adjacent market segments and that, as a result, spatial pricing theory must ... and Ohta 1975) and that their pricing behaviors can be asymmetric, but do not.
Gordon F. Mulligan and Timothy J. Fik
Price Variation in Spatial Oligopolies The “geography of price” is being given renewed attention by geographers awl economists. This paper examines spatial price variation in both unbounded (circular) and banded (linear) one-dimensional markets. Firms compete for consumers in thc short run b y adjusting price until the Bertrand equilibrium is reached in thc market. While firms act as spatial oligopolists in specific market segments, thc profit-maximizing price of any given firm depends directly and indirectly upon the spatial-economic properties (locations, marginal costs) of all other firms in the market. 1. INTRODUCTION
While spatial pricing continues to be of great interest to economists and geographers (Beckmann and Thisse 1986), remarkably little is known about nondiscriminatory pricing behavior in spatial markets having “several” firms. This paper examines (mill) price variation in both unbounded (circular) and bounded (linear) markets, where numerous firms located at given points compete in price while selling a homogeneous good to rational consumers distributed evenly over those markets. The paper agrees (as is well known) that distance necessarily restricts the number of firms that can directly compete in a specific market segment, but then points out that a spatial market is actually comprised of a set of adjacent market segments and that, as a result, spatial pricing theory must recognize that the behaviors of all firms can be directly and indirectly linked through this adjacency property. In other words, the chain-like structure of demand in a spatial market means that the profit-maximizing price of any one firm is inextricably linked to the spatialeconomic properties of all other firms in that spatial market. The analysis of the paper is confined in several ways and this deserves some comment at the outset. First, the model follows from Bertrand who suggested that spatial oligopolists compete by varying price and not output. In this paper all firms are assumed to have the same zero conjectural variation with respect to nearest rivals’ prices, an assumption Dften called Hotelling-Smithies competition in the literature (Hotelling 1929; Smithies 1941; Eaton 1976; Capozza and Van Order The authors thank three anonymous referees for their very helpful comments. Gordon F. Mulligan is associate projessur and Timothy J. Fik is a graduate student geography and regional deoelopment, The University of Arizonu.
Ojf
Geographical Analysis, Vol. 21, No. 1 (January 1989) 0 1989 Ohio State University Press Submitted 1/88. Revised version accepted 5/88.
Gordon F . Mulligan and Timothy I. Fik / 33
1977; Benson and Faminow 1985; Faminow and Benson 1985). We recognize that firms can have other price conjectural variations (Lijsch 1954; Greenhut, Hwang, and Ohta 1975) and that their pricing behaviors can be asymmetric, but do not entertain these complicating issues in this paper (Mulligan 1988). Second, we confine our interests to the short run where firms can neither relocate nor enter/exit the market. Long-run solutions, based on firms’ conjectural variations with respect to rivals’ locations, will be identified in future research. Given this, we should stress that Bertrand models are thought to be most appropriate for such short-run market adjustments (Jacquemin 1987). Third, in order to phrase the argument in a linear model we assume that consumer demand is inelastic. While perversities in spatial pricing sometimes occur under this assumption, we note that prices do not blow up with Hotelling-Smithies behavior as they do with Lijsch behavior (Capozza and Van Order 1977) and that long-run concerns about induced clustering (Smithies 1941) are unwarranted because we actually restrict our analysis (see below) to those cases where firms are not located close together. In any case we have already initiated research in order to clarify how demand elasticity modifies (if in fact it does) our results about firms’ behavioral interdependencies (Fik and Mulligan 1988). Fourthly, we confine our analysis to one-dimensional markets. Since this paper stresses the behavioral underpinnings of spatial competition, and not the sizes and shapes of firms’ market areas, we feel justified in not looking at the very complicated two-dimensional case at this time. Two further issues require comment before the analysis can proceed. First, the paper does make an assumption that a price equilibrium exists in the market without providing a complete characterization of the conditions (locations, etc.) under which such an equilibrium exists. The paper simply assumes that all firms occupy separate locations and that they are sufficientlyfar apart that fluctuating prices, recently identified by d’Aspremont, Gabszewicz, and