Mixed oligopoly, sequential entry, and spatial price discrimination.

Heywood, John S. ; Ye, Guangliang

I. INTRODUCTION

The idea that a public firm can regulate oligopolistic behavior dates at least to Merrill and Schneider (1966), and researchers since then have spent considerable effort isolating whether or not the presence of the public firm actually improves welfare. Despite the fact that the public firm maximizes social welfare, the result of its interaction with private profit maximizing firms does not routinely increase welfare. Thus, DeFraja and Delbono (1989) show that the presence of a public firm increases welfare when there are only a few Cournot rivals, but decreases welfare when there are a larger number of rivals. While recent models consider foreign firms (Fjell and Pal 1996), subsidies and taxes (Pal and White 1998; Poyago-Theotoky 2001;Sepahvand 2004), and the role of leadership (Fjell and Heywood 2004), all of these extensions remain outside the spatial context and, in total, present mixed results on whether or not the presence of the public firm improves welfare.

Within the spatial context, Cremer, Marchand, and Thisse (1991) examine Hotel ling (1929) pricing and location on a line segment. They show that the presence of a public firm actually harms welfare when the total number of firms is more than two and less than six, the type of tight oligopolies governments might be most interested in controlling. While a variety of other studies expand on these results, they retain the assumption of no spatial price discrimination, and they do not consider the welfare consequences of the public firm. Nilssen and Sorgard (2002) examine location choices in a duopoly with fixed prices and asymmetric transport costs. Matsushima and Matsumura (2003a) show that agglomeration of private firms occurs on the opposite side of a circular market from the public firm's location. Matsumura and Matsushima (2003) examine a sequential move duopoly with quadratic transport costs in which either the public or private firm may move first. Lu (2006) characterizes the equilibrium of a mixed duopoly with linear transport costs in a location-then-price model.

This paper is the first to investigate the consequences of a welfare maximizing public firm in a model of spatial price discrimination (delivered pricing). We show that the presence of a public firm generally improves welfare, and never reduces welfare, as it often does outside the spatial context or in spatial models without price discrimination.

This showing stands as important because spatial price discrimination remains an important economic phenomenon for markets with large transport costs or with differentiated products. Indeed, Greenhut (1981) presents survey evidence showing that among firms for which transport is at least 5% of costs, such discrimination is "nearly ubiquitous" in the United States, Europe, and Japan. In addition to the actual delivery of bulky physical products, such location models may apply to many industries with horizontal differentiation. To the extent that firms can identify the location of consumers, the possibility for discrimination exists. Certainly, newspapers locate along a political spectrum from left to right, cereals vary in sweetness, and airlines choose times of day for flights. (1) Thisse and Vives (1988) show that spatial price discrimination emerges as a natural consequence of profit maximization and that such pricing will be adopted instead of uniform mill pricing whenever conditions allow. In turn, this has resulted in the application of such models to a variety of policy questions. For example, Rothschild, Heywood, and Monaco (2000) examine the consequences of horizontal merger and Gupta, Kats, and Pal (1994) examine pricing within a vertical chain, and both examinations are set in a model of spatial price discrimination. However, no one has examined the consequences of a mixed oligopoly in the context of the spatial price discrimination. This failure does not flow from a lack of relevance as mixed oligopolies in industries from steel and automobiles to airlines produce products with either substantial transport costs or extensive elements of horizontal differentiation. (2)

In what follows, we examine three circumstances. In the first, and as a benchmark, n firms locate simultaneously along a unit line segment. In the second, a private leader locates first in anticipation of the remaining n - 1 firms locating in a second stage (see Gupta 1992 for an early model of such leadership in spatial price discrimination). In the third, three firms locate in a full sequence. In each case, we compare the equilibria both with and without a public firm assumed to maximize social welfare. In the first case, the presence of the public firm has no influence on locations or welfare. This should be anticipated as simultaneous location choices by private spatial price discriminators are well known to generate first-best welfare. The heart of the analysis consists of the next two cases. In the second case, the presence of the public firm always increases welfare. In the third case, the presence of the public firm increases welfare as long as it does not locate last. We provide intuition for these results and discuss implications. We conclude that when a public firm makes a location decision, this acts as a powerful tool to improve welfare. Indeed, the role of the public firm in improving welfare appears far more robust in models of spatial discrimination than in other contexts, a point we return to in the conclusion.

