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Spatial competition with free entry, Chamberlinian tangencies, and social efficiency
JOURNAL
OF URBAN
ECONOMICS
15,
210-286 (1984)
Spatial Competition with Free Entry, Chamberlinian Tangencies, and Social Efficiency’ BRUCE L. BENSON Department of Agricultural
Economics and Economics, Montana State University, Bozeman, Montana 59717
Received April 15,1982; revised November 22,1982 Free entry in Liischian spatial competition leads to a tangency between each firm’s negatively sloped average revenue and the downsloping portion of average costs-as in Chamberlin’s monopolistic competition. It is generally concluded that this equilibrium involves too many inefficiently small firms. However, this conclusion is incorrect. ‘Ihe difference between price and firm marginal production costs in spatial equilibrium is just sufficient to cover the additional marginal cost of output resulting from availability of multiple locations. This Chamberlinian tangency does not imply inefficiency, because it does not include all the social costs and benefits resulting from spatial competition.
The demand functions which spatially competitive firms face are not perfectly elastic, even when large numbers of firms are involved in competition [28]. Therefore, a zero profit long-run equilibrium in spatial competition involves a tangency between each spatial firm’s demand and the negatively sloped section of the firm’s average cost curve, a la Chamberlin [11].2 Consequently, it is generally contended that free entry to zero profits in an industry characterized by spatially distributed buyers and sellers and costly distance, must provide an inefficient allocation of resources, since price exceedseach firm’s marginal production costs. Mills’ and Lav’s classic paper on Lbsch’s model of spatial competition, for example, maintained that The model presented in this paper bears a family resemblance to models of monopolistic competition. In particular, free entry results in the familiar tangency condition between the negatively sloped demand curve and the average cost curve, and therefore f.o.b. price equals average cost. The f.o.b. price thus exceeds marginal production cost, and delivered price exceeds marginal produc‘I thank M. L. Greenhut, Thomas Cosimano, and Hiroshi Ohta for helpful comments as I developed this model. ‘Spatial competition does not really fit the textbook monopolistic competition model that is most closely associated with Chamberlin [28]. Rather, spatial competition more closely fits the linked oligopoly model which Chamberlin discussed[ll, pp. 103,104]. However, with entry to zero profits these oligopoly firms end up with demand curves tangent to the downsloping section of their average cost curves. 270 0094-1190/84 $3.00 Copyright All rights
8 1984 by Academic Press, Inc. of reproduction in any form reserved
SPATIAL COMPETITION
271
tion plus transportation cost. Therefore, in any welfare model that requires the equality of marginal cost and price for a social optimum, our firms misallocate resources. Thus, our model indicates misallocation in the standard sense that monopolistic firms produce too little for a social optimum. This is an inevitable consequence of the zero-profit condition and declining average cost. [38, p. 2861
Mills and Lav contended further that there is a price and market size which would yield a more efficient use of resources than the free entry outcome provides, and that this more efficient solution cannot occur when firms are profit maximizers. Denike and Parr, in light of similar inefficiency claims, concluded that restriction of entry to increase the market area of individual firms would increase social welfare [18, p. 631. In a similar vein, Capozza and Van Order contended that “spatial competition is inefficient because there are too many firms; but having a barrier to entry in the form of large adjustment costs may improve efficiency” [lo, p. 10521. Such welfare implications are inappropriate, however, when we consider the second-best nature of spatial competition.3 A competitive spatial firm in a long-run zero profit equilibrium is not inefficiently small. It is true that the spatial firm has a negatively sloped demand curve and entry to zero profits leads to a Chamberlinian tangency. Therefore, f.o.b. mill price exceeds the firm’s marginal production costs and a first-best solution cannot arise. However, in long-run equilibrium the difference between price and firmspecific marginal production cost is just sufficient to cover a social cost-the additional marginal cost of output resulting from the availability of multiple locations. In other words, spatial competition implies that we are in a second-best situation where the underallocation of resources associated with typical spacelessmodels of imperfect competition tends to be offset because of external (to the firm) social costs and benefits which arise in a spatial market. Mills and Lav appropriately hedged their welfare conclusions with the statement: “in any welfare model that requires the equality of marginal cost and price for a social optimum, our firms misallocate resources” [38, p. 2861.This statement is correct of course, but the appropriate welfare model for analysis of spatial competition is one in which price should exceed 3Not all economists concerned with space accept the welfare implications Mills and Lav [38], Denike and Parr [18], Capozza and Van Order [lo], Villegas [46,47], and others have ascribed to spatial competition. Starrett, for example, demonstrated that efficiency in a spatial setting requires production on the downsloping segment of average cost [43]. However, Starrett wrongly argued that “we cannot expect a freely competitive world to generate optimal solutions” [43, p. 4341,and discussed the optimal location problem from a planning perspective. Tullock, on the other hand, maintained that a zero profit spatial equilibrium is Pareto efficient when producer welfare is taken into account (451. Furthermore, Greenhut has consistently contended that spatial competition leads to an efficient allocation of resources [27-301. The Greenhut argument concerning the efficiency of the spatial firm has been accepted by Ohta [39,40].
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BRUCE L. BENSON
marginal cost-that is, a second-bestmodel. A Chamberlin type of model is a useful analytical tool for examination of the pricing policy of individual spatial ~~MLS.However, it should not be used to infer social werfare implications becausethe model does not consider all of the social costs and benejts
resulting from spatial competition. The same is true of nonspatial imperfectly competitive markets, of course, when selling cost [13-171 or queuing and the cost of waiting [19-211 are considered.Thus, the presentationwhich follows can be viewed as an extension of the Demsetz-DeVany type of argument to include spatial competition. Such an extension seems appropriate in light of claims that spatial competition must be inefficient-claims which continue to appear in the spatial literature (i.e., Capozza and Van Order [lo] Villegas [46,47]). SPATIAL COMPETITION Assume that buyers possess identical demands, and are evenly and continuously distributed along a linear market at a density of 1. The linear market is circular and has a total circumferenceof T. Also let the freight rate per unit of distance be a positive constant t. Consequently P = Pm + tu,
0)
where P denotes the full (or delivered) price a consumer pays, Pm is the f.o.b. mill price, and u representsthe units of distancea consumer’slocation is from a seller. Thus, the quantity demanded at any buying point can be defined as 4 = f(Pm + tu),
(2)
which establishesthe quantity demanded,q, in terms of mill price.4 Eq. (2) is the net (of transportation costs) individual demand function at a buying site u distanceunits from a firm’s location. Let a single tirm begin producing this product, as in Losch [37], and assumethat this firm sells in both directions from his location. Aggregate demand is obtained from (2) by summing individual net demands Q(Pm) = Z/‘f(Pm 0
+ tu) du,
(3)
where U is the maximum distance over which the single firm sells,as given by
4!ke Greenhut aad Ohta [33] for further discussion of this demand representation.
273
SPATIAL COMPETITION
f-l(o) is the price intercept of demand, and T representsthe exogenously given total market space.Thus, this spatial firm obtains profits
(5)
7 = PmQ(Pm) - C(Q),
where C(Q) representsthe firm’s production cost function. If one location is profitable, then the assumptionof an even,continuous distribution of consumersover a linear market indicates that other profitable locations also exist. This is guaranteedunder assumption of identical costs functions for production at every possible location. Manifestly, it is worthwhile for other entrepreneursto establish plants at diverse locations. Aggregate demand of all firms in the market can be calculated by Q(Pm, N) = 2NlUf(Prn
+ tu) du,
0
(6)
where N denotesthe number of firms. This equation presumesfirm locations are evenly spacedthroughout the circular market, sd
u’I f’W - pm t
’
U’IG’
T
We shall assumethat entry is sufficientto serveall consumersso U’ = T/2 N. Since the portion of total salesmade by each firm (S = Q/N) is S = 21U>(Pm + tu) du,
(8)
S = S&m, N; T),
(9)
industry-wide quantity demandedcan be stated as a function of the number of &ms in the market and the mill price. Thus Q = NS(Pm, N).
