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An Oligopoly Model for Analyzing and Evaluating (Re)-Assignments of Spectrum Licenses

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Abstract

The Communications Act requires the Federal Communications Commission to assess whether proposed spectrum license transactions serve the public interest, convenience, and necessity. We review the FCC’s implementation of this component of the Act. We provide a tractable economic model of competition among wireless service providers in which spectrum licenses are a cost-reducing input. This model allows us to evaluate the effects of (re-)assigning spectrum licenses on economic outcomes and to define operational measures of “warehousing” licenses. Calibrating the model, we find little evidence of warehousing and that the approved Verizon-T-Mobile-SpectrumCo transaction increased social surplus.

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Notes

  1. Memorandum Opinion and Order and Declaratory Ruling in the Matter of Applications of Cellco Partnership d/b/a Verizon Wireless and SpectrumCo LLC et al., WT Docket Nos. 12-4, 12-175, Adopted August 21, 2012 (hereafter Verizon-T-Mobile-SpectrumCo MO&O), available at http://hraunfoss.fcc.gov/edocs_public/attachmatch/FCC-12-95A1.

  2. See, e.g., Verizon-TMobile-SpectrumCo MO&O, para. 76.

  3. See, e.g., Verizon-TMobile-SpectrumCo MO&O, paras. 68–72.

  4. More precisely, we consider a theoretical environment in which the market supply of capacity is perfectly price elastic and perfectly competitive. This means that firms can purchase as much spectrum as they want at a fixed market price for spectrum.

  5. In the model, the price for capacity is taken as given by the firms. To derive an implied price for capacity, we calculate the price that minimizes the sum of squared deviations between equilibrium capacities and empirically observed capacities. The dollar prices per MHz*Pop are, respectively, $0.45 and $0.53 for the model and for Auction 66.

  6. Available at http://transition.fcc.gov/Reports/1934new.

  7. Order in the Matter of Application of AT&T Inc. and Qualcomm Incorporated, WT Docket No. 11-18, Adopted December 22, 2011 (hereafter AT&T-Qualcomm Order).

  8. In our model formulation, capacity enters through the firms’ cost functions; however, one might consider an alternative formulation in which a firm’s capacity enters the demand function for its product by altering the marginal utility to the consumer of that firm’s product.

  9. In Sect. 4.3, we explore two alternative calibrations. First, we assume that firms differ with respect to the slope parameters \(b_i\) rather than the intercept \(a_i\). Second, we assume that some revenue accrues to firm \(i\) from consumers whose prices and consumption levels are predetermined (while again imposing heterogeneity with respect to the \(a_i\) ’s and setting \(b_i=1\) for all \(i\)).

  10. To see that the denominator is positive, note that it is minimized at \(s=1\) and is positive when evaluated at \(s=1\).

  11. U.S. Department of Justice, “Ex Parte Submission of the United States Department of Justice,” WT Docket No. 12-269, April 11, 2013 (DOJ ex parte).

  12. It should be noted that this result hinges in part on the assumption of the bounded marginal unit that underlies the system of linear demand functions. If the marginal utility that is derived from the consumption of a firm’s output became very large as this output approaches zero, it would never be optimal to shut down such a firm.

  13. See, e.g., DOJ ex parte, 2013.

  14. Riordan (1998) and Loertscher and Reisinger (2014a) analyze the intermediate case by assuming an upward sloping, price elastic supply of capacity. Further work that makes the notion of warehousing operational in a model in which oligopolistic firms exert market power on both input and output markets would seem valuable. For example, one could suppose that the inverse market supply of capacity \(K\) is \(R(K)=r_{0}+r_{1}K\) and calibrate based on \( (r_{0},r_{1})\) as opposed to just \(r_{1}\) as in the exercise we describe.

  15. A natural conjecture is that the value of \(\hat{r}\) that is used to derive the theoretical equilibrium capacities affects the size (and eventually the sign) of the various \(\hat{k}_{i}-k_{i}^{*}(\hat{r})\)’s but will not affect the ranking of the firms according to \(\hat{k}_{i}-k_{i}^{*}(\hat{ r})\). This conjecture is consistent with the exercises we perform below, where we show that the ranking is the same when equilibrium values are computed using \(\hat{r}\) or the empirically observed price of capacity (see Tables 8, 9 for details).

  16. The exceptions include Maggi (1996) and Boccard and Wauthy (2000, 2004), who assume that the marginal costs of production is higher beyond the constraint but still constant, and Tirole (1988), who discusses the competitiveness of price competition in relation to the convexity of the cost function.

