# Multinational Firm’s Production Decisions under Overlapping Free Trade Agreements: Rule of Origin Requirements and Environmental Regulation

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## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Analysis

**Proposition 1**

**(Type I of the Spaghetti Bowl Effect).**

- (i)
- $\underset{\_}{{e}_{2}}<{e}_{2}^{*}\text{}and\text{}\underset{\_}{{e}_{1}}\text{}\in \left({\psi}_{1a},\text{}{\psi}_{1b}\right),$ where$${\psi}_{1a}=\frac{\left({\alpha}_{2}-c-{t}_{2}\right)\left(1-\sqrt{\frac{4\gamma -{\beta}^{2}}{4\gamma}}\right)+{t}_{2}}{\beta}\text{}and\text{}{\psi}_{1b}=\frac{\left({\alpha}_{2}-c-{t}_{2}\right)\left(1+\sqrt{\frac{4\gamma -{\beta}^{2}}{4\gamma}}\right)+{t}_{2}}{\beta}$$
- (ii)
- $\underset{\_}{{e}_{2}}\ge {e}_{2}^{*}\text{}$ and $\underset{\_}{{e}_{2}}\text{}\notin \left[{\psi}_{2a},{\psi}_{2b}\right]$, where
- ${\psi}_{2a}=\frac{-\beta \left({\alpha}_{2}-c\right)+4\gamma \underset{\_}{{e}_{1}}-\sqrt{{\left(\beta \left({\alpha}_{2}-c\right)-4\gamma \underset{\_}{{e}_{1}}\right)}^{2}+\left(4\gamma -{\beta}^{2}\right)\left({t}_{2}\left(2{\alpha}_{2}-2c-{t}_{2}\right)-4\gamma {\underset{\_}{{e}_{1}}}^{2}\right)}}{4\gamma -{\beta}^{2}}$ and
- ${\psi}_{2b}=\frac{-\beta \left({\alpha}_{2}-c\right)+4\gamma \underset{\_}{{e}_{1}}+\sqrt{{\left(\beta \left({\alpha}_{2}-c\right)-4\gamma \underset{\_}{{e}_{1}}\right)}^{2}+\left(4\gamma -{\beta}^{2}\right)\left({t}_{2}\left(2{\alpha}_{2}-2c-{t}_{2}\right)-4\gamma {\underset{\_}{{e}_{1}}}^{2}\right)}}{4\gamma -{\beta}^{2}}$.

Then, Type I of the Spaghetti Bowl Effect is quantified by ${\Pi}_{2}^{t}\left({t}_{2}\right)-{\Pi}_{2}^{f}\left(\underset{\_}{{e}_{2}}\right)$.

**Proposition 2**

**(Type II of the Spaghetti Bowl Effect).**

**Proposition**

**3.**

**Proposition**

**4.**

- The amount for Type I of the Spaghetti Bowl Effect increases as the conversion efficiency of a firm decreases (i.e., as $\gamma $ increases) when $\underset{\_}{{e}_{1}}\ne \underset{\_}{{e}_{2}}$.
- The amount for Type II of the Spaghetti Bowl Effect increases as the conversion efficiency of a firm decreases (i.e., as $\gamma $ increases).

