# Location of Firms and Outsourcing

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## Abstract

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## 1. Introduction

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## 2. The Model

- Case I: Both the suppliers and the firms are within the segment.
- Case II: The suppliers are within the segment, whereas the firms are outside the segment.
- Case III: The firms are within the segment, whereas the suppliers are outside the segment.
- Case IV: Both the suppliers and the firms are outside the segment.

## 3. Equilibrium Outcome with Complete Outsourcing

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**Proposition**

**1.**

- (i)
- firms locate outside (within) the suppliers when $s\ge (\le )-\frac{7}{4}$;
- (ii)
- firms locate within the segment between 0 and 1 when t is low enough, and/or s and τ are high enough;
- (iii)
- the distance between Firms A and B decreases with s, increases (decreases) with t if $s\ge (\le )-\frac{7}{4}$, and increases (decreases) with τ if $s\le (\ge )-\frac{7}{4}$.

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

## 4. Equilibrium Outcome with Bi-Sourcing

**Proposition**

**5.**

- (i)
- the distance between Firms A and B increases with s, decreases (increases) with t if s is high (low), and increases (decreases) with τ if s is high (low).
- (ii)
- the equilibrium prices and profits of the firms and the suppliers increase with t, increase with s, and increase (decrease) with τ if s is high (low).

**Proposition**

**6.**

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## 5. Extensions

#### 5.1. The Suppliers Are Located at the Same Place of the Firms

#### 5.2. The Suppliers Choose Where to Locate

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | |

2 | Interestingly, input procurement by a spatially localized firm has received some attention in a different tradition. Indeed, the literature adopting the Weber triangle (Weber, 1909) [12] focuses on the problem of a monopolistic firm localized in a point aiming to procure inputs (located in other places) in order to produce goods to be sold in a market at a certain distance from the firm’s plant (see for instance Moses, 1958 [13], Sakashita, 1967 [14], Shieh and Mai, 1997 [15], Tan, 2001) [16]. Therefore, there are transportation costs to procure the inputs and transportation costs to sell the goods. However, as clearly pointed by Lai and Tabuchi (2012) [17], “Weber (1909) [12] is more realistic in terms of the fact that manufacturing firms use inputs in producing a final product. However, Hotelling (1929) [14] is more realistic in terms of the fact that there is competition between firms” (p. 1017). In this paper, we incorporate the issue of input procurement in the Hotelling model because we want to analyze its implications for firms competing both in locations and prices. |

3 | For example, asset specificity might be due to irreversible R&D expenditures or sunk marketing expenditures that, by increasing the value of trade between the downstream firm and the input supplier, generate a switching cost that might preclude any outside option for both the firm and the supplier. Joskow (1991) [20] provides an empirical survey that widely documents the existence of firm-specific input suppliers. |

4 | For example, Sony internally produces display panels—which are an input for the final goods (namely, LCD TVs)—but it also procures display panels from professional panel suppliers such as AU Optronics (see Lin et al., 2016) [24]. |

5 | This situation is better suited to describe the relationship between the labor force (“input” supplier) and the downstream firm. By contrast, our framework, where the suppliers might be far apart from the firms, describes those cases where the input supplier is a firm too. |

6 | |

7 | Lai and Tabuchi (2012) [17] also consider input procurement within the context of a traditional location-price Hotelling game with quadratic transportation costs. However, in that paper, the inputs are simply raw materials which are located in space and are freely available to the firms. Therefore, there are no input suppliers which set the prices of the inputs. |

8 | Another somehow related paper is by Liang and Mai (2006) [34]. In a spatial competition model, they consider two firms, and one of them could subcontract the production of inputs to the (more efficient) rival. Therefore, differently from our paper, procurement of input is not from an upstream supplier, but from a rival operating at the same market level. |

9 | There is a study analyzing equilibrium product differentiation in non-spatial models. See Han et al. (2022) [35] and the references therein. However, that study neither shows the effects of the distance between the input suppliers and different types of transportation costs nor considers the effects of bi-sourcing. |

10 | Alternatively, one might assume that each supplier incurs its own transportation costs. The results would be qualitatively the same. |

11 | Following a consolidated tradition in the Hotelling models, we consider both the consumers’ transportation cost, t, and the firms’ transportation costs, τ, as exogenous variables. In a recent paper, Kucera and Kaderabkova (2023) [36] treat the transportation costs as an endogenous variable. However, they do not consider the problem in hand. |

