# Dynamic Vertical Foreclosure with Learning-by-Doing Production Technologies

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Static Model

**Lemma 1.**

**Lemma 2.**

**Proposition 1.**

## 3. The Dynamic Model

#### 3.1. Period Two

#### 3.2. Period One

**Lemma 3.**

**Lemma 4.**

**Proposition 2.**

**Remark 1.**

**Remark 2.**

## 4. Consumer Surplus and Total Welfare

**Remark 3.**

**Proposition 3.**

**Proposition 4.**

## 5. The Social Planner’s Solution

#### 5.1. Static Social Optimum

**Lemma 5.**

#### 5.2. Dynamic Social Optimum

**Lemma 6.**

**Lemma 7.**

**Proposition 5.**

## 6. The Role of the Intermediary

**Remark 4.**

**Remark 5.**

## 7. Conclusions and Further Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Notes

1 | Exclusivity contracts or refusal to deal are often viewed with suspicion by the Antitrust Authorities. Note that exclusion may arise in a market through vertical integration too. |

2 | Mathewson and Winter (1987) [5] argue that manufacturers under exclusive dealing arrangements compete on the basis of wholesale prices for the right to be selected by the retailer. Besanko and Perry (1993) [6], in a differentiated products oligopoly, support that exclusive dealing can eliminate inter-brand externalities due to increased promotional investments. O’Brien and Shaffer (1993) [7] examine corner solutions in an oligopolistic vertical setting. |

3 | |

4 | If the bargaining power was more evenly distributed across the upstream and downstream firms, additional tensions would arise, partly reflecting results from earlier research on downstream ‘bottleneck’ models. A short discussion on such further research is delegated to the Conclusion. |

5 | The representative consumer is characterized by the quadratic and strictly concave utility function $U({q}_{A},{q}_{B})=a({q}_{A}+{q}_{B})-\frac{{q}_{A}^{2}+{q}_{B}^{2}+2b{q}_{A}{q}_{B}}{2}$ (see Singh and Vives (1984)) [28]. |

6 | An alternative way to foreclosure an input supplier is the retailer to explicitly sign an exclusive dealing contract with the rival manufacturer. |

7 | We assume that if the manufacturer is indifferent between obtaining zero profits from production and not producing at all, they will choose to produce. |

8 | A more detailed discussion about the consumers’ surplus and the total welfare is deferred for Section 4. |

9 | The analysis here contains the subcase (i) non-exclusivity in the first period, i.e., no subsequent cost asymmetry, and non-exclusivity in the second period, and (ii) exclusivity in the first period, i.e., subsequent cost asymmetry, and non-exclusivity in the second period. Nevertheless, a corner solution may arise in the latter case. |

10 | The analysis here contains the subcase (i) non-exclusivity in the first period and exclusivity in the second period, and (ii) exclusivity in the first period and exclusivity in the second period. Production cost ${c}_{A2}$ is a function of ${q}_{A1}$. |

11 | Since $\frac{a(2a-{c}_{A2}-{c}_{B2})}{4(1+b)}+\frac{(a-{c}_{A2})(b{c}_{B2}-{c}_{A2})+(a-{c}_{B2})(b{c}_{A2}-{c}_{B2})}{4(1-{b}^{2})}\ge \frac{{\left(a-{c}_{A2}\right)}^{2}}{4}.$ |

12 | The subcase non-exclusivity in period one, i.e., subsequent cost symmetry, and exclusivity in period two, cannot be an equilibrium outcome. |

13 | We obtain positive quantities and the second-order conditions are satisfied when $4{(1+b)}^{2}-{\lambda}^{2}\delta >0,4{\left(1-b\right)}^{2}-{\lambda}^{2}\delta 0,4\left(1-{b}^{2}\right)-{\lambda}^{2}\delta 0$. |

14 | All relevant expressions obtained under non-exclusivity are in the Appendix A. This subcase corresponds to non-exclusivity in both periods. |

15 | This subcase corresponds to exclusivity in period one and non-exclusivity in period two. |