00)
‘Firm locations are assumed to be evenly dispersed over space in long-run equilibrium, with the realization that this need not be the case under certain circumstances [35]. However, when the space is large, demand slopes downward, and there are no endpoints for the market (as is the case with the circular-shaped linear market assumed here) an even spacing of firms should result in the long run. In fact, in a model such as the one developed here Eaton and Lipsey [25] point out that if demand has some price elasticity firms would always choose to be in the center of their market area and this, in turn, implies that firms will be evenly spaced.
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BRUCE
L. BENSON
Then, following Leibniz’s rule6
$!$ = S + IV?% = S - 2U’f(Pm + tvl).
As more firms are dispersed over the market space, aggregate demand increases.7Initially, when there are only a few firms, additional sites should increase market demand by substantial amounts, but eventually additional plants must mean that the market share going to existing plants declines. That is, U may be defined by an equality from the first part of (4) for a while, but eventually the inequality in the first part of (or equality from the second part of) (7) (and U’) becomes relevant. Thereafter, additional firms increase total industry demand by smaller and smaller amounts. The reason for the above description of industry aggregatedemand is that it demonstrates that the expenditures for alternative locations allow increased benefits to consumers. As the number of sites increases more consumers are served and/or transportation costs for many existing customers fall. Consequently, effective (net) market demand increases.There is obviously a benefit from additional firms that cannot be seen in an individual firm’s revenues. Furthermore, of course, there is a social cost of providing additional locations which is not directly observable in any one firm’s cost relationships. N is not controlled by a single entrepreneur, so the cost of providing alternative locations does not enter into any firm’s profit-maximizing decision. Each firm considers its revenue conditions and its site-specific marginal cost of production in order to choose a profit-maximiring price. Of course, this does not mean that the cost of providing alternative locations does not exist for society to bear. 6Leibniz’s rule implies
-= aN
aN
= gf(Pm
+ TV’) - gf(Pm
au! -= dN
- T
a0 -=o
as -= aN
- -$f(Pm
2N2’ aN
and
+ HI’) + L”‘-$$du g
= 0, it follows that
+ fU) = - $f(Pm
+ W),
and aQ/aN is given by (11). ‘A similar demonstration of the impact of entry on market demand can be found in Benson [4,5], but the proof is repeated here because several of the equations are needed later.
SPATIAL COMPETITION
275
Specialized production and spatial competition will occur only if the average production cost curve of a firm has a downsloping section 191.A sufficient condition for downsloping average cost is assumption of a fixed capital (and/or land) requirement to establish a location and produce. Therefore, assume that each firm’s production function is
s = g(L, 7i).
(12)
L represents the labor (or variable input) usage, and x denotes the fixed level of capital (and land) required for production. Therefore, a firm’s total production costs (TC) are TC = WL + rk
(13)
with w (wages) and r (the rental return to the tixed capital and land needed to produce) representing the market prices for inputs. The &m’s demand for labor depends upon the prices of both inputs, and on S, but the demand for capital is not variable with firm production levels. Therefore, an established firm’s internal marginal production cost is simply ax aL as=Was,
04)
Entry may also alter the cost conditions faced by each existing firm since aTC -=w aN
aL ar --+--+-i ar aN
aL aw aw aN
aL as e+,-g. as aN i +LaN
(15)
Thus, if either wages or rental returns are effected by entry and increased competition for inputs, each firm’s production cost conditions are altered.* Part of the cost of the establishment of an additional firm is borne by existing firms (and their customers), since existing firms’ cost conditions may be altered with entry of a new competitor (Eq. (15)). Therefore, entry may create external adjustment costs. Thus, the following general function is used to represent each firm’s production costs, TC = C(N, S(Pm, N)).
(16)
Of course, there is nothing unique to spatial competition about the above cost function. However, this cost representation is useful since we shah be comparing the costs and benefits of entry (changes in N) in a spatial market. Now let us turn to consideration of the firm’s price setting decision. 80f course, a constant cost industry is possible if ar/aN
= 0 and aw/aN = 0.
BRUCE L. BENSON
276
Each spatial competitor’s profits are B = PmS(Pm, N, V) - C( N, S(Pm, N, U’)).
(17)
U’ now appears in the firm’s cost and demand functions because firms maximize profits over their supply area of radius U’. This U’ is variable in competition in that it depends on the mill price charged by a firm, relative to the mill prices of its nearest competitors. Thus, profit maximization involves
as
ar
-=S+PmK+PmmaPm-------= aPm
as au!
ac as as aPm
ac as av, as au ah
o .
The resulting competitive price is characterizedby ac
Pm==-
S
as aPm
+
--as au av i3Pm
++
1 [
esPm
+
eScl’eU’Pm
1 -l
(19)
where
as Pm
%m=j&yy-’
as u esu =--aul ST
and
avl Pm eupm=-JPm U *
In order for (19) to represent a unique equilibrium price for all spatial competitors an assumption must be made concerning each entrepreneur’s expectationsof competitor reactions to changesin his mill price 1311.Such assumptions place a value on aU’/aPm [9]. The Loschian behavioral assumption is employed in this analysis. That is, each firm assumesits market area is hxed and prices as a monopolist within that area. This assumption is employed for two reasons.First, it is the behavioral assumption used by Mills and Lav [38] (as well as by Denike and Parr [18] and many others).9 Second, Liischian behavior appears to be the appropriate assumptionto characterizethe spatial entrepreneurin the long run [l].” The Loschian assumption implies aU/aPm = O.ll Consequently, (19) simpli91n fact, this probably is the most widely used behavioral assumption in the spatial pricing and market area literature. This behavioral assumption is not the only possibility, however, as shall be noted below. loThe L&chian assumption has been criticized elsewhere [8,9]. However, the criticisms have been shown to be invalid [3,26]. “Other potential behavioral assumptions include the Hotelling-Smithies assumption (%/‘/aPm = - (1/2r)) which is used by Capozza and Van Order [lo], and the Greenhut-Ohta assumption ( W’/aPm = -(l/f)) [9,31].
277
SPATIAL COMPETITION
fies to
ac
Pm==-7.
s
(20)
Under the assumptions of this model, all firms (including new entrants) charge identical mill prices in the long run. This results because (1) consumers with identical gross demands are uniformly and continuously distributed over the linear space; (2) production cost functions are the same at each point in space; (3) firm locations are evenly dispersed over the line market; and (4) entrepreneurs have identical behavioral expectations [2,9]. Now assumeentry continues until profits are driven to zero. This assumption actually requires an assumption about entrepreneurial locational conjectural variation [lo]. It is assumedhere that entering entrepreneurs believe existing firms to be portable so they will move when entry occurs, as in Capozza and Van Order [lo, pp. 104%10491. Without such a locational conjectural variation, profits may exist in an equilibrium setting with no firms entering. In addition, we must assume that total market space (T) is very large [lo, pp. 1046-10471. This allows us to avoid the problem which could arise in a relatively small market space where pure profits could exist but the profits might not be sufficient to support another firm and allow all producers to break even. Given the above assumptions, the long-run equilibrium for a firm occurs when the firm’s perceived (Loschian) demand is tangent to site-specific P, 1
LMC
Pdj.LMC Pm-Q. LWCS MR’CS
/-
< \-
P’ . N’S
FIG. 1. Long-run zero profit equilibrium and margin cost pricing
278
BRUCE L. BENSON
averageproduction cost (C/S) along the downsloping segmentof average cost. Price (and averagecost) is greater than firm-specificmarginal production cost (C, = K/L&) as in Fig. 1, panel i. Therefore, each firm’s total revenue(PmS) equalsits total cost (C). Those who contend that spatial competition leads to an inefficient allocation of resourcestypically end their analysis(if they go this far) by pointing out that price exceedsmarginal production cost (since &S/aPm < 0) and production does not take place at minimum averagecost [38,18, lo]. However, for examination of the welfare implications of spatial competition we must go further. The cost of providing alternativelocations has not yet been considered. The long-run zero profit equilibrium can also be characterizedby noting that industry-wide total revenuesequal industry-wide total cost NPmS = NC(N, S).
