LIBRARY OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY WORKING PAPER ALFRED P. SLOAN SCHOOL OF MANAGEMENT A GEOGRAPHIC MODEL OF AN URBAN AUTOMOBILE MARKET Theodore E. Hlavac, Jr. John D. C. Little 180-66 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 50 MEMORIAL DRIVE CAMBRIDGE, MASSACHUSETTS 02139 A GEOGRAPHIC MODEL OF AN URBAN AUTOMOBILE MARKET Theodore E. Hlavac, Jr. John D. C. Little 180-66 MASK. INST. T£CH~ 'JVN 20 fo- April 1966 C .1^ 7 RECEIVED JUN 29 1966 M. 1. T. LIBRARIES Abstract A person chooses a store to shop at partly on the basis of the difficulty of getting there. An understanding of the relation- ships involved is important in marketing strategy, particularly for site selection purposes. We study an urban new-car market. As a measure of shopping difficulty, the distance from a person's residence to the car dealer is used. Assumptions about buyer behavior lead to a model of competitive interaction among dealers and car makes. The model is fitted to three months of sales data for metropolitan Chicago. An interactive computer system has been programmed to make the model available for on-line investi- gation of a variety of site selection and inter-brand competition questions. For example, a user can add, eliminate, or move a dealer and then ask for the model-predicted changes in the sales and penetration of any dealer or make. A GEOGRAPHIC MODEL OF AN URBAN AUTOMOBILE MARKET Theodore E. Hlavac, Jr. Cambridge Research Institute John D. C. Little Sloan School of Management Massachusetts Institute of Technology 1. Introduction When a person buys a car, he takes into account either explicitly or implicitly, the distance he must travel to the prospective place of purchase. He is likely to be attracted to a nearby dealer because of easy accessibil^ity for shopping trips' and tuture service visits. He is less likely to be attracted to a distant dealer, but some attraction will exist, especially in the case of a well-advertised dealer having an image of low price and high throughput. Within an urban market, where there is considerable choice of dealehrs, the probability of purchase from a given dealer can be expected to fall off with distance; and, in fact, such a relationship is easily established empirically. In the present paper we develop £a model in which a customer's probability of purchase at a given dealer is affected by dealer location and customer make preference, as well as the locations and strengths of all other dealers. Aggregation of the customer model gives a dealer market share (penetration) model, which may also be viewed as a model of competitive interaction. Such a model is fit to data for metropol- itan Chicago. After fitting, the model permits estimation of the sales of a dealership with specified strength and location. The most obvious practical use of the model relates to market strategy for new dealerships in the automobile industry, but the model appears to be adaptable to site location problems in other fields as we 1 1 . 1 This work was supported in part by Project MAC, an MIT research program sponsored by the Advanced Research Projects Agency, Department of Defense, under Office of Naval Research Contract Number Nonr-A102(01) Part of the research was done at the MIT Computation Center. 2. Model 2.L Dealer Pull. To some extent all dealers are attractive to a person planning to buy a car. The attractiveness of a specified dealer presumably is a function of dealer characteristics (such as make of car sold, extent of advertising), buyer characteristics (such as make preference), and distance from dealer to buyer. The attrac- tiveness of a dealer will be called his "pull". Pull will not be a directly observable quantity but will be used to develop expressions that are. We shall assume that the number of car purchases is fixed for the time period under consideration. We have not modeled the effect of a dealer in