Thisse (1979) for duopolies, do not exist. The approach of the paper is to constrain the analysis to those price and locational configurations where (i) no firm adopts a policy which drives the revenue of a nearest rival to zero and (ii) the transportation cost between contiguous firms always exceeds the difference in their mill prices. Each firm is in short-run equilibrium when there exists no price (given its location) that will increase that firm’s level of expected profit. The market is in short-run equilibrium when this condition holds for each and every firm (Eaton 1976). This is a Nash equilibrium where firms hold the view that the behaviors of their competitors cannot be manipulated. Finally, it is a central point of concern of this paper that both unbounded and bounded spatial markets be examined and their price equilibria be compared. Boundaries are not phenomena that can be assumed away in a generalized location theory-at least not by geographers who consistently recognize and try to accommodate boundary conditions in a variety of analytical contexts. This paper shows that some properties of price equilibria are independent of the length of the spatial market when the market is unbounded (circular), while these same properties are fully dependent upon the extent of the spatial market (and firms’ locations) when that market is bounded (linear). We anticipate that future research will shed light on how the degree of permeability of market boundaries affects spatial price variation. 2. ASSUMPTIONS AND NOTATION
Suppose identical consumers are uniformly and continuously distributed over a one-dimensional market of length A. Assume each of these consumers exhibits inelastic demand for the good within appropriate (delivered) price levels; that is, no
34 /
Geogruphicul Analysis
consumers face prices sufficiently high to be forced out of the market. Assume further, without loss of generality, that consumers only purchase one unit of the good during the short run. Then consumer demand can be visualized as a continuous density function of unit height. Along this market n firms are located at X,, X , , . . . , and X , . All firms have cost conditions taking the form C = F + kQ, where C is total cost, F is fixed (overhead) cost, k is the constant marginal cost of production, and Q is output of the firm. Since all firms adhere to a mill (f.0.b.) pricing system, consumers must always bear the burden of transportation costs. Let pp denote the (delivered) price to a consumer purchasing at firm i (1 < i Q n), where p y is comprised of the mill price p, and the transportation cost t x (where t denotes the constant transportation rate and x denotes distance), and n >, 3. Furthermore, suppose that agents behave according to the following assumptions ( 1) all consumers practice price-minimizing behavior; that is, consumers purchase
the good only where delivered,price is lowest; and (2) all firms practice profit-maximizing behavior in the short run; that is, firms establish their equilibrium prices while holding their locations constant. This ondimensional market of length A can assume two different geometrical configurations, a circle or a line. The former represents an unbounded spatial market while the latter represents a bounded spatial market. An unbounded market has only interior firms because each firm has a neighboring competitor on all possible sides; however, a bounded market, by its very nature, has one or more exterior firms located near the market’s spatial extremity(ies). In this paper the circular market has n interior firms, while the linear market has two exterior firm:; (numbered 1 and n ) and n - 2 interior firms. 3. CIRCULAR (UNBOUNDED) MARKET
Any given firm must be located between two neighboring competitors and, as ii result, a left-hand component and a right-hand component of that firm’s market area can always be identified. Consider the case of the ith firm, located at X i and producing an amount Q, at (mill) price pi. Denote the left-hand component of thi:; firm’s market area by u i L and its right-hand component by u i R .It follows that the spatial extent a , of this firm’s market area is a , = u I I . + u i R , where u i = Qi because of the unit demand density assumption. From the consumer coverage assumption it also follows that n
A
=
Cu,. i=l
Now the ith firm shares a market area boundary with the (i - 1)th firm (located at X i - , ) on the left-hand side and the ( i + 1)th firm (located at X i + l ) on the right-hand side. The first and nth firms also share a market area boundary simply because the market is unbounded. Each market area boundary occurs where the delivered price from adjacent firms is identical; i.e., at those locations where pfp1 = pf and pd = ~ f +Thus ~ . the boundary between firms i - 1 and i (denoted by B i p1, or B i L ) is
1 B ~ ,= E ( p i - p i P l
+ tX, + tXiWl)
where X i - l < BiI, < XI
Gordon F . Mulligan and Timothy I. Fik / 35 \
P.
