Spatial pricing in interdependent markets: testing

Environment and Planning A 1998, volume 30, pages 67 - 84

Spatial pricing in interdependent markets: testing assumptions and modeling price variation. A case study of gasoline retailing in St Cloud, Minnesota P Plummer Department of Geography, University of Wisconsin-Madison, 384 Science Hall, 550 North Park Street, Madison, WI 53706-1491, USA; e-mail: [email protected] R Haining Department of Geography, University of Sheffield, Geography Building, Winter Street, Sheffield S10 2TN, UK; e-mail: [email protected] E Sheppard Department of Geography, University of Minnesota, Twin Cities, Minneapolis, MN 55455, USA; e-mail: sheppOOl @maroon.tc.umn.edu Received 18 January 1996; in revised form 3 May 1996

Abstract. In this paper we present an empirical evaluation of assumptions about consumer purchasing behavior for gasoline. Previous research has developed a theoretical model of spatial pricing in oligopolistically competitive markets in which it is hypothesized that retail prices vary because of both consumer price sensitivity and the choice sets available to consumers as well as awareness of prices at competing locations. With the use of household survey data collected from St Cloud, Minnesota we evaluate the plausibility of these assumptions, finding evidence to support the consumer purchasing behavior assumed in the theoretical model. By means of a spatial time series of gasoline price data for the same metropolitan area, we develop an empirical model of spatial price variation that incorporates some of the hypotheses of the original model. The results suggest support for the proposition that spatial price variations depend on the service characteristics of individual retailers and the accessibility or locational advantage of individual gasoline stations within the spatial configuration of the urban market. There also is empirical support for the conjecture that those sites which are more accessible, have larger choice sets, and charge lower prices tend to be those which attract the most sales from other retailers. 1 Introduction There has been a long-standing research interest in h u m a n geography in modeling the link between spatial structure and spatial interaction (Bennett et al, 1985). This research area has included modeling the impact of spatial structure on the nature and degree of competition between firms operating in spatially differentiated u r b a n markets (Haining, 1985). Recently, a n u m b e r of researchers have attempted to explain both spatial and temporal variations in intraurban prices by means of models of oligopolistic competition in spatially interdependent markets (Anderson and de Palma, 1989; Sheppard et al, 1992). These models represent specific applications of a wider class of game-theoretic models of spatial price competition in which the competitive strategies utilized by retailers when operating in a n u r b a n market depend u p o n the strategies pursued by their competitors, where the nature and degree of interaction between firms is reflected in the d e m a n d conditions facing each firm (Fik a n d Mulligan, 1991; Harker, 1986; Nagurney, 1993). T h o u g h much of this research has been oriented towards establishing the theoretical properties and computational efficiency of these models of spatial competition, there has been rather less research effort directed towards empirical evaluation. Exceptions to this are the work of Haining (1983; 1986), Robinson a n d H e b d o n (1973), a n d Slade (1992), w h o explore the influence of local neighbourhood retailer competition on b o t h the spatial pattern of gasoline prices a n d the nature a n d duration of gasoline price wars.

68

P Plummer, R Haining, E Sheppard

The empirical research presented in this paper is based upon a previously developed theoretical model of spatial pricing. In this model, it is hypothesized that retail prices vary spatially because of price sensitivity and choice sets (information) available to mobile consumers and the local price awareness of retailers with respect to their market competitors (Sheppard et al, 1992). The model was considered particularly relevant for those retail goods that are frequently purchased and with little real product differentiation, gasoline retailing fits these conditions. The purpose of the paper is twofold. First, using results from a household questionnaire survey in St Cloud, Minnesota we attempt to assess the assumptions about consumer purchasing behavior contained in the Sheppard et al model in the case of gasoline. Second, we model gasoline price data for all fifty-three service stations in St Cloud collected over a period of twenty-four weeks. This modeling extends earlier work by incorporating theoretical terms that are contained within the Sheppard et al model. It is necessary to find a way of operationalizing these theoretical terms and we describe one approach to this problem based on a method owed to Nystuen and Dacey (1961). 2 Modeling intraurban spatial competition In this section, we outline the basic theoretical structure of a model of intraurban spatial competition between retailers that compete directly for consumer demand in spatially interdependent markets. In this model, market interdependencies between retailers are hypothesized to be contingent upon the daily activity patterns of consumers, the price awareness of consumers, and the spatial configuration of the urban market. The theoretical properties of this model are developed and extended elsewhere (Haining et al, 1996; Plummer, 1996a; 1996b; Sheppard et al, 1992). Consider the case of N retail sites selling gasoline, distributed at fixed locations in an urban area, which consumers must visit to make their purchases (figure 1). The urban area which is effectively bounded by the edge of the metropolitan region is partitioned into N submarkets, where market j is defined as that submarket of the city whose residents habitually consider retailer j first when purchasing gasoline. Submarkets and firm sites are both indexed by locations i, j = 1, ..., N. Consumers residing in any submarket j are conceptualized as having familiarity with a subset of the retail sites. This subset represents the effective choice set considered by the residents of j and may have anywhere from 1 to N elements. Choice sets may be geographically determined by patterns of individual travel within the city or may be built up through other sources of information. These choice sets are effectively a measure of spatial variation in the information available to consumers about gasoline prices.

