Delivered pricing and merger with demand constraints.

Monaco, Kristen ; Heywood, John S. ; Rothschild, R. 等

I. INTRODUCTION

Studies of merger activity in models of spatial price discrimination conventionally assume that pricing by firms is not constrained by demand. Instead, the only constraints on the discriminatory prices charged are the cost structures of the adjacent rivals. The issue of demand constraints has received attention by Wang and Yang (1999) in related literature but not in examining the consequences of a spatial merger.

We demonstrate that when discriminatory prices are constrained by willingness to pay (demand), the conclusions of existing models must be substantially modified. First, although introducing a demand constraint does not alter firm locations in the absence of merger, it does alter firm locations in the presence of merger. Second, the demand constraint results in equilibrium locations that depend on transport cost, which is not the case in previous models. Transport cost plays an important role in spatial models, and its absence to date from models of spatial merger is unrealistic. Moreover, the dependence of location on transport costs invites the possibility that a tax on transport costs can change locations in an efficient fashion. Third, introducing a demand constraint reverses previous results relating to the merger paradox. In particular, in the absence of a demand constraint, models of spatial price discrimination provide instances in which the parties to a merger gain profit and the excluded firms lose profits. There are no such instances in the presence of demand constraints.

In what follows, the second section places our work within the literature and highlights the issue of demand constraints. The third section presents the model, contrasting it with earlier models. The fourth section identifies the equilibrium locations that arise when there is the prospect of merger and contrasts these locations with those that arise in the absence of demand constraints. The section also demonstrates that when demand is constrained the model of spatial price discrimination yields no cases that resolve the merger paradox, a result contrasting with that offered by the model in which demand constraints are absent. The fifth section demonstrates that a tax on transport can increase welfare. Indeed, the optimal tax often completely eliminates spatial price competition by creating local monopolies. The sixth section considers the full range of cases in which some (but not all) firms may face constrained demand and also proves that in the fully constrained case considered earlier, the entire market will be served. A seventh section considers a partially constrained case, and the eighth section concludes.

II. SPATIAL PRICE DISCRIMINATION AND CONSTRAINED DEMAND

Spatial models have proven especially useful in capturing aspects of competition in horizontally differentiated markets as emphasized by Tirole (1988). In particular, they provide a general method for examining markets in which an ordered product characteristic differentiates output. (1) Among the more popular models is that of discriminatory pricing in which the price a firm is able to charge depends upon how "close" it is to its rivals. Thisse and Vives (1988) show that such pricing is the preferred alternative when firms are able to adopt it. Lederer and Hurter (1986) demonstrate that under such pricing rivals locate symmetrically along a linear market, thus minimizing total transport costs.

Early studies of merger in a model of spatial discrimination assumed that location choices do not anticipate the merger. Reitzes and Levy (1995) show that such a merger increases the prices and profits of the participants but leaves those of all other firms unchanged. Gupta et al. (1997) show that an anticipated horizontal merger alters the locations of spatially discriminating duopolists and increases transport cost, thereby reducing efficiency. Rothschild et al. (2000) confirm this finding in the case of two firms merging in a three firm market and also examine the effects of merger on the excluded rival. They identify a range of outcomes in which the merging firms benefit and the excluded firm is harmed by the merger. These outcomes help resolve the "merger paradox," the name given to the surprising result obtained from nonspatial models by Salant et al. (1983) that rivals excluded from a merger usually benefit and often benefit more than the merging firms. This is a paradox because it points to the free-rider aspect of merger. As Pepall et al. (1999, 406) put it for a three-firm model, "the real beneficiary of the merger will be the third firm that did not participate in the merger." Heywood et al. (2001) examine a two-firm merger in an N-firm spatial market and demonstrate that for a critical class of mergers, the range in which the merging firms benefit and the excluded rival is harmed increases with the number of rivals. Thus it would seem that models of spatial price discrimination are fruitful as an avenue of further inquiry into the effects of merger.

The particular contribution of this article is to consider merger in markets in which demand is constrained. This is the situation in which the reservation price is low relative to the combined production and transport costs. Considered a realistic case for heavy, low-value commodities, this possibility has proven important in other contexts. Economides (1984) first considered the implications of a low reservation price in a Hotelling model, and Wang and Yang (1999) show that in that model, a low reservation price will be associated with firm locations that are closer together than they would be otherwise. Thus, as a baseline, the first issue herein is to identify equilibrium locations in a model of spatial price discrimination that is demand constrained. The second issue is to examine the effects of anticipated merger to see whether the locations differ from those already derived for the unconstrained case. The third issue is to isolate the role that constrained demand plays on the ability of the model to help resolve the merger paradox.