(21)
This equation is a defining characteristic of the long-run equilibrium but it gives no information about the marginal conditions which must prevail in the equilibrium-information which is needed in order to examine the efficiency of the market. In order to determine the relevant marginal conditions, we shall employ a procedure often used in models which examinethe impact of an exogenouschangeon an equilibrium.12This forces the systemto move to a new equilibrium and we can seewhat the marginal conditions are in the new equilibrium. Therefore, assumesome exogenous variable, e, changes(e may be demand density, for example),and differentiate (21) with respectto this change. as as c)@de S+PmaPm+PmYW~Pm f3e
+(PmS+ PrnNg)gde
= N ac as as aPm +
(22) Equation (22) is required if a zero profit equilibrium is disturbed, to 12The “differential modeling” which follows is standard practice in several areas of policy analysis in which constraints are used to impose conditions on the movement from one equilibrium to another. Such movements are analyzed by differentiating the constraint equation or equations as (21) is differentiated here. One example of such analysis is the literature on the government budget constraint which results in restrictions on the fiscal policy multiplier. See, for a few examples, Christ [12], Hansen [34], Blinder and Wow [7], and Tobin and Buiter [44].
SPATIAL COMPETITION
279
guarantee that the total changes balance in establishing a new equilibrium. The profit-maximizing effort of each firm (18) implies that the terms multiplying (aPm/ae) de in (22) sum to zero. Therefore,
i
PmS+PmN~)~de=(C+N!&+N~~)~de.
(23)
As long as aN/ae # 0 (and it clearly is not if e is demand density), it must be that the sum of the terms which are multiplied by (JN/ae) de equal zero. Thus, recalling that S + N( k!Y/aN) = (@/aN), the long-run zero profit equilibrium price must be
The long-run equilibrium value of N with free entry results when mill price is equal to what DeVany has called the long-run marginal cost of output (LMC) [21].13The numerator of (24) is the industry-wide full incremental cost of increasing the number of locations.14 This full incremental cost of increasing the number of locations is multiplied by l/( JQ/aN), the increment to the number of firms required to induce a unit change in output.15 Thus, the right-hand side of (24) is the industry-wide full marginal cost of output when output is changed through an expansion in the number of locutions, or LMC. The necessaryand sufficient conditions for obtaining this
marginal cost pricing result are (1) profit-maximizing behavior on the part of individual entrepreneurs (18); and (2) zero profits in any long-run free 13This long-ruri marginal cost is, in fact, not identical to the long-run marginal cost in DeVany [19]. The difference arises because DeVany assumesan industry cost function such that TCI = C(N, Q) whileTC, = NC( N, S) here. 14To see that the numerator of the right-hand side of (24) is the industry-wide full incremental cost of increasing the number of firms, recall the firm’s cost function from (16). Firm cost is multiplied by N to derive the total industry costs of production, TC, = NC( N, S(Pm, N)). Consequently, the change in industry costs resulting from a change in the number of firms is aTc, -= aN
C+N%+N??% aN
as aN
“In order to convert the industry-wide marginal cost of the additional location (see footnote 14) into the full marginal cost of output, the full incremental cost of increasing the number of locations must be multiplied by the increment to the number of firms required to induce a unit change in output [21].
280
BRUCE L. BENSON
entry equilibrium (22). (This industry equilibrium is depicted in Fig. 1, panel ii.) Price set equal to the site-specific marginal cost of output would result in inefficiency. Efficient pricing demands that price equal the marginal cost of output with the efficient number of locations, as in (24).16Thus, spatial firms are marginal cost pricers in long-run equilibrium. This price is greater than the plant-specific marginal cost of production, but the difference between price and this site-specific marginal production cost is the firm’s share of a social cost. This social cost is a charge for the use of capital (and other inputs) to establish alternative locations. One other point should be made. Since firms all break even in the long run, and all charge identical prices, it follows that price equals industry-wide long-run average cost (LAC), where LAC is the average cost of output when output is altered by a change in the number of locations. Consequently, UC is at its minimum. Price equals LMC (26), so LMC cost equals LAC [21]. Thus, LAC is neither rising nor falling. Industry-wide aggregate demand (Q) is not tangent to the downsloping segment of industry-wide LAC then, even though firm demand (S) is tangent to firm-specific average cost (C/S) in the long-run zero profit equilibrium (as in Fig. 1, panel ii). The profit-m axinking procedure of a spatial firm leads to a competitive price given by (20). However, this price must equal the zero profit price from (23) in long-run equilibrium. Therefore,
c ac -=pm=as-as= s
""+Ngg C+NaN
s
aQ
'
(25)
aN
aPm or
--= S as
dPm
ac CfNm-Sz
-aQ aN
ac .
(26)
Once again, the difference between price and site-specific marginal cost is simply the difference between industry LMC and firm marginal cost.17 ‘6Price equal to such a long-run marginal cost has been used as an efficiency condition in a number of recent papers concerned with situations where expansion in the number of firms, plants, or plant capacity impact market demand. See, for example, DeVany [19-211, DeVany and Saving [22,23], and Benson [5,6]. 171t is important to note that even if this industry is characterized by constant cost (aC/aN = 0), LMC is greater than private firm-specific marginal production cost (K/as) since C > S X/as, given the zero profit Chamberlinian-type tangency.
SPATIAL COMPETITION
281
Pm
FIG. 2. Marginal benefits of an additional location.
The marginal cost pricing result obtained here is necessaryfor an efficient allocation of resourcesin a spatial model, but is it sufficient?Market-wide marginal benefits are the increasein consumer surplus associatedwith an increasein N. The changein consumersurplus shah be approximated by the changein the area under the market-wide aggregatedemand (Q) that arises with entry.l* Recall (6) where total industry output is stated as a function of the number of firms and mill price:
Q = Q(Pm,N).
(27)
The total differential of this equation is dQ = -$$dPm
+ $$dN,
(28)
but if we assumethat only very small changes(i.e., dQ - 0) are involved,
dQ KdPm
= - aQ,, aN
or dPm -= dN
JQ
-- m
aQ’
(30)
aPm “We must be cautious in using the area under the demand curve to measure consumer surplus, of course, since it is only au exact measure if the marginal utility of income is constant in the senseof Marshall or Hicks.
282
BRUCE L. BENSON
Equation (30) tells us the change in mill price as the number of firms change,given the changein output is very small, as in Fig. 2. One possible measureof the changein consumersurplus is the shadedarea in Fig. 2. This area is very small when dQ = 0, so the area is approximated by (dPm/dN)Q, ori
-aQ
MB=-i!!!i Q=- Q(S+Nm as) JQ
(31)
JPm Since Q = NS the substitution suggestedby (26) can be made, changing (31) to
MB=(s+Nas)‘~+N~-s~ . (34 I aN 1 Now recall (11) which changes(32) to
ac
MB=C+NaN-S,,.