Submarket j from which customers may patronize i

Figure 1. U r b a n market structure.

Spatial pricing in interdependent markets

69

The nature and degree of competitiveness between firms thus depend on a prior specification of the spatial structure of consumer choice sets. Analytically, choice sets are defined by a binary variable, \vjh which equals one if and only if residents of submarket j include retail site / in their choice set. The variable Wji may be arranged as a binary connectivity matrix with ones on the main diagonal. The interdependencies between firms can be represented as a map of the fixed locations of retail outlets together with an imposed network structure linking the set of sites which arise as a consequence of the nature of the consumer choice sets. In previous research one of the authors constructed these interdependencies based on two rules of linkage: proximity of gas stations to one another along principal radial routeways or proximity at important road intersections (Haining, 1983; 1990). This was intended to reflect intersite competition arising from consumer travel behavior (and hence choice) but is undoubtedly a considerable simplification of the real choice set structure underlying the patterns of intersite competition. Such a network is probably a more accurate reflection of who retailers perceive their competitors to be than of a process of intersite competition driven by consumer behavior and consumer choice sets. In this paper we examine consumer choice sets directly based upon responses to a survey of consumer purchasing and travel behavior in St Cloud, Minnesota. The probability that a consumer will purchase gasoline from any submarket in their choice set is hypothesized to depend on the price of gasoline in that submarket relative to the prices charged at the other sites in that choice set. The price of gasoline is assumed to include any transportation costs incurred by the consumer in visiting the retail site, implying that the marginal cost of consumer travel to gas stations is negligible. This can be justified in cases where distances are short, where several commodities are being purchased, or where purchases are generally made during trips taken for other purposes. We hypothesize that this is the case for most gasoline purchases, because typically they are made either in a consumer's immediate neighborhood during the course of other activities, or during other trips around the city such as the journey to work, rather than as a result of special purpose trips from home. Formally, the probability that customers resident in market j will purchase at site /, Pji, is hypothesized to depend on the price at i relative to the prices at other sites in the choice of consumers from submarket j (j = k). This probability depends on three factors: the geography of the choice set; spatial price differentials; and an index (/?) reflecting the probability that cheaper sites will be favored (Sheppard et al, 1992): =

Hfrexp(-/fo) 2^H^exp(-ft? A .)

9

(1)

k

where /?, is the price charged by the retailer at site /. If /? equals zero, demand will be partitioned equally among all sites in the choice set irrespective of price because consumers are indifferent to price variation; as /? approaches infinity, virtually all purchases will be made from the cheapest competitor. This equation, then, has the convenient property that within a single formulation it allows for a range of consumer behavior from cost minimization under perfect information about prices to completely random behavior. The size of ft reflects the degree to which cheaper sites are favored. It cannot, however, be given a behavioral interpretation as some index of economic rationality because the ratio [in equation (1)] is not independent of general gasoline price levels (Leonardi, 1982). The demand function in each market is assumed to be rectangular: demand in submarket j equals a constant, dj, as long as prices are below some large value and zero if prices exceed this value. This demand is probabilistically allocated amongst

70


those retail sites within the search space or choice set of the residents of this submarket as described in equation (1). Thus, each urban submarket is served by several firms because consumers from one subarea of the city patronize different locations. This has commonly been observed for shopping behavior (Wrigley, 1988). At prices pt (i = 1, ..., 7), expected sales to consumers from submarket j at site z, qjt, are: W//exp(-fe?7.) ?/'• =

d

jPji

=

d

J

]^H>exp(- ppk)

(2)

k

The total expected sales at site i, q„ are therefore:

= £4 j

WjiQxpi-PPi) ^Twjkexp(-Ppk)

(3)

Underlying this theoretical model is the presumption that price competition between retailers is contingent upon the existence of spatial interdependencies derived from variations in the purchasing behavior of consumers. Variations in consumer purchasing behavior arise from differences in the configuration of demand in each submarket and the spatial structure of how that demand is distributed across retail sites (converted into sales), as defined by the geography of consumer choice sets. Further, it is expected that consumer purchasing behavior is in turn determined primarily by spatial variation in gasoline prices within the set of locations that are considered when purchasing gasoline. The key assumptions underpinning this model of purchasing behavior are that: (1) consumers make gasoline purchases as part of multipurpose rather than specialpurpose trips; (2) consumers adjust their purchasing behavior based in part upon the price of gasoline; and (3) consumers possess well-defined choice sets that can be aggregated. In section 3, the plausibility of these assumptions about consumer purchasing behavior is assessed by reference to the St Cloud case study. Though it is our purpose in section 3 of the paper to establish the plausibility of the assumptions of our theoretical model of consumer purchasing behavior in relation to gasoline, this model of consumer purchasing behavior embodies assumptions that are typical of a more general class of spatial choice models (Leonardi, 1982; Wrigley, 1988). Based upon a consumer survey, we examine the following questions with regard to our assumptions about consumer purchasing behavior and market demand: (1) Is there a significant level of spatial variability in mean household gasoline consumption across the St Cloud urban market? (2) Do consumers typically purchase gasoline as part of other trips and if so what types of trips? (3) What are the important determinants of consumers' decisions to purchase gasoline? (4) How sensitive are consumers to gasoline retail price differences? (5) Do consumers possess well-defined choice sets for gasoline? If so, are these choice sets sufficiently similar for subsets of the population that they can be aggregated on a geographical or some other basis? Following from this theoretical model, spatial variation in prices will be contingent upon variations in the spatial pattern of consumer purchasing behavior. In section 4, we use the information derived from the consumer survey to model spatial and temporal variations in gasoline prices in St Cloud. This involves fitting, and subsequently validating, an empirical model of temporal and spatial price variation based upon retail site characteristics and a surrogate measure of retail site accessibility derived from individual

