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Market prices, spatial distribution of consumers and firms’ optimal locations in a linear city

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Abstract

We study a game of spatial competition in prices. In particular, we focus on the linear-city duopoly model to see what we can learn about the distribution of consumers, which is not required to be uniform—as in the original Hotelling model. Using variation in firms’ prices and costs, we identify points of the distribution of consumers. Based on these points, we estimate the spatial distribution of consumers along the linear city. We apply our methodology to a dataset of prices of two gas stations on a straight highway. By estimating the distribution of consumers, we are able to find the optimal location of an entrant gas station. Using our estimated distribution of consumers and the entrant’s optimal point, we simulate welfare gains under counterfactual locations of an entrant.

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Notes

  1. To make our strategy more realistic, as Bajari and Benkard (2005) do in their seminal paper, we assume that firms choose the “right” (equilibrium) prices each period with some (zero-mean) measurement error.

  2. As we explain later, the estimated distribution of consumers along the highway does not reflect a static distribution of consumers’ residence locations or addresses. It reveals rather the pattern of traffic distribution in different sections of the highway. That is, the peak of the distribution reflects the section of the highway which is usually congested. We carefully selected a straight section of a highway with entries and exits. Entries and exits along a highway generate different traffic flows and congestion zones in different sections of the highway.

  3. The fact that gasoline is a homogeneous good is challenged by Lewis (2008). However, his argument relies on the price differences that arise between “high-brand” and “low-brand” retailers in highly dense urban areas considering the number and type of the surrounding competitors.

  4. We acknowledge that the relatively simple and well-defined structure of the linear-city model (which allows us to develop our empirical strategy) comes at the cost of imposing some of these “simple” assumptions to the empirical applications that we may have in mind (e.g., constant transportation cost across all consumers, unit-demand, etc.). As we discuss below, some of these “simple” assumptions that are present in the theoretical model can also be challenged in the particular empirical application that we propose in this paper (see Sect. 4).

  5. Similar requisites are required in many other countries, such as Australia (see the technical note on the “Investigation of Service Station Sites,” that describes industry best practice in the assessment of service station sites in NSW) and Italy (Decreto legislativo 11 febbraio 1998, n. 32).

  6. This is a valid assumption in the context of our empirical application: wholesale gasoline prices (typically formed in supranational markets, such as the Mediterranean market or the North-Western Europe market; see Sect. 4 for further details) impact gasoline retail prices, but not the other way around. That is, the retail gasoline prices of a pair of (vertically integrated) gas stations at t are not likely to impact supranational wholesale gasoline prices at \(t+1\) (unless some kind of sophisticated collusive agreement at the international level is in play).

  7. This is also a valid assumption in the context of our empirical application, since retail gasoline prices change every day and they are posted electronically (the cost of changing them is virtually zero).

  8. For the sake of simplicity, we normalize it to [0, 1] in the upcoming sections. However, we keep the general form throughout this section.

  9. See Caplin and Nalebuff (1991) for a more in-depth discussion on this Theorem.

  10. In our empirical application, \(\theta \) is the (common) fuel wholesale price and \(\epsilon \) captures gas station-specific marginal costs (labor efficiency, management issues, etc.)—see Thomadsen (2005).

  11. As discussed above, there are no “dynamics” from one period to the next one in the model that justify the use of more sophisticated (time-dependent) solution concepts.

  12. For such a baseline case, we assume a symmetric beta distribution, with shape parameters \(\alpha =\beta =2\), firms’ symmetric locations—in particular \(x_1=\frac{1}{3}\) and \(x_2=\frac{2}{3}\)—and transportation cost \(\tau =25\).

  13. A different question concerns what the observable and the nonobservable variables are. We deal with this issue in the following section.

  14. Shuai (2017) explains that, under perfect inelastic demand (which is the case in the linear-city model), “maximizing welfare is equivalent to minimizing consumers transportation cost.”

  15. If we think in our empirical application, the goal of the social planner (the General Direction of Highways) is to authorize the construction of a new gas station in such a way that the total probability of vehicles running out of gas is minimized.

  16. In this case, if \(f(\cdot )\) is strictly positive in all its support, the optimal location of the entrant is at \(x^*=0\).