ac
Industry-wide marginal benefits of a new location equal the difference between LMC and each firm’s marginal production cost as in Fig. 1. This is the efficient result.*’ The necessaryand sufficientconditions for this efficient 19dPm/dN may be positive in Liischian competition (price may rise with entry) [31,9]. However, d Pm/dN > 0 arises only under some demand and cost conditions [3,41,42]. Furthermore, as Donvard notes in paraphrasing Loasby [36]: “given free entry, there are grounds for expecting long-run market demand curves to be much more elastic than short run instantaneous demand curves” [24, pp. 251,252]. Thus, the dPm/dN > 0 result of Lbschian competition should typically only arise in the relatively short run. It arises, after all, because L&churn firm demand becomes less elastic with increases in distant competition when the individual demand function is positive exponential in form (i.e., Pm + tu = a - (b/x)q” x > 0) [3]. But the tendency over time for demand to become more elastic (as it tends to be over the relevant range of a negative exponential demand under Los&an competition) should prevent the surprising results from arising in long-run situations. “The industry-wide marginal cost of an additional location ( 8TC1/8N) is derived in footnote 14. Part of this industry marginal cost of new locations simply involves a movement along producing firms’ marginal production cost curves, however. This portion of the industry marginal cost of locations is, therefore, a private (or internal) marginal production cost which is associated with direct consumption benefits. Thus, to find the industry (or external social)
283
SPATIAL COMPETITION
result include the two conditions noted above for long-run marginal cost pricing, and a third-Loschian behavior on the part of entrepreneurs.‘l The long-run LSschian equilibrium is efficient, in contrast to the Mills-Lav [38], and Den&e-Parr [18] claims. Thus, the conventional view that this Chamberlinian tangency implies that the firm is operating at too small a scale does not hold in spatial competition. The Chamberlinian tangency result of Liischian competition occurs when the optimum number of efficient-sized firms exist. CONCLUSIONS Free entry and profit-m aximizkg behavior, in contrast to what has often been claimed, leads to an efficient use of resources in Los&an spatial competition. The excess of f.o.b. mill price over site-specific marginal production cost is the additional marginal cost of output due to the marginal cost which is not directly accounted for in any firm’s decision and which should, as a consequence, be compared to the benefits of a new location, we must net out this private marginal production cost. The industry marginal cost of a new location net of private cost (MCN) is obtained by subtracting (X/as)( 6’Q/aN) from the equation for aTCl/aN. Thus,
It is this difference between industry (or social) marginal cost of new locations and firm-specific (or private) marginal production cost that should be compared to the marginal benefits of an additional location from (31) (as in (33)). Note that since Pm = C/S this equation can be rewritten as MB=MCN=(Pm-$$)S+Ng, which approximates the change in producer surplus as N changes. *lit should be stressed that the efficiency results obtained here arise only under Lijschian behavior. Other entrepreneurial behavior models (see footnote 11) do not yield the optimal number of locations (although they do yield price equal to long-run marginal cost) because in these cases
-
‘(SNS “)
aQ X --CC+=-Sas,
X
where W’/aPm < 0 and equal to the value for Hotelling-Smithies or Greenhut-Ohta competition from footnote 11. Thus, the marginal benefits of additional locations exceed the marginal costs under these alternative behavioral models, and more locutions would increase social welfare. This result is diametrically opposite the conclusions reached by Capozza and Van Order [lo] who assumed Hotelling-Smithies behavior.
284
BRUCE L. BENSON
existence of alternative production sites. This is a social cost which should be added to the production cost of each firm to obtain the mill price, since consumers receive the benefits in terms of lower transportation costs. Furthermore, the marginal cost of providing the last location equals the marginal benefits obtained from the location. These conclusions are reached even though each firm’s demand curve is tangent to the downsloping portion of the average production cost curve. The mere fact that this tangency condition describes the individual firm in the spatial long-run competitive equilibrium is not sufficient to conclude that spatial competition leads to too many inefficient small firms as Mills and Lav [38], Denike and Parr [18], Capozza and Van Order [lo], and others have argued. These inefficiency arguments are based on an incomplete picture of the social costs and benefits of spatial competition. The basis for the conclusion that Liischian competition is socially efficient is that consumers pay a higher price than producers receive. Consumers incur transportation costs above the price charged by individual firms, but the cost of transportation is reduced when increasing numbers of tlrms are spatially dispersed. It is this spatial dispersion of firms which leads to consumers’ differentiation of firms (by location), downsloping demand, and ultimately to the Chamberlinian tangency result. So, while individual firms may appear to be inefficient, the benefits of multiple firms imply that the market as a whole may be efficient (as in Fig. 1). Chamberlin argued that the existence of “excess capacity” does not necessarily imply an inefficient allocation of resources. Product differentiation (i.e., spatial differentiation) is desired per se; and it inevitably gives rise to negatively sloped demand curves for individual fums. Differentness (location alternatives) is a quality of the product and it entails a cost just as any other quality. The presence of competitive alternatives guarantee that buyers can select the amount of differentness they wish to purchase (the location from which they wish to purchase). Thus, in the case of spatial competition, Chamberlin would regard a tangency to the left of minimum average production cost as a “sort of ideal” for a market in which there is product differentiation [ll, p. 941. This tangency represents more than a “sort of ideal” in the case of Loschian spatial competition. It is a social ideal, since marginal benefits and marginal costs of multiple locations are equated. The same desirable characteristics of long-run equilibrium in the theory of perfect competition are associated with long-run zero profit equilibrium in Loschian competition. This claim will not surprise readers who are familiar with writings by Demsetz [13-171, DeVany [19-211, DeVany and Saving [22,23], Tullock [45], or Greenhut [27-301, but continued claims to the contrary (i.e., Capozza and Van Order [lo], Villegas [46,47]) in the context of spatial pricing models justifies the above presentation.
SPATIAL COMPETITION
285
REFERENCES 1. B. Benson, Spatial microeconomics: Implications for the use of concentration ratios to represent market power, Rev. Regional Studies, 9, l-15 (1979). 2. B. Benson, An examination of U.S. v. Philadelphia National Bank in the context of spatial microeconomics, Ind. Organization Rev., 8, 27-65 (1980a). 3. B. Benson, Lbschian competition under alternative demand conditions, Amer. Econ. Rev., 70,1098-1105 (198Ob). 4. B. Benson, The impact of entry at a distance on market demand, Rev. Regional Studies, 10, 62-68 (198Oc). 5. B. Benson, The optimal sire and number of market areas, Southern Econ. J., 47,1080-1085 (1981). 6. B. Benson, The optimal sire and number of market areas: Reply. Southern Econ. J., 49, 267-271 (1982). 7. A. Blinder and R. Solow, “Does fiscal policy matter?“J. Public Econ., 2, 318-337 (1973). 8. D. Caporza and R. Van Order, Pricing under spatial competition and spatial monopoly, Econometrica, 45, 1329-1338 (1977). 9. D. Capozza and R. Van Order, A generalized model of spatial competition, Amer. Econ. Reu., 68, 896-908 (1978). 10. D. Capozza and R. Van Order, Unique equilibria, pure profits, and efficiency in location models, Amer. Econ. Rev., 70, 1046-1053 (1980). 11. E. Chamberlin, “The Theory of Monopolistic Competition,” 8th ed. Harvard Univ. Press, Cambridge, Mass. (1933). 12. C. Christ, A short-run aggregate demand model of the interdependence and effects of monetary and fiscal policies with Keynesian and classical interest elasticities, Amer. Econ. Rev. Proc., 57, 434-443 (1967). 13. H. Demsetz, The nature of equilibrium in monopolistic competition, J. PO/. Econ., 67, 21-30 (1959). 14. H. Demsetz, The welfare and empirical implications of monopplistic competition, Econ. J., 74, 623-641 (1964). 15. H. Demsetz, Monopolistic competition: A Reply, Econ. J., 77,412-420 (1967). 16. H. Demsetz, Do competition and monopolistic competition differ? J. Pal. Econ., 76, 146-148 (1968). 17. H. Demsetz, The inconsistency in monopolistic competition: A Reply, J. Pal. Econ., 80, 592-597 (1972). 18. K. Denike and J. Parr, Production in space, spatial competition, and restricted entry, J. Regional Sci., 10, 49-63 (1970). 19. A. DeVany, Capacity utilization under alternative regulatory restraints: An analysis of taxi markets, J. Pol. Econ., 83, 83-95 (1975a). 20. A. DeVany, The effect of price and entry regulation on airline output, capacity and efficiency, Bell J. Econ., 6, 327-345 (1975b). 21. A. DeVany, Uncertainty, waiting time, and capacity utilization: A stochastic theory of product quality, J. Pal. Econ., 84, 523-541 (1976). 22. A. DeVany and T. Saving, Product quality, uncertainty, and regulation: The trucking industry, Amer. Econ. Rev., 67, 583-594 (1977). 23. A. DeVany and T. Saving, Competition and highway pricing for stochastic traffic, J. Business,53, 44-60 (1980). 24. N. Dorward, Impacts of distance on microeconomic theory: A critique, Manchester School, 49,245-258 (1981). 25. B. Eaton and R. Lipsey, Freedom of entry and the existence of pure profit, Econ. J., 88, 455-469 (1978). 26. J. Greenhut, M. L. Greenhut, and W. Kelly, “A Spatial-Theoretical Perspective for Bank
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27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.