III. THE MODEL

The market is a line of unit length with consumers uniformly distributed with density one. Each consumer has inelastic demand for one unit of the good, with reservation price r. If two firms offer the same delivered price, the consumer will purchase the good from the nearer firm. The cost of transportation is t per unit of distance, and, without loss of generality, production cost is normalized to zero.

We model a three-stage game as in Rothschild et al. (2000). In stage one, three firms enter the market simultaneously and choose locations. In stage two, the two firms on the left consider merger to capture the profits that would otherwise be lost through price discrimination at a later stage. In stage three, both the merging firms and the excluded firm engage in spatial discriminatory pricing and announce delivered price schedules.

Unlike all previous work in this area, we do not assume that for all locations r is greater than the limit price set by the delivered cost of adjacent rivals. Figure 1 portrays the case from the previous literature. Figure 2 portrays the case in which all firms are constrained by the reservation price. Thus in the previous literature the bound on the delivered price is given entirely by the adjacent rival's cost structure, but in the constrained case the upper bound on the delivered price is given in part by the rival's delivered cost structure but in part by the reservation price. These respective upper bounds define the profit for each firm, identified as [[PI].sub.i] in the figures. The extent to which the bound is given by r depends on the relative size of t compared to r, and we follow the literature normalizing r to a value of 1 so that all values of t will be expressed implicitly in terms of r. Also following the literature, we assume firms 1 and 2 split the merger profit, with [alpha] representing the share of firm 1 and (1-[alpha]) representing the share of firm 2. The profit from merger is identified as [[PI].sub.M] in the Figures 1 and 2.

[FIGURE 1-2 OMITTED]

IV. CONSTRAINED DEMAND

The profits of the three firms in the constrained case illustrated in Figure 2 are as follows (recognizing the normalization of r to 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where b=(1/2)([L.sub.1]+[L.sub.2]), c=(1/2)([L.sub.1]+[L.sub.3]) and d=(1/2)([L.sub.2]+[L.sub.3]). Maximizing each firm's profit with respect to its own location generates location reaction functions. The functions are solved simultaneously to yield equilibrium locations. Unlike the case without constrained demand, the locations are a function of transportation cost as well as firm 1's share of the profit from merger:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Absent the possibility of merger firms adopt the cost minimizing locations: 1/6, 1/2, 5/6. Thus in contrast to the Hotelling model explored by Wang and Yang (1999), the imposition of a demand constraint does not alter equilibrium locations in the model of spatial price discrimination in which no merger can occur.

Comparative statics of the locations in (2) yields the following:

PROPOSITION 1. When mergers are allowed and demand is constrained:

a. [delta][L.sub.i]/[delta][alpha] > 0, [for all]i. Thus all firms move to the right as [alpha] increases.

b. [delta][L.sub.1]/[[delta]t >0 or [less than or equal to] 0 as [alpha] < 0.375 or [greater than or equal to] 0.375 and [delta][L.sub.i]/[delta]t > 0 or [less then or equal to] 0 as [alpha] < 0.750 or [greater than or equal to] 0.750 for i=2 or 3. Thus locations move in response to changes in transportation cost, but the direction depends on [alpha].

c. ([delta][L.sub.3]/[delta][alpha]-[delta][L.sub.1]/[delta][alpha]) < 0, [for all][alpha], t and ([delta][Lsub.3]/ [delta]t-[delta][L.sub.1]/[delta]t) > 0, [for all][alpha], t. Firms locate closer together as [alpha] increases and locate further apart as t increases.

Table 1 presents equilibrium locations assuming merger for two illustrative constrained cases (t = 4 and t = 5) and for the range of [alpha] as derived from (2) as well as locations for the unconstrained case. (2) The table illustrates the general propositions. In particular, note that as firm 1's share of the merger profit increases, all firms move to the right, with firm 3 moving against a fixed point, causing firm 3's market share and profits to decrease. Also note that as t increases, the market becomes more constrained by willingness to pay (analogous to decreasing r) and the distance between firms 1 and 3 increases. The constraint on the market is a function of r/t (see Figure 2). As t increases, the firms spread out and locate nearer the "monopoly" positions achieved when the market is split into thirds. This result contrasts directly with that derived for the Hotelling model by Wang and Yang (1999).