16 | This subcase corresponds to exclusivity in both periods. |

17 | We need $4\left(1-{b}^{2}\right)-\delta {\lambda}^{2}>0$ for the second-order conditions to be satisfied and to obtain positive quantity in period one. Otherwise, no production occurs. All remaining expressions for this Lemma are in the Appendix A. |

18 | (N)E stands for (non-)exclusivity. We have already proved that NE in period one and E in period two, cannot be an equilibrium outcome. |

19 | We have $1>\overline{b}=\frac{\sqrt{{\lambda}^{4}{\delta}^{4}-2{\lambda}^{2}{\delta}^{3}\left({\lambda}^{2}+2\right)+{\delta}^{2}\left(\lambda \left(24\lambda +{\lambda}^{3}+32\right)+16\right)+\delta \left(4\left(\lambda \left(8-\lambda \right)+8\right)\right)+16}-\lambda \delta \left(\lambda \left(1+\delta \right)+4\right)}{4\left(\delta +\lambda \delta +1\right)}\ge \frac{4-{\lambda}^{2}\delta}{2\left(\lambda +2\right)}>0$ |

20 | For $b\le \frac{4-\delta {\lambda}^{2}}{2\left(\lambda +2\right)}$, we obtain $\Delta \Pi =\left({\Pi}_{R1}^{E}+\delta {\Pi}_{R2}^{E}\right)-\left({\Pi}_{R1}^{NE}+\delta {\Pi}_{R2}^{NE}\right)<0$. |

21 | For $b\in (\frac{4-{\lambda}^{2}\delta}{2\left(\lambda +2\right)},\overline{b})$ we obtain $\Delta \Pi <0$ while for $b\in (\overline{b},1)$ we obtain $\Delta \Pi >0$. Therefore, it is never an equilibrium outcome to purchase product A in the first period and both products in the second period. |

22 | See Singh and Vives, 1984 [28]. |

23 | We prove that $\frac{d\left(C{S}_{t}^{E}-C{S}_{t}^{NE}\right)}{db}={q}_{it}^{NE}\left(-{q}_{it}^{NE}-2\left(1+b\right)\left(\frac{d{q}_{it}^{NE}}{db}\right)\right)>0$, with $\frac{d{q}_{it}^{NE}}{db}<0,\phantom{\rule{3.33333pt}{0ex}}t=1,2$, that is, $\Delta CS$ is increasing in b. Furthermore, we obtain that $\Delta CS\left(b=0\right)<0$ and $\Delta CS\left(b=1\right)>0$. |

24 | To obtain positive quantities and to satisfy the second-order conditions, we should have ${\left(1+b\right)}^{2}-\delta {\lambda}^{2}\ge 0,\phantom{\rule{3.33333pt}{0ex}}{\left(1-b\right)}^{2}-\delta {\lambda}^{2}>0$ and $\left(1-{b}^{2}\right)-\delta {\lambda}^{2}>0$. All relevant expressions obtained in this case are in the Appendix A. |

25 | We need $\left(1-{b}^{2}\right)-\delta {\lambda}^{2}>0$ for the second-order conditions to be satisfied and to obtain positive quantity in period one. All relevant expressions obtained in this case are in the Appendix A. |

26 | For the second period, we have ${q}_{i2}^{C}=\frac{a-c}{2+b}+\frac{\lambda (a-c)\left((2-b){(2+b)}^{2}+4\delta \lambda \right)}{(2+b)\left((2-b){(2+b)}^{3}-4\delta {\lambda}^{2}\right)}$. |

27 | With the presence of the retailer, we also obtain ${q}_{B1}^{*}=\left\{\begin{array}{ccc}\frac{(a-c)\left(2\right(1+b)+\delta \lambda )}{4{(1+b)}^{2}-\delta {\lambda}^{2}}& if& b\in \left(-1,\overline{b}\right)\\ -& if& b\in \left(\overline{b},1\right)\end{array}\right.$. |