  17. In this case, if \(f(\cdot )\) is strictly positive in all its support, the optimal location of the entrant is at \(x^*=1\).

  18. Formally, tie-breaking rules are as follows:

    1. (d)

      (Tie-breaking rule #1) If \(T_1= T_{\text {mid}} > T_2\), the optimal location of the entrant is determined by the following rule: \(x^*={{\,\mathrm{\mathrm{arg\,min}}\,}}_{x\in [\underline{x},\overline{x}]} f(x)\) such that \(f(x^*)>0\) with probability \(\frac{1}{2}\) and \(x^*=\hat{x}\) with probability \(\frac{1}{2}\).

    2. (e)

      (Tie-breaking rule #2) If \(T_{\text {mid}} = T_2 > T_1\), the optimal location of the entrant is determined by the following rule: \(x^*=\hat{x}\) with probability \(\frac{1}{2}\) and \(x^*={{\,\mathrm{\mathrm{arg\,max}}\,}}_{x\in [\underline{x},\overline{x}]} f(x)\) such that \(f(x^*)>0\) with probability \(\frac{1}{2}\).

    3. (f)

      (Tie-breaking rule #3) If \(T_2 = T_1>T_{\text {mid}}\), the optimal location of the entrant is determined by the following rule: \(x^*={{\,\mathrm{\mathrm{arg\,max}}\,}}_{x\in [\underline{x},\overline{x}]} f(x)\) such that \(f(x^*)>0\) with probability \(\frac{1}{2}\) and \(x^*={{\,\mathrm{\mathrm{arg\,max}}\,}}f(x)\) such that \(f(x)>0\) with probability \(\frac{1}{2}\).

  19. More realistic is to assume that entry and exit exist in the market instead, but we leave this issue aside. Estimation of entry and exit decisions in a (general) oligopolistic market is by Bajari et al. (2007).

  20. Some authors have previously analyzed Hotelling’s linear-city model assuming a more general class of distribution of consumers. See Neven (1986), Tabuchi and Thisse (1995), Anderson et al. (1997), Torrisi (2011), Shuai (2017).

  21. In principle, if \(\tau \) is identical for all consumers (as the theoretical model assumes), one could argue that it should be easily calculated (and, therefore, observable). For this reason, we develop the empirical strategy for the case in which the transportation cost is known in Sect. 3.1. However, one could also argue that the calculation of \(\tau \) is not entirely obvious (even for the case of identical consumers). That is why we extend our empirical strategy for the case in which the transportation cost is unknown in Sect. 3.2.

  22. As we have detailed extensively above, in our empirical application we cannot assume that individual gas stations’ posted prices have an impact on wholesale prices and Brent crude oil prices: no gas station has enough power to determine the evolution of the price of fuel commodities. However, this assumption may not be realistic if there we study dominant-market firms (which may influence wholesale prices).

  23. These measurement errors could arise due to modeling error, random idiosyncratic facts and/or firm optimization error.

  24. Previous authors have proposed different methodologies to estimate transportation cost parameters—see, for instance, Englin and Shonkwiler (1995). However, we need to think that \(\tau \) (the transportation cost) acts as an “equilibrium parameter” that helps us to fit the model equilibrium into the observed data.

  25. This happens when both shape parameters, \(\alpha \) and \(\beta \), are equal to 1.

  26. If \(\tau \) is the true transportation cost parameter, then \(x^t=\lambda ^t(\tau )\).

  27. In particular, using the conditions above, the set is bounded below by \(\max \left\{ \min \left\{ 0, \frac{p_{1}^{t,*}-p_{2}^{t,*}}{(x_2-x_1)^2}\right\} , \min \left\{ 0, \frac{p_{2}^{t,*}-p_{1}^{t,*}}{(x_2-x_1)(2-x_2-x_1)}\right\} \right\} \) and the set can be bounded above (if necessary) considering that the sum of the (unique) \(\nu ^t\)’s must be less than or equal to 1—which is a requirement of the distribution itself–, i.e., \(\tau \le \frac{\sum _t p_{1}^{t,*}-c(\cdot )+p_{2}^{t,*}-c(\cdot )}{2(x_2-x_1)}\) for every unique pair of equilibrium prices and costs.