Following the tradition of using total transportation cost as a measure of social welfare, the model implies:

PROPOSITION 2. Merger reduces social welfare (that is, increases transportation cost) for all [alpha].

Proof. The transport costs in the constrained market, the lower envelope of the delivered cost curves in Figure 2, are calculated using firm locations and are simplified as a function of [alpha] and t:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking the derivative of this with respect to [alpha] yields a cost-minimizing value of [alpha]= 0.643. Substituting [alpha] = 0.643 in (3) results in a function of t only:

(4) [C.sub.c](t) = 0.027(11.02-3.67t+3.367[tsup.2])/t.

For all t this minimum cost associated with constrained demand exceeds that which arises when there exists no possibility of merger.

For example, when t = 4, the cost under merger is 0.342 and that with no merger is 0.333. When t = 5, the merger and no merger costs are 0.418 and 0.4165, respectively. As t continues to grow, separate local monopolies eventually arise (more detail on this is given in section VI) and obviously the possibility of merger becomes irrelevant.

PROPOSITION 3. Merger with constrained demand reduces social welfare relatively less than does merger with unconstrained demand.

Proof. Define relative efficiency as the ratio of costs under merger to those without merger (0.08330t). (3) For the constrained case this comes from (3) and is [C.sub.c]/0.0833t. For the constrained case it is [C.sub.uc]/0.0833t, where [C.sub.uc]=(t[30-36[alpha] + 20[apha.2]-8[[alpha].sup.3] + 3[[alpha].sup.4]/2[[12-6[alpha]+[[alpha.sup.2].sup.2). The unconstrained ratio depends only on [alpha] and is always less than the constrained ratio for relevant values of t and [alpha].

This result follows intuitively from the recognition that one consequence of merger, in both the constrained and unconstrained cases, is generally to bunch the firms together. Yet the imposition of the demand constraint causes firms to behave more like monopolies and spread out. Thus in the face of a demand constraint merger is relatively less inefficient than it is when the constraint is absent.

PROPOSITION 4. Both merging firms individually benefit from merger when 0.48 < [alpha]< 0.62, [for all]t.

Proof. Setting firms profit with merger equal to that without yields critical values of [alpha]. (4) Firm 1's profits increase under merger when [alpha] > 0.48, and firm 2's profits increase under merger when [alpha] < 0.62.

PROPOSITION 5. Firm 3 is harmed by merger when [alpha] > 0.75.

Proof. Set firm 3's profit functions with and without merger equal to each other and solve for [alpha].

COROLLARY 1. There exists no range of [alpha] where merger results in increased individual profit for the merging firms and decreases the profit for the excluded firm.

Following Propositions 4 and 5, the range of [alpha] where merger is rational for firms 1 and 2 lies outside the range of [alpha] where the excluded firm is harmed by the merger. The 'merger paradox' exists for all [alpha] in the case where demand is constrained. Thus, for those products characterized by high transport cost relative to willingness to pay, the resolution of the merger paradox presented by Rothschild et al. (2000) vanishes.

Intuitively, when firms operate subject to a demand constraint, the cost (in terms of forgone revenue) to the leftmost firm as it moves rightward with merger is greater than it would be in the absence of the constraint. This means that, for any given [alpha], this firm's location will lie to the left of the location it would otherwise occupy. Firm 2 will consequently also be located further to the left than it would be in the absence of the demand constraint. This reduces the degree to which firm 3's profits are squeezed by the merger. The result is that the participants' relative profits are lower and the excluded firm's profits higher than would be the case if there were no constraint.

V. EFFICIENCY AND A TAX ON TRANSPORT

The results of the last section are now used to examine the tax policy implications for a government that is, for whatever reason, unable to simply prevent mergers from taking place. As was made clear by Proposition 2, such a prohibition eliminates the inefficient location choices, but an outright ban on mergers constitutes a considerably more severe policy intervention than does the imposition of taxes. Here we demonstrate that the two approaches can achieve similar objectives.

