Optimal Rules of Origin with Asymmetric Compliance Costs under International Duopoly

Abstract

We examine the optimal rules of origin (ROO) in a free trade area/agreement (FTA) by employing a stylized three-country partial equilibrium model of an international duopoly. We incorporate compliance costs of the ROO into the model. In particular, compliance costs are higher for a firm located in a non-member country of the FTA than for a firm (an internal firm) located in an FTA member country, whereas marginal production costs are lower for the former. The FTA member countries set the optimal level of ROO to maximize their joint welfare. An importing country within the FTA imposes tariffs on imports that do not comply with the ROO. We show that the optimal ROO may have a protectionist bias in the sense that they are set for only the internal firm to comply. ROO may also cause low utilization of FTAs when they are set such that even the internal firm does not comply with them. These cases arise depending on parameter values.

Appendices

Appendix A

Proof of Proposition 1

In stage 3, each firm chooses compliance, C, or non-compliance, N. Thus, the game in stage 3 can be represented by the normal form game shown in Table 1.

Table 1 A normal form representation of stage 3

Firms’ best responses can be found in the following way. First, for firm I, Eqs. 11 and 17 yield

$$\pi^{I}_{cc}-\pi^{I}_{nc} =\frac{4(a-2c-t+(k-1)\phi)(t-\phi)}{9}. $$

Thus, $\pi ^{I}_{cc} \geq \pi ^{I}_{nc}$ holds if and only if ϕ≤t. Moreover, from Eqs. 4 and 15, we have

$$\pi^{I}_{nn}-\pi^{I}_{cn} =\frac{4(a-2c-\phi)(\phi-t)}{9}, $$

which implies that $\pi ^{I}_{nn} \geq \pi ^{I}_{cn}$ holds if and only if ϕ≥t. Second, for firm E, Eqs. 11 and 15 yield

$$\pi^{E}_{cc}-\pi^{E}_{cn} =\frac{4(a+c-t-(k-1)\phi)(t-k\phi)}{9}. $$

Thus, $\pi ^{E}_{cc} \geq \pi ^{E}_{cn}$ holds if and only if ϕ≤t/k. Similarly, from Eqs. 4 and 17, it follows that

$$\pi^{E}_{nn}-\pi^{E}_{nc} =\frac{4(a+c-k\phi)(k\phi-t)}{9}, $$

yielding that $\pi ^{E}_{nn} \geq \pi ^{E}_{nc}$ if and only if ϕ≥t/k.

Then, when 0≤ϕ<t/k, C is the dominant strategy for both firms; hence, {C,C} is a unique NE as long as ε>0 is sufficiently small. When t/k≤ϕ<t, C is still the dominant strategy for firm I, and N is the dominant strategy for firm E in the presence of ε>0. Thus, {C,N} is a unique NE. Finally, when t≤ϕ, N becomes the dominant strategy for both firms in the presence of ε>0. Thus, {N,N} is a unique NE. □

Appendix B

Proof of Lemma 5

(i) For 5/2<α≤43/14, ϕ _{c n} =t holds for ϕ _{c n} =ϕ _b =(2a−c)/8, and t=t _b =(2a−c)/8, and ϕ _{c n} =t/k holds for ϕ _{c n} =ϕ _{c n2}=(a+4c)/(11k−4) and t=t _{c n} (ϕ _{c n2})=(a+4c)k/(11k−4). Comparing joint welfare of the FTA members in these cases, Eqs. 27 and 30 yield

$$\begin{array}{@{}rcl@{}} W^{AB}_{cn}(\phi_{b}) - W^{AB}_{cn}(\phi_{cn2})&=&-\frac{1}{8(11k-4)^{2}}[(2a-3c)k\{(53a-19c)k-8(8a+11c)\}\\ & &+12(a+2c)(3a-2c)]. \end{array} $$

(41)

It can be shown that the term within the curly brackets on the RHS of Eq. 41 is negative if Eq. 32 holds, which implies that $W^{AB}_{cn}(\phi _{b}) > W^{AB}_{cn}(\phi _{cn2})$. Otherwise, the this term is non-negative; hence, $W^{AB}_{cn}(\phi _{b}) \leq W^{AB}_{cn}(\phi _{cn2})$ holds. As the RHS of Eq. 32 becomes one when α=3 and a complex number for α<3, ϕ _{c n} =ϕ _b =(2a−c)/8 never holds for α≤3.