creating sales that would not have occurred In hla absence. Our data do not seem to lend themselves to the estimation of this effect, and perhaps it is small in today's well developed markets. Buyers will be separated into market segments. The pull of a dealer on a buyer in a given segment will be broken into two parts; (1) an intrinsic pull independent of the make sold by the dealer and (2) the make preference of the buyer. Let S(^>j) = ^^'^ pull of dealer j on a buyer in market segment i. i = 1, . . .,S j = 1, . . .,D. ^^C^/J) = the intrinsic pull of dealer j on a buyer in segment i. q(i,m) = the make preference of a buyer in segment i for make m. m = 1,...,M. We specify that q(i,m) > 0 and S^W q(i,m) -- 1. Let m(j) denote the make sold by dealer j. We stipulate that the above quantities be related by g(i,j) = h(i,j) q(i, m(.i)) (1) Thus, the pull cf a dealer on a buyer is the dealer's intrinsic pull weighted by the buyer's brand preference. 2.2 Purchase probability. The probability that a buyer purchases at a given dealer will be taken as the pull of that dealer on the buyer divided by the total pull on the buyer. Let P(i^j) = the probability that a buyer in market segment i purchases at dealer j. p(i,j) = g(i,j) / L^l^ g(i,k) (2) \-lc note that make preference can be interpreted as the probability of purchase of the make under the conditicns that the sum of the intrinsic pulls on the buyer is the same for each make. This result can be deduced from (1) and (2). 2.3 Geographic effect. We hypothesize that pull falls off expo- nentially with the distance between dealer and buyer. This is only one of many possible relations but it appears to work well. Let x(i;j) = distance of the buyers in market segment i to dealer J. h(ij) = a. e-^J^^^'J^ (3) Here a and b, are constants specific to dealer j. The constant a. expresses the dealer's strength in his own immediate neighborhood.-' The constant b tells how fast his sales fall off with distance. Using (1), (2), and J (3), we get p(i j) _ q(l,m(:,1)) a c-^^<^^-^) J ^k=l ^^ ' "^W) \ c '^ ^ ' ' 2.4 Dealer sales and penetration. Let N(i) = number of buyers in market segment 1 (called the potential of the segment) in a given time period. 3(j) = expected sales of dealer j in the given time period. n(j) = expected penetration of dealer j In the whole city. Then s(j) = L^l^ N(i) p(i,j) (5) n(j) = s(j) / l^J^ N(i) (6) 3. Fitting the Model to Data 3.1 D.3 1 a . The model has been fit to R.L.Polk new car registrations for April, May, and June 1963 in Cook County, Illinois (fleet sales elim- inated) . In order to maintain reasonable sample sizes with three months of data, the analysis has been confined to 3 major makes (Ford, Chevrolet, iiambler, Pontiac, PlymoaCh, Buicx, Oldsmobile, and Dodge) and to dealers selling SO or more cars in the cimc period. This involves 47,670 cars and 1&4 dealers, or 91. 5% of the carj and 627. of the dealers of the makes conBidcred. The county has been divided into 1^*0 marketing areas or cells. The rationale for this is discussed by James [I]. The location of each dealer has been determined as has been the sales of each dealer in each cell. The market has been segmented by geographic area, i.e., by cell. For make preference we have used market share. Since the city is laid out rectangularly, distance has been measured rectangularly instead of diagonally. Some alternatives to these choices will be suggested below. 