In what follows, the next section sets up the model. The third section briefly presents the simultaneous location equilibria and the fourth section presents equilibria with leadership. Section V investigates equilibria with fully sequential location of three firms. In each case, we focus attention on the locations and the welfare implications of having a public firm. Section VI provides concluding remarks.

II. SETTING UP THE MODELS

Consumers are uniformly distributed over a unit line segment. Each consumer has inelastic demand for one unit of the good, with reservation price r. We assume r is sufficiently large that it is profitable to serve all customers. All n firms have identical production technologies with the marginal cost of production assumed to be constant and normalized to be zero without loss of generality. The number of firms is fixed in order to focus on issues of market power. Transport cost is t per unit of distance. We adopt the linear market both because it is common and also because it provides a distinct advantage for leadership (compared to a circular market). The ability to move first and locate in the corner allows the leader to earn greater profit than those firms locating in the interior (not the corner) of the market.

Let [L.sub.i], i = 1, 2, 3 ... n, denote the location of firm i on the unit line segment, where [L.sub.i] < [L.sub.i+1] (see Figure 1). Thus, the cost for firm i to supply all consumers in the line segment [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [c.sub.i] = t[absolute value of x - [L.sub.i]]. The equilibrium delivered price schedule for any consumer located at x is as follows:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As shown in Figure 1, the price schedule is the lower envelope of rival's cost curves as shown by EGF. We follow convention by assuming that customers who are indifferent (they face the same delivered price from more than one firm) purchase from the closest firm.

Consequently, the profit of an interior firm i is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The profits of corner firms 1 and n are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The total cost for the industry is

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The welfare function becomes

(5) W([L.sub.1],[L.sub.2], ..., [L.sub.n]) = r - TC.

Thus, with constant willingness to pay, r, total industry cost becomes the measure of social welfare.

[FIGURE 1 OMITTED]

III. MIXED OLIGOPOLY WITH SIMULTANEOUS LOCATION CHOICE

In each of the next two sections, we compare sequential location models for a mixed oligopoly with those for a private oligopoly that has no public firm. Before moving to those comparisons, we remind readers of the solution to the simultaneous location model with only private firms and use this solution to present the analogous solution for a mixed oligopoly model with simultaneous location.

LEMMA. A private oligopoly engaging in spatial price discrimination and locating simultaneously generates symmetric locations.

Given [L.sub.i-1] and [L.sub.i+ ] from (2), the optimal location for any interior firm i is [L.sub.i] = [L.sub.i-1] + [L.sub.i-1]/2 yielding [L.sub.i] - [L.sub.i-1] = [L.sub.i+1] = [L.sub.i]. Similarly, from (3) the optimal locations of the corner firm will be [L.sub.1] = (1/3)[L.sub.2] and [L.sub.n] = 2/3 + (1/3) [L.sub.n-1]. Simultaneous solution yields that all firms locate symmetrically. This is the well-known result that spatial price discrimination results in symmetric locations and lowest total cost, highest welfare. See Lederer and Hurter (1986) for the original two-firm proof.

The simultaneous location equilibrium with a public firm remains identical to that without the public firm. This demonstration serves to emphasize the importance of location timing in generating the inefficiency that the presence of a public firm would hope to reduce. The firms in the industry play a two-stage game. In stage 1, firms enter and choose a location in the market. In the second stage, firms simultaneously announce the delivered price schedule. (3) We assume that the public firm locates at [L.sub.1].

PROPOSITION 1. In spatial price discrimination, the presence of a public' firm does not change equilibrium locations and social welfare.

Proof: From the Lemma, Li - (2n-2i+1) [L.sub.1] + 2i-2/2n-1, and so [L.sub.2] = ((2n-3)[L.sub.1] + 2)/2n-1. The public firm minimizes cost (maximizes welfare) along the best response function [L.sub.1] = [L.sub.2]/3 implying locations [L.sup.*.sub.2] = 1/2n and [L.sup.*.sub.i] = 2i-1/2n, identical to those without a public firm.