  17. For a summary of transactions between April 2012 and April 2013, see “The merger and acquisitions fever in the US telecommunications industry, at a glance,” Associated Press, April 15, 2013, http://www.washingtonpost.com/business/technology/the-merger-and-acquisitions-fever-in-the-us-telecommunications-industry-at-a-glance/2013/04/15/1d8d773a-a600-11e2-9e1c-bb0fb0c2edd9_story.html.

  18. Memorandum Opinion and Order and Declaratory Ruling in the Matter of Applications of Cellco Partnership d/b/a Verizon Wireless and SpectrumCo LLC et al., WT Docket Nos. 12-4, 12-175, Adopted August 21, 2012 (hereafter Verizon-TMobile-SpectrumCo MO&O), available at http://hraunfoss.fcc.gov/edocs_public/attachmatch/FCC-12-95A1.

  19. Sixteenth Report In the Matter of Implementation of Section 6002(b) of the Omnibus Budget Reconciliation Act of 1993; Annual Report and Analysis of Competitive Market Conditions With Respect to Mobile Wireless, Including Commercial Mobile Services, WT Docket No. 11-186, Adopted March 19, 2013 (Sixteenth Mobile Competition Report), available at http://hraunfoss.fcc.gov/edocs_public/attachmatch/FCC-13-34A1.

  20. As shown in Table 5, the total quantity in the model in the “Verizon at 110” scenario is 0.449223. The FCC’s Sixteenth Mobile Competition Report (p. 9) reports that there were 298.3 million subscribers to mobile telephone, or voice, service at the end of 2011. If modeled output is in units of billions of users, then the model has 449.223 million users.

  21. The FCC’s Sixteenth Mobile Competition Report (p. 15) reports that the total revenue generated by the mobile wireless industry was $171.28 billion in 2011.

  22. To be more precise, let quantities be in units of (298.3 million / 0.449223 users) = 664 million users. Let prices be in units of (171.28/0.14145 $billion / 664 million users) = 1820 $/per user. Then revenue is in units of (\(171.28/0.14145=\)) \(1211\) $billion. Since \(k_{i}r\) must be in 1211 $billion, the units for \(r\) are 40.37 $/MHz*Pop. Thus, \(r=1\) corresponds to $40.37 per MHz*Pop, and \(\hat{r}=0.0168\) corresponds to 0.68 $/MHz*Pop.

  23. This calculation is based on FCC data, which are available at http://wireless.fcc.gov/auctions/default.htm?job=auction_summary&id=66. Net winning bids were $13,700,267,150 and total MHz*Pops sold were 25,705,840,050.

  24. Additional effects are possible in models with increasing returns to scale technologies. For example, Horstmann and Markusen (1986) show in a model of international trade with increasing returns to scale that protectionism can reduce welfare by encouraging greater domestic firm entry and therefore lowering production levels (and increasing costs) for individual firms.

  25. The model, which is a slight generalization of the downstream market of the differentiated Bertrand model in Loertscher and Reisinger (2014b), combines the demand side of Singh and Vives (1984) and Häckner (2000) with a short-run cost function that represents constant returns to scale in the long-run, which has been used by Perry (1978); Riordan (1998) and Loertscher and Reisinger (2014a).

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Acknowledgments

We thank Michelle Connolly, Rosemary Humberstone, Federico Mini, Larry White, and participants at the 2013 CMD workshop on spectrum economics at the University of Melbourne for valuable comments.

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Correspondence to Leslie M. Marx.

Appendix

Appendix

Table 12 provides firm-level details for the three scenarios described in Table 5.

Table 12 Firm-level details of equilibrium outcomes for the three scenarios that are associated with the Verizon-T-Mobile-SpectrumCo transaction as derived from the calibrated model

Table 13 provides firm-level details for the scenarios in which 2.5 MHz is transferred away from Verizon but not allocated to any other bidder. This scenario is used in the calculation of use and foreclosure values in Table 6.

Table 13 Firm-level details of equilibrium outcomes for the case in which capacities are: Verizon–107.5, AT&T–88, Sprint–53, and T-Mobile–56 using the calibrated model

Table 14 provides details related to the results in Sect. 4.3.3 . The first row is the same as the corresponding row in Table 5, while the second row provides the corresponding numbers for the calibration that accounts for 10 % pre-existing contracts.

Table 14 Comparison of equilibrium welfare associated with the Verizon-T-Mobile-SpectrumCo transaction as derived from the original versus the adjusted calibration to allow for pre-existing contracts

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Loertscher, S., Marx, L.M. An Oligopoly Model for Analyzing and Evaluating (Re)-Assignments of Spectrum Licenses. Rev Ind Organ 45, 245–273 (2014). https://doi.org/10.1007/s11151-014-9427-y

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