45. 46. 47.
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Merger Regulation.” Federal Reserve Bank of Chicago Proceedings of a Conference on Bank Structure and Competition, pp. 210-254 (1977). M. Greenhut, “Microeconomics and the Space Economy.” Scott, Foresman, Glenview, Ill. (1963). M. Greenhut, “A Theory of the Firm in Economic Space.” Lone Star Publishers, Austin, Tex. (1971). M. Greenhut, A theoretical mapping from perfect competition to imperfect competition, Southern Econ. J., 42,177-192 (1975). M. Greenhut, Mr. Dorward and impacts of distance on microeconomic theory, Manchester School, 49,259-265 (1981). M. Greenhut, M. Hwang, and H. Ohta, Observations on the shape and relevance of the spatial demand function, Econometrica, 43, 669-682 (1975). M. Greenhut and H. Ohta, Spatial configurations and competitive equilibrium, Weltwirtschajliches Archiu, 109, 87-104 (1973). M. Greenhut and H. Ohta, “Theory of spatial pricing and market areas,” Duke Univ. Press, Durham, N.C. (1975). B. Hanson, On the effect of fiscal and monetary policy: A taxonomic discussion,” Amer. Econ. Rev., 63, 546-571 (1973). H. Hotelling, Stability in competition, Econ. J., 39, 41-57 (1929). B. Loasby, “Choice, Complexity and Ignorance.” Cambridge Univ. Press, Cambridge, Mass. (1976). A. Msch, “The Economics of Location.” Yale Univ. Press, New Haven, Corm. (1954). E. Mills and R. Lav, A model of market area with free entry, J. Pal. Econ., 72, 278-288 (1964). H. Ohta, On efficiency of production under conditions of imperfect competition, Southern Econ. J., 43,1124-1135 (1976). H. Ohta, On the excess capacity controversy, Econ. Inquiry, 15, 153-165 (1977). H. Ohta, Spatial competition, concentration and welfare, Regional Sci. Urban Econ., 10, 3-16 (1980). H. Ohta, The price effect of spatial competition, Reu. Econ. Studies, 48, 317-325 (1981). D. Starrett, Principles of optimal location in a large homogeneous area, J. Econ. Theory, 9, 418-448 (1974). J. Tobin and W. Buiter, Longrun effects of fiscal and monetary policy on aggregate demand, in “Monetarism” (.I. Stein, Ed.), North-Holland, Amsterdam, pp. 273-309 (1976). G. Tullock, Optimality with monopolistic competition, Western Econ. J., 4, 41-48 (1965). D. ViIIegas, Comparative performance of spatial models of economic markets with linear demand, Southern Econ. J., 48, 893-908, (1982a). D. Villegas, The optimal size and number of market areas: Comment, Southern Econ. J., 49,262-266 (1982b).
OF URBAN
ECONOMICS
15,
210-286 (1984)
Spatial Competition with Free Entry, Chamberlinian Tangencies, and Social Efficiency’ BRUCE L. BENSON Department of Agricultural
Economics and Economics, Montana State University, Bozeman, Montana 59717
Received April 15,1982; revised November 22,1982 Free entry in Liischian spatial competition leads to a tangency between each firm’s negatively sloped average revenue and the downsloping portion of average costs-as in Chamberlin’s monopolistic competition. It is generally concluded that this equilibrium involves too many inefficiently small firms. However, this conclusion is incorrect. ‘Ihe difference between price and firm marginal production costs in spatial equilibrium is just sufficient to cover the additional marginal cost of output resulting from availability of multiple locations. This Chamberlinian tangency does not imply inefficiency, because it does not include all the social costs and benefits resulting from spatial competition.
The demand functions which spatially competitive firms face are not perfectly elastic, even when large numbers of firms are involved in competition [28]. Therefore, a zero profit long-run equilibrium in spatial competition involves a tangency between each spatial firm’s demand and the negatively sloped section of the firm’s average cost curve, a la Chamberlin [11].2 Consequently, it is generally contended that free entry to zero profits in an industry characterized by spatially distributed buyers and sellers and costly distance, must provide an inefficient allocation of resources, since price exceedseach firm’s marginal production costs. Mills’ and Lav’s classic paper on Lbsch’s model of spatial competition, for example, maintained that The model presented in this paper bears a family resemblance to models of monopolistic competition. In particular, free entry results in the familiar tangency condition between the negatively sloped demand curve and the average cost curve, and therefore f.o.b. price equals average cost. The f.o.b. price thus exceeds marginal production cost, and delivered price exceeds marginal produc‘I thank M. L. Greenhut, Thomas Cosimano, and Hiroshi Ohta for helpful comments as I developed this model. ‘Spatial competition does not really fit the textbook monopolistic competition model that is most closely associated with Chamberlin [28]. Rather, spatial competition more closely fits the linked oligopoly model which Chamberlin discussed[ll, pp. 103,104]. However, with entry to zero profits these oligopoly firms end up with demand curves tangent to the downsloping section of their average cost curves. 270 0094-1190/84 $3.00 Copyright All rights
8 1984 by Academic Press, Inc. of reproduction in any form reserved
SPATIAL COMPETITION
271
tion plus transportation cost. Therefore, in any welfare model that requires the equality of marginal cost and price for a social optimum, our firms misallocate resources. Thus, our model indicates misallocation in the standard sense that monopolistic firms produce too little for a social optimum. This is an inevitable consequence of the zero-profit condition and declining average cost. [38, p. 2861
Mills and Lav contended further that there is a price and market size which would yield a more efficient use of resources than the free entry outcome provides, and that this more efficient solution cannot occur when firms are profit maximizers. Denike and Parr, in light of similar inefficiency claims, concluded that restriction of entry to increase the market area of individual firms would increase social welfare [18, p. 631. In a similar vein, Capozza and Van Order contended that “spatial competition is inefficient because there are too many firms; but having a barrier to entry in the form of large adjustment costs may improve efficiency” [lo, p. 10521. Such welfare implications are inappropriate, however, when we consider the second-best nature of spatial competition.3 A competitive spatial firm in a long-run zero profit equilibrium is not inefficiently small. It is true that the spatial firm has a negatively sloped demand curve and entry to zero profits leads to a Chamberlinian tangency. Therefore, f.o.b. mill price exceeds the firm’s marginal production costs and a first-best solution cannot arise. However, in long-run equilibrium the difference between price and firmspecific marginal production cost is just sufficient to cover a social cost-the additional marginal cost of output resulting from the availability of multiple locations. In other words, spatial competition implies that we are in a second-best situation where the underallocation of resources associated with typical spacelessmodels of imperfect competition tends to be offset because of external (to the firm) social costs and benefits which arise in a spatial market. Mills and Lav appropriately hedged their welfare conclusions with the statement: “in any welfare model that requires the equality of marginal cost and price for a social optimum, our firms misallocate resources” [38, p. 2861.This statement is correct of course, but the appropriate welfare model for analysis of spatial competition is one in which price should exceed 3Not all economists concerned with space accept the welfare implications Mills and Lav [38], Denike and Parr [18], Capozza and Van Order [lo], Villegas [46,47], and others have ascribed to spatial competition. Starrett, for example, demonstrated that efficiency in a spatial setting requires production on the downsloping segment of average cost [43]. However, Starrett wrongly argued that “we cannot expect a freely competitive world to generate optimal solutions” [43, p. 4341,and discussed the optimal location problem from a planning perspective. Tullock, on the other hand, maintained that a zero profit spatial equilibrium is Pareto efficient when producer welfare is taken into account (451. Furthermore, Greenhut has consistently contended that spatial competition leads to an efficient allocation of resources [27-301. The Greenhut argument concerning the efficiency of the spatial firm has been accepted by Ohta [39,40].