71


consumer choice sets. This reference to consumer choice sets is a cornerstone of the model of Sheppard et al (1992). The theoretical model predicts that the effect of accessibility on equilibrium spatial price variation depends on the profit objective that is pursued by retailers. Specifically, in less well-connected locations, equilibrium rates of profit-maximizing prices are lower, whereas total profit-maximizing prices are higher, than elsewhere. By contrast, in well-connected locations rates of profitmaximizing prices are lower, whereas total profit-maximizing prices are higher than elsewhere (Haining et al, 1996). Within the context of the empirical model the surrogate measure of retail site accessibility (with its origins in the theoretical model) is considered important in defining which sites are expected to have high and which are expected to have low prices. 3 The St Cloud case study 3.1 Survey details There are three major components to the St Cloud dataset: (1) a weekly time series for all the fifty-three gas stations in St Cloud for the twenty-four week period between 17 January 1991 and 21 June 1991; (2) gas station site characteristics, including brand name, types of gasoline, locational environment, number of pumps, type of service, auto-related services (including car accessories, car wash, garage services, and car sales), and non-auto-related services (including travel related goods, general groceries, and restaurant); and (3) questionnaire responses from 301 St Cloud households about their gasoline purchasing behavior. The questionnaire responses are derived from a spatially stratified random sample of 4% of the households in St Cloud, drawn from the street directory for the St Cloud metropolitan area. The sampling scheme was based upon random drawings of individual households from a commercially based directory listing of street locations. This produced maximal covering of the metropolitan area weighting the longer and more densely housed streets yet still assuming randomness within the survey. The households sampled were asked to provide information on their demand for different types of gasoline, where they had purchased gasoline in the last month, the factors they consider important in choosing where to buy gasoline, and their demand for different types of gasoline.

0.3 •

15

0.2

10

0.1

0

£4-

6 12 18 24 Number of responses per submarket

Figure 2. Frequency of responses.

1

72


Empirical analysis is based upon an overall 30% response rate to the survey questionnaire. Recalling that the theoretical model assumes that the location of each individual retailer is associated with the existence of a submarket, individual household responses were then aggregated into submarkets based upon which of the fifty-three gas stations in St Cloud are identified by a consumer as the retailer that they consider first when purchasing gasoline. Typically, this retail site was also the one closest to the residence of each household. Based upon this sample information, the number of responses varies widely between the different submarkets. The frequency of responses by submarket is shown in figure 2. Note that the frequency of response is positively skewed, with a large number of submarkets having only a small number of respondents. Indeed a number of retailers were not identified at all in the sample collected. 3.2 Testing model assumptions Model prices are sensitive to submarket differences in aggregate demand. Unfortunately, the number of responses is small for each gas station, so that empirical estimates of average demand per gasoline station are unlikely to be stable. Also, as responses to the spatially stratified sample did not adequately cover the whole of the St Cloud metropolitan area, it is not possible to derive estimates for some submarkets. Consequently, it is not possible to derive significant empirical results by the use of maps of either demand per household or average demand per gas station for each submarket of the total metropolitan area. However, it is possible to summarize the overall spatial structure of demand in St Cloud by means of data that are spatially aggregated within the St Cloud metropolitan area. These aggregated spatial units are the city of St Cloud, and the villages of Sauk Rapids and Waite Park. In figure 3 (page 76), the village of Waite Park is location west of the city of St Cloud area and the village of Sauk Rapids is located in the northeast of the St Cloud metropolitan area. (1) Demand for gasoline One-way ANOVA (analysis of variance) was used to compare mean household gasoline consumption between respondents who live in either St Cloud, Sauk Rapids, or Waite Park (table 1). 'Effect' represents the variation in mean household consumption between spatially aggregated units and 'error' represents the variation in purchasing behavior between consumers who live in the same spatially aggregated unit. The probability that there is no statistically significant difference between mean household gasoline consumption between aggregated spatial units is 0.1957 which implies that there is no significant difference in mean household gasoline consumption between respondents who live in St Cloud, Sauk Rapids, and Waite Park. One-way ANOVA was also used to compare demand levels between submarkets with sample sizes of ten or more respondents (table 2). 'Stat' represents the variation in mean household purchases between submarkets and 'error' represents the variation in purchasing behavior between consumers who live in a given submarket. The probability that there is no significant difference between mean household gasoline consumption between the six submarkets is 0.526. This suggests that there are no significant differences in mean household gasoline consumption at least for those submarkets that could be included in the test. Based upon the results from the ANOVA and tables 1 and 2, there is no empirical evidence of significant spatial variation in the structure of household demand in the St Cloud SMSA. In the context of the theoretical model, this suggests that variation in the level of demand originating in each submarket (dj) across the St Cloud metropolitan area can be represented as being proportional to the number of households in each submarket.