3.2 Fitting procedure. The model has been fit to the data by a modified maximum likelihood procedure. We observe that, if the denomin- ator of (4) is regarded as a known constant, each dealer becomes a separate estimation problem involving two parameters, a. and b., and 140 data points, the dealer's sales in each cell. Given all the a. and b. the denominator of (4) can be calculated. This suggests an iterative procedure. Suppressing the dealer index j, let n. = sales of dealer J in cell i. w 1 q(l; n,(j))/E^^^ q(i, m(k)) a^^ e'^k '^(^^k) ^^j bxj p = w a e -= probability a buyer in cell 1 purchases at dealer j. Then assuming Independence of purchase, the likelihood function for the observed sales figures is L = 'ill ("^'^Pi'^d-P^)"*"'"' w Wo start with a trial set of w . Values of a and b are then chosen to maximize L. This is done by setting the derivatives of log L with respect to a and b to zero and solving the resulting equations by Nowton's method. The a and b are calculated for each dealer. These are then used to recompute the w, from (7). The process is repeated ■ ;ntil the values of the ri's and B's converge. J. 3 Results . Figure 1 shows the difference between actual sales and model-predicted sales for the 184 dealers. The two are very close. Thlr; is of course desirable, but it is only a moderately good measure of model fit. Although the fitting is done on cell sales rather than dealer sales, and there is no requirement that dealer sales fit exactly, it would be expected from the nature of the calculation that the fit would be close. Figure 2 compares actual and predicted penetration by coll for one dealer. The discrepancies are considerable but, again, they are only a fair indication of fit. Most of these discrepancies are random fluctuations resulting from the small sample sizes Involved in any given cell. Figure 1. Error (Predicted Sales - Actual Sales) vs. Dealer Size (Actual Sales) Error +200 • 200 ¥* ■ ^11 ■■>mWii ^■•i^*^* ■•«<»ii ^iil Dlr. Size gO 100 200 400 600 1000 2000 Figure 2. Penetration vs. Distance from Dealer for a Large Chevrolet Dealer . penetration 10 - OS - 0 •••*^^-^*^'*- dist (miles) 15 30 45 penetration 15 - . 10- .05 0 dist (miles) 15 30 45 2a. Predicted Penetration vs. Distance 2b. Actual Penetration vs. Distance Figure 3. POTENTIAL DENSITY one dot equal s approx I natel y 100 cars For a better appraisal of fit, wu have hypothesized that, for a given dealer, the probability of purchase in cell i is not p but is p.' where Pi' = Pi^^ ^ ^t^ and € is a random error term. The standard deviation, a of c. may be viewed as a type of a coefficient of variation. The value of a has been estimated for each dealer and has a median value of .32. This seems quite good, considering how few factors are included in the model. 4. An Interactive Computer System Conceivably, one could use the model to work out a mathematically optimal pattern of dealers over the city or, more modestly, the optimal location of a new dealer. Such an optimization is probably sterile. A decision on a dealership involves many factors not included in the model: the availability of property, financing, the micro-geography of the location, etc. Perhaps some of these factors can be modeled but as of now they are not. Yet we do not want to wait to take advantage of what we can learn about macro-geography and competitive interaction. What is needed is a convenient way to make the information available to a person working on dealership problems. To demonstrate how this can be done, an interactive computer syptem was programmed for the model on the Project MAC time-shared computer at MIT. With the system a user can sit at a remote console of the com- puter, make hypothetical changes in dealerships, and learn immediately the model's prediction of the effects. See Exhibit I for an example of the system in action. Changes that a user can make include adding a dealer, moving a dealer, eliminating a dealer, changing a dealer's a and b parameters, and changing the potential of a cell. 5. Possible Improvements Experimentation with the functional form of the distance relations would be of interest. Similarly, different measures of "distance" might be tried. Travel time has strong intuitive appeal. Much better market segmentation is possible. The natural one would be to break down the cell population by make-model year of car owned. Then make preference could be