272
BRUCE L. BENSON
marginal cost-that is, a second-bestmodel. A Chamberlin type of model is a useful analytical tool for examination of the pricing policy of individual spatial ~~MLS.However, it should not be used to infer social werfare implications becausethe model does not consider all of the social costs and benejts
resulting from spatial competition. The same is true of nonspatial imperfectly competitive markets, of course, when selling cost [13-171 or queuing and the cost of waiting [19-211 are considered.Thus, the presentationwhich follows can be viewed as an extension of the Demsetz-DeVany type of argument to include spatial competition. Such an extension seems appropriate in light of claims that spatial competition must be inefficient-claims which continue to appear in the spatial literature (i.e., Capozza and Van Order [lo] Villegas [46,47]). SPATIAL COMPETITION Assume that buyers possess identical demands, and are evenly and continuously distributed along a linear market at a density of 1. The linear market is circular and has a total circumferenceof T. Also let the freight rate per unit of distance be a positive constant t. Consequently P = Pm + tu,
0)
where P denotes the full (or delivered) price a consumer pays, Pm is the f.o.b. mill price, and u representsthe units of distancea consumer’slocation is from a seller. Thus, the quantity demanded at any buying point can be defined as 4 = f(Pm + tu),
(2)
which establishesthe quantity demanded,q, in terms of mill price.4 Eq. (2) is the net (of transportation costs) individual demand function at a buying site u distanceunits from a firm’s location. Let a single tirm begin producing this product, as in Losch [37], and assumethat this firm sells in both directions from his location. Aggregate demand is obtained from (2) by summing individual net demands Q(Pm) = Z/‘f(Pm 0
+ tu) du,
(3)
where U is the maximum distance over which the single firm sells,as given by
4!ke Greenhut aad Ohta [33] for further discussion of this demand representation.
273
SPATIAL COMPETITION
f-l(o) is the price intercept of demand, and T representsthe exogenously given total market space.Thus, this spatial firm obtains profits
(5)
7 = PmQ(Pm) - C(Q),
where C(Q) representsthe firm’s production cost function. If one location is profitable, then the assumptionof an even,continuous distribution of consumersover a linear market indicates that other profitable locations also exist. This is guaranteedunder assumption of identical costs functions for production at every possible location. Manifestly, it is worthwhile for other entrepreneursto establish plants at diverse locations. Aggregate demand of all firms in the market can be calculated by Q(Pm, N) = 2NlUf(Prn
+ tu) du,
0
(6)
where N denotesthe number of firms. This equation presumesfirm locations are evenly spacedthroughout the circular market, sd
u’I f’W - pm t
’
U’IG’
T
We shall assumethat entry is sufficientto serveall consumersso U’ = T/2 N. Since the portion of total salesmade by each firm (S = Q/N) is S = 21U>(Pm + tu) du,
(8)
S = S&m, N; T),
(9)
industry-wide quantity demandedcan be stated as a function of the number of &ms in the market and the mill price. Thus Q = NS(Pm, N).
00)
‘Firm locations are assumed to be evenly dispersed over space in long-run equilibrium, with the realization that this need not be the case under certain circumstances [35]. However, when the space is large, demand slopes downward, and there are no endpoints for the market (as is the case with the circular-shaped linear market assumed here) an even spacing of firms should result in the long run. In fact, in a model such as the one developed here Eaton and Lipsey [25] point out that if demand has some price elasticity firms would always choose to be in the center of their market area and this, in turn, implies that firms will be evenly spaced.
274
BRUCE
L. BENSON
Then, following Leibniz’s rule6
$!$ = S + IV?% = S - 2U’f(Pm + tvl).
As more firms are dispersed over the market space, aggregate demand increases.7Initially, when there are only a few firms, additional sites should increase market demand by substantial amounts, but eventually additional plants must mean that the market share going to existing plants declines. That is, U may be defined by an equality from the first part of (4) for a while, but eventually the inequality in the first part of (or equality from the second part of) (7) (and U’) becomes relevant. Thereafter, additional firms increase total industry demand by smaller and smaller amounts. The reason for the above description of industry aggregatedemand is that it demonstrates that the expenditures for alternative locations allow increased benefits to consumers. As the number of sites increases more consumers are served and/or transportation costs for many existing customers fall. Consequently, effective (net) market demand increases.There is obviously a benefit from additional firms that cannot be seen in an individual firm’s revenues. Furthermore, of course, there is a social cost of providing additional locations which is not directly observable in any one firm’s cost relationships. N is not controlled by a single entrepreneur, so the cost of providing alternative locations does not enter into any firm’s profit-maximizing decision. Each firm considers its revenue conditions and its site-specific marginal cost of production in order to choose a profit-maximiring price. Of course, this does not mean that the cost of providing alternative locations does not exist for society to bear. 6Leibniz’s rule implies
-= aN
aN
= gf(Pm
+ TV’) - gf(Pm
au! -= dN
- T
a0 -=o
as -= aN
- -$f(Pm
2N2’ aN
and
+ HI’) + L”‘-$$du g
= 0, it follows that
+ fU) = - $f(Pm
+ W),
and aQ/aN is given by (11). ‘A similar demonstration of the impact of entry on market demand can be found in Benson [4,5], but the proof is repeated here because several of the equations are needed later.
SPATIAL COMPETITION
275
Specialized production and spatial competition will occur only if the average production cost curve of a firm has a downsloping section 191.A sufficient condition for downsloping average cost is assumption of a fixed capital (and/or land) requirement to establish a location and produce. Therefore, assume that each firm’s production function is
s = g(L, 7i).
(12)
L represents the labor (or variable input) usage, and x denotes the fixed level of capital (and land) required for production. Therefore, a firm’s total production costs (TC) are TC = WL + rk
(13)
with w (wages) and r (the rental return to the tixed capital and land needed to produce) representing the market prices for inputs. The &m’s demand for labor depends upon the prices of both inputs, and on S, but the demand for capital is not variable with firm production levels. Therefore, an established firm’s internal marginal production cost is simply ax aL as=Was,
04)
Entry may also alter the cost conditions faced by each existing firm since aTC -=w aN
aL ar --+--+-i ar aN
aL aw aw aN
aL as e+,-g. as aN i +LaN
(15)
Thus, if either wages or rental returns are effected by entry and increased competition for inputs, each firm’s production cost conditions are altered.* Part of the cost of the establishment of an additional firm is borne by existing firms (and their customers), since existing firms’ cost conditions may be altered with entry of a new competitor (Eq. (15)). Therefore, entry may create external adjustment costs. Thus, the following general function is used to represent each firm’s production costs, TC = C(N, S(Pm, N)).
(16)
Of course, there is nothing unique to spatial competition about the above cost function. However, this cost representation is useful since we shah be comparing the costs and benefits of entry (changes in N) in a spatial market. Now let us turn to consideration of the firm’s price setting decision. 80f course, a constant cost industry is possible if ar/aN
= 0 and aw/aN = 0.
BRUCE L. BENSON
276
Each spatial competitor’s profits are B = PmS(Pm, N, V) - C( N, S(Pm, N, U’)).
(17)
U’ now appears in the firm’s cost and demand functions because firms maximize profits over their supply area of radius U’. This U’ is variable in competition in that it depends on the mill price charged by a firm, relative to the mill prices of its nearest competitors. Thus, profit maximization involves
as
ar
-=S+PmK+PmmaPm-------= aPm
as au!
ac as as aPm
ac as av, as au ah
o .