  28. For further details on diesel cars versus gasoline cars, see Verboven (2002).

  29. Hybrid cars, electric cars and cars powered with alternative fuels are still residuals in Spain. According to the European Automobile Manufacturers’ Association, just 2% among new passenger cars in 2015 were hybrid or electric, and less than 2.2% of the cars among the passenger car fleet in Spain in 2014 were powered by alternative fuels.

  30. The data are available at http://geoportalgasolineras.es/#/Inicio [last access: 2020/05/08 14:05:29].

  31. Data are not available for October 12, 2014, which is the Spanish National day.

  32. Marion and Muehlegger (2011) show that gasoline taxes are indeed fully passed onto consumers and are incorporated fully into the tax-inclusive price, under typical supply and demand conditions.

  33. Cepsa, the owner of the gas stations considered in this study, is a partner of this organization.

  34. This formula is used by the Spanish Association of Operators of Oil Products (AOP) to calculate the wholesale cost faced by gas stations in Spain. The formula is included in the annual AOP reports—see, for instance, Asociación Española de Operadores de Productos Petrolíferos (2015). Likewise, the Spanish National Competition Commission (CNC) uses the same formula to calculate gas stations’ markups. See, for instance, the 2015 special report on the Spanish retail petroleum products market—Comisión Nacional de los Mercados y la Competencia (2015). A similar weighted average of the relative importance of Genoa and Rotterdam prices (66.1% the former, 33.9% the latter) was employed by Rodrigues (2009) to study asymmetries in the adjustment of pump prices for the Spanish case.

  35. In Appendix C, we include a sensitivity analysis of our main estimations for different distribution cost parameters.

  36. Additional congestion maps are included in Appendix D. See Dirección General de Carreteras (2015) for further information and maps.

  37. We call a point between gas station #1 and gas station #2 a midpoint location.

  38. We have considered the following as “locations based on geographical distances” for an entrant firm: first, the geographical midpoint between town A and gas station #1; second, the geographical midpoint between gas station #1 and gas station #2; and third, the geographical midpoint between gas station #2 and town B.

  39. The results included in this table are preliminary.

  40. It seems also reasonable to consider professional truck drivers as a group of consumers that have a substantially different transportation cost in comparison with “regular” (nonprofessional) drivers. However, gas stations usually have alternative (lower) prices for professional truck drivers (typically settled through bilateral agreements between gas stations and freight companies). Unfortunately, these (lower) prices that gas stations charge to truck drivers are not observable for the econometrician. Therefore, it is not possible for us to perform a separate estimation for this well-differentiated segment of the market.

  41. For instance, following Bjornerstedt and Verboven (2016), one possible extension could be to consider a constant expenditures demand specification. Obviously, this alternative specification is also potentially subject to a similar critique (as they assume that the fraction of the budget spent is constant across all consumers), and it would also require the econometrician to observe individual consumers’ income data.

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Authors and Affiliations

  1. Department of Economics, Edificio Amigos, University of Navarra, 31009, Pamplona (Navarra), Spain

    Raúl Bajo-Buenestado

  2. Baker Institute for Public Policy, Rice University, Houston, TX, USA

    Raúl Bajo-Buenestado

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  1. Raúl Bajo-Buenestado
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Correspondence to Raúl Bajo-Buenestado.

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I acknowledge helpful comments and suggestions by Marcus Berliant, Dodge Cahan, Gilles Duranton, Hülya Eraslan, Nicholas Frazier, Peter Hartley, Jeffrey Lin, Kenneth Medlock, Shaun McRae, Xun Tang, and two anonymous referees, and seminar participants at the X EGSC Conference at Washington University in St. Louis and at the XXX Spanish Economic Association meeting. Thanks also to María García, who kindly helped me with the data. I gratefully acknowledge financial support from the Social Sciences Research Institute at Rice University and Fundación Ramón Areces (Grant Number CISP16A4779). The remaining errors are solely mine. The dataset and the MATLAB and R codes are available upon request from the author.

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Bajo-Buenestado, R. Market prices, spatial distribution of consumers and firms’ optimal locations in a linear city. Empir Econ 61, 443–467 (2021). https://doi.org/10.1007/s00181-020-01871-x

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