73

Table 1. Mean household gas consumption.

Waite Park Sauk Rapids Saint Cloud

count

Mean

Standard deviation

SE of mean

Min.

Max.

19 32 203

13.42 14.42 11.37

8.85 10.99 9.47

2.03 1.94 0.67

1.0 2.0 1.0

30.0 50.0 75.0

Analysis of variance Source

Sum of squares

Degrees of freedom

Mean square

F-ratio

P

Effect Error

304.5989 23282.9769

2 251

152.2994 92.7609

1.6419

0.1957

SE, Standard error. Table 2. Demand levels of residents closest to specified sites. Station number

Count

Mean

Standard deviation

SE of meanL

Min.

Max.

17 18 15 + 16 30 45 52

14 18 15 14 10 19

8.86 11.83 11.07 12.14 7.65 12.63

7.13 11.21 6.77 7.38 5.40 6.50

1.91 2.64 1.75 1.97 1.71 1.49

3.0 2.0 2.0 2.0 1.5 1.0

30.0 50.0 25.0 25.0 20.0 24.0

Analysis of variance Source

Sum of squares

Degrees of freedom

Mean square

i^ratio

P

Stat Error

257.939265 5167.307957

5 84

51.587853 61.515571

0.838615

0.526047

SE, Standard error.

That is,

4 ==

ac- 5

(4)

where a represents the mean consumption level, and Nj is the number of households whose residents live in submarkety. (2) Trip purpose and gasoline purchasing In the St Cloud survey, consumers were asked to rank order ( 1 - 7 or not ranked) the importance of different types of trips (jo u r n e y to work, shopping, school, social or recreational, personal business, special trips to purchase gasoline, or other) when deciding to purchase gasoline (1 denotes most important, 7 denotes least important). Table 3 (see over) shows that gasoline purchasing is most frequently made on journeys to work and journeys to shop. Gasoline is not often purchased as a result of a special trip. It also indicates that journey to work (68%) and shopping trips (80%) are clearly

74


Table 3. Ranking of importance of type of trip when buying gasoline (number of times ranked). Ranked a

Work Shop

School Social or recreational Personal business Special trip Other a b

1

2

3

4

153 78

33 95

7 46

8 17

9 40 22 33 3

12 77 18 15 2

14 73 24 26 5

5

6

1 5

8 30 26 28 0

2 0

9 8 13 27 1

15 0 12 12 2

7

0 0

1 0 0 0 6

Not ranked

98 61

234 74 187 161 283

Respondents13

(%) 68 80

23 25 38 47 6

1 denotes most important, 7 denotes least important. Percentage of individual respondents ranking type of trip when buying gasoline.

the most important in terms of the percentage of respondents who chose to rank these types of trip when purchasing gasoline. These findings are consistent with the theoretical assumption that gasoline purchases are generally made either where several commodities are being purchased, or where purchases are made during trips taken for other purposes. As a result, transport costs are unlikely to be considered a factor in purchasing gasoline as they will be absorbed within the overall costs of the trip. (3) Service station characteristics and gasoline purchasing Consumers were also asked to rank order ( 1 - 8 or not ranked) the relative importance of convenience (location), price, brand, quality of service, or types of services (extra services, credit cards, special offers, or other) in determining whether to purchase gasoline from a particular station. Table 4 shows that for St Cloud consumers, convenience of location (89%) and price (87%) are clearly the most important factors in determining where to purchase gasoline and that they are ranked highly. The quality of services offered (64%) and the brand of gasoline sold at that station (58%) are important but less frequently cited factors in deciding where to purchase gasoline. Table 4. Factors important in determining whether to buy gasoline from a station (number of times ranked). Ranked a

Price Location Service Brand Extra service Credit

1

2

3

4

5

6

7

119 118 43 39 15 12

83 82 30 28 20 19

35 43 47 28 29 16

8 22 42 27 27 16

9 2 17 28 38 12

4 1 9 16 28 15

3 2 4 8 11 39

Not ranked

Respondents13

40 32 110 128 134 169

87 89 64 58 56 44

8 1 0 0 0 0 4

(%)

Special 9 15 18 15 17 33 24 5 166 45 Other 19 3 7 4 0 0 2 13 254 16 a 1 denotes most important, 8 denotes least important. b Percentage of individual respondents ranking different factors when determining whether to buy gasoline from a station. (4) Price sensitivity and gasoline purchasing In order to ascertain the sensitivity of consumers to price differences, consumers were asked to estimate the price difference per gallon that would be necessary for them to switch from their current purchasing behavior to another station. The number of


75

Table 5. Price sensitivity of consumers (price difference per gallon that would be necessary for a consumer to switch to another station). Price difference per gallon (cent) 1

2

Count 17 36 na, not applicable.

