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. 