We modify the model of the last section to allow a regulatory authority a first-stage move that consists of setting a transport tax, k, which is proportional to t, the existing transport cost. (5) Thus the firms face a gross per unit transport cost of gt, where g=(1 + k). The authority is presumed to maximize welfare (minimize inefficiency) and the use of the tax instrument is presumed to come at an efficiency cost that we designate as a proportion, [lambda], of the total tax. This could represent the cost of raising the tax or be a measure of the allocative inefficiency associated with the tax (and eventual spending). The remainder of the game proceeds as before. We suppose, for the purposes of this exercise, that [alpha] is common knowledge. Then, given [alpha], the resulting objective function for the authority is to minimize with respect to g (from which k can be solved) the following:

(5) L = [C.sub.c]([alpha],gt) - (1 - [lambda]) (g-1)[C.sub.c]([alpha],gt).

The first term is the gross transport cost, including the portion that is tax. The second term is the value of the tax revenue diminished by the inefficiency cost associated with using the tax scheme. The constrained cost function, [C.sub.c], is obtained from (3).

The authority imposes the tax on transport, an action that causes the firms to move toward the symmetric and cost minimizing locations. This is accomplished, however, at the inefficiency cost of using the tax scheme.

PROPOSITION 6. For all values of t that constrain demand, there exists a [lambda] > 0 such that inefficiency is reduced by setting k so as to create local monopolies.

Proof. Local monopolies are created when gt = T-6. Thus, it is sufficient to show that for [lambda]>0, [C.sub.c]([alpha], gt)-(1-[lambda])(g-1)[Csub.c]([alpha],gt)< [Csub.c]([alpha], t) when gt = T. When [lambda] = 0, this becomes (t/T)[Csub.c]([alpha], T) < [C.sub.c]([alpha], t). This is always true as the left-hand-side is the resource cost associated with transport from the local monopoly locations, which are efficient, and the right-hand-side is the transport cost associated with constrained demand locations which are inefficient (by Proposition 2). Thus, there exists a small enough [lambda] > 0 for which the inequality holds.

This surprising result indicates that a policy of setting taxes so as to create local monopolies and eliminate price competition can lead to a pareto improvement. By substituting (3) into (5), the range of values for which this is true can be derived and these are illustrated in Table 2. (6) These values show that even when the extent of tax inefficiency is very great, it often remains preferable to create local monopolies. For example, when [alpha]=0.5 and t =4.0, as long as the inefficiency cost is less than 37.5% of the tax revenue, it remains preferable to create local monopolies.

In general, the creation of local monopolies not only is pareto superior to a policy of not taxing but also maximizes welfare. The exception arises for very low values of [alpha] and t. Recall that this is the situation in which the firms are most bunched to the left and small increases in the tax generate large gains in efficiency from relocation. In this case an intermediate tax rate can be optimal.

PROPOSITION 7. For small [alpha] and t, there exists a range of [lambda] such that 0 < k* < (T/t - 1). For all other [alpha] and t, k* = 0 or k* = ( T/t - 1 ).

Proof. Substitute (3) into (5), choose values of [alpha] and t from Table 2, and take the derivative with respect to g and set equal to zero. Solving the resulting equation yields three roots, two of which are generally either imaginary or fall outside the range 0 < k* < (T/t-1), but one that falls inside the range for low values of [alpha] and t is a local minimum.

These results follow from the simulation presented in Table 2, which reveals the existence of only five cases with interior maxima. (7) Table 3 shows the ranges of tax inefficiency for which an interior maximum would be adopted for these five cases. In all other cases the optimal policy involves either no taxation or the creation of local monopolies.

The cases generating an interior solution merit a little more attention. As an extreme example consider the case when [alpha]=0 and t=4. Now imagine that [lambda] = 1. Thus any tax revenue raised has such a high inefficiency cost that it adds nothing to welfare. Minimizing (5) is now just minimizing [C.sub.c]([alpha] = 0, g[t = 4] ) with respect to g. The optimal value is k*=0.082. Thus, even though every dollar of tax is completely wasted, an 8.2% tax on transport costs actually lowers total transport cost. Such a result is completely dependent on the demand constraint and cannot be generated in its absence.