I
FIG. 1. Prices, Locations, and Market Area Boundaries in a Onedimensional Spatial Market (Note that the market is not in equilibrium.)
while the boundary between firms i and i 1 BiR = -2t( p i + l
- pi
+ 1 (denoted by BiR or B i + l ,L ) is
+ tX,+l + tX,)
where X i < Bi,
, 5. Using these results it is an easy matter to specify the equilibrium price p,?' of the ith firm in a market having n firms:
p,n'
= -
IC"l
2bi",tA + (b; - br2)tX1 +
n-1
c (b;j-,
-
b;j+l)tXj
j=2
n-1
- byn)tXn
+ b;kl + 2 c bGkj + h,P,k, j=2
1
(43)
where all elements b1nj can be derived from the known first-row elements through equation (40). Equation (43) shares certain similarities with equations (26) and (29) earlier in the discussion. Again, at the market equilibrium a firm's price depends directly and indirectly upon the transportation rate, the location of all other firms (for n odd or even), and the marginal costs of all firms. However, in the bounded market a firm's equilibrium price also depends upon the firm's own location (except for the case of the median firm when n is odd) and the spatial extent A of the entire market. Firms' locations and marginal costs are now weighted asymmetrically in space, except again for the [(n 1)/2]th firm when n is odd. Tables 3 and 4 provide numerical solutions for equation (43) for bounded markets having relatively few firms. Some salient features of these tables deserve comment. First, note that the market length A has a much greater positive effect on price when the firm has an interior (Table 4) as opposed to an exterior (Table 3) location. Second, note in Table 3 that the locational effect on price due to a first, second, etc. nearest neighbor quickly stabilizes as the number of firms in the
+
44
/ Geographical Analysis ~~~
TABLE 3 Components of Equilibnum Pnce of First Firm in Bounded Market of Length A _ _ _ _ _ ~ ~ ~~~. ~ _ _ _ _ ~ ~
~
~
~~
_
_
_
n-3
fA
0 16667
tx, tX, tX3 tX4 tX5 tX, tX 7
0 41667 050000 0 08333 -
_
~
11
~~
-
4
,I
=
5
Market Length 0 04444 0 01190 Firms’ Locations 0 42222 0 42262 053333 0 53571 0 13333 0 14286 0 03571 0 02222 0 00595
-
Firms’ Marginal Costs 0.57778 0.57738 0.31111 0.30952 0.08889 0.08333 0.02222 0.02381 0.00595 -
0.58333 0.33333 0.08333
-
~~
II =
6
,I =
7
0 00319
0.0008:i
0 42265 0 53588 0 14354 0 03828 0 00957 0 00159
0.42265 0.53590 0.14359 0.038463 0.010263 0.002!3i 0.00043
0.57735 0.30941 0.08293
0.57735 0.30940 0.08291 0.02222 0.00598 0.0017 1 0.00043
-
0.022.33
0.00638 0.00159 -
~~
TABLE 4 Components of Equilibnum Price of Median Firm wth an Odd Number of Firms in Bounded Market of Length A n =7
tA
1st nearest 2nd nearest 3rd nearest 4th nearest OWn
1st nearest 2nd nearest 3rd nearest 4th nearest
0.33333
Market Length 0.08333 Firms’ Locations
0.02222
0.25OOO (tX4 - tX,) 0.04167 (tX.5 - t X i )
-
Firms’ Marginal Costs 0.58333 (k3) 0.16667 (k, + k4) 0.04167 ( k i + k.5) -
market increases (e.g., when the seventh firm is added to the right-hand side of the market and all six existing firm locations remain constant, then the incremental positive effect on price at the first firm is only O.oooO2 due to firm 2’s location, only O.oooO5 due to firm 3’s location, and so on). This result holds for interior firms a:$ well; in the case of the median firm (Table 4) note that the locational effects due to the pairs of second, third, etc. nearest neighbors also stabilize (but not as quickly). This stability feature is also exhibited by the marginal cost effects, both for exterior and interior firms. Of course, the prevailing feature of the two tables is the same distancedecay effect uncovered earlier for the unbounded market: the greatest impact on a firm’s equilibrium price is due to the spatialeconomic properties of its
Gordon F . Mulligan and Timothy 1. Fik
/ 45
first nearest neighbor(s), while the properties of its second, third, etc. nearest neighbor(s) have an increasingly smaller impact on that price.
