# The Connection between Imported Inputs and Exports: The Importance of Strategic Interdependence

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{1}. However, ignoring strategic interactions among final goods producers, the extant theoretical literature explains only the negative relationship between the costs of imported inputs and the export of the final goods using those inputs. The lower costs of imported inputs increase the exports of the final goods either by improving the productivities of the final goods producers or by diffusing knowledge about modern technologies ([1,2])

^{2}.

^{3}.

^{4}. The firms differ in terms of input coefficients, and they import inputs from a competitive world market. In this framework, we discuss how a lower cost of imported inputs affects the firms’ equilibrium outputs (i.e., export volumes) and the incentive for export.

## 2. Literature Review

## 3. The Model

**Lemma 1**.

^{5}.

**Proof**.

## 4. The Effects of a Lower Cost of Imported inputs

^{6}shows how the ranges of G over which different equilibria occur change. The solid lines in Figure 2 show the situations under initial c, and the dashed lines show the situations after a reduction in c. Figure 2 helps to prove the following result.

**Proposition 1**.

- (a)
- A lower cost of imported inputs, i.e., a lower c
- (i)
- Increases the possibility of export by firm 1, but
- (ii)
- May increase or decrease the possibility of export by firm 2.

- (b)
- A lower cost of imported inputs
- (i)
- Increases the volume of exports for firm 1, but
- (ii)
- May either increase or decrease the volume of exports for firm 2.

**Proof.**

## 5. The Implications for Consumer Surplus, Total Profits, and World Welfare

^{7}. This happens because if a lower cost of imported inputs changes the equilibrium from (NE, E) to (E, NE), it creates production inefficiency in the industry by shifting the exporting decision from the high-productive firm to the low-productive firm.

^{8}. For equilibrium (E, E), we have seen a negative (ambiguous) relationship between the cost of imported inputs and export volume for firm 1 (firm 2). If we look at the effect of a lower c on the total outputs of firms 1 and 2, we get $\frac{\partial q}{\partial c}=\frac{\partial {q}_{1}}{\partial c}+\frac{\partial {q}_{2}}{\partial c}=\frac{-1-\lambda}{3}$ < 0. Hence, a lower cost of imported inputs will reduce the price in the export market by increasing the total outputs of the exporters and, therefore, will make the consumers better off, although it may reduce the equilibrium output of firm 2.

^{9}. However, $\frac{\partial {\pi}^{*}}{\partial c}<0$ if either $a>5c$ or $2c\le a<5c$ and $\frac{a+8c}{10c}-\frac{1}{10}\sqrt{\frac{{a}^{2}+36ac-36{c}^{2}}{{c}^{2}}}<\lambda <1$.

**Proposition 2**.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Lemma 1

## Appendix B. Proof of Proposition 1

**,**$\frac{\partial {q}_{2}}{\partial c}=\frac{1-2\lambda}{3}>0$ for $\lambda <\frac{1}{2}$. Hence, there is a negative relationship between the costs of imported inputs and export volume for firm 1, but the relationship between the costs of imported inputs and export volume is positive for firm 2 if $\lambda <\frac{1}{2}$.

## Appendix C. Bertrand Competition

**Lemma A1**.

**Proof:**

**Proposition A1**.

**Proof.**

^{10}.

**Proposition A2**.

## Notes

1 | Using firm-level Indian data, Goldberg et al. [11] discuss a related issue. Rather than considering the effects on exports, they look at the effects on new product development. They show that a lower tariff on imported inputs increases new product development by Indian firms. |

2 | |

3 | There is a vast literature on international trade using oligopoly models. However, that literature did not address the question discussed here. We are not going to review that literature here but refereeing to [12], which shows the role of oligopoly models in international trade and the references therein. |

4 | |

5 | We focus on the pure strategy equilibria. However, there is a mixed strategy equilibrium for ${\frac{\left(a-2\lambda c+c\right)}{9}}^{2}<G<\frac{{\left(a-c\right)}^{2}}{4}$. Since the consideration of the mixed strategy equilibrium will not add much to our purpose, we concentrate on the pure strategy equilibria only. |

6 | This condition holds for $c<\frac{a}{3}$. |

7 | For simplicity, this discussion did not consider the tariff rate in the cost of imported inputs. Otherwise, world welfare will also have a component of tariff revenue. |

8 | If the equilibrium is either (E, NE) or (NE, E), only one firm exports, and it follows from Proposition 1 that a lower c increases the output of that firm. |

9 | If the equilibrium is either (E, NE) or (NE, E), only one firm exports, and it is immediate that a lower c increases the profit of the exporter. |

10 | The demand functions mentioned here can be found from a representative consumer’s utility function $U=\frac{1}{1-{\epsilon}^{2}}\left(a\left(1+\epsilon \right)\left({q}_{1}+{q}_{2}\right)-\frac{{q}_{1}{}^{2}+{q}_{2}{}^{2}+2\epsilon {q}_{1}{q}_{2}}{2}\right)$. Hence, the consumer surplus is $CS=U-{p}_{1}{q}_{1}-{p}_{2}{q}_{2}$, and world welfare is $WW=U-c{q}_{1}-\lambda c{q}_{2}$. |

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**Figure 1.**Export decisions when either $c<\frac{a}{5}$ or $c>\frac{a}{5}$ and $\lambda \frac{5c-a}{4c}$.

**Figure 2.**Changes in the export decisions due to a lower c when either $c<\frac{a}{5}$ or $c>\frac{a}{5}$ and $\lambda \frac{5c-a}{4c}$, and $\lambda <\frac{1}{2}$.

Firm 2 (High-Productive Firm) | |||

E(Export) | NE(No Export) | ||

Firm 1(Low-productive firm) | E(Export) | ${\pi}_{1}^{*},{\pi}_{2}^{*}$ | ${\pi}_{1}^{M},0$ |

NE(No Export) | $0,{\pi}_{2}^{M}$ | 0, 0 |

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**MDPI and ACS Style**

Mukherjee, A.; Liu, Y.
The Connection between Imported Inputs and Exports: The Importance of Strategic Interdependence. *Games* **2023**, *14*, 6.
https://doi.org/10.3390/g14010006

**AMA Style**

Mukherjee A, Liu Y.
The Connection between Imported Inputs and Exports: The Importance of Strategic Interdependence. *Games*. 2023; 14(1):6.
https://doi.org/10.3390/g14010006

**Chicago/Turabian Style**

Mukherjee, Arijit, and Yao Liu.
2023. "The Connection between Imported Inputs and Exports: The Importance of Strategic Interdependence" *Games* 14, no. 1: 6.
https://doi.org/10.3390/g14010006