VI. ALTERNATIVE CASES AND THE NATURE OF EQUILIBRIUM

The preceding two sections have examined the case of constrained demand as illustrated in Figure 2, but whether the appropriate Figure is 1 or 2 depends on the relationship between the exogenous variables r, [alpha], and t. This section isolates that relationship and identifies cases of partial constraint in which some but not all firms face a demand constraint or in which the profit from merger is constrained even if the individual firm profits are not. To isolate these cases in two dimensions, the reservation price, r, is again normalized to 1. The relationships between [alpha] and t that identify the different cases are depicted in Figure 3. First, assuming [alpha] approaching zero, as transport cost increases, the market initially appears as shown in Figure 1. As t reaches approximately 1.4, the merger profit becomes constrained by r. As t increases further, firm 3's profit is demand constrained, with both firms 2 and 3 demand constrained when t [member of] (3, 4) and [alpha] = 0. Finally, for t [member of] (4, 6) and [alpha] = 0, all firms are demand constrained, as shown in Figure 2.

[FIGURE 3 OMITTED]

Given that firm locations shift to the right when [alpha] increases, a similar pattern of change in the appearance of the market can be identified for values of [alpha] close to one. Again, as t initially increases only the profit from merger is constrained by r. As t increases further, firm 1 becomes demand constrained with both firms 1 and 3 constrained when t increases beyond two. Finally, when t > 3.5, all firms are demand constrained.

Regardless of the value of [alpha], t [greater than or equal to] 6 results in "monopolies" with each firm serving one-third of the market. In this case, firms each locate in the center of their market segment, resulting in firm locations of 1/6, 1/2, 5/6, the cost-minimizing locations.

The earlier sections contrast the results when all firms are constrained (upper portion of Figure 3) with those from the unconstrained model. Although other possible pictures of demand constrained markets exist (e.g., only two constrained firms), the broad conclusions drawn for the case where all three firms are constrained hold when only one or two firms are constrained. (8)

A related concern is whether equilibrium when all firms are constrained includes allowing part of the market not to be served. (9) There are four possibilities: The left hand of the market is not served, market between firm 1 and firm 2 is not served, market between firm 2 and firm 3 is not served, and finally, the right-hand-side of the market is not served. All four can be ruled out. We consider here the right-hand-side of the market not served and market not served between firms 2 and 3. The other two cases follow analogously. (10) Take first the case of the constrained market as illustrated in Figure 2. We recognize that it can be easily drawn so that part of the market on the right is not served as shown in Figure 4. Yet,

[FIGURE 4 OMITTED]

PROPOSITION 8. When the market is fully constrained and t [less than or equal to] 6, the equilibrium will serve the entire market on the right.

Proof. Assume there exists market not served of amount [epsilon] on the right of the location of firm 3. Given the location of firm 3, firm 1 and 2 adopt Cournot-Nash locations resulting in the intersection of the cost structures for firm 2 and firm 3. This intersection is sufficient to show that firm 3 can earn greater profit moving to the right and reducing [epsilon]. In the limit [epsilon] = 0 and in equilibrium there is no market not served on the right. Details are in the appendix.

The fact that firms 1 and 2 earn greater profit locating so that the cost structures of firms 2 and 3 intersect is sufficient to prove

COROLLARY 2. When the market is fully constrained and t [less than or equal to] 6, the equilibrium will serve the entire market between firms 2 and 3.

The fact that the market is fully served when demand is constrained is sensible because a segment of market not served invites a change in location by one or more firms in an effort to increase profit. When the ratio of t to r (r normalized to one) is as shown in the upper portion of Figure 3, no firm can be a monopoly in equilibrium, emphasizing the incentive to serve the entire market. (11)

VII. A CASE OF PARTIALLY CONSTRAINED DEMAND

In this section we examine the partially constrained case in which individual firms are not constrained by consumers' reservation price, r, however, the area of merger profit is constrained by this upper bound. This situation is depicted in Figure 5, and the relevant values of [alpha] and t for which it holds are shown in Figure 3.

[FIGURE 5 OMITTED]

The profits of the three firms and the merger profit are as follows:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Maximizing these profits in the case of merger again yields firm locations solely as a function of [alpha].

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These locations are identical to those in an unconstrained market, presented in Table 1. Comparative statics on the locations are thus identical to those presented for firms in unconstrained markets (Rothschild et al. [2000]). Profit, however, is a function of t as well as [alpha].

PROPOSITION 9. Merger reduces social welfare.