12 | Alternatively, the transportation costs might not depend on the demand. This case refers to a situation in which what matters is the freight cost per se, for instance that of a truck, irrespective of its load. |

13 | Under the alternative specification of the transportation costs, where transportation costs do not depend on the demand, the profit functions would be ${\pi}_{A}=({p}_{A}-{w}_{A})k-\tau {(s-{x}_{A})}^{2}$ and ${\pi}_{B}=({p}_{B}-{w}_{B})(1-k)-\tau {(1-s-{x}_{B})}^{2}$. It can be shown that there is no qualitative difference in the results under the two specifications. Hence, we only focus on the case where the transportation costs depend on the demand. |

14 | It can be observed that the second-order conditions are satisfied at (10). |

15 | When the transportation costs of the firms do not depend on the demand, the equilibrium locations could be found as ${\tilde{x}}_{A}*=1-{\tilde{x}}_{B}*=\frac{36\tau s-7t}{4(9\tau +t)}$. Therefore, the results are qualitatively similar to those emerging from the model discussed in the text. |

16 | We get $\frac{\partial {x}_{A}*}{\partial t}=-\frac{\tau (7+4s)}{4{(t+\tau )}^{2}}$. |

17 | We get $\frac{\partial {x}_{A}*}{\partial \tau}=\frac{t(7+4s)}{4{(t+\tau )}^{2}}$. |

18 | In a model where total demand is inelastic, higher input prices can be easily passed on to the consumers in the form of higher output prices, implying that the downstream firms benefit from higher input prices in a symmetric equilibrium. |

19 | Due to symmetry, we omit the subscripts when not necessary. |

20 | Due to symmetry, we consider just the left-hand side of the market. |

21 | The transportation costs of the consumers are minimized when ${x}_{A}*=1/4$. |

22 | As for Section 3, the results in this section would be similar by assuming that the transportation costs of the firms do not depend on the demand. Details are available upon request. |

23 | The equations for the firms’ equilibrium locations are available upon request. |

24 | Only the impact of t is the same (and positive), as usual in the Hotelling models. |

25 | Due to the complexity of the equations involved, the proof of Proposition 6 has been performed by using Mathematica software (Mathematica, Version 5) and it is available on request. The upper (lower) graphs in Figure 3 are drawn for t =1 (τ = 1/10). |

26 | Note that the complete in-house production result depends on our assumption of zero production costs of the downstream firms. However, it can be shown that introducing positive production costs of the firms does not affect the linearity of firms’ profits with respect to the in-house quantity level. In particular, if the production costs are sufficiently low, complete in-house production emerges in equilibrium, whereas if they are sufficiently high, complete outsourcing occurs. In any case, bi-sourcing never arises. |

27 | |

28 | We do not explicitly consider consumer surplus and welfare under bi-sourcing. However, because the impacts of s and τ on the firms’ distance, prices and profits, are opposite to those under complete outsourcing, it is intuitive that their impacts on consumer surplus and welfare are also opposite. When considering t, we know that the impact of t on the firms’ distance is opposite in the two frameworks. Therefore, the impact on welfare (which depends only on the firms’ distance) is likely to be opposite too. However, no clear prediction can be made for the impact of t on consumer surplus, which depends on both firms’ distance and the equilibrium prices and profits. |

29 | For example, one might imagine that the firms and suppliers are vertically integrated. Firm J is the headquarters of the vertically-integrated firm, whereas supplier J is the internal division producing the inputs and charges a transfer price to the headquarters. |

30 | Indeed, without bi-sourcing, the equilibrium profits of the firms are $9t/4$. |

31 | The results would be the same if the suppliers and firms choose sequentially, with the suppliers choosing first and the firms choosing second. |

32 | The details are available upon request. |

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Colombo, S.; Mukherjee, A.
Location of Firms and Outsourcing. *Games* **2023**, *14*, 70.
https://doi.org/10.3390/g14060070

**AMA Style**

Colombo S, Mukherjee A.
Location of Firms and Outsourcing. *Games*. 2023; 14(6):70.
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**Chicago/Turabian Style**

Colombo, Stefano, and Arijit Mukherjee.
2023. "Location of Firms and Outsourcing" *Games* 14, no. 6: 70.
https://doi.org/10.3390/g14060070