In a simultaneous location game, all firms are identical and locate symmetrically, thus generating optimal social welfare. There is no room for a public firm to improve welfare.

IV. A PRIVATE STACKELBERG LEADER

The idea of sequential entry has a long history in location models as firms are rarely seen to enter simultaneously (Prescott and Visscher 1977; Hay 1976). We emphasize that if the public firm is the entry leader,

Appendix 1 shows that it locates efficiently, and all followers will locate symmetrically recovering the equilibrium outlined in the previous section. Thus, we now compare equilibrium locations for a mixed oligopoly with a private Stackelberg leader to a fully private oligopoly with a Stackelberg leader. In doing so, we again investigate the welfare consequences.

A. Private Oligopoly

The firms play a three-stage game. In stage 1, the leader chooses a location in the market with perfect foresight. In stage 2, n - 1 firms make simultaneous location choices. In stage 3, all firms simultaneously announce the delivered price schedule.

Given a line segment, a corner firm has no rival on one side and so can move toward the middle earning profit not available to interior firms. Thus, the leader locates at the corner, and we assign it [L.sub.1] (obviously an otherwise identical equilibrium could be derived by assigning the leader [L.sub.n]). All other firms locating right of [L.sub.1] do so symmetrically:

(6) (2n- 2i + 1)[L.sub.1] + 2i - 2/2n-1

Consequently, an interior firm i's profit becomes

(7) [[pi].sub.i] 2t [([L.sub.1] - 1).sup.2]/[(2n- 1).sup.2]

In order to maintain its profitable corner location, the leader must locate such that no firm earns greater profit locating on its left. When there is only one follower (n = 2) , the leader need not worry about jumping as its unconstrained optimal location leaves the second firm better off on the right side (Gupta 1992). For n > 2, if firm j does jump to the left side of the [L.sub.1], it maximizes profit at [L.sub.j]' = [L.sub.1]/3 with [[pi].sub.j]' = t[L.sup.2.sub.1]/3. Thus, the leader must locate such that firm j is no better off jumping. (4) Solving [[pi].sub.j] = [[pi].sub.j]' yields the location for [L.sub.1] and (6) gives the location for all other firms.

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The total transport cost of the industry becomes

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As an illustration, if n = 4, [L.sup.*.sub.1] = 0.26, [L.sup.*.sub.2] = 0.47, [L.sup.*.sub.3] = 0.68, [L.sup.*.sub.4] = 0.89, [TC.sup.prv*] = 0.073t. These locations are far from symmetric and have far greater costs. The symmetric locations would be (0.125, 0.375, 0.625, 0.875) with a cost of TC = 0.0625t. Thus, allowing for a private leader results in a 17% increase in transport costs. The fact that the private firm equilibrium is no longer welfare maximizing gives rise to the possibility that the presence of a public firm may improve welfare.

B. Public Follower

Introducing the public firm, even as a follower, fundamentally changes the equilibrium locations. The public firm requires a lower market share and profit to jump to the left of the leader's corner position than does a private firm. In essence, it values the symmetry and its lower cost. Thus, having even a single firm with a welfare objective dramatically reduces the strategic advantage of leadership.

As before, the leader locates at [L.sub.1], and all other firms locate symmetrically on the right of [L.sub.1]. (5) The location of firm i becomes

(10) (2n - 2i + 1)[L.sub.1] + 2i - 2/2n - 1.

Consequently, an interior firm i's profit becomes

(11) [[pi].sub.i] 2t[([L.sub.1] - 1).sup.2]/[(2n- 1).sup.2].

The total cost of the industry becomes

(12) [TC.sup.pub]= (2n[L.sup.2.sub.1] - 2[L.sub.1] + 1)t/4n - 2.

For the leader to keep the profitable corner location, neither a representative private firm nor the public firm can be better off jumping to the left of [L.sub.1]. If the public firm does jump, it locates at [L.sub.j] = [L.sub.1]/3, generating the following total industry cost:

(13) [TC.sup.pub.sub.j] = (2n[L.sup.2.sub.1] - 6[L.sub.1] + 3)t/6(2n - 3).

