1. Economic Geography 1bis – Spatial interaction and the gravity models 121EC A.Y. 2014/2015 Dr. Giuseppe Borruso Faculty of Economics University of Trieste E-mail. giuseppe.borruso@econ.units.itgiuseppe.borruso@econ.units.it Ph. +39 040 558 7008 Skype: giuseppe.borruso

2. Learning Objectives  In this lesson we will: Introduce the concept of spatial interaction Introduce gravity models and particularly: Reilly’s law of gravitation Huff’s probabilistic analysis of shopping centre trade areas  Start visualizing some of the connections between economic geography modeling, GIS and geodemographics

3. Topics 1. Spatial interaction 2. Gravity models 1. Reilly 2. Huff 3. Steps in consumer’s potential for a shopping centre

4. Spatial interaction (Conti, 1996) (1/2)  = the easiest way to understand the relations between different places. The non-availability of a product in one place is a sufficient condition to generate a movement towards another place where it is available The presence of two or more places specialized in the production of different goods / services will set an interaction between them  A functional distance substitutes a physical distance The relationship is not a linear one, but it depends on the different functions played by the involved places  Condtitions for a proper functionality of an economic system: High rate of interactions; Need to reduce the costs related to the flows of goods, services, people.  Targets: Maximisation of interactions Minimization of costs  Places and interactions between places are the building blocks of the different kinds of spatial organization  Places in the Earth space: Do not have ‘innate qualities’; Are characterized by:  The position they occupy with respect to other places they are related to;  The intensity of mutual contacts and exchanges.

5. Spatial interaction (Conti, 1996) (2/2)  Necessary conditions for interaction (Ullman, 1956):  Complementarity: If the demand of a given good / commodity / service cannot be satisfied locally, there is a need to move to other places where the good is available – such locations will be complementary with respect to the first one; Complementary locations will import from other places those goods they will not be able to directly produce; I.e. commuters’ flows of working populations moving to a city=> complementarity between urban centre and its hinterland.  Transferrability: Each movement is a tributary of distane and relative costs; Transferrability takes place when the transferring costs pays the utility expressed in economical terms; I.e. oil: it travels for (thousand of) miles before reaching the final markets; bread: movements limited to neighbourhood.  Opportunity: Is referred to those factors that can foster the movement of people and goods from one place to another one, chosen between different alternatives virtually complementary I.e., the advantages of an urban agglomeration can generate long-range movements of economic activities and population, even if centres closer to the origin place exist. I.e., ‘important’ university cities attract student people also from cities fitted with universities.

6. Spatial interaction models  Newton's law of universal gravitation  “every massive particle in the universe attracts every other massive particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them”  Travels of people and goods tend to follow patterns helping the comprehension of relations between flows and cities.  Reilly’s law of gravitation  Huff’s (probabilistic) law of gravitation  Market areas  Catchment areas

7. Distance decay  The level of spatial interaction – flows between nodes / regions – is related to distance by an inverse relation;  Closer regions have a higher rate of interaction than those located far apart  Empirical evidence shows that spatial interaction is related to a relation inverse to the squared distance between settlements.  Pareto function – relation between flows and distance  F = flow; D = distance; a and b = constant  Low values of b => low slope curve – flows on a wide area  High values of b => higher slope curve – flows on a restricted area  Studies on migrations b = -2  (a similar structure deals with gravitation of urban areas / shopping centres) Distance Exponential function Spatial interaction Distance Linear function Spatial interaction F = a*D -b F = a*(1/D 2 )

8. Urban functions and gravitation  Reilly’s law (1929) on commercial areas  The interaction between a city and its hinterland can be expressed as: F = a * D – b  a = mass of a city (population or other indices of ‘centrality’)  D = distance between places outside and within a city  b = parameter that shows slope of the curve (=> extension of the gravity area) – measures the frictional effect of distance

9. Gravitation and spatial interaction  Newton’s law F ij = amount of interaction between body i and j M i = size of body i M j = size of body j d ij = distance separating body i and body j  Spatial interaction I ij = amount of interaction between place i and j P i = population size of place i P j = population size of place j d ij = distance separating place i and place j