The resulting competitive price is characterizedby ac
Pm==-
S
as aPm
+
--as au av i3Pm
++
1 [
esPm
+
eScl’eU’Pm
1 -l
(19)
where
as Pm
%m=j&yy-’
as u esu =--aul ST
and
avl Pm eupm=-JPm U *
In order for (19) to represent a unique equilibrium price for all spatial competitors an assumption must be made concerning each entrepreneur’s expectationsof competitor reactions to changesin his mill price 1311.Such assumptions place a value on aU’/aPm [9]. The Loschian behavioral assumption is employed in this analysis. That is, each firm assumesits market area is hxed and prices as a monopolist within that area. This assumption is employed for two reasons.First, it is the behavioral assumption used by Mills and Lav [38] (as well as by Denike and Parr [18] and many others).9 Second, Liischian behavior appears to be the appropriate assumptionto characterizethe spatial entrepreneurin the long run [l].” The Loschian assumption implies aU/aPm = O.ll Consequently, (19) simpli91n fact, this probably is the most widely used behavioral assumption in the spatial pricing and market area literature. This behavioral assumption is not the only possibility, however, as shall be noted below. loThe L&chian assumption has been criticized elsewhere [8,9]. However, the criticisms have been shown to be invalid [3,26]. “Other potential behavioral assumptions include the Hotelling-Smithies assumption (%/‘/aPm = - (1/2r)) which is used by Capozza and Van Order [lo], and the Greenhut-Ohta assumption ( W’/aPm = -(l/f)) [9,31].
277
SPATIAL COMPETITION
fies to
ac
Pm==-7.
s
(20)
Under the assumptions of this model, all firms (including new entrants) charge identical mill prices in the long run. This results because (1) consumers with identical gross demands are uniformly and continuously distributed over the linear space; (2) production cost functions are the same at each point in space; (3) firm locations are evenly dispersed over the line market; and (4) entrepreneurs have identical behavioral expectations [2,9]. Now assumeentry continues until profits are driven to zero. This assumption actually requires an assumption about entrepreneurial locational conjectural variation [lo]. It is assumedhere that entering entrepreneurs believe existing firms to be portable so they will move when entry occurs, as in Capozza and Van Order [lo, pp. 104%10491. Without such a locational conjectural variation, profits may exist in an equilibrium setting with no firms entering. In addition, we must assume that total market space (T) is very large [lo, pp. 1046-10471. This allows us to avoid the problem which could arise in a relatively small market space where pure profits could exist but the profits might not be sufficient to support another firm and allow all producers to break even. Given the above assumptions, the long-run equilibrium for a firm occurs when the firm’s perceived (Loschian) demand is tangent to site-specific P, 1
LMC
Pdj.LMC Pm-Q. LWCS MR’CS
/-
< \-
P’ . N’S
FIG. 1. Long-run zero profit equilibrium and margin cost pricing
278
BRUCE L. BENSON
averageproduction cost (C/S) along the downsloping segmentof average cost. Price (and averagecost) is greater than firm-specificmarginal production cost (C, = K/L&) as in Fig. 1, panel i. Therefore, each firm’s total revenue(PmS) equalsits total cost (C). Those who contend that spatial competition leads to an inefficient allocation of resourcestypically end their analysis(if they go this far) by pointing out that price exceedsmarginal production cost (since &S/aPm < 0) and production does not take place at minimum averagecost [38,18, lo]. However, for examination of the welfare implications of spatial competition we must go further. The cost of providing alternativelocations has not yet been considered. The long-run zero profit equilibrium can also be characterizedby noting that industry-wide total revenuesequal industry-wide total cost NPmS = NC(N, S).
(21)
This equation is a defining characteristic of the long-run equilibrium but it gives no information about the marginal conditions which must prevail in the equilibrium-information which is needed in order to examine the efficiency of the market. In order to determine the relevant marginal conditions, we shall employ a procedure often used in models which examinethe impact of an exogenouschangeon an equilibrium.12This forces the systemto move to a new equilibrium and we can seewhat the marginal conditions are in the new equilibrium. Therefore, assumesome exogenous variable, e, changes(e may be demand density, for example),and differentiate (21) with respectto this change. as as c)@de S+PmaPm+PmYW~Pm f3e
+(PmS+ PrnNg)gde
= N ac as as aPm +
(22) Equation (22) is required if a zero profit equilibrium is disturbed, to 12The “differential modeling” which follows is standard practice in several areas of policy analysis in which constraints are used to impose conditions on the movement from one equilibrium to another. Such movements are analyzed by differentiating the constraint equation or equations as (21) is differentiated here. One example of such analysis is the literature on the government budget constraint which results in restrictions on the fiscal policy multiplier. See, for a few examples, Christ [12], Hansen [34], Blinder and Wow [7], and Tobin and Buiter [44].
SPATIAL COMPETITION
279
guarantee that the total changes balance in establishing a new equilibrium. The profit-maximizing effort of each firm (18) implies that the terms multiplying (aPm/ae) de in (22) sum to zero. Therefore,
i
PmS+PmN~)~de=(C+N!&+N~~)~de.
(23)
As long as aN/ae # 0 (and it clearly is not if e is demand density), it must be that the sum of the terms which are multiplied by (JN/ae) de equal zero. Thus, recalling that S + N( k!Y/aN) = (@/aN), the long-run zero profit equilibrium price must be
The long-run equilibrium value of N with free entry results when mill price is equal to what DeVany has called the long-run marginal cost of output (LMC) [21].13The numerator of (24) is the industry-wide full incremental cost of increasing the number of locations.14 This full incremental cost of increasing the number of locations is multiplied by l/( JQ/aN), the increment to the number of firms required to induce a unit change in output.15 Thus, the right-hand side of (24) is the industry-wide full marginal cost of output when output is changed through an expansion in the number of locutions, or LMC. The necessaryand sufficient conditions for obtaining this
marginal cost pricing result are (1) profit-maximizing behavior on the part of individual entrepreneurs (18); and (2) zero profits in any long-run free 13This long-ruri marginal cost is, in fact, not identical to the long-run marginal cost in DeVany [19]. The difference arises because DeVany assumesan industry cost function such that TCI = C(N, Q) whileTC, = NC( N, S) here. 14To see that the numerator of the right-hand side of (24) is the industry-wide full incremental cost of increasing the number of firms, recall the firm’s cost function from (16). Firm cost is multiplied by N to derive the total industry costs of production, TC, = NC( N, S(Pm, N)). Consequently, the change in industry costs resulting from a change in the number of firms is aTc, -= aN
C+N%+N??% aN
as aN
“In order to convert the industry-wide marginal cost of the additional location (see footnote 14) into the full marginal cost of output, the full incremental cost of increasing the number of locations must be multiplied by the increment to the number of firms required to induce a unit change in output [21].
280
BRUCE L. BENSON
entry equilibrium (22). (This industry equilibrium is depicted in Fig. 1, panel ii.) Price set equal to the site-specific marginal cost of output would result in inefficiency. Efficient pricing demands that price equal the marginal cost of output with the efficient number of locations, as in (24).16Thus, spatial firms are marginal cost pricers in long-run equilibrium. This price is greater than the plant-specific marginal cost of production, but the difference between price and this site-specific marginal production cost is the firm’s share of a social cost. This social cost is a charge for the use of capital (and other inputs) to establish alternative locations. One other point should be made. Since firms all break even in the long run, and all charge identical prices, it follows that price equals industry-wide long-run average cost (LAC), where LAC is the average cost of output when output is altered by a change in the number of locations. Consequently, UC is at its minimum. Price equals LMC (26), so LMC cost equals LAC [21]. Thus, LAC is neither rising nor falling. Industry-wide aggregate demand (Q) is not tangent to the downsloping segment of industry-wide LAC then, even though firm demand (S) is tangent to firm-specific average cost (C/S) in the long-run zero profit equilibrium (as in Fig. 1, panel ii). The profit-m axinking procedure of a spatial firm leads to a competitive price given by (20). However, this price must equal the zero profit price from (23) in long-run equilibrium. Therefore,
c ac -=pm=as-as= s
""+Ngg C+NaN
s
aQ
'
(25)
aN
aPm or
--= S as
dPm
ac CfNm-Sz
-aQ aN
ac .