Proof. Total transport cost for the three firms under merger can be expressed as a function of [alpha] and t thus:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For all [alpha] and t this is greater than the total cost when there is no possibility of merger, 0.0833t. Given merger, the value of a that minimizes firm cost is 0.895.

PROPOSITION 10. There exists a range of [alpha] such that merger is individually rational for the merging firms. This range decreases as t increases.

Proof. The profits of firms 1 and 2 when there is no possibility of merger are 0.0833t and 0.0555t, respectively. Set these profits equal to firm profit under merger in (6) and solve for [alpha]. The critical level of [alpha] is a function of t. For both firms 1 and 2, as t decreases the range of [alpha] that yields profit at least equal to the no merger level increases. When t= 1.5, firm 1 benefits from merger when [alpha] > 0.347 and firm 2 benefits from merger when [alpha] < 0.886. When t = 1.3, firm 1 benefits from merger when [alpha] > 0.336 and firm 2 benefits from merger when [alpha] < 0.895.

PROPOSITION 11. Firm 3 is harmed by merger when [alpha]>0.873, [for all] t.

Proof. Set profit of firm 3 with merger equal to its profit without merger, 0.0833t. Firm 3's profit decreases due to merger when [alpha] > 0.873, regardless of the value of t.

COROLLARY 3. For t < 1.67 there exists a range of [alpha] such that the merging firms individually benefit from merger, the excluded firm is harmed by merger, and merger decreases social welfare.

In the partially constrained case, this range of [alpha] is a function of transportation cost and decreases as t increases. When t = 1.3 this range is [0.873, 0.895] and decreases to [0.873, 0.886] when t-1.5. When t= 1.67, firm 2 benefits from merger when [alpha] <0.873 and firm 3 is hurt by merger when [alpha] > 0.873, a fact that generates the merger paradox. Thus the merger paradox is only resolved for relatively high levels of r/t--where individual firm's profits are unconstrained--represented by areas 1 and 2 of Figure 3. Even here the range of [alpha] that resolves the paradox appears trivially small. In this way the partially constrained case yields results that lie between the unconstrained and the constrained cases.

VIII. CONCLUDING COMMENTS

Although the imposition of a demand constraint in a Hotelling framework can result in relocation, it does not do so in a model of spatial price discrimination. The introduction of the possibility of merger does, however, result in locations that are influenced by such a constraint. Specifically, the efficiency distortion generated by the possibility of merger is reduced by the introduction of constrained demand as firms spread out in response to lower willingness to pay. Nonetheless, merger always reduces efficiency even in the case of constrained demand.

In the face of constrained demand, the resolution of the merger paradox obtained for the unconstrained case vanishes. There is no sharing rule for which the parties to the merger gain and the excluded firm is harmed. In the partially constrained case, mergers remain inefficient and there exists a very small range of [alpha] in which the merging firms gain and the excluded firm is harmed.

The introduction of the constraint results in locations that depend on transport cost. This gives rise to the possibility that transport taxes might increase welfare. Moreover, the optimal tax is often that which eliminates spatial price competition and creates local monopolies. Whether this is true depends on the extent of the tax inefficiency. Yet even when the entire tax revenue is wasted, there remain cases in which it is optimal to levy transport taxes because they reduce total transport cost by moving firms toward more symmetrical locations.

We note, in regard to the analysis of the tax question, that in our earlier formulation, [alpha] was taken to be common knowledge. The implication of this assumption is that the participants in the merger are not in a position to select [alpha] in anticipation of the tax. Although the idea of selecting [alpha] lies outside of the central focus of the present article, we have already observed there exist critical values of [lambda] above which, depending on [alpha] and t, the government may not be able to effectively implement the tax. One consequence of this is that, given t, in choosing to merge, firms might avoid those particular values of [alpha] for which the imposition of the tax might be practicable. This will obviate the prospect that the merger will be taxed out of existence. Thus, for example, if t = 5.8, then if [lambda] is greater than 3.6%, of tax revenue, no merger will be discouraged; but if [lambda] is greater than 4.0% then only mergers involving [alpha] in the range [0, 0.3.] will be viable.