5. CONCLUSIONS
This paper has examined price variation in spatial markets having both numerous firms and numerous consumers. The main intention of the paper has been to demonstrate that the pricing behavior of a spatial oligopolist (a firm competing in a specific market segment) is not just dependent upon the attributes of its nearest rival(s), but is ultimately linked to the attributes of all rivals in the spatial market. This result holds because oligopolists compete in different yet adjacent market segments so that the effects of chainlike demand can be transmitted (via contiguous firms) throughout the entire spatial market. The analysis was confined in several respects (zero price conjectural variation, short-run adjustments, inelastic demand, and one-dimensional markets) but we are already extending our research to overcome these constraints (Fik and Mulligan 1988; Mulligan 1988). The paper has analyzed price variation in both unbounded and bounded markets. In both cases the location(s) and marginal production cost(s) of a firm’s first nearest neighbor(s) have the greatest influence on that firm’s equilibrium price, while the location(s) and marginal production cost(s) of a firm’s second, third, etc. nearest neighbor(s) have increasingly less influence on that firm’s equilibrium price. In addition, boundary conditions themselves clearly affect a variety of pricing properties at the market equilibrium For instance, equations (8), (ll), (26), and (29) show that the average equilibrium price (unweighted by variable market areas) in an unbounded market is actually independent of the length A of that spatial market; equations (35),(38),and (43)indicate, on the other hand, that the average equilibrium price in a bounded market not only depends upon length A, but also upon the locations X,, X , , . . . , X , of the various n firms in the spatial market (in addition to the marginal production costs of the n firms, which influence average price in both cases). Future research should clarify how different sorts of boundaries affect spatial price variation, particularly when long-run adjustments are accommodated. One final remark should be made about the stability of the price equilibrium in multifirm spatial markets. It is well known that (mill) price instabilities persist in spatial duopolies whenever the two firms are located near one another. Further research must clarify to what degree-if any-such instabilities depend upon the presence of so few firms in the market. In any case we confine our results in this paper only to those cases where the difference between the (mill) prices of all contiguous firms never exceeds the cost of transportation between those two firms (Hotelling 1929);that is, l p r l , - p1.1 < t ( X , + , - X i ) for 1 < i < n. Unfortunately we do not, at this time, characterize all firms’ locations that would ensure this condition is not violated at the multifirm price equilibrium.
LITERATURE CITED d’Aspremont, C., Gabszewicz, and J.-F. Thisse (1979). “On Hotelling’s Stability in Competition.” Econometrica 4J;Ii45-50. (1986). “The Location of Production Activities.” In flandbook of Beckmann, M. J., a n d i -F. Thisse , Regional and Urban conomtcs, edited by Peter Nijkamp, pp. 1-95. Amsterdam: North-Holland. Benson, B. L., and M. D. Faminow (1985). “An Alternative View of Pricing in Retail Food Markets.” American Jouml of Agricultural Economics 67, 296-306. Capozza, D. R., and R. Van Order (1977). “Pricing under Spatial Competition and Spatial Monopoly.” Econometrica 45, 1329-38.
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/ Geographical Analysis
Eaton, B. C. (1976). “Free Entry in One-Dimensional Models: h e Profits and Multiple Equilibria.” Journal of Regional Science 16,2143. Faminow, M. D., and B. L. Benson (1985 “Spatial Economics: Implications for Food Market Response to Retail Price. Reporting.” The Jouma/of Consumer Affairs 19, 1-19. Fik, T. J., and G . F. Mulligan (1988). “Price Patterns in Spatial Markets.” Discussion Paper 88-5, Department of Geography and Regional Development, University of Arizona, Tucson. Greenhut, M. L., M. Hwang! and H. Ohta (1975). “Observations on the Shape and Relevance of the Spatial Demand Function, Economehica 43, 669-82. Hotelling, H. (1929). “Stability in Competition.’’ Economic Joumol 39, 41-57. Jacquemin, A. (1987). The New Industrial Organization: Market Forces and Strategic Behavior. Cambridge, Mass.: MIT. Liisch, A. (1954). The Economics of Location. New Haven: Yale. Mulligan,G. F. (1988). “Price Reaction Functions: A Conceptual Framework for Spatial Competition.” Discussion Paper 882, Department of Geography and Regional Development, University of Arizona, Tucson. Smithies, A. (1941). “Optimum Location in Spatial Competition.’’ Journal of Political Economy 49, 423-39.
