Abstract
We analyze equilibrium locations of downstream retailers assuming transport cost from a monopoly input supplier. When the upstream transport costs equal those of retailers, a downstream monopoly may locate efficiently and two downstream firms never locate inefficiently. Even with discriminatory pricing upstream, two downstream firms locate efficiently. When assuming downstream transport costs are greater than upstream costs, routinely inefficient locations reemerge in keeping with previous literature.
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Notes
Thus, Courey (2016) shows that this hallmark inefficiency remains when assuming convex production costs downstream of the sort imagined by Gupta (1994). Moreover, it cannot be eliminated by the upstream firm assigning exclusive territories (Fjell and Heywood 2005) and it serves as an incentive for vertical integration (Scholer 2001). The inefficient locations will also not be eliminated by introducing a public or partially public downstream firm (Heywood et al. 2020). In addition, transport cost itself can be similarly set inefficiently high to force the same reduction of the input price (Gupta et al. 1995, 1999) and analogous results have been shown in the context of governmental taxation (Heywood and Pal 1996).
The upstream firm cannot increase profit by increasing \({p}_{0}\) and so cutting the market when an infinitely small increase \(\varepsilon\), causes net profit gain, \(\varepsilon \left(1-\frac{\varepsilon }{t}\right)-{p}_{0}\left(\frac{\varepsilon }{t}\right)\), to fall. Specifically, \(\varepsilon /t\) is the market lost from increasing \({p}_{0}\) by \(\varepsilon\) and \((1-\varepsilon /t)\) is the remaining market with the increased price. Substituting in (3) and (4) and solving for \(r\) shows that the upstream does not cut the market if \(r>-tU+2t\) if \({L}_{1}>U\) and \(r>tU-2t{L}_{1}+2t\) if \({L}_{1}<U\). With the equilibrium location \({L}_{1}=0\), this condition becomes \(r>tU+2t\).
The total transport cost is \(TC=\frac{1}{2}t{L}_{1}^{2}+\frac{1}{2}t{(1-{L}_{1})}^{2}+t(U-{L}_{1})\), the first two terms are the transport cost of serving the customers located to the left and to the right of \({L}_{1}\) and the third term is the transport cost from the upstream monopolist. If \({L}_{1}=U\), \(TC=t{U}^{2}+\frac{t}{2}-tU\). If \({L}_{1}=0\), \(TC=tU+\frac{t}{2}\) so the surplus loss is \(tU(2-U)\).
If the indifferent consumer buys from firm 1, then \({p}_{d1}\left(x\right)=\) \({p}_{0}+t({L}_{2}-U)+t{(L}_{2}-x)\). If the indifferent consumer buys from firm 2, then \({p}_{d2}(x)=\) \({p}_{0}+t(U-{L}_{1})+t{(x-L}_{1})\). Setting these two prices equal to each other and solving for \(x\) yields the location of the indifferent consumer.
We say that firm \(i\in \mathrm{1,2}\) is a critical firm on its left (right) if \({p}_{0}+t{L}_{i}=r\) (\({p}_{0}+t(1-{L}_{i})=r\)).
These are derived by solving \(r={p}_{0}+t\left({L}_{2}-U\right)+t\left({L}_{2}-a\right)\), \(r={p}_{0}+t\left(U-{L}_{1}\right)+t\left(b-{L}_{1}\right)\) and \({p}_{0}+t({L}_{2}-U)+t{(L}_{2}-x)={p}_{0}+t(U-{L}_{1})+t{(x-L}_{1})\).
The upstream firm cannot increase profit by increasing \({p}_{0}\) and so cutting the market when an infinitely small increase \(\varepsilon\), causes the net profit gain, \(\varepsilon \left(1-\frac{\varepsilon }{t}\right)-{p}_{0}\left(\frac{\varepsilon }{t}\right)\), to fall. Specifically, \(\varepsilon /t\) is the market lost from increasing \({p}_{0}\) by \(\varepsilon\) and \((1-\varepsilon /t)\) is the remaining market with the increased price. Substituting into (8) and solving for \(r\) shows that the upstream does not cut the market if \(r>2t-tU\).
Alternatively, and analytically identical, it could simply discriminate by charging the two firms different prices at the mill.
We note that if for some reason it was the case that t1>t2, the results of the earlier section continue to hold with colocation.
The direction of movement reverses when \(\alpha\) and U are both small. Specifically, \(\frac{\partial {L}_{2}^{SW}}{\partial \alpha }>0\) if \(0<\alpha <0.414\) and \(\frac{{(1+\alpha )}^{2}}{4}<U<\frac{-1-8\alpha +4{\alpha }^{2}+4{\alpha }^{3}+{\alpha }^{4}}{-4-8\alpha +4{\alpha }^{2}}\) or \(0.414<\alpha <0.632\) and \(\frac{1-\alpha +3{\alpha }^{2}+{\alpha }^{3}}{4\alpha +4{\alpha }^{2}}<U<\frac{-1-8\alpha +4{\alpha }^{2}+4{\alpha }^{3}+{\alpha }^{4}}{-4-8\alpha +4{\alpha }^{2}}\). In these cases, as \(\alpha\) increases, firm 2 needs to move to the right to decrease the asymmetry in location. This movement saves on transport cost generated from downstream delivery.
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Acknowledgements
The authors thank seminar participants at Hainan University School of Economics. Zheng Wang’s research was supported by the National Natural Science Foundation of China (NSFC) [Grant Numbers: 71803137 and 71733001].
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Heywood, J.S., Wang, Z. An upstream monopoly with transport costs. J Econ 139, 159–176 (2023). https://doi.org/10.1007/s00712-023-00820-3
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DOI: https://doi.org/10.1007/s00712-023-00820-3