APPENDIX

Firm Profits in the Constrained Market

Without merger:

[[PI].sub.1]=0.5-0.5(1/t)-0.04167t

[[PI].sub.2]=0.667-(1/t)-0.05556t

[[PI].sub.3]=0.5-0.5(1/t)-0.04167t

With merger:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Details of Proof for Proposition 8

Remembering that r=1, set [L.sub.3] = 1-(1/t)-[epsilon]. Consider two cases: the cost structure of firms 2 and 3 intersect, and they do not intersect. In the first case substituting the new definition of [L.sub.3] into the profit expressions of firms 1 and 2 in (2), maximizing each firm's profit with respect to its own location, and solving the resulting equations generates:

(A-1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the second case assume that the leftmost point of firm 3"s cost structure is separated by [gamma] [greater than or equal to] 0 from the rightmost point of firm 2's cost structure. Thus, [L.sub.3]=1-(1/t)-[epsilon] and [L.sub.2]=1-[epsilon]-[gamma]-(3/t). Firm 1 maximizes, given these locations, which yields

(A-2) [L.sub.1]=([alpha]-1)[t([epsilon]+[gamma])]+ t-3+5[alpha]-[alpha]t/(3-[alpha])t.

Returning the locations from both cases into the relevant profit functions yields two potential equilibrium profit functions for firms 1 and 2 associated with allowing an intersection between the cost functions of firm 2 and 3, identified by I, and not allowing the intersection, identified by NI: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Comparing these profit functions shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus firms 1 and 2 locate so as to allow an intersection in the cost structures. Recognizing this, firm 3 can earn greater profit by moving to the right, away from the intersection and into the market not being served. This proves that the location of firm 3 associated with the original [epsilon] is not an optimal response to the location choices of firms 1 and 2. This is true for any [epsilon] > 0, so that in equilibrium it must be the case that [epsilon] = 0.

Taking the derivative of [[pi].sup.NI.sub.1] and [[pi].sup.NI.sub.2] with respect to [gamma] yields the expressions

(A-3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Both of these are less than zero for the relevant ranges of t, [alpha], [epsilon], [gamma]. Thus, firms 1 and 2 earn greater profit by eliminating any market not served between firms 2 and 3.

ABBREVIATIONS

CA: Conventional Arbitration

FOA: Final-Offer Arbitration

NHL: National Hockey League

DOI: 10.1093/ei/cbh044

TABLE 1
Equilibrium Locations

 Alpha 0 0.1 0.2 0.3 0.4 0.5 0.6

Unconstrained
market
 L1 0.083 0.104 0.125 0.147 0.168 0.189 0.210
 L2 0.250 0.264 0.280 0.300 0.324 0.351 0.384
 L3 0.750 0.754 0.760 0.766 0.775 0.783 0.795

Constrained
market
t=4 L1 0.083 0.109 0.132 0.153 0.171 0.188 0.202
 L2 0.025 0.297 0.338 0.375 0.408 0.438 0.464
 L3 0.075 0.766 0.779 0.792 0.802 0.813 0.821
t=5 L1 0.133 0.144 0.153 0.161 0.168 0.175 0.181
 L2 0.400 0.419 0.435 0.450 0.463 0.475 0.486
 L3 0.800 0.806 0.812 0.817 0.821 0.825 0.829

 Alpha 0.7 0.8 0.9 1

Unconstrained
market
 L1 0.230 0.250 0.269 0.285
 L2 0.421 0.464 0.514 0.571
 L3 0.807 0.821 0.838 0.857

Constrained
market
t=4 L1 0.216 0.228 0.240 0.250
 L2 0.489 0.511 0.531 0.550
 L3 0.829 0.837 0.844 0.850
t=5 L1 0.186 0.191 0.196 0.200
 L2 0.495 0.504 0.513 0.520
 L3 0.832 0.835 0.838 0.840

TABLE 2
Critical Values of Tax Inefficiency
 t

 4.0 5.0 5.8

[alpha] 0.0 0.505 0.333 0.066
 0.1 0.495 0.277 0.055
 0.2 0.466 0.239 0.049
 0.3 0.425 0.213 0.042
 0.4 0.393 0.197 0.039
 0.5 0.375 0.187 0.037
 0.6 0.367 0.184 0.036
 0.7 0.367 0.184 0.036
 0.8 0.374 0.187 0.037
 0.9 0.386 0.193 0.038
 1.0 0.400 0.800 0.040

Note: Values of [lambda] * such that when [lambda] < [lambda] * local
monopolies are welfare superior to delivered price competition.