3

4

5

10

20

na

58

29

68

50

21

23

respondents in each category are shown in table 5, where na means that price differences are not relevant to the respondent and the other categories refer to the price differences that would be necessary in order for respondents to switch their current purchasing behavior. Based upon the evidence of this survey, the majority of respondents are sensitive to some price differential, with about 70% responsive to relatively small price differentials (between lc and 5c). In table 6 we compare consumer price sensitivity (where it has been declared by the respondent) with the rank importance which consumers attach to price when deciding where to purchase gasoline. The expected values in parentheses are computed under the null hypothesis that there is no relationship between sensitivity and rank. A %2 test shows a statistically significant relationship between respondents' sensitivity to price and the rank importance they attach to price in deciding where to purchase gasoline (x2 = 13.42; critical value with one degree of freedom is 3.8 at the 5% level). This test confirms the consistency of respondents' answers. That is, those respondents that rank price (low) high when deciding where to purchase gasoline tend to be price (in)sensitive. Table 6. Price awareness of consumers. Price ranked

Sensitivity to price l-5c

10-20c

1 or 2 143 (131.7) 50 (61.2) 3 or higher 27 (38.2) 29 (17.7) Note: expected values computed under the null hypotheses are given in parentheses. (5) Choice sets and gasoline purchasing The Sheppard et al (1992) model assumes households have well-defined choice sets. Not only were choice sets assumed to exist but the model also assumed that all the households that consider site j first will also consider the same set of alternatives. This is clearly a strong assumption but in the survey it becomes possible to examine how far real gasoline choice set structures depart from this property. In addition to establishing the existence of choice sets, this information is of interest as an exploration of the issues of the similarity of gasoline choice sets in different parts of the city, and the similarity of choice sets amongst respondents in the same submarket. Overall, the survey has yielded information indicating that at least forty four out of the fifty-three stations were selected by at least one consumer as the place they habitually consider first, with 99% of individual respondents indicating that they consider one or more gasoline stations when deciding where to buy their gasoline. In general, aggregate choice sets vary in both size and spatial configuration between different submarkets of the urban area. Similarly, choice set sizes and configurations vary between individuals within the same submarket. As an example, consider both the individual and aggregate choice sets for retail site 3 and retail site 14 (figures 3 and 4, see over). Retail site 3 is identified below as a nodal gas station, located on a principal

76


Wv

%

,1

26.

»•%.

-ll'^E-

s 19»

2f^r 2

.X •18

Ji HA

15«>®16

9

•13

fl

5 =>v. ®4

46 J ^ * " 5?«

-49

\ . A

..

•50

\

Gas station Gas station number Frequently chosen site Principal routeway

3 ® =

Q_

-5 MILE

Figure 3. Choice sets for retail site 3.

5

• 3

Gas station Gas station number Stations considered by more than one person Stations considered by one person = : : : = Principal routeway

.5 MILE

Figure 4. Choice sets for retail site 14.

t


77

routeway into the St Cloud Metropolitan area. In contrast, retail site 14 occupies a relatively remote location within the network of routeways. Of the eleven respondents who consider station 3 as their nearest gas station, only seven consider that station when deciding to purchase gasoline. Of the remaining sites that are considered to be part of the aggregate choice set, only sites 4 (four respondents), 43 (three respondents), 44 (four respondents), and 45 (three respondents) are considered by multiple respondents. This suggests that there is variability in both the size and configuration of individual choice sets. Within this submarket, individual choice set sizes range from no choice preference indicated to four sites in an individuals' search space. For each choice set size, there exists a range of choice set configurations. This confirms that for these respondents, their size of the choice sets is small and the spatial configuration of choice sets differs between individuals within the same submarket (table 7). Table 7. Choice set configurations, site 3. Choice set size

configurations

1 2 3 4

3, 41, 43 (3, 4), (3, 44), (4, 45),, (44, 45) (3, 43., 44), (3, 4,46) (3, 42., 43, 44), (3, 4, 15, 45)

In contrast, residents living near retail site 14 have larger aggregate choice set sizes that are more widely dispersed across the St Cloud metropolitan area. In this instance, eight of the ten respondents indicated that they consider site 14 when deciding where to buy gasoline, and there were multiple respondents for eight of the ten sites in the aggregate choice set. However, as in the case of site 3, there exists considerable variability in both the size and configuration of individual choice sets. The individual choice sets for retail site 14 are shown in table 8. These results are hardly surprising. If gasoline is bought on trips for other purposes, the choice set is likely to depend on the number of trips individuals take and their geographical range. Although individuals living in the same area of town may have similar activity patterns and lifestyles they are unlikely to be similar enough to generate identical choice sets. Individual choice sets are important in defining how consumer demand is distributed as retail site sales, but their main importance within the context of the model is in defining which sites are expected to have high prices and which are expected to have low prices. This arises from the connection, demonstrated in Haining et al (1996) between choice set size and price level. Table 8. Choice set configurations, site 14. Choice set size

configurations

1 2 3 4 5

15, 39 (9, 14), (14, 20), (14, 41), (14, 43) (14, 15, 41), (14, 15, 48) (6, 9, 10, 14) (3, 10, 14, 18, 39)

