta_{6} \equiv (8 - 10k - 3k^{2} + 4k^{3} )a - (4 - 10k - 2k^{2} + 4k^{3} )c_{x} , $$

$$ \Delta_{7} \equiv (16 - 4k - 8k^{2} + k^{3} )a + (8 + 16k - 2k^{2} - 6k^{3} )c_{x} . $$

Again, by utilizing $c_{x} \, < \, (2 - k)a/2$ and $k \in \left( {0,1} \right)$, we find that $\Delta_{5}$, $\Delta_{6}$, and $\Delta_{7}$ are all positive. Consequently, we can derive that $\pi_{x}^{DM} \, > \, \pi_{x}^{IM}$, $\pi_{y}^{DM} \, < \, \pi_{y}^{IM}$, $SW^{DM} \, < \, SW^{IM}$, and $WW^{DM} \, < \, WW^{IM}$ if $c_{x} \, < \, ka/2(1 + k) \equiv c_{x}^{U}$. This is the result in Sect. 5.1.2.

We now show the detailed algebra for the results in Sect. 5.2 dealing with Bertrand competition. By routine calculations, we can derive the differences in terms of the profit of the domestic firm, the domestic social welfare, the world welfare, and the profit of the foreign firm under the two AD policies as follows:

$$ \pi_{x}^{T} - \pi_{x}^{U} = - \frac{{k^{2} ( 2- k^{2} ) ( ( 2+ k )a - 2c_{x} )\Delta_{8} }}{{16(4 - k^{2} )^{4} ( 1- k^{2} )}}, $$

$$ {\text{SW}}^{T} - {\text{SW}}^{U} = \frac{{k ( ( 2+ k )a - 2c_{x} )\Delta_{9} }}{{32(4 - k^{2} )^{3} ( 1- k^{2} )}}, $$

$$ {\text{WW}}^{T} - {\text{WW}}^{U} = \frac{{k(2 - k^{2} ) ( ( 2+ k )a - 2c_{x} )\Delta_{10} }}{{32(4 - k^{2} )^{4} ( 1- k^{2} )}}, $$

$$ \pi_{y}^{T} - \pi_{y}^{U} = \frac{{k ( 2a + ka - 2c_{x} )(\bar{c}_{x}^{DM} - c_{x} )}}{{8(4 - k^{2} )^{4} ( 1- k^{2} )}} \, > ( \le ) \, 0,{\text{ if }}c_{x} \, < ( \ge ) \, \bar{c}_{x}^{DM} , $$

where,

$$ \Delta_{8} \equiv (64 - 32k - 36k^{2} + 14k^{3} + 6k^{4} - k^{5} )(a - c_{x} ) - (32 - 14k^{2} + k^{4} )kc_{x} , $$

$$ \Delta_{9} \equiv (192 - 184k - 172k^{2} + 96k^{3} + 40k^{4} - 10k^{5} - k^{6} )a + (184k - 96k^{3} + 10k^{5} )c_{x} , $$

$$ \Delta_{10} \equiv (128 - 80k - 104k^{2} + 52k^{3} + 34k^{4} - 10k^{5} - 5k^{6} )a + (80k - 52k^{3} + 10k^{5} )c_{x} . $$

Note that the equilibrium output $x^{T}$ under the DM regime is positive, which requires that $c_{x} \, < \, {{\left( {(2 + k)(8 - 8k - k^{2} + 2k^{3} )a} \right)} \mathord{\left/ {\vphantom {{\left( {(2 + k)(8 - 8k - k^{2} + 2k^{3} )a} \right)} {2(8 - 5k^{2} + k^{4} )}}} \right. \kern-0pt} {2(8 - 5k^{2} + k^{4} )}}$. Utilizing this condition together with the assumption $k \in \left( {0,1} \right)$, we can derive that $\Delta_{8}$, $\Delta_{9}$, and $\Delta_{10}$ are all positive. Therefore, we can find that $\pi_{x}^{T} \, < \, \pi_{x}^{U}$, ${\text{SW}}^{T} \, > {\text{ SW}}^{U}$, and ${\text{WW}}^{T} \, > {\text{ WW}}^{U}$ unconditionally and $\pi_{y}^{T} \, > ( \le ) \, \pi_{y}^{U} ,{\text{ if }}c_{x} \, < ( \ge ) \, \bar{c}_{x}^{DM} .$ This is the result in Sect. 5.2.1.

Finally, we can derive the profit and welfare differences in Sect. 5.2.2 as follows:

$$ \pi_{x}^{T} - \pi_{x}^{U} = - \frac{{kc_{x} \Delta_{1 1} }}{{(2 + k)^{4} (2 - k )^{2} (1 - k)}}, $$

$$ {\text{SW}}^{T} - {\text{SW}}^{U} = \frac{{c_{x} \Delta_{12} }}{{2(2 + k)^{3} (2 - k )(1 - k^{2} )}}, $$

$$ {\text{WW}}^{T} - {\text{WW}}^{U} = \frac{{c_{x} \Delta_{13} }}{{2(2 + k)^{4} (2 - k )^{2} (1 - k)}}, $$

$$ \pi_{y}^{T} - \pi_{y}^{U} = \frac{{c_{x} (c_{x} - \bar{c}_{x}^{IM} )}}{{(2 + k)^{4} (2 - k)^{2} (1 - k^{2} )}} \, > ( \le ) \, 0,{\text{ if }}c_{x} \, > ( \le ) \, \bar{c}_{x}^{IM} , $$

where

$$ \Delta_{1 1} \equiv (8 - 2k^{3} - 6k^{2} ) (a - c_{x} )- ( 1 { + }k )kc_{x} , $$

$$ \Delta_{12} \equiv (12 + 2k - 10k^{2} - 4k^{3} ) (a - c_{x} )+ (5 + 6k - k^{2} - 2k^{3} )c_{x} > 0, $$

$$ \Delta_{13} \equiv (16 - 8k - 20k^{2} + 2k^{3} + 8k^{4} { + }2k^{5} )a + (12 + 20k - k^{2} - 11^{3} - 4k^{4} )c_{x} > 0. $$

It is straightforward to show that $\Delta_{12}$ and $\Delta_{13}$ are positive as $k \in \left( {0,1} \right)$. Furthermore, it is worth noting that the equilibrium output $x^{T}$ under the IM regime in Sect. 5.2.2 is positive, which requires that $c_{x} \, < {{ \, \left( {(1 - k)(2 + k)^{2} a} \right)} \mathord{\left/ {\vphantom {{ \, \left( {(1 - k)(2 + k)^{2} a} \right)} {\left( {(4 + k - 2k^{2} - k^{3} )} \right)}}} \right. \kern-0pt} {\left( {(4 + k - 2k^{2} - k^{3} )} \right)}}$. Utilizing this condition together with the assumption of $k \in \left( {0,1} \right)$, we can derive that $\Delta_{1 1} \, > \, 0$. Therefore, we have $\pi_{x}^{T} \, < \, \pi_{x}^{U}$, ${\text{SW}}^{T} \, > {\text{ SW}}^{U}$, ${\text{WW}}^{T} \, > {\text{ WW}}^{U}$, and $\pi_{y}^{T} \, > ( \le ) \, \pi_{y}^{U} ,{\text{ if }}c_{x} \, > ( \le ) \, \bar{c}_{x}^{IM} .$ This is the result in Sect. 5.2.2.