10. Gravitation and spatial interaction  Newton: the power of attraction between two bodies is proportional to the product of their masses and inversely proportional to the squared distance beteen them;  The sum of all the characteristic data of each body allows to determine its potential attractiveness;  Each body influences all the other ones, given that the power of attraction decreases as distance increases.  In spatial interaction the mass measurement is expressed by the power of mutual attraction played by two centres  The volume of interaction between two centres is directly proportional to the product of the different populations and inversely proportional to the (power of) distances separating them

11. Gravity model of flows between centres  Dimension of population (a) and distance (b) used to estimate the spatial interaction between four centres.  (c) real distance;  (d) squared distance;  D 13 = 2 (= distance between City 1 and City 3)  In (d), F 13 = 64 derives from:  Population 1 * Population 3 / squared distance between 1 and 3  F 13 = (64 x 4) / (2 x 2) = 64 (Haggett, 2001, p. 249) P 1 = 64 P 3 = 4 P 4 = 1 P 2 = 16 D 13 = 2 D 34 = 3 D 24 = 3 D 12 = 2 D 14 = 4 D 23 = 3 1 3 4 2 128 1,3 5,3 512 16 21 1 2 3 4 64 0,4 1,8 256 4,0 7,1 (a) Measure of population (P)(b) Measure of distance (D) (c) Estimate of flows (F) by distance F ij = P i P j /D ij (d) Estimate of flows (F) by distance F ij = P i P j /D ij 2

12. Gravity models in retailing  Prerequisite in determining the potential market demand for the products or services of a prospective retail firm (or of agglomerations of prospective retail firms) within an urban area is a geographical delineation of the region containing the probable customers for such good.  Such a region is called a retail trade area  Key to success: Knowledge of the retail trade area provide a basis for estimating potential sales it also makes it possible to determine investment requirements for land, buildings, and fixtures As well as the kinds and extent of merchandise offerings, promotional activities, etc.  Methods: Subjective judgements based on observations of similar conditions from which generalizations have been drawn Or use of empirically derived mathematical formulations (Gravity models) Huff. D., A probabilistic analysis of Shopping Center trade areas

13. Historical background of gravity models in retailing  W. J. Reilly (1929)  Focus: determining the relative retail pulling power of two competing cities on an intervening area rather than on intra-urban retail area determination  Hypothesis: two cities attract retail trade from an intermediate city in the vicinity of the breaking point approximately in direct proportion to the populations of the two cities and In inverse proportion to the square of distances from the two cities to the intermediate town.

14. Reilly’s law and market areas Adaptation of Reilly’s formula determining the breaking point between two cities =>

15. Reilly gravitation law and market areas 35 30 55 20 15 30 km 15 km 35 km km abcde a 19,4719,719,3617,26 b 15,53 c 10,29 d 5,64 e 12,74 a b c d e

16. Reilly’s law of gravitation  Curtis Publishing Company (1947): adaptation of Reilly’s Law to determine the breaking point between two cities:  Ba = Bb (point of equilibrium) represents the point up to which one city exercises a dominant trading influence and beyond which another city dominates Or: the breaking point represents the 0.5 probability position between two cities.  Also, trade areas are estimated within cities,  And parameters changed: Square meters of each retail centre substitute population measures; Travel time substitutes physical distance èIf: City A = 400,000; City B = 200,000; Dab = 80 miles => Bb = 33.3 miles

17. a b 250,000 100,000 75 km 29.1 km 45.9 km M ab Reilly’s law of gravitation

18. Limitations of Reilly’s gravity model  Calculations of breaking points conveys an impression that a trading area is a fixed boundary circumscribing the market potential of a retail facility population distribution (and variability) should be considered  Vast majority of market analysts naively assumed that the exponent 2 estimated for intra-urban trade movements would possess the same value within urban areas. Studies showed that the exponent range from 1.5 to 3 depending on the trip type  Gravity model possesses little theoretical content It does not reveal why observed regularity occur as they do.