The public no-jump condition requires [TC.sup.pub] = [T.sup.pub.sub.j], yielding equilibrium locations:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The fact that [L.sup.pub*.sub.1] from (14) is less than [L.sup.prv*.sub.1] from (8) for all n > 2 (shown in Appendix 2) reflects that the public firm requires a smaller market to jump than does a private firm. The total cost associated with (14) becomes

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The total cost without a public firm, [TC.sup.prv], is given by (9) and is larger than (15).

PROPOSITION 2. In spatial price discrimination with a Stackelberg leader, the presence of a public firm follower always improves social welfare. Therefore, privatizing a public firm always decreases social welfare.

Proof: [TC.sup.prv*] > [TC.sup.pub*] for all 17 > 2 and substituting into Equation (5) yields [W.sup.prv*] < [W.sup.pub*]--shown formally in Appendix 2.

Again, for our illustration of n = 4, the locations and total cost become [L.sup.*.sub.1] = 0.23, [L.sup.*.sub.2] = 0.45, [L.sup.*.sub.3] = 0.67, [L.sup.*.sub.4] = 0,89, [TC.sup.pub*] = 0.68t. These locations are more nearly symmetric, and the total costs fall almost halfway between those without a public firm, 0.0073 and the first best symmetric locations, 0.0625.

We recognize that if barriers to relocation are too great, privatization will obviously not result in short-run relocation. Even in such a case, we still contrast two situations, one with a public firm and one without a public firm. Proposition 2 remains important as it contrasts with past results on mixed oligopoly outside the spatial context. Those results emphasize that whether or not welfare increases with the presence of a public firm depends on the number of firms in the market and the presence of government tax and subsidy tools. (6) Our result shows that the presence of a public firm in a model of spatial price discrimination with a private leader always improves efficiency.

V. FULLY SEQUENTIAL LOCATIONS

Fully sequential location models with spatial price discrimination have been difficult to generalize to large numbers of firms. Gupta (1992) presents the case of a private firm oligopoly of three firms in a model of spatial price discrimination. Using the critical "no jump" constraints, she presents the equilibrium: [L.sup.1*] = 0.275, [L.sup.2*] = 0.725, [L.sup.3*] = 0.5, TC = 0.1010t, where the superscript denotes the order of location. In this section, we consider the introduction of a public firm into this three-firm market. We alter the order in which the public firm locates considering it to initially locate first, then second, and finally third. This allows us both to identify a "best" stage at which the public firm can enter theoretically and to comment on the natural variations observed in actual markets. In the typical view of "regulation by participation," the public firm enters in a later stage hoping to regulate markets that initially lack sufficient competition (Merrill and Schneider 1966). Yet, frequently a monopolistic public firm has been given first entry by a government that later allows competitive private entry (examples include the privatization and deregulation of transportation and utilities in many countries).

A. The Public Firm Locates First

Equilibrium locations are solved by backward induction. With three firms, the public firm locates exactly in the middle forcing the other two firms to either side so as to maximize their profit in the remaining markets. This returns first-best locations: [L.sup.1*] = 0.5, [L.sup.2*] = 0.167, [L.sup.3*] = 0.833, TC = 0.0834t. Because each of the followers is best off moving to their respective sides of the market, there is no scope for strategic behavior to develop because of the sequential entry.

If we continue to assume that the first mover locates on the left of the market (admittedly because of some exogenous constraint), we would first identify the location of the third entrant conditional on the first two. As the second entrant will want the third to locate in the middle, we identify locations that ensure this. We do so by equating the profits that the third mover would earn locating in the middle and on the right corner. From this equality comes the best response function of the second entrant in terms of first. With this, the total cost can be written as a function of only the first entrant's location, and the public firm can locate so as to minimize this cost.

The public firm locates in the first move so as to minimize this cost. This optimal location and the reaction functions above yield the final equilibrium (as detailed in Appendix 3):

(16) [L.sub.1] = 0.1938, [L.sub.2] = 0.6938, [L.sup.3] = 0.4438, [TC.sup.publ] = 0.0969t.