(ii) For 43/14<α, ϕ _{c n} =t holds when ϕ _{c n} =ϕ _b =(2a−c)/8 for α≤13/2 and ϕ _{c n} =ϕ _{c n1}=(a+4c)/7 for 13/2<α. On the other hand, ϕ _{c n} =t/k holds when ϕ _{c n} =ϕ _b /k=(2a−c)/8k for 43/14<α<13/2 and k<4(2α−1)/(14α−43), and ϕ _{c n} =ϕ _{c n2}=(a+4c)/(11k−4) for k≥4(2α−1)/(14α−43). Note that 4(2α−1)/(14α−43)=1 holds at α=13/2. Thus, for 43/14<α≤13/2, we compare joint welfare of the FTA members at ϕ _{c n} =ϕ _b =(2a−c)/8 and ϕ _{c n} =ϕ _b /k=(2a−c)/8k. From Eqs. 27 and 31, we have

$$W^{AB}_{cn}(\phi_{b}) - W^{AB}_{cn}(\phi_{b}/k) = -\frac{(k-1)(2a-c)\{(26a-45c)k-3(2a-c)\}}{384k^{2}}. $$

Since 3(2α−1)/(26α−45)<1 holds for α>5/2, it follows that $W^{AB}_{cn}(\phi _{b}) < W^{AB}_{cn}(\phi _{b}/k)$; hence, ϕ _{c n} =ϕ _b =(2a−c)/8 does not hold. Moreover, a comparison of joint welfare of countries A and B under ϕ _{c n} =ϕ _b =(2a−c)/8 and ϕ _{c n} =ϕ _b /k=(2a−c)/8k has been made in Eq. 41. However, it is shown that

$$\frac{4(2\alpha -3)(8\alpha +11)+2(\alpha +4)\sqrt{35(2\alpha -3)(\alpha -3)}}{(2\alpha -3)(53\alpha -19)}<\frac{4(2\alpha -1)}{(14\alpha - 43)} $$

holds for all α∈(43/14,13/2). Thus, $W^{AB}_{cn}(\phi _{b}) < W^{AB}_{cn}(\phi _{cn2})$ holds. It follows that ϕ _{c n} =ϕ _b =(2a−c)/8 is never optimal in this case. Finally, for 13/2<α, comparing joint welfare of countries A and B under ϕ _{c n} =ϕ _{c n1}=(a+4c)/7 and ϕ _{c n} =ϕ _{c n2}=(a+4c)/(11k−4), Eqs. 26 and 30 yield

$$W^{AB}_{cn}(\phi_{cn1}) - W^{AB}_{cn}(\phi_{cn2}) = -\frac{5(k-1)(a+4c)\{(25a-54c)k-(11a-12c)\}}{14(11k-4)^{2}}. $$

Thus, if

$$k<\frac{11a-12c}{25a-54c}=\frac{11\alpha-12}{25\alpha-54}, $$

then $W^{AB}_{cn}(\phi _{cn1})> W^{AB}_{cn}(\phi _{cn2})$ holds. However, since (11α−12)/(25α−54)<1 holds for α>3, $W^{AB}_{cn}(\phi _{cn1})< W^{AB}_{cn}(\phi _{cn2})$ holds in the relevant range of the parameter values. □

Appendix C

Proof of Proposition 2

(i) From Eqs. 20 and 30, we have

$$\begin{array}{@{}rcl@{}} W^{AB}_{cn}(\phi_{cn2})-W^{AB}_{nn}(t_{b})&=&\frac{1}{64(11k-4)^{2}}\{(364a^{2}-124ac-149c^{2})k^{2}\\ &&-8(84a^{2} +116ac-319c^{2})k\\ & &+224a^{2}+576ac-464c^{2}\}. \end{array} $$

(42)