related to the buying rates of each segment for each make. Another subject for investigation is the effect of clustering of dealers. Perhaps there is a special advantage to being in an "automobile row" because the row itself attracts customers. If so, the model might be modified to account for this. A highly desirable line of research would be to investigate how the a and b parameters are related to various observable characteristics of the dealer. Reference 1. J. W. James, "An Analysis of the Optimum Market Representation Policies Relating to a Large Metropolitan Market," MS Thesis, MIT, 1964. EXHIBIT I A TYPICAL CONSOLE SESSION AN INTERACTIVE MODEL OF THE CHICAGO AUTOMOBILE MARKET T. f. HLAVAC, JR. J. n. C. LITTLE RERIN PLEASE mov^ filr 222 to eel I IJl OOnr, DEALER NUMBER 222 HAS BEEN MOVED TO CELL IJl. -the computer prints a title (User commanHa etc always In lower case, coQiputer rrspnnsefl In upper caaa) -a signal that Initialization ts complete and the tiser may begin -the u«er moves a dealer Crom his present location •the aysteo executes the Instruction, CONTINUE change for dir 222 nODG DLR 222 FORMFR SYSTEM PRESFNT SYSTEM NFT LOSS SALES PENETRATION 82 .17126 73 .15391 9 .01735 and requests the user to continue -the user wishes to know how this move affects the dealer's porCormance -the dealer suffered a net loss by the move, CONTINliE bej; I n -so the user returns the system t<> its inltlnl ntate CfNTINl F -ready for a fresh start pot»»ntlal of cell 5lt, not ■ 600 IMF MOTFNTIAL OF CFLL 5b HAS RFFN CHArinEO FROM 8 UNITS TO 600 UNITS. TMF TCTAL POTENTIAL OF THE SYSTEM IS NOV; UR262 UNITS. -the user wishes to Increase the potential uf a cell, possibly because of an anticipated new housing deve 1 opment -the result of the change CONTINUE chanp.p In dodc sales 8 liOnr, OFALFRS Fl)l*i;rH SYSTEM PHFSFMT SYSTEM NFT GAItl SALFS PENETRATION im 2.5Ufi71 1128 2.5579U 9 -.00877 -how did the increase in potential aCfect Dodge sales? -Dodge dealers obtained 9 of Che 5''2 new units CONTINUE add dodf; In cell 5U, n • h.O, b • -.52 THIS DEALER HAS BEEN ASSIGNED NUMBER 226. THFRF ARE NOW 9 DOOR DEALERS. ■an additional Dodge dealer Is added in the newly developed cell -the reiult CONTINUE EXHIBIT I (continued) sflip* of dlr 22C •the user wishes to know the sales of Che dealer he has just added SALES OF nODG DEALER NUMBER 226 ARE 537 UNITS FOK A PENETRATION OF .69870 PERCENT. -the answer CONTINUE chanr.p In Hortc, sales 9 nonr, dealers FORriER SYSTEM present SYSTEM NET GAIN CDNTINUE rPi'firt SALES PENETRATION 1119 2.5U671 USg 5.02305 5U0 .6765l« -the user realizes that acme of the 337 units represent sales formerly belonging to other Dodge dealers. How did Dodge fare on the whole? -the net gain of 340 units comprises 9 units gained by an Incraasa In potential and 331 units gained by the addition of a new dealer. Other Dodge dealers lost a total of 6 units to the new dealer -what has been the effect on each of the other makes? MAKF FORn CMEV RAMd PONT PLYM BUIC OLDS DOOO TOTAL •SALES- FORMER PRESENT CHANGF 8137 20907 1010 7307 500 1*862 3828 1119 l«7670 81fil< 21016 1020 7355 501 i*R93 38511 1U59 l>8262 27 109 10 it8 1 31 26 3U0 592 FORMFP 17.07010 1(5.85713 2.11928 15.32770 1.014959 10,19915 8.02921 2.51*671 •PEMFTRATION- PPESENT 16.915U5 U5.5I*61U 2.11266 15.21*008 1.05752 10.1580U 7.q857U 5.02305 CHANCE DEALERSHIPS SALES PFR DLR .151*65 ,51099 .00662 ,08762 ,01187 .06111 .01*51*7 .67651* FMR PRS CHG 58 1*7 8 28 5 2U 26 8 58 1*7 8 28 5 2U 26 9 FMR PRS CHG 181* 185 211* 215 ltl*5 l*i*7 126 127 261 265 100 100 205 201* 11*7 1U8 lUO 162 259 261 1 2 1 2 0 1 1 22 CONTINUE quit -the user terminates the program RUN TERMINATED. R 27. 550*11. 153 GOOD-BYE. -an exit message from the program. The session has usad 28 seconds of running time and 11 seconds of swap time. Cost at SlO/mlnute cooios to about $6.50 ate Uue ^ MAR 0 9 '78. 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