It may be surprising that the public leader moves away from the cost-minimizing symmetrical location of 1/6 toward the middle. Moving toward the middle limits the extent to which the second mover can use its timing advantage over the third private firm. Yet, the total cost exceeds that of the public firm locating in the middle demonstrating that (16) is not a subgame-perfect equilibrium absent an exogenous constraint requiring the leader to locate on the left.

B. The Public Firm Locates Second

Using backward induction, we first identify the locations of the third and second entrants conditional on the first. This begins by recognizing that the private leader locates such that neither the public nor the private firm will be better off jumping to its left side. We identify this first entrant's location by equating the total costs that the second (public) mover would generate locating on the left to that generated by it locating on the right corner given that the final entrant locates in the middle. Thus, assuming [L.sup.3] = [L.sup.1] + [L.sup.2]/2 the public firm chooses its optimal location at the right corner yielding the following best response for the second entrant and total cost in terms of [L.sup.1]:

(17) [L.sub.2] = 4/5 + 1/5 [L.sup.1], TC = 1/10 t -1/5 t[L.sup.1] + 3/5 + [(L.sup.1).sup.2]

If the public firm jumps to the left side of the leader, the locations and total cost would be

(18) [L.sup.2,j] = 1/3[L.sup.1], [L.sup.3,j] = 2/3 + 1/3 [L.sup.1], [TC.sup.j] = 1/6 t - 1/3t [L.sup.1] + 1/3t[([L.sup.1]).sup.2]

The no-jump condition requires TC = [TC.sup.j], implying an equilibrium location of the first entrant [L.sup.1] = 0.3090. The remaining locations and costs follow:

(19) [L.sup.1] = 0.3090, [L.sub.2] = 0.8618, [L.sup.3] = 0.5854, [TC.sup.pub2] = 0.0955t.

It can be easily checked that the third entrant earns less in either corner than it does in the middle. This fits the general point that a profit maximizing firm requires at least as large a market segment to jump as would the public cost minimizing firm. Note that the public firm moving second generates higher social welfare than does the public firm moving first but constrained to enter on the left.

C. The Public Firm Locates Third

We now consider the public firm locating third. As in Gupta (1992) and the earlier cases in this section, the first two movers locate symmetrically so as to force the public firm into the middle. They do this by making it impossible for the public firm to improve welfare by jumping to either corner. The total costs when the public firm locates in the middle and right side of the second firm are

(20) TC = 9/8t - 5/2t[L.sup.2] + 2/3t[(L.sup.2).sup.2], [TC.sup.j] = 11/12t - 7/3[L.sup.2] + 5/3t[([L.sup.2]).sup.2].

The no-jump condition requires TC = [TC.sup.J]. The resulting equilibrium location of the second firm is [L.sub.2] = 0.7247 giving the following:

(21) [L.sub.1] = 0.2753, [L.sub.2] = 0.7247, [L.sup.3] = 0.50, [TC.sup.pub3] = 0.1010t.

The equilibrium locations are exactly that of Gupta (1992). In the earlier two cases, the public firm could locate knowing how a subsequent entrant would locate. This information could be used to generate a more symmetric equilibrium. As the final entrant, the best the public firm can do is locate symmetrically in the largest available market segment. This is precisely what a private firm would do if it were the third entrant.

We summarize the results of this section in a proposition and a corollary.

PROPOSITION 3. In spatial price discrimination with three firms locating sequentially, the presence of a public firm never decreases welfare and increases welfare when the public firm locates either first or second. Thus, privatizing a public firm will never increase welfare.

Whenever the public firm has a timing advantage (moves before at least one private firm), privatization decreases social welfare. Put somewhat differently, the public firm can use its timing advantage to force the private firms into more symmetric, lower cost locations.

COROLLARY. In spatial price discrimination with three firms locating sequentially. a public' firm locating first generates first-best social welfare.