Then, it can be shown that, if Eq. 37 holds, the term within the curly brackets on the RHS of Eq. 42 is negative and, hence, it follows that $W^{AB}_{cn}(\phi _{cn2})<W^{AB}_{nn}(t_{b})$. In Eq. 37, 28α ²−108α+87=0 when $\alpha =(27+2\sqrt {30})/14 \approx 2.71$ and 28α ²−108α+87<0 for $\alpha < (27+2\sqrt {30})/14$. Thus, Eq. 37 does not hold for $\alpha \leq (27+2\sqrt {30})/14$. From Eqs. 20 and 27, we have

$$ W^{AB}_{nn}(t_{b})-W^{AB}_{cn}(\phi_{b})=\frac{(2a-c)(2a-5c)}{64}. $$

(43)

Thus, $W^{AB}_{cn}(\phi _{b})< W^{AB}_{nn}(t_{b})$ always holds for α>5/2.

(ii) For 43/14<α≤9/2, from Lemma 5, ϕ _{c n} =ϕ _{c n2} if Eq. 34 holds, and ϕ _{c n} =ϕ _b /k if Eq. 33 holds. Equations 20 and 31 yield

$$W^{AB}_{cn}(\phi_{b}/k)-W^{AB}_{nn}(t_{b})=\frac{(2a-c)\{(14a-15c)k^{2}-16(2a-3c)k+3(2a-c)\}}{384k^{2}}. $$

It can be shown that if Eq. 38 holds, it follows that $W^{AB}_{cn}(\phi _{b}/k)>W^{AB}_{nn}(t_{b})$. On the other hand, Eqs. 20 and 30 yield

$$\begin{array}{@{}rcl@{}} W^{AB}_{cn}(\phi_{cn2})-W^{AB}_{nn}(t_{b})&=&\frac{1}{64(11k-4)^{2}}\{(364a^{2}-124ac-149c^{2})k^{2}\\ &&-8(84a^{2}+ 116ac-319c^{2})k\\ & &+224a^{2}+576ac-464c^{2}\}. \end{array} $$

Thus, if Eq. 39 holds, it follows that $W^{AB}_{cn}(\phi _{cn2})>W^{AB}_{nn}(t_{b})$. Note that for 43/14<α<9/2, 4(2α−1)/(14α−43) is monotonically decreasing, and the RHS of both Eqs. 38 and 39 is monotonically increasing, and all three take the same value at $\alpha =(323+16\sqrt {205})/126 \approx 4.38$.

(iii) For α>9/2, from Lemma 5, regime CN has two possible optimal ROO solutions: ϕ _{c n} =ϕ _b /k and ϕ _{c n} =ϕ _{c n2}. Equations 30 and 36 yield

$$W^{AB}_{cn}(\phi_{cn2})-W^{AB}_{cc}(\phi_{cc}=0)=\frac{(a+4c)\{(19a-12c)k^{2}-2(26a-33c)k+19a-12c\}}{6(11k-4)^{2}}. $$

Thus, if Eq. 40 holds, it follows that $W^{AB}_{cn}(\phi _{cn2})>W^{AB}_{cc}(\phi _{cc}=0)$. Note that the RHS of Eq. 40 becomes $(8+\sqrt {15})/7 \approx 1.70$ at α=9/2 and is increasing for α>9/2. Moreover, Eqs. 31 and 36 yield

$$W^{AB}_{cn}(\phi_{b}/k)-W^{AB}_{cc}(\phi_{cc}=0)=\frac{1}{384k^{2}}(2a-c)\{(10a+3c)k^{2}-16(2a-3c)k+3(2a-c)\}. $$

Thus, if

$$ k \geq \frac{8(2\alpha-3)+\sqrt{(14\alpha-15)(14\alpha-39)}}{10\alpha+3} $$

(44)

holds, it follows that $W^{AB}_{cn}(\phi _{b}/k) \geq W^{AB}_{cc}(\phi _{cc}=0)$. Note that the RHS of Eq. 44 becomes $1+\sqrt {1152}/48 \approx 1.71$ at α=9/2, and it is increasing for α>9/2. On the other hand, from Lemma 5, if Eq. 34 holds, then ϕ _{c n} =ϕ _{c n2}. Note that the RHS of Eq. 34 becomes 4(2α−1)/(14α−43)=1.6 at α=9/2 and is decreasing for α>9/2. □