78


In the case of the total profit-maximizing scenario, those sites that are associated with larger choice sets tend to have lower prices and higher levels of demand. Large choice sets tend to include highly accessible sites such as at intersections or along principal routeways. In any empirical context, therefore, we would expect a knowledge of choice sets to lead to the definition of an index of flows that should help explain price variation. The survey information on individual choice sets could be aggregated into a matrix of choice sets that would then be manipulated in order to arrive at just such an index. In practice, precise empirical derivation of the geography of consumer choice sets by means of an accessibility matrix is somewhat problematic because of the lack of survey responses for some parts of the St Cloud metropolitan area. Any empirical estimate of the choice set structure in St Cloud is likely to be sensitive to the fact that estimated flows are very small. Further, the final structure will be compromised by the variable levels of response from different areas within the metropolitan area. Here we have employed a modified form of a technique developed by Nystuen and Dacey (1961) to identify nodes on a network. These nodes represent the most dominant or accessible sites given the structure of a network. The flow structure here is the 53 by 53 matrix of aggregated choice sets. The resulting nodal (or terminal) sites are then used in conjunction with local knowledge of commuting patterns in St Cloud to derive a surrogate measure of accessibility which is then included within an empirical model of spatial price variation. No doubt other techniques could be employed to identify the most accessible sites given choice set data. The Nystuen and Dacey procedure for calculating nodal points in a network structure is outlined here for clarity and convenience. A more detailed exposition is contained in Wilson and Kirkby (1980). Specification of the measure of gas station accessibility in the St Cloud network used in the analysis involves the following computational procedure: (1) With data derived from the St Cloud survey, set up the choice set data as a trip flow matrix (T)Y) with site origins for consumer purchases as rows and destination gas stations as columns. (2) With the trip matrix TJt identify the origin of the largest inflow for each retail gasoline station. That is, let Mf = max{T)7} be the largest element of the zth row. j

(3) Rank order the gas stations in terms of total inflows from all origin submarkets. Let Di = ^2 Ty be the total estimated inflow from all submarkets j to retail site i. j

Use D; to measure the rank order of the submarkets. (4) Identify all nodal or terminal retail sites. The Nystuen and Dacey rule states that terminal gasoline stations are those stations whose largest flow comes from a subordinate site in the rank ordering of gasoline inflows. The terminal or most frequently visited sites in St Cloud are shown in figure 5 (see over). These terminal sites can be interpreted as those sites in the St Cloud metropolitan area that attract a relatively large number of consumer trips for gasoline under the implicit assumption that the data contained in the trip flow matrix (T}z) is an unbiased sample of the flows. Although far from being a perfect match, comparing the location of terminal sites with the network of roadways in St Cloud confirms the plausibility of using rules of linkage which emphasize proximity along principal routeways or proximity at important road intersections. In the analysis of spatial price variation, these terminal sites define a surrogate measure of accessibility which may be useful in explaining spatial variation in prices in St Cloud.

79


= =

Principal routeway

\\ 2

A\ -53

Q

.5

1

MILE

Figure 5. The terminal or most frequently visited sites in St Cloud. 4 St Cloud gasoline price data Given the limitations of the results from the survey of consumer purchasing behavior, any attempt to model spatial and temporal variations in prices in St Cloud is necessarily tentative and exploratory. However, we do possess an extensive spatial time series consisting of 1219 observations (53 retail sites collected over a 24-week period). Price changes may be m o r e t h a n once a week b u t the sampling interval was chosen because it was n o t practical to collect the data any m o r e often. It is noticeable that there is little spatial variation in gasoline prices for each week. O n average, about 40 of the 53 stations charge the same price each week. However, from week to week it is not always the same stations that are charging higher or lower prices t h a n average, and for this reason we believe there is sufficient variability in the data set providing modeling proceeds by taking sufficiently large blocks of the s p a c e - t i m e d a t a set. The observations were split into two subsets: a set consisting of weeks 1 - 1 2 and a set consisting of weeks 13 - 24. The first subset is used to estimate a plausible empirical model for the price series after subtracting the weekly m e a n price. The second subset represents extra-sample information that is set aside before model specification a n d estimation in order to validate our fitted model. Model validation consists of testing for the stability of empirical model parameter estimates by means of nonnested F-tests (Kennedy, 1992). Empirical modeling of the price d a t a proceeds as follows: (a) fit an initial model of time-dependent d u m m y variables to remove time trends from the spatial price series; and (b) model price deviations by m e a n s of the residual price variations from this time trend. This involves fitting an empirical model to determine which stations charge higher or lower t h a n average prices based u p o n available data, including the empirically derived measure of accessibility and retail site characteristics derived from the survey questionnaire.

80


0.1k;

&

0.0 L

-0.1 L

_0.2"

1

'

1

'

1

0

5

10

15

20

25

Time

Figure 6. Mean price versus time.