19. Trade area analysis  Retail trade area analysis focuses on locating and describing the target market. marketing and merchandising purposes choosing new retail locations  techniques on how to delimit and analyze trade areas: simple ones, such as an application of rings, to more sophisticated, such as utilizing probabilistic trade area surfaces  All techniques represent either the spatial monopoly or market penetration approaches. Spatial monopoly:  The concentric rings method, drive time/distance polygons or Thiessen (Voronoi) polygons are examples of this type of approach  they assume that a store has a monopoly over the area – that all households in the trade area relate to the store and no households outside the trade area visit the store.  Once the trade area is delimited geographically as a ring, Thiessen or other type of polygon, it is easy to prepare a market profile by extracting and aggregating data using GIS software.  the methods representing the spatial monopoly approach involve a lot of simplification because they do not account for the existence of competing stores Market penetration:  assumes that there is a spatial variation in the proportion of households served by a store due to competition  Huff trade area model:  The trade area is conceptualized as a probability surface, which represents the likelihood of customer patronage  What is the probability that a customer will decide to shop at a particular store, given the presence of competing stores?

20. Huff’s law  A variation of the general model of gravitation;  Interaction is based on the probability of an interaction between the origin i and a set od destinations j;  I.e., probability of purchasing in a shopping centre  Substitution of population with a measure of the attraction power of that centre (i.e., square meters of retail surface; revenues)

21. Huff, gravity models, GIS, Geodemographics  Huff’s model. Combines: Theoretical modeling; GIS data and software; Geodemographics  Gravity model as ‘probabilistic surface’  Customer’s choice (probability for each customer to reach a given shopping centre)  Shopping centre location  Spatial distribution of population  Spatial characters of population (geodemographic data)

22. Geospatial technologies: GIS (1/2)  A GIS is a an information system that allow: collecting; modelling; handling; retrieval; analysis of geographically referred (georeferenced) data Reminder:

23. Geospatial technologies: GIS (2/2)  “A GIS is a set of tools to collect, store, retrieve, transform, display spatial data from the real world for particular purposes” (Burrough e McDonnell, 1998)  “A GIS is a database in which the most of data has a spatial index, and on which procedures can be executed to answer requests and queries on spatial entities in the database.” (Smith et al., 1987) Reminder:

24. GI & data integration Market data èProspects èReal Estate èDemographics èCompetition èConsumer Lifestyle Infrastructure èBuildings èRoads èTelecommunications Facilities èATM Locations èBranch Locations èCorp. Facilities Environmental èNatural Disasters èTopographic èBio Hazards èToxic Sites Sales Data èProducts/Services èClient locations èRevenue data Administrative Boundaries èBranch Territories èTrade Areas èService Areas èZip code èCensus Reminder:

27. The Huff model  The model is based on the premise of the probability that a person confronted with a set of alternatives will select a particular item is directly proportional to the perceived utility of each alternative  Choice behaviour can be viewed as probabilistic  It is unlikely that any given alternative will be selected exclusively unless no other alternative exist  The probability that an individual (i) will select the alternative (j) given the utility of j is relative to the sum of the utilities of all other choices (n) considered by individual (i).  The utility of each alternative must be defined empirically.

28. The Huff model  The probability (Pij) that a consumer located at i will choose to shop at store j is calculated according to the following formula  Aj is a measure of attractiveness of store j, such as square footage  Dij is the distance from i to j  α is an attractiveness parameter estimated from empirical observations  β is the distance decay parameter estimated from empirical observations  n is the total number of stores including store j.

29. The Huff model  The quotient received from dividing by is known as the perceived utility of store j by a consumer located at i.  The α parameter is an exponent to which a store’s attractiveness value is raised, and enables the user to account for nonlinear behavior of the attractiveness variable.  The β parameter models the rate of decay in the drawing power of the store as potential customers are located further away from the store.  Increasing the exponent would decrease the relative influence of a store on more distant customers.