Finally, it is worth contrasting this case with that of the private leader discussed in the previous section. In that earlier case, all followers locate symmetrically, and welfare is driven solely by the location of the leader. With fully sequential location, the position of the first firm does not determine welfare. Thus, the leader moves more toward the middle in second case than that in the third case, but the social welfare is higher in the second case than that in the third one. This follows because every firm in the fully sequential location game has a relative strategic advantage over any remaining followers.

VI. CONCLUSION

The idea of using a public firm to improve performance in an oligopoly has traditionally been based on the assumption that a welfare maximizing public firm will increase its output beyond that of profit maximizing oligopolists. Yet, the literature presents malay circumstances in which the interaction of the firms in the market results in welfare reductions being generated by the presence of the public firm. At issue in this paper has been whether or not a public firm will locate in model of spatial price discrimination in such a way as to improve welfare.

The context of price discrimination is an important one because deviations from first-best cost minimization happen as a result of the timing of location decisions. With simultaneous location, cost is minimized. With an order of location, as might be anticipated in the real world, early entrants locate in such a way as to put followers at a disadvantage, increasing total costs and so decreasing welfare. We've shown that the presence of a public firm can reduce this misallocation. A welfare maximizing (cost minimizing) public firm will usually accept a smaller market share and reduced profit in order to achieve more symmetric overall locations. This willingness forces earlier private entrants to locate in a more nearly welfare maximizing fashion.

The contrast between the location choice of our public firm and quantity choice of the public firm in the nonspatial model deserves emphasis. The contrast is not in the behavioral assumption of welfare maximization, but rather in the structures of the models. With sequential entry, our public firm can limit the extent to which earlier movers can use their first locator advantage relative to later movers. The public firm in the nonspatial model typically increases output to improve welfare, but does so by increasing the asymmetry of output with the private firms, and so increasing the total cost of supplying the market. The typical convex production costs that generate this cost increase are absent from our model. Similarly, the fact that an increase in price lowers consumer surplus is also absent in our model.

When compared to other spatial models with a public firm, our assumption of spatial price discrimination eliminates the agglomeration issues that stand at the center of those models. Thus, our model retains relevance in those cases in which demand is largely inelastic and in which transportation costs represent a large share of total costs. As shown, with these conditions and the resulting assumption of spatial price discrimination, the location of the public firm can preempt otherwise welfare diminishing location choices by private firms.

We recognize that the welfare effects associated with privatization should be taken as illustrative. If relocation costs are sufficiently high, privatization may change nothing as the locations of the firm will not change. Nonetheless, the results can be taken as comparing cases with and without public firms or giving guidance when relocation costs are small. While the line segment is the most common type of spatial market, it would be interesting to explore how robust the conclusions are to alternatives.

APPENDIX 1

A. The mixed oligopoly with a public Stackelherg leader

The public firm locates first with perfect foresight at [L.sub.1]. The private firms locate second and symmetrically on the right side of [L.sub.1] with a distance between them of d = 2[L.sup.2.sub.1] - 2/2n -1. The total cost of the industry becomes

(A.1) C = (2n[L.sup.2.sub.1] - 2[L.sub.1] + 1)t/4n -2,

and the cost minimizing first-order condition for the public firm yields: [L.sup.*.sub.1] = 1/2n. Since [L.sub.i] = [L.sub.1] + (i - 1)d, the general solution for firm i becomes [L.sup.*.sub.i] = 2i - 1/2n, the symmetric, cost-minimizing solution.

APPENDIX 2

A. Proof of [L.sup.prv*.sub.1] > [L.sup.pub*.sub.1] for n > 2.

Substituting from (14) and (8) it must be shown that [2n[square root of 6] - [square root of 6] - 6/4[n.sup.2] - 4n - 5] > [-3 + [square root of 12[n.sup.2] - 24n+9]/ 4n(n - 2)]. For all n, the middle expression is larger than that on the right: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the original inequality must hold if the expression on the left is larger than that in the middle. Collecting common positive denominators, the first inequality holds if f(n) = 4n(n - 2)(2n [square root of 6] - [square root of 6] - 6) - (-3 + (n - 1) [square root of 12])(4[n.sup.2] - 4n - 5) > 0. As f'(n) > 0 and f(3) = 60[square root of 6] - 76[square root of 6] - 15 > 0, f (n) > 0 for all n > 2 and [L.sup.prv*.sub.1] > [L.sup.pub*.sub.1]