Figure 6 shows the deviation of individual weekly mean prices from the mean price for week 24. That is, the mean price level for any week except week 24 is a constant (1.198811) plus a corresponding time coefficient. The mean price for week 24 is given by the constant. These results indicate that the mean price of gasoline in St Cloud is falling from week 1 to 9 of the survey, rising from week 10 to 17, and is relatively stable for weeks 18 through 24. A tentative empirical model of the residuals from the price data suggests that weekly price deviations from the mean value can be explained statistically by use of a subset of the information on retail site characteristics derived from the survey questionnaire (AUT01, AUT02, PUMPS, BRANDED, see table 9) and information on station accessibility derived by the Nystuen and Dacey criterion (ACCESS). Variables AUTOI and AUTO2 are site characteristics for each gas station, which simultaneously represent both costs to the gasoline retailer and attributes designed to encourage consumer patronage. From table 5 we know that the sorts of services represented by these variables were valued by some respondents to the survey questionnaire. Though such services are provided to encourage consumer patronage, they add to the costs of the gasoline station. Consequently, it is expected that both variables will generally result in higher prices at those stations offering such services. The number of pumps at a gasoline station (PUMPS) can be conceptualized as a surrogate variable designed to measure potential scale economies. It might be expected that such scale economies will result in lower prices for gasoline at those stations with more pumps. From the perspective of consumer demand, branded gasoline (BRANDED) connotes a form of product differentiation, where branded is supposed to imply guaranteed quality and hence higher prices. Last, terminal stations are those stations that are most commonly cited in consumer choice sets and tend to be the most accessible (ACCESS). Based upon the survey results, a combination of easy access and low price satisfy the two primary decision criteria cited by respondents as important determinants of gasoline purchases and, hence, those stations that are more accessible (as defined here) might be expected to charge lower prices. This follows from the predictions in the Sheppard et al (1992) model under the assumption of total profit maximization (Haining et al, 1996). The results from fitting these variables to the price deviation series by means of OLS (ordinary least squares) estimation are presented in table 9. Although the overall level of statistical explanation of price deviations is not large (R2 = 22.5%), the overall fit of the empirical model does not include variations in mean levels of prices through time. Furthermore, the signs of the regression


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Table 9. Price deviation series regression. Coefficient

Explanatory variable CONSTANT AUT01 AUT02 PUMPS BRANDED ACCESS

Standard error

-0.017178 0.019961 0.008669 -0.000272 0.004531 -0.003128

T

0.002266 0.002358 0.002007 0.000064 0.001517 0.001459

-7.58062 8.46632 4.31965 -4.26840 2.98753 -2.14485

P (two-tail)

0.00000 0.00000 0.00002 0.00002 0.00292 0.03235

Dependent variable: PRICEDEV N 636 Adjusted R2 0.225 Standard erroi* of estimate 0.015562 Analysis of variance Source

Sum of squares

Degrees of freedom

Mean square

i^-ratio

P

Regression Residual

0.045875 0.152574

5 630

0.009175 0.000242

37.885200

0.0

Notes: PRICEDEV—price deviations from time-dependent mean; AUT01 —sales of car accessories (dummy variable = 1 if sold by retailer); AUT02—car wash services (dummy variable = 1 if facilities available); PUMPS—number of pumps at gasoline station; BRANDED—branded gasoline (dummy variable = 1 if station gas is branded); ACCESS—accessibility measure (dummy variable = 1 if a terminal point according to Nystuen and Dacey criterion). coefficients are consistent with theoretical expectations. Though the explanatory variables in this statistical model are theoretically consistent, there are two data issues that should be flagged. First, throughout the sample period (weeks 1 -12) site 1 has consistently large leverage.(1) In this case, the large leverage for retail site 1 is the result of the very large number of gas pumps at this location. The large number of pumps at this location is possibly the result of the proximity of this retail site to a major interstate highway. As a consequence, this site might be expected to compete with retailers along the interstate highway as well as in the St Cloud metropolitan area. Second, the fitted model consistently underestimates the price deviation at gas station 18 by between three and four standard deviational units. Based upon local knowledge of St Cloud, this is possibly a result of the fact that this site is the sole representative of the Mobil corporation in the St Cloud area in addition to its peripheral location and hence lack of local neighborhood competition. Given these estimates, we can use the information set aside in the extra-sample period (weeks 13-24) to validate the fitted model. Our modeling procedure is to start with the estimated model from the sample period and fit this model to the full set of price deviation information (weeks 1 - 24). Subsequently, we compare this model with more general models that include additional variables to represent the constant and slope parameters between the sample and extra-sample periods. Specifically, we include (1) Leverage measures are used to detect cases that have extreme values in the independent variable domain. Further information on the use of leverage measures as diagnostic tools is available in Weisberg (1985, chapters 5 and 6). Applications to spatial econometrics are available in Haining (1994).