## References

- Brenkers, R.; Verboven, F. Liberalizing a Distribution System the European Car Market. J. Eur. Econ. Assoc.
**2005**, 4, 216–251. [Google Scholar] [CrossRef] - Rey, P.; Tirole, J. A Primer on Foreclosure. In Handbook of Industrial Organization III; Armstrong, M., Porter, R.H., Eds.; North Holland: Amsterdam, The Netherlands, 2023; Available online: https://econpapers.repec.org/bookchap/eeeindchp/3-33.htm (accessed on 7 February 2024).
- Krattenmaker, T.; Salop, S. Competition and cooperation in the market for exclusionary rights. Am. Econ. Rev.
**1986**, 76, 109–113. [Google Scholar] - Aghion, P.; Bolton, P. Contracts as a barrier to entry. Am. Econ. Rev.
**1987**, 77, 388–401. [Google Scholar] - Mathewson, F.; Winter, R. The competitive effects of vertical agreements: Comment. Am. Econ. Rev.
**1987**, 77, 1057–1062. [Google Scholar] - Besanko, D.; Perry, M.K. Equilibrium incentives for exclusive dealing in a differentiated products oligopoly. Rand J. Econ.
**1993**, 24, 646–667. [Google Scholar] [CrossRef] - O’Brien, D.P.; Shaffer, G. On the dampening-of-competition effect of exclusive dealing. J. Ind. Econ.
**1993**, 41, 215–221. [Google Scholar] [CrossRef] - McAfee, P.; Schwartz, M. Opportunism in multilateral vertical contracting: Nondiscrimination, exclusivity, and uniformity. Am. Econ. Rev.
**1994**, 84, 210–230. [Google Scholar] [CrossRef] - Rey, P.; Stiglitz, J. The role of exclusive territories in producers’ competition. RAND J. Econ.
**1995**, 26, 431–451. [Google Scholar] [CrossRef] - Reisinger, M.; Tarantino, E. Vertical integration, foreclosure, and productive efficiency. RAND J. Econ.
**2015**, 46, 461–479. [Google Scholar] [CrossRef] - Kourandi, F.; Pinopoulos, I. Vertical Contracting between a Vertically Integrated Firm and a Downstream Rival. Econ. Theory
**2023**. [Google Scholar] [CrossRef] - Fumagalli, C.; Motta, M. Dynamic Vertical Foreclosure. J. Law Econ.
**2020**, 63, 763–812. [Google Scholar] [CrossRef] - Sandiumenge, B. Vertical Foreclosure: A Dynamic Perspective. 2023. Available online: https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=JEI2023&paper_id=80 (accessed on 7 February 2024).
- Martimort, D. Exclusive dealing, common agency, and multiprincipals incentive theory. Rand J. Econ.
**1996**, 27, 1–31. [Google Scholar] [CrossRef] - Bernheim, D.B.; Whinston, M.D. Exclusive dealing. J. Political Econ.
**1998**, 106, 64–103. [Google Scholar] [CrossRef] - Segal, I.R.; Whinston, M.D. Exclusive contracts and protection of investments. Rand J. Econ.
**2000**, 31, 603–633. [Google Scholar] [CrossRef] - Marx, L.; Shaffer, G. Upfront payments and exclusion in downstream markets. Rand J. Econ.
**2007**, 38, 823–843. [Google Scholar] [CrossRef] - Chen, Y. Vertical disintegration. J. Econ. Manag. Strategy
**2005**, 14, 209–229. [Google Scholar] [CrossRef] - Yong, J.S. Exclusionary vertical contracts and product market competition. J. Bus.
**1999**, 72, 385–406. [Google Scholar] [CrossRef] - Taylor, T.A.; Plambeck, E.L. Supply chain relationships and contracts: The impact of repeated interaction on capacity investment and procurement. Manag. Sci.
**2007**, 53, 1577–1593. [Google Scholar] [CrossRef] - Spence, M. The learning curve and competition. Bell J. Econ.
**1981**, 12, 49–70. [Google Scholar] [CrossRef] - Fudenberg, D.; Tirole, J. Learning-by-doing and market performance. Bell J. Econ.
**1983**, 14, 522–530. [Google Scholar] [CrossRef] - Cabral, M.B.; Riordan, M.H. The learning curve, market dominance, and predatory pricing. Econometrica
**1994**, 62, 1115–1140. [Google Scholar] [CrossRef] - Besanko, D.; Doraszelski, U.; Kryukov, Y. How Efficient Is Dynamic Competition? The Case of Price as Investment. Am. Econ. Rev.
**2019**, 109, 3339–3364. [Google Scholar] [CrossRef] - Lewis, T.; Yildirim, H. Managing dynamic competition. Am. Econ. Rev.
**2002**, 92, 779–797. [Google Scholar] [CrossRef] - Lewis, T.; Yildirim, H. Learning by doing and dynamic regulation. Rand J. Econ.
**2002**, 33, 22–36. [Google Scholar] [CrossRef] - Andrew, S.; Jia, D.; Hui, S.; Yao, X. Dynamic Price Competition, Learning-by-Doing, and Strategic Buyers. Am. Econ. Rev.
**2022**, 112, 1311–1333. [Google Scholar] - Singh, N.; Vives, X. Price and quantity competition in a differentiated duopoly. Rand J. Econ.
**1984**, 15, 546–554. [Google Scholar] [CrossRef]