## 4. Policy Implications

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Proofs

**Proof of Proposition**

**1.**

- (i)
- When $\underset{\_}{{\mathrm{e}}_{2}}<{\mathrm{e}}_{2}^{*}$, ${\Pi}_{2}^{\mathrm{t}}\left({\mathrm{t}}_{2}\right)>{\Pi}_{2}^{\mathrm{f}}\left(\mathrm{max}\left\{\underset{\_}{{\mathrm{e}}_{2}},{\mathrm{e}}_{2}^{*}\right\}\right)={\Pi}_{2}^{\mathrm{f}}\left({\mathrm{e}}_{2}^{*}\right)$ and using Equations (7), (8), and (11), we obtain:$${\Pi}_{2}^{\mathrm{t}}\left({\mathrm{t}}_{2}\right)\text{}{\Pi}_{2}^{\mathrm{f}}\left({\mathrm{e}}_{2}^{*}\right)$$$$\frac{{\left({\mathsf{\alpha}}_{2}-\mathrm{c}-{\mathrm{t}}_{2}\right)}^{2}}{4}>\frac{\mathsf{\gamma}{\left({\mathsf{\alpha}}_{2}-\mathrm{c}-\mathsf{\beta}\underset{\_}{{\mathrm{e}}_{1}}\right)}^{2}}{4\mathsf{\gamma}-{\mathsf{\beta}}^{2}},$$$$\frac{\left({\mathsf{\alpha}}_{2}-\mathrm{c}-{\mathrm{t}}_{2}\right)\left(1-\sqrt{\frac{4\mathsf{\gamma}-{\mathsf{\beta}}^{2}}{4\mathsf{\gamma}}}\right)+{\mathrm{t}}_{2}}{\mathsf{\beta}}<\underset{\_}{{\mathrm{e}}_{1}}<\frac{\left({\mathsf{\alpha}}_{2}-\mathrm{c}-{\mathrm{t}}_{2}\right)\left(1+\sqrt{\frac{4\mathsf{\gamma}-{\mathsf{\beta}}^{2}}{4\mathsf{\gamma}}}\right)+{\mathrm{t}}_{2}}{\mathsf{\beta}},$$
- (ii)
- When $\underset{\_}{{\mathrm{e}}_{2}}\ge {\mathrm{e}}_{2}^{*}$, ${\Pi}_{2}^{\mathrm{t}}\left({\mathrm{t}}_{2}\right)>{\Pi}_{2}^{\mathrm{f}}\left(\mathrm{max}\left\{\underset{\_}{{\mathrm{e}}_{2}},{\mathrm{e}}_{2}^{*}\right\}\right)={\Pi}_{2}^{\mathrm{f}}\left(\underset{\_}{{\mathrm{e}}_{2}}\right)$ and using Equations (7) and (11):$${\Pi}_{2}^{\mathrm{t}}\left({\mathrm{t}}_{2}\right)\text{}{\Pi}_{2}^{\mathrm{f}}\left(\underset{\_}{{\mathrm{e}}_{2}}\right)$$$$\frac{{\left({\mathsf{\alpha}}_{2}-\mathrm{c}-{\mathrm{t}}_{2}\right)}^{2}}{4}>-\frac{1}{4}({\left({\mathsf{\alpha}}_{2}-\mathrm{c}\right)}^{2}-4\mathsf{\gamma}{\underset{\_}{{\mathrm{e}}_{1}}}^{2}-\underset{\_}{{\mathrm{e}}_{2}}\left(2\mathsf{\beta}\left({\mathsf{\alpha}}_{2}-\mathrm{c}\right)-8\mathsf{\gamma}\underset{\_}{{\mathrm{e}}_{1}}+\left(4\mathsf{\gamma}-{\mathsf{\beta}}^{2}\right)\underset{\_}{{\mathrm{e}}_{2}}\right))$$$$\underset{\_}{{\mathrm{e}}_{2}}<\frac{-\mathsf{\beta}\left({\mathsf{\alpha}}_{2}-\mathrm{c}\right)+4\mathsf{\gamma}\underset{\_}{{\mathrm{e}}_{1}}-\sqrt{{(\mathsf{\beta}\left({\mathsf{\alpha}}_{2}-\mathrm{c}\right)-4\mathsf{\gamma}\underset{\_}{{\mathrm{e}}_{1}})}^{2}+\left(4\mathsf{\gamma}-{\mathsf{\beta}}^{2}\right)\left({\mathrm{t}}_{2}\left(2{\mathsf{\alpha}}_{2}-2\mathrm{c}-{\mathrm{t}}_{2}\right)-4\mathsf{\gamma}{\underset{\_}{{\mathrm{e}}_{1}}}^{2}\right)}}{4\mathsf{\gamma}-{\mathsf{\beta}}^{2}}$$$$\underset{\_}{{\mathrm{e}}_{2}}>\frac{-\mathsf{\beta}\left({\mathsf{\alpha}}_{2}-\mathrm{c}\right)+4\mathsf{\gamma}\underset{\_}{{\mathrm{e}}_{1}}+\sqrt{{\left(\mathsf{\beta}\left({\mathsf{\alpha}}_{2}-\mathrm{c}\right)-4\mathsf{\gamma}\underset{\_}{{\mathrm{e}}_{1}}\right)}^{2}+\left(4\mathsf{\gamma}-{\mathsf{\beta}}^{2}\right)\left({\mathrm{t}}_{2}\left(2{\mathsf{\alpha}}_{2}-2\mathrm{c}-{\mathrm{t}}_{2}\right)-4\mathsf{\gamma}{\underset{\_}{{\mathrm{e}}_{1}}}^{2}\right)}}{4\mathsf{\gamma}-{\mathsf{\beta}}^{2}}$$

**Proof of Proposition**

**2.**

**Proof of Proposition**

**3.**

**Proof of Proposition**

**4.**

## Appendix B. Duopoly Case

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**Figure 1.**The illustration of the Spaghetti Bowl Effect (source: [25]).

**Figure 2.**Firm’s optimal strategy in Country 2 (We use ${\mathsf{\alpha}}_{2}=1,\text{}{\mathrm{t}}_{2}=0.1,\text{}\mathrm{c}=0.3,\text{}\mathsf{\gamma}=4\text{}\mathrm{and}\text{}\mathsf{\beta}=1$. An analysis of the duopoly case is illustrated in Figure 4).

**Figure 4.**Firm’s optimal strategy in Country under the duopoly case (We use ${\mathsf{\alpha}}_{2}=1,\text{}{\mathrm{t}}_{2}=0.1,\text{}\mathrm{c}=0.3,\text{}\mathsf{\gamma}=4\text{}\mathrm{and}\text{}\mathsf{\beta}=1$).

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**MDPI and ACS Style**

Lee, S.H.; Park, K.S.; Seo, Y.W. Multinational Firm’s Production Decisions under Overlapping Free Trade Agreements: Rule of Origin Requirements and Environmental Regulation. *Sustainability* **2017**, *9*, 42.
https://doi.org/10.3390/su9010042

**AMA Style**

Lee SH, Park KS, Seo YW. Multinational Firm’s Production Decisions under Overlapping Free Trade Agreements: Rule of Origin Requirements and Environmental Regulation. *Sustainability*. 2017; 9(1):42.
https://doi.org/10.3390/su9010042

**Chicago/Turabian Style**

Lee, Sung Hee, Kun Soo Park, and Yong Won Seo. 2017. "Multinational Firm’s Production Decisions under Overlapping Free Trade Agreements: Rule of Origin Requirements and Environmental Regulation" *Sustainability* 9, no. 1: 42.
https://doi.org/10.3390/su9010042