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A Public Firm in a Vertically Linked Spatial Duopoly

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Published: Tue, 28 Nov 2017

A Public Firm in a Vertically Linked Spatial Duopoly with Price Discrimination

We show that, in a vertically linked duopoly where neither firm can produce all varieties demanded, spatial competition between a public and private firm induces them to deviate from the socially optimum location. We identify specific conditions under which a change in the degree of privatization induces one firm to move toward, while the other moves away from, the socially optimal location. There exists a critical level of privatization, above (below) which the public and private firms will come close (drift apart) with a rise in the degree of privatization.

  1. Introduction

Braid (2008) established that the equilibrium locations of any two firms are partially centralized, when none of them can supply all varieties demanded, to the socially optimal extent if there is spatial price discrimination. Beladi et al. (2008) demonstrated that a vertical merger with a monopoly upstream will tempt each downstream firm (inside and out of the merger), engaged in spatial competition for a market where neither of the downstream firms can produce all varieties demanded, to deviate from Braid’s (2008) socially optimal location. With interest in the role of a public firm in location decisions continuing to mount, since the dramatic financial events of this millennia brought about the creation of new sectors where private and public organizations vie to supply the same customers, Beladi et al. (2014, 2015) showed that the equilibrium locations of two spatially price discriminating firms (none of which can produce all varieties demanded) are invariant with the degree of privatization, when firms move simultaneously, but are sensitive to the degree of privatization when the public and private firms move sequentially.[1]

In this paper, we build on Braid (2008) and Beladi et al. (2008, 2014) to capture the responsiveness of equilibrium locations of public and private firms, selling different varieties of a product in a vertically related industry, to a change in the degree of privatization. We demonstrate that the Nash equilibrium locations, of a public and a private firm competing spatially in a vertically structured mixed duopoly, are not socially optimal and can vary with the degree of privatization when no firm can produce all varieties demanded and the demand for all product varieties are not identical. When the degree of privatization rises the private firm will move toward, while the public firm moves away from, the socially optimal location if the fraction of consumers wanting to buy the commonly produced good falls short of the fraction of those wanting to buy one of the goods produced exclusively by either firm. The public firm moves toward, while the private firm moves away from, the socially optimal location if the degree of privatization rises when the fraction of consumers wanting to buy the commonly produced good exceeds the fraction of those wanting to buy one of the goods produced exclusively by either firm. There exists a critical level of privatization, below which the public and private firms will drift apart, and above which the firms will come closer with a rise in the degree of privatization.

  1. Model and Results

Visualize, following Beladi et al. (2008), a stylized representation of a vertically related industry where an upstream manufacturer () produces an intermediate good and sells this good to 2 downstream retailers (:). The downstream retailers transform each unit of the intermediate good into one unit of a differentiated final good. The final good is sold to consumers uniformly distributed with unit density on a linear (uni-dimensional) market interval. The location of R1 and R2 are denoted by x and y, respectively, on this market interval with support [0, 1]. R1 sells products A and C, and R2 sells products B and C. A fraction c of buyers demand good A; a fraction c of buyers demand good B; and a fraction b buyers demand good C.[2] Suppose, as in Beladi et al. (2014, 2015), one of the downstream retailers (say, R2, without loss of any generality) is publicly owned with parameterizing the proportion of privately held shares in R2.

We assume that there is spatial price discrimination for good C of the sort originally examined by Lerner and Singer (1937), where a Nash equilibrium exists in delivered price schedules. Consumers are willing to pay maximum reservation price (k) that is sufficiently high so that it becomes relevant only when there is no competition between the two firms. Transportation costs are measured by td, where t is a constant and d is distance shipped. Monopoly goods, A and B, are priced at a uniform delivered price that is infinitesimally below k.

As in Beladi et al. (2008), the downstream firms simultaneously choose their locations in the retail market, while the upstream manufacturer’s offer takes the form of a two-part tariff. Decisions are taken in stages with perfect monitoring, that is, all past actions become common knowledge at the end of each stage. In the first stage, a take-it-or-leave-it two-part tariff offer is made by the upstream manufacturer to each of the downstream retailers: ’s offer takes the form , extracting all of the profits from , where is a uniform wholesale price and is a fixed fee. At this stage of the game, R1 and R2 simultaneously choose their locations in the retail market. In the second stage, R1 and R2 simultaneously decide whether or not to accept or decline’s offer. The fixed fee () is collected by at this stage, only if decides to accept the contract offered. In the next stage, R1 and R2 engage in spatial price discrimination. In the final stage, consumers reveal their demand for goods. The downstream retailers pay the wholesale price (), for each unit that is ordered from the upstream manufacturer, and then sell the final goods to the consumers. A solution is reached by backward induction. R1’s (located at x) profits from a) selling A, at a uniform delivered price (), are ; b) selling C, to consumers located in the market interval from to , are ; and c) selling C, to consumers located in the market interval from to , are . R1 chooses to maximize its own profits