**Figure 2.**$\Delta \Pi =\left({\Pi}_{R1}^{E}+\delta {\Pi}_{R2}^{E}\right)-\left({\Pi}_{R1}^{NE}+\delta {\Pi}_{R2}^{NE}\right)$, for $\lambda =1$.

b | ${\Pi}_{R1}^{E}+\delta {\Pi}_{R2}^{E}$ | ${\Pi}_{R1}^{NE}+\delta {\Pi}_{R2}^{NE}$ | |

$b\le \frac{4-\delta {\lambda}^{2}}{2\left(\lambda +2\right)}$ | $\begin{array}{c}\frac{{\left(a-c\right)}^{2}\left(8\delta -8b\delta -{\lambda}^{2}{\delta}^{2}+4\lambda \delta -4{b}^{2}-4b\lambda \delta +4\right)}{4\left(4\left(1-{b}^{2}\right)-\delta {\lambda}^{2}\right)}\\ \mathrm{E}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{period}\phantom{\rule{4.pt}{0ex}}\mathrm{one}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{NE}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{two}\end{array}$ | vs. | $\begin{array}{c}\frac{2{\left(a-c\right)}^{2}\left(\left(\delta +1\right)\left(b+1\right)+\lambda \delta \right)}{8b-{\lambda}^{2}\delta +4{b}^{2}+4}\\ \mathrm{NE}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{both}\phantom{\rule{4.pt}{0ex}}\mathrm{periods}\phantom{\rule{4.pt}{0ex}}\end{array}$ |

$b>\frac{4-\delta {\lambda}^{2}}{2\left(\lambda +2\right)}$ | $\begin{array}{c}\frac{\left(\delta +\lambda \delta +1\right){\left(a-c\right)}^{2}}{4-\delta {\lambda}^{2}}\\ \mathrm{E}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{both}\phantom{\rule{4.pt}{0ex}}\mathrm{periods}\end{array}$ | vs. | $\begin{array}{c}\frac{2{\left(a-c\right)}^{2}\left(\left(\delta +1\right)\left(b+1\right)+\lambda \delta \right)}{8b-{\lambda}^{2}\delta +4{b}^{2}+4}\\ \mathrm{NE}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{both}\phantom{\rule{4.pt}{0ex}}\mathrm{periods}\phantom{\rule{4.pt}{0ex}}\end{array}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kourandi, F.; Vettas, N.
Dynamic Vertical Foreclosure with Learning-by-Doing Production Technologies. *Games* **2024**, *15*, 9.
https://doi.org/10.3390/g15020009

**AMA Style**

Kourandi F, Vettas N.
Dynamic Vertical Foreclosure with Learning-by-Doing Production Technologies. *Games*. 2024; 15(2):9.
https://doi.org/10.3390/g15020009

**Chicago/Turabian Style**

Kourandi, Frago, and Nikolaos Vettas.
2024. "Dynamic Vertical Foreclosure with Learning-by-Doing Production Technologies" *Games* 15, no. 2: 9.
https://doi.org/10.3390/g15020009