(26)
Once again, the difference between price and site-specific marginal cost is simply the difference between industry LMC and firm marginal cost.17 ‘6Price equal to such a long-run marginal cost has been used as an efficiency condition in a number of recent papers concerned with situations where expansion in the number of firms, plants, or plant capacity impact market demand. See, for example, DeVany [19-211, DeVany and Saving [22,23], and Benson [5,6]. 171t is important to note that even if this industry is characterized by constant cost (aC/aN = 0), LMC is greater than private firm-specific marginal production cost (K/as) since C > S X/as, given the zero profit Chamberlinian-type tangency.
SPATIAL COMPETITION
281
Pm
FIG. 2. Marginal benefits of an additional location.
The marginal cost pricing result obtained here is necessaryfor an efficient allocation of resourcesin a spatial model, but is it sufficient?Market-wide marginal benefits are the increasein consumer surplus associatedwith an increasein N. The changein consumersurplus shah be approximated by the changein the area under the market-wide aggregatedemand (Q) that arises with entry.l* Recall (6) where total industry output is stated as a function of the number of firms and mill price:
Q = Q(Pm,N).
(27)
The total differential of this equation is dQ = -$$dPm
+ $$dN,
(28)
but if we assumethat only very small changes(i.e., dQ - 0) are involved,
dQ KdPm
= - aQ,, aN
or dPm -= dN
JQ
-- m
aQ’
(30)
aPm “We must be cautious in using the area under the demand curve to measure consumer surplus, of course, since it is only au exact measure if the marginal utility of income is constant in the senseof Marshall or Hicks.
282
BRUCE L. BENSON
Equation (30) tells us the change in mill price as the number of firms change,given the changein output is very small, as in Fig. 2. One possible measureof the changein consumersurplus is the shadedarea in Fig. 2. This area is very small when dQ = 0, so the area is approximated by (dPm/dN)Q, ori
-aQ
MB=-i!!!i Q=- Q(S+Nm as) JQ
(31)
JPm Since Q = NS the substitution suggestedby (26) can be made, changing (31) to
MB=(s+Nas)‘~+N~-s~ . (34 I aN 1 Now recall (11) which changes(32) to
ac
MB=C+NaN-S,,.
ac
Industry-wide marginal benefits of a new location equal the difference between LMC and each firm’s marginal production cost as in Fig. 1. This is the efficient result.*’ The necessaryand sufficientconditions for this efficient 19dPm/dN may be positive in Liischian competition (price may rise with entry) [31,9]. However, d Pm/dN > 0 arises only under some demand and cost conditions [3,41,42]. Furthermore, as Donvard notes in paraphrasing Loasby [36]: “given free entry, there are grounds for expecting long-run market demand curves to be much more elastic than short run instantaneous demand curves” [24, pp. 251,252]. Thus, the dPm/dN > 0 result of Lbschian competition should typically only arise in the relatively short run. It arises, after all, because L&churn firm demand becomes less elastic with increases in distant competition when the individual demand function is positive exponential in form (i.e., Pm + tu = a - (b/x)q” x > 0) [3]. But the tendency over time for demand to become more elastic (as it tends to be over the relevant range of a negative exponential demand under Los&an competition) should prevent the surprising results from arising in long-run situations. “The industry-wide marginal cost of an additional location ( 8TC1/8N) is derived in footnote 14. Part of this industry marginal cost of new locations simply involves a movement along producing firms’ marginal production cost curves, however. This portion of the industry marginal cost of locations is, therefore, a private (or internal) marginal production cost which is associated with direct consumption benefits. Thus, to find the industry (or external social)
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SPATIAL COMPETITION
result include the two conditions noted above for long-run marginal cost pricing, and a third-Loschian behavior on the part of entrepreneurs.‘l The long-run LSschian equilibrium is efficient, in contrast to the Mills-Lav [38], and Den&e-Parr [18] claims. Thus, the conventional view that this Chamberlinian tangency implies that the firm is operating at too small a scale does not hold in spatial competition. The Chamberlinian tangency result of Liischian competition occurs when the optimum number of efficient-sized firms exist. CONCLUSIONS Free entry and profit-m aximizkg behavior, in contrast to what has often been claimed, leads to an efficient use of resources in Los&an spatial competition. The excess of f.o.b. mill price over site-specific marginal production cost is the additional marginal cost of output due to the marginal cost which is not directly accounted for in any firm’s decision and which should, as a consequence, be compared to the benefits of a new location, we must net out this private marginal production cost. The industry marginal cost of a new location net of private cost (MCN) is obtained by subtracting (X/as)( 6’Q/aN) from the equation for aTCl/aN. Thus,
It is this difference between industry (or social) marginal cost of new locations and firm-specific (or private) marginal production cost that should be compared to the marginal benefits of an additional location from (31) (as in (33)). Note that since Pm = C/S this equation can be rewritten as MB=MCN=(Pm-$$)S+Ng, which approximates the change in producer surplus as N changes. *lit should be stressed that the efficiency results obtained here arise only under Lijschian behavior. Other entrepreneurial behavior models (see footnote 11) do not yield the optimal number of locations (although they do yield price equal to long-run marginal cost) because in these cases
-
‘(SNS “)
aQ X --CC+=-Sas,
X
where W’/aPm < 0 and equal to the value for Hotelling-Smithies or Greenhut-Ohta competition from footnote 11. Thus, the marginal benefits of additional locations exceed the marginal costs under these alternative behavioral models, and more locutions would increase social welfare. This result is diametrically opposite the conclusions reached by Capozza and Van Order [lo] who assumed Hotelling-Smithies behavior.
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BRUCE L. BENSON
existence of alternative production sites. This is a social cost which should be added to the production cost of each firm to obtain the mill price, since consumers receive the benefits in terms of lower transportation costs. Furthermore, the marginal cost of providing the last location equals the marginal benefits obtained from the location. These conclusions are reached even though each firm’s demand curve is tangent to the downsloping portion of the average production cost curve. The mere fact that this tangency condition describes the individual firm in the spatial long-run competitive equilibrium is not sufficient to conclude that spatial competition leads to too many inefficient small firms as Mills and Lav [38], Denike and Parr [18], Capozza and Van Order [lo], and others have argued. These inefficiency arguments are based on an incomplete picture of the social costs and benefits of spatial competition. The basis for the conclusion that Liischian competition is socially efficient is that consumers pay a higher price than producers receive. Consumers incur transportation costs above the price charged by individual firms, but the cost of transportation is reduced when increasing numbers of tlrms are spatially dispersed. It is this spatial dispersion of firms which leads to consumers’ differentiation of firms (by location), downsloping demand, and ultimately to the Chamberlinian tangency result. So, while individual firms may appear to be inefficient, the benefits of multiple firms imply that the market as a whole may be efficient (as in Fig. 1). Chamberlin argued that the existence of “excess capacity” does not necessarily imply an inefficient allocation of resources. Product differentiation (i.e., spatial differentiation) is desired per se; and it inevitably gives rise to negatively sloped demand curves for individual fums. Differentness (location alternatives) is a quality of the product and it entails a cost just as any other quality. The presence of competitive alternatives guarantee that buyers can select the amount of differentness they wish to purchase (the location from which they wish to purchase). Thus, in the case of spatial competition, Chamberlin would regard a tangency to the left of minimum average production cost as a “sort of ideal” for a market in which there is product differentiation [ll, p. 941. This tangency represents more than a “sort of ideal” in the case of Loschian spatial competition. It is a social ideal, since marginal benefits and marginal costs of multiple locations are equated. The same desirable characteristics of long-run equilibrium in the theory of perfect competition are associated with long-run zero profit equilibrium in Loschian competition. This claim will not surprise readers who are familiar with writings by Demsetz [13-171, DeVany [19-211, DeVany and Saving [22,23], Tullock [45], or Greenhut [27-301, but continued claims to the contrary (i.e., Capozza and Van Order [lo], Villegas [46,47]) in the context of spatial pricing models justifies the above presentation.
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