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a time varying constant DCONST that takes on a value of 1 if the fth observation is in the sample time period and zero for the extra-sample period. Similarly, in the case of testing for varying slope coefficients, each variable is multiplied by the time varying constant. This is equivalent to using a nonnested F-test procedure to compare different model specifications (Charemza and Deadman, 1992). In the manner of Weisberg (1985, pages 179-183), we compare four generic types of model: (1) Model 1: a general formulation in which all parameters vary between the sample and extra-sample period; (2) Model 2: parallel regressions in which the constant (intercept) is a time-varying parameter but the slope parameters are assumed to be invariant; (3) Model 3: concurrent regressions in which the slope coefficients are timevarying parameters but the intercept term is assumed to be time invariant; and (4) Model 4: a coincident model in which all parameters are time invariant. Model 4 is equivalent to our initial model specification but fitted to all the sample information. The estimates for each of these models, which form the basis for model comparison are summarized in table 10. As we are interested in testing for the stability of parameter estimates between the sample and extra-sample period, we use nonnested .F-tests to compare the smaller models (null hypothesis) with a larger model (alternative hypothesis), where the smaller or more restricted model can be obtained from the larger or less-restricted model by setting some parameters in the larger model equal to zero. In this case, the .F-test comparing models gives evidence against the null hypothesis. That is, if the computed F-walue is small, then this indicates that there is no evidence that the larger model variance dominates our original model specification, and hence there is no evidence that there are statistically significant parameter differences between the sample and the extra-sample period. The computed F-statistics for the nonnested tests are defined as Fab, where the subscript a represents the null model and the subscript b represents the alternative model, and the critical values at the 5% significance level are in parentheses. If we use the general model (Model 1) as the alternative hypothesis, F2l = 3.759 (2.21), F3l = 3 . 3 7 (3.84), and F4l = 3.13 (2.1). In this case, there is evidence to suggest that the general model variance dominates both the parallel (Model 2) and coincident models (Model 4). Conversely, there is no evidence in the sample to suggest that the general model variance dominates the concurrent model (Model 3). Thus, using the Table 10. Model comparison by means of parameter estimates (t -values are given in parentheses). Variable CONSTANT DCONST AUT01 AUT02 PUMPS BRANDED ACCESS D*AUTO1 D*AUTO2 D*PUMPS D*BRAND D*ACCESS RSS DF

Model 1 -0.01266 (-6.37) -0.00492 (-1.84) 0.01356 (6.77) 0.00382 (2.24) -0.00012 (-2.44) 0.00333 (2.59) -0.00308 (-2.49) 0.0064 (2.31) 0.00485 (2.05) -0.00016 (-2.08) 0.0012 (0.67) -0.00005 (-0.03) 0.19327 1207

Model 2 -0.01483 (11.00) -0.000 (-0.00) 0.0169 (12.00) 0.00635 (5.36) -0.0002 (-5.24) 0.00396 (4.41) -0.0031 (-3.6)

0.19628 1212

Model 3

Model 4

-0.01483 (11.11)

-0.01483 (11.00)

0.01569 0.00334 -0.00008 0.00335 -0.00304 0.00231 0.00577 -0.00022 0.00116 -0.00013

0.0169 (12.00) 0.00635 (5.36) -0.0002 (-5.25) 0.00396 (4.42) -0.0031 (-3.61)

0.19381 1208

(9.59) (1.98) (-1.636) (2.6) (-2.45) (1.39) (2.50) (-3.24) (0.65) (-0.8)

0.19628 1213

Notes: DF, degrees of freedom associated with each model; RSS, residual sum of squares


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logic of nonnested testing, Model 3 is the most appropriate empirical specification for the combined sample and extra-sample period. This implies that there is evidence of instability of the parameter estimates of the fitted model once both slope and intercept time varying parameters are included. This evidence together with the low R2 value suggests that this tentative model is far from explaining the pattern of gasoline price variation in St Cloud. 5 Summary In this paper we have attempted to evaluate empirically, by the use of gasoline data from St Cloud, Minnesota, the plausibility of the assumptions of a theoretical model of spatial price competition attributed to Sheppard et al (1992), in which local neighborhood competition is a significant factor in determining both the spatial and temporal patterning of gasoline prices. The model hypothesized that spatial and temporal variations in gasoline prices depend, at least in part, on the demand conditions facing each of the gas stations that compete in this spatial market. In turn, the demand conditions facing each gasoline retailer depend on the daily activity patterns of consumers, the price awareness of consumers, and the spatial configuration of demand within the urban market. Based upon the research presented in this paper, there is some evidence to support the plausibility of the consumer purchasing behavior assumed in the theoretical model. First, the price awareness of consumers and the accessibility of gasoline stations appear to be significant influences in deciding where to purchase gasoline. Second, few special trips are being made to purchase gasoline. Third, choice sets are small and vary between locations. Because of a highly variable spatial response rate throughout the St Cloud metropolitan area, the possibility of deriving empirically meaningful spatially aggregated choice sets for each retail site proved to be problematic. However, with the use of the survey results, it has been possible to examine what choice sets look like and to use these results to derive a surrogate measure of retail site accessibility. These results are consistent with the theoretical model insofar as they suggest that consumer choice sets exist and that they vary between different areas of St Cloud. However, these results question the likelihood that individuals located in the same submarket will possess similar choice sets. It is not clear to what extent this last finding undermines the empirical usefulness of the original model. The survey data show that just because a consumer lives closest to a particular retailer it does not mean that the retailer will be included in the choice set but this is not in conflict with the original model. However, taken together these observations underline the difficulty of incorporating choice set data into any empirical study and hence the difficulty of approaching the problem of explaining price variation from the perspective of consumer behavior, as opposed to more direct measures of whom retailers themselves consider they are competing against. In addition, of course, choice set data are both time consuming and difficult to collect. Last, it is possible to derive some tentative conclusions with regard to the overall determinant of spatial and temporal price variations. Based upon these results, there is some empirical evidence to suggest that spatial price variations depend on both the service characteristics of the individual gasoline retailers and the accessibility or locational advantage that each station enjoys within the spatial configuration of the urban market. The empirical evidence lends some support to the theoretical conjecture that those sites which are more accessible, have larger choice sets, and charge lower prices tend to be those which attract the most sales from other retailers in St Cloud.

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