30. Huff’s law and market areas Huff (probability; travel times)

31. Huff’s Law a b 250,000 100,000 75 km P(C a )= 0.71 P(C b )= 0.29 35.5 km

32. Huff’s probabilistic analysis of shopping centre trade areas  Improved formulation for delineating trade areas of proposed shopping centres  Conceptual properties of the gravity model  The focus is on the consumer rather than on the retail firm  Measuring a shopping centre’s utility  Utility of a shopping centre to a customer: Number of items of the kind a consumer desires that are carried by various shopping centres; The travel time that is involved in getting from a consumer’s travel base to alternative shopping centres  Shopping decisions made under conditions of uncertainty (a consumer does not know in advance if a shopping centre will fulfil a specified purchase desire)  A consumer has an a priori knowledge of the probability that various shopping centres might satisfy his/her shopping demands.  Such probability is based on the number of items that he/she desires and feels are carried by various shopping centres (presumably, the greater the number of items carried in such centres, the greater the consumer’s expectation that his shopping trip will be successful) This factor can be approximated using the square footage of selling space devoted to sale.  Consumers will show a willingness to travel further distances for various goods and services as the number of such items available at various shopping centres increases.  Travel time: effort and expense perceived to be involved in traveling to various shopping centres.  The anticipated cost of transportation, the effort involved and making the trip tend to detract from the utility of a shopping centre the further it is from the consumer’s travel base  Consumer must choose how much time devoting to different activities located in space.  A shopping centre’s utility to consumer is inversely related to the effort and expense involved in getting from the consunmer’s point of origin to a given shopping centre.

33. Huff’s law and market areas Huff (probability; travel times)

34. Huff’s probabilistic analysis of shopping centre trade areas The use of the model in determining a shopping centre’s trade area - steps 1. Divide the area surrounding any existing shopping centre into small statistical units (i.e., census enumeration districts; grid) 2. Determine the square footage of retail selling space of all shopping centres (j’s) within the area of analysis 3. Ascertain the travel time involved in getting from a particular statistical unit i to a each of the specified shopping centres j 4. Calculate the probability of consumers in each of the statistical units going to the particular shopping centre under investigation 5. Map the trading area of the shopping centre in question by drawing all statistical units having like probabilities Probability of consumers in each of the statistical units going to the particular shopping centre

35. Huff’s probabilistic analysis of shopping centre trade areas The use of the model in determining a shopping centre’s trade area - steps 6. Calculate the number of households within each statistical units. Then multiply each of these figures by their appropriate probability values to determine the expected number of households who will patronize the shopping centre in question

36. Huff’s probabilistic analysis of shopping centre trade areas The use of the model in determining a shopping centre’s trade area - steps 7. Determine the annual average per household incomes of each of the statistical units. Compare such figures to corresponding annual household budget expenditure figures in order to determine the average expected amounts spent by sich families on various classes of products

37. Trade area analysis for a single site using a single variable for site attractiveness  Two data sets were used in this example: shopping centers with their characteristics, and small census units with some related data.  The purpose of this example was to create a market profile for a single mall.  The Gross Leasable Area (GLA) was used as an attractiveness variable. The patronage probability surface (a grid) was created for a selected mall (Figure 1).  A potential customer is assumed to be located at every grid cell.  The probability of a customer patronizing a selected mall is positively related (directly proportional) to the attractiveness of the mall and negatively related (inversely proportional) to the distance between the mall and the customer, given the presence of all competing malls.

38. Customer patronage probability map for Micmac Mall, Halifax-Dartmouth, Nova Scotia, using Gross Leasable Area (GLA) as an attractiveness variable.

39. Trade area analysis for a single site using a single variable for site attractiveness  patronage probability surface was converted to regions of probability, delineated using contours  The data was then extracted for each region from underlying census polygons  values of probabilities used as weights for scaling down the numbers from table Market profile (unweighted data) for regions of patronage probability Market profile – weighted data for regions of patronage probability

40. References  Gross, B., 1997. Dynamic Trade Areas. Business Geographics, September, 30-33  Hooper, H., 1997. Who’s Really Shopping My Store? Business Geographics, September, 34- 36  Huff, D.L., 1963. A Probabilistic Analysis of Shopping Center Trade Areas. Land Economics 39: 81-90  Huff, D.L., 2003. Parameter Estimation in the Huff Model. ArcUser, October-December, 34- 36  Huff, D.L., 2005. The Use of Geographic Information Systems and Spatial Models in Market Area Analysis. ESRI GeoInfo Summit, April 18-19, Chicago  Jones, K., Simmons, J., 1993. Location, Location, Location. Nelson Canada  Peterson, K., 1997. A Trade Area Primer. Business Geographics, September, 18-21  Pick, J. B., 2005. Geographic Information Systems in Business. Idea Group Publishing  Simmons, W., 1998. Defining Trade Areas. Business Geographics, September, 28-30  HARRIS R., SLEIGHT P., WEBBER R., Geodemographics, GIS and Neighbourhood Targeting, Chichester, Wiley, 2005  CONTI S., Geografia Economica, Torino, Utet, 1996