B. Proof of [TC.sup.prv*] > [TC.sup.pub*] for n > 2.

Total cost with or without a public firm remains a function of the leader's location:

TC (2n[L.sup.2.sub.1] - 2[L.sub.1] + l)t/4n - 2] with [partial derivative]TC/[partial derivative][L.sub.1] = (4n[L.sub.1] - 2)t/4n - 2. As the leader locates right of the simultaneous location equilibrium, [L.sub.1] > 1/2n [??][partial derivative]TC/[partial derivative][L.sub.1] > 0 Thus, [L.sup.prv*.sub.1] > [L.sup.pub*.sub.1] [??] [TC.sup.prv*] > [TC.sup.pub*].

APPENDIX 3

A. Sequential Entry with the Public Firm Locating First and on the Left

The optimal location and the associated profits for the third entrant in the middle are

(A.2) [L.sup.3] = [L.sup.1] + [L.sup.2]/2, [[pi].sup.3] = ([L.sup.1] - [L.sup.2]).sup.2]/8,

while those for the third entrant jumping to the right corner are

(A.3) [L.sup.3,j] = 2/3 + 1/3 [L.sup.2], [[pi].sup.3,j] = 1/3t - 2/3t[L.sup.2] + 1/3t([L.sup.2]).sup.2].

Setting [[pi].sup.3] = [[pi].sup.3,j] implies the best response function for firm two under the constraint of the no jump condition, [BR.sup.2] = 0.62 + 0.38[L.sup.1]. Substitute [L.sup.2] = [BR.sup.2] into [L.sup.3] yields best response function for firm three, [BR.sup.3] = 0.31 + 0.69[L.sup.1]. Substituting [L.sup.2] = [BR.sup.2] and [L.sup.3] = [BR.sup.3] into total social cost function yields TC = 0.62t([L.sup.1]).sup.2] - 0.24t[L.sup.1] + 0.12t, which is minimized to yield the public firm's optimal location.

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(1.) See Brueckner and Flores-Filio (2006) for a recent review of attempts to model departure times in a location context.

(2.) Thus, Doganis (2001) identifies 85 passenger airlines with controlling public ownership as of 2000 and several European and Asian steel and automakers that have critical public ownership. As one example of the latter, Volkswagen's largest shareholder is the German state of Lower Saxony. Even in North America, the provincial governments of Quebec and Nova Scotia own steel mills (Tupper 2006).

(3.) Note that if the public firm prices at its cost rather than at the envelope of its neighbor's costs, it will merely increase consumer surplus an amount identical to the lost profit. Such a change in pricing by the public firm does not alter the objective functions or location choices of any of the firms.

(4.) The authors have proven that the jump constraint is always binding for n > 2 and the proof is available upon request.

(5.) We have confirmed that in the subgame-perfect Nash equilibrium the private leader always locates at the comer to gain the first mover advantage. While this might be expected, a proof by contradiction is available from the authors.

(6.) DeFraja and Delbono (1989) examine a simultaneous move mixed oligopoly and Heywood and Ye (2005) examine a mixed oligopoly with a private leader. In both cases, when n is large the presence of the public firm reduces welfare. Pal and White (1998) examine a model with foreign firms and a subsidy showing that, even with small n, the presence of the public firm decreases welfare. Cremer, Marchand, and Thisse (1991) examine a Hotelling model showing that for markets with either 3.4, or 5 firms privatization reduces welfare.

JOHN S. HEYWOOD and GUANGLIANG YE *

* The authors express thanks to R. Rothschild and seminar participants at the University of Lancaster. Heywood. Professor, Department of Economics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201. Phone 414-229-4310, Fax 414-229-5915, E-mail heywood@uwm.edu

Ye: Associate Professor, Research Institute of Economics and Management, Southwestern University of Finance and Economics, Chengdu, Sichuan 610074 China. Fax 86-28-87099300, E-mail gye@swufe.edu.cn