TABLE 3
Ranges of Tax Inefficiency
 4.0 5.0 5.8

[alpha] 0.0 0 to 0.500 0.285 to 0.461 0.065 to 0971
 0.1 0.491 to 0.850 0.273 to 0.310

Note.: The range of [lambda] such that 0 < k * < (T/t) - 1 is
optimal.

(1.) Thus airline flights between city pairs differ by departure time from early morning to late evening, the editorial policies of newspapers differ from liberal left to conservative right, and breakfast cereals differ in their sugar content.

(2.) The unconstrained case follows from Figure 1 and is taken from Rothschild et al, (2000).

(3.) The value of 0.0833t is identical for both the constrained and unconstrained case as the locations are identical from equations (2).

(4.) The merger profits for the two firms are obtained by substituting the equilibrium locations in (2) into the original profit expressions in (1).

(5.) In a very different application Heywood and Pal (1996) consider the efficiency consequences of transport taxes for firms in spatial markets.

(6.) The critical values of [lambda] are calculated by taking specific values of [alpha] and t and calculating the associated transport cost from (3). This cost is set equal to L from (5), after substituting the same values of [alpha] and t and also substituting 6/t for g. As is evident from (5), the result is a linear expression in [lambda] that allows a unique solution.

(7.) The simulation is done in MapleV and is available from the authors.

(8.) Proofs are available from the authors.

(9.) Thanks are expressed to a reviewer who suggested this line of inquiry.

(10.) These are available from the authors.

(11.) With arbitrary locations it is obviously possible to generate market segments not served as in Figure 4, but there are infinitely many arbitrary locations and we focus only on the equilibrium locations.

REFERENCES

Economides, N. "The Principle of Minimum Differentiation Revisited." European Economic Review, 24(2), 1984, 345-68.

Gupta, B., J. S. Heywood, and D. Pal. "Duopoly, Delivered Pricing and Horizontal Mergers." Southern Economic Journal, 63(3), 1997, 585-93.

Heywood, J., and D. Pal. "How to Tax a Spatial Monopolist." Journal of Public Economics, 61 (1), 1996, 107-18.

Heywood, J. S., K. Monaco, and R. Rothschild. "Spatial Price Discrimination and Merger: The N-Firm Case." Southern Economic Journal, 67(3), 2001, 672-84.

Lederer, P., and A. Hurter. "Competition of Firms: Discriminatory Pricing and Location." Econometrica, 54(3), 1986, 623-40.

Pepall, L., D. Richards, and G. Norman. Industrial Organization: Contemporary Theory and Practice. Cincinnati: South-Western College Publishing, 1999.

Reitzes, J., and D. Levy. "Price Discrimination and Mergers." Canadian Journal of Economics, 28(2), 1995, 427-36.

Rothschild, R., J. S. Heywood, and K. Monaco. "Spatial Price Discrimination and the Merger Paradox." Regional Science and Urban Economics, 30(5), 2000, 491-506.

Salant, S., S. Switzer, and R. Reynolds. "Losses from Horizontal Merger; the Effects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium." Quarterly Journal of Economics, 98(2), 1983, 185-213.

Thisse, J., and X. Vives. "On the Strategic Choice of Spatial Price Policy." American Economic Review, 78(1), 1988, 122-37.

Tirole, Jean. The Theory of Industrial Organization. Cambridge MA: MIT Press, 1988.

Wang, H., and B. Yang. "On Hotelling's Location Model with a Restricted Reservation Price." Australian Economic Papers, 38(3), 1999, 259-75.

KRISTEN MONACO, JOHN S. HEYWOOD, and R. ROTHSCHILD *

* Affiliations of the authors are respectively Department of Economics, Long Beach State University, Department of Economics, University of Wisconsin-Milwaukee and Department of Commerce, University of Birmingham and Department of Economics, Lancaster University.

Monaco: Associate Professor, Department of Economics, California State University-Long Beach, Long Beach, CA 90840. Phone 1-562-985-5076, Fax 1-562-985-5804, E-mail [email protected]

Heywood: Professor, Department of Economics, University of Wisconsin-Milwakee, P.O. Box 413, Milwakee, WI 53201. Phone 1-414-229-4310, Fax 1-414-229-5915, E-mail [email protected]

Rothschild: Professor, Department of Economics, Lancaster University, Lancaster LAI 4YX, UK. Phone +44 1524 594217, Fax +44 1524 594244, E-mail [email protected]