The objective of the publicly-owned firm (R2, located at y), as in Beladi et al. (2014), is to maximize a weighted average of its own producer’s surplus and social welfare, where the weight is the degree of privatization. Social welfare comprises the profits of both firms as well as the consumer surplus. An underlying model of bargaining between the public and the private shareholders, where the board of this firm consists of the government’s representatives who advocate welfare (consumer and producer surplus) and the representatives of the private shareholders who advocate profit, can be used to rationalize such a welfare function: bargaining will involves percent of representatives who have an objective of maximizing profits and percent of representatives who have an objective of maximizing welfare since is the proportion of publicly held shares in the R2 and the rest is privately owned.[3] The monopoly goods A and B are priced to leave zero consumer surplus, while the spatial duopoly good C generates consumer surplus that consists of a) for consumers located in the market interval from to , where is the delivered price and b) for consumers located in the market interval from to , where is the delivered price. Thus R2 choosesto maximize

The first order conditions for profit-maximization, yields

In comparison, Braid (2008) had shown that the socially optimal locations are

Our main propositions follow.

Proposition I. The Nash equilibrium locations are not socially optimal, with or without privatization.

Proof. for any .

Proposition II. The private firm moves toward (away from), while the public firm moves away from (toward), the socially optimal location when privatization rises if .

Proof. if and if .

Proposition III. A rise in the degree of privatization, above (below) a critical level, induces the private and public firms to come close (drift apart).

Proof.  if.

In sum, when a publicly owned firm competes with a private firm with neither firm producing all varieties demanded, firms do not locate at the socially optimal Nash equilibrium. Except when demand for all product varieties are identical (i.e. ), the Nash equilibrium locations of both firms are sensitive to the degree of privatization. A rise in the degree of privatization induces the private (public) firm to move toward (away from) the socially optimal location if the fraction of consumers wanting to buy the commonly produced good falls short of the fraction of those wanting to buy one of the goods produced exclusively by either firm. When the degree of privatization rises, the public (private) firm moves toward (away from) the socially optimal location if the fraction of consumers wanting to buy the commonly produced good exceeds the fraction of those wanting to buy one of the goods produced exclusively by either firm. There exists a critical level of privatization, below which the public and private firms will drift apart, and above which the firms will come closer with a rise in the degree of privatization.

  1. Conclusion

The role of privatization in the location choice of vertically linked firms engaged in spatial competition has been attracting significant academic attention. We show that, when a publicly owned firm competes with a private firm in a vertically related industry where neither firm can produce all varieties demanded, firm locations are not socially optimal as long as the demand for all product varieties are not identical. The private firm moves toward, while the public firm moves away from, the socially optimal location if the degree of privatization rises when the fraction of consumers wanting to buy the commonly produced good falls short of the fraction of those wanting to buy one of the goods produced exclusively by either firm. The public firm will move toward, while the private firm moves away from, the socially optimal location if the degree of privatization rises when the fraction of consumers wanting to buy the commonly produced good exceeds the fraction of those wanting to buy one of the goods produced exclusively by either firm. A rise in privatization, above (below) a critical level, will induce the public and private firms to come close (drift apart). Some interesting extensions, of this paper, may involve the allowance for informal wage á la Marjit (2003), trade barriers á la Oladi (2005), and/or mergers á la Mukherjee and Davidson (2007).

Endnotes

1


[1] More specifically, Beladi et al. (2015) have shown that a rise (fall) in the degree of privatization will induce the public and private firms to move closer to (farther from) the socially optimal Nash equilibrium when the public firm leads.

[2] It is possible to contemplate an equivalent scenario where one of the downstream firms sells one variety while the other sells a different variety, and some consumers want to buy only one of the two varieties and some are indifferent between the two. Following Braid (2008), if neither firm can price discriminate, it is possible to assume mixed price strategies. Unlike Dasgupta and Maskin (1986), who had a single mixed-strategy Nash equilibrium in mill prices for any given set of firm locations, there would be a different mixed-strategy Nash equilibrium in delivered prices for any given locations of firms.

[3] Such a bargaining, following Chao and Yu (2006), will yield a mixed objective between profits and welfare in